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Sample records for gyrokinetic vlasov-poisson equations

  1. Exact momentum conservation laws for the gyrokinetic Vlasov-Poisson equations

    SciTech Connect

    Brizard, Alain J.; Tronko, Natalia

    2011-08-15

    The exact momentum conservation laws for the nonlinear gyrokinetic Vlasov-Poisson equations are derived by applying the Noether method on the gyrokinetic variational principle [A. J. Brizard, Phys. Plasmas 7, 4816 (2000)]. From the gyrokinetic Noether canonical-momentum equation derived by the Noether method, the gyrokinetic parallel momentum equation and other gyrokinetic Vlasov-moment equations are obtained. In addition, an exact gyrokinetic toroidal angular-momentum conservation law is derived in axisymmetric tokamak geometry, where the transport of parallel-toroidal momentum is related to the radial gyrocenter polarization, which includes contributions from the guiding-center and gyrocenter transformations.

  2. On Higher-order Corrections to Gyrokinetic Vlasov-Poisson Equations in the Long Wavelength Limit

    SciTech Connect

    W.W. Lee and R.A. Kolesnikov

    2009-02-17

    In this paper, we present a simple iterative procedure for obtaining the higher order E x B and dE/dt (polarization) drifts associated with the gyrokinetic Vlasov-Poisson equations in the long wavelength limit of k⊥ρi ~ o(ε) and k⊥L ~ o(1), where ρi is the ion gyroradius, L is the scale length of the background inhomogeneity and ε is a smallness parameter. It can be shown that these new higher order k⊥ρi terms, which are also related to the higher order perturbations of the electrostatic potential Φ, should have negligible effects on turbulent and neoclassical transport in tokamaks, regardless of the form of the background distribution and the amplitude of the perturbation. To address further the issue of a non-Maxwellian plasma, higher order finite Larmor radius terms in the gyrokinetic Poisson's equation have been studied and shown to be unimportant as well. On the other hand, the terms of o(k2⊥ρi2) ~ o(ε) and k⊥L ~ o(1) can indeed have impact on microturbulence, especially in the linear stage, such as those arising from the difference between the guiding center and the gyrocenter densities due to the presence of the background gradients. These results will be compared with a recent study questioning the validity of the commonly used gyrokinetic equations for long time simulations.

  3. Renormalized perturbation theory - Vlasov-Poisson system, weak turbulence limit, and gyrokinetics

    NASA Astrophysics Data System (ADS)

    Zhang, Y. Z.; Mahadjan, S. M.

    1988-10-01

    The self-consistency of the renormalized perturbation theory of Zhang and Mahajan (1985) is demonstrated by applying it to the Vlasov-Poisson system and showing that the theory has the correct weak turbulence limit. Energy conservation is proved to arbitrary high order for the electrostatic drift waves. The theory is applied to derive renormalized equations for a low-beta gyrokinetic system. Comparison of this theory with other current theories is presented.

  4. Gyrokinetic Vlasov-Poisson simulation in slab geometry using the conservative IDO scheme

    NASA Astrophysics Data System (ADS)

    Imadera, Kenji; Kishimoto, Yasuaki; Li, Jiquan; Saito, Daisuke; Utsumi, Takayuki

    2008-11-01

    We have introduced the IDO-CF (Conservative Form of Interpolated Differential Operator) scheme [1], which is one of the multi-moment schemes and has been applied to various CFD problems, in solving a Vlasov-Poisson system. The IDO scheme is found to be efficient in capturing a sharp domain interface like shock propagation, and in introducing dissipations like particle collision and also external source/sink terms. Furthermore, the IDO-CF scheme has exact mass conservation properties, so that we can apply it to the problems that need long time scale simulations. We first apply the scheme in studying the nonlinear Landau damping and two-stream instability. We have investigated the conservation property of the total mass, energy and entropy, and found that the IDO-CF scheme allows stable simulation over many bounce periods keeping higher accuracy than other multi-moment schemes. We have also developed a gyrokinetic full-f Vlasov code with the IDO-CF scheme in studying the slab ITG driven turbulence. [1] Y.Imai et al., J. Comput. Phys. 227, 2263(2008).

  5. Two coupled particle-finite volume methods using Delaunay-Voronoie meshes for the approximation of Vlasov-Poisson and Vlasov-Maxwell equations

    SciTech Connect

    Hermeline, F. )

    1993-05-01

    This paper deals with the approximation of Vlasov-Poisson and Vlasov-Maxwell equations. We present two coupled particle-finite volume methods which use the properties of Delaunay-Voronoi meshes. These methods are applied to benchmark calculations and engineering problems such as simulation of electron injector devices. 42 refs., 13 figs.

  6. AP-Cloud: Adaptive Particle-in-Cloud method for optimal solutions to Vlasov-Poisson equation

    NASA Astrophysics Data System (ADS)

    Wang, Xingyu; Samulyak, Roman; Jiao, Xiangmin; Yu, Kwangmin

    2016-07-01

    We propose a new adaptive Particle-in-Cloud (AP-Cloud) method for obtaining optimal numerical solutions to the Vlasov-Poisson equation. Unlike the traditional particle-in-cell (PIC) method, which is commonly used for solving this problem, the AP-Cloud adaptively selects computational nodes or particles to deliver higher accuracy and efficiency when the particle distribution is highly non-uniform. Unlike other adaptive techniques for PIC, our method balances the errors in PDE discretization and Monte Carlo integration, and discretizes the differential operators using a generalized finite difference (GFD) method based on a weighted least square formulation. As a result, AP-Cloud is independent of the geometric shapes of computational domains and is free of artificial parameters. Efficient and robust implementation is achieved through an octree data structure with 2:1 balance. We analyze the accuracy and convergence order of AP-Cloud theoretically, and verify the method using an electrostatic problem of a particle beam with halo. Simulation results show that the AP-Cloud method is substantially more accurate and faster than the traditional PIC, and it is free of artificial forces that are typical for some adaptive PIC techniques.

  7. Linear stability analysis of the Vlasov-Poisson equations in high density plasmas in the presence of crossed fields and density gradients

    NASA Technical Reports Server (NTRS)

    Kaup, D. J.; Hansen, P. J.; Choudhury, S. Roy; Thomas, Gary E.

    1986-01-01

    The equations for the single-particle orbits in a nonneutral high density plasma in the presence of inhomogeneous crossed fields are obtained. Using these orbits, the linearized Vlasov equation is solved as an expansion in the orbital radii in the presence of inhomogeneities and density gradients. A model distribution function is introduced whose cold-fluid limit is exactly the same as that used in many previous studies of the cold-fluid equations. This model function is used to reduce the linearized Vlasov-Poisson equations to a second-order ordinary differential equation for the linearized electrostatic potential whose eigenvalue is the perturbation frequency.

  8. Nonlinear gyrokinetic equations

    SciTech Connect

    Dubin, D.H.E.; Krommes, J.A.; Oberman, C.; Lee, W.W.

    1983-03-01

    Nonlinear gyrokinetic equations are derived from a systematic Hamiltonian theory. The derivation employs Lie transforms and a noncanonical perturbation theory first used by Littlejohn for the simpler problem of asymptotically small gyroradius. For definiteness, we emphasize the limit of electrostatic fluctuations in slab geometry; however, there is a straight-forward generalization to arbitrary field geometry and electromagnetic perturbations. An energy invariant for the nonlinear system is derived, and various of its limits are considered. The weak turbulence theory of the equations is examined. In particular, the wave kinetic equation of Galeev and Sagdeev is derived from an asystematic truncation of the equations, implying that this equation fails to consider all gyrokinetic effects. The equations are simplified for the case of small but finite gyroradius and put in a form suitable for efficient computer simulation. Although it is possible to derive the Terry-Horton and Hasegawa-Mima equations as limiting cases of our theory, several new nonlinear terms absent from conventional theories appear and are discussed.

  9. On the singularity of the Vlasov-Poisson system

    SciTech Connect

    Zheng, Jian; Qin, Hong

    2013-09-15

    The Vlasov-Poisson system can be viewed as the collisionless limit of the corresponding Fokker-Planck-Poisson system. It is reasonable to expect that the result of Landau damping can also be obtained from the Fokker-Planck-Poisson system when the collision frequency ν approaches zero. However, we show that the collisionless Vlasov-Poisson system is a singular limit of the collisional Fokker-Planck-Poisson system, and Landau's result can be recovered only as the ν approaches zero from the positive side.

  10. On the Singularity of the Vlasov-Poisson System

    SciTech Connect

    and Hong Qin, Jian Zheng

    2013-04-26

    The Vlasov-Poisson system can be viewed as the collisionless limit of the corresponding Fokker- Planck-Poisson system. It is reasonable to expect that the result of Landau damping can also be obtained from the Fokker-Planck-Poisson system when the collision frequency v approaches zero. However, we show that the colllisionless Vlasov-Poisson system is a singular limit of the collisional Fokker-Planck-Poisson system, and Landau's result can be recovered only as the approaching zero from the positive side.

  11. Two-Dimensional Current Carrying Bernstein-Greene-Kruskal (BGK) Modes for the Vlasov-Poisson-Ampere System

    NASA Astrophysics Data System (ADS)

    Ng, C. S.

    2014-10-01

    Electrostatic structures have been observed in many regions of space plasmas, including the solar wind, the magnetosphere, the auroral acceleration region. One possible theoretical description of some of these structures is the concept of Bernstein-Greene-Kruskal (BGK) modes, which are exact nonlinear steady-state solutions of the Vlasov-Poisson system of equations in collisionless kinetic theory. We generalize exact solutions of two-dimensional BGK modes in a magnetized plasma with finite magnetic field strength to cases with azimuthal magnetic fields so that these structures carry electric current as well as steady electric and magnetic fields. Such nonlinear solutions now satisfy exactly the Vlasov-Poisson-Ampere system of equations. This work is supported by a National Science Foundation Grant PHY-1004357.

  12. Global Existence for the Vlasov-Poisson System in Bounded Domains

    NASA Astrophysics Data System (ADS)

    Hwang, Hyung Ju; Velázquez, Juan J. L.

    2010-03-01

    In this paper we prove global existence for solutions of the Vlasov-Poisson system in convex bounded domains with specular boundary conditions and with a prescribed outward electrical field at the boundary.

  13. Verification of gyrokinetic microstability codes with an LHD configuration

    SciTech Connect

    Mikkelsen, D. R.; Nunami, M.; Watanabe, T. -H.; Sugama, H.; Tanaka, K.

    2014-11-01

    We extend previous benchmarks of the GS2 and GKV-X codes to verify their algorithms for solving the gyrokinetic Vlasov-Poisson equations for plasma microturbulence. Code benchmarks are the most complete way of verifying the correctness of implementations for the solution of mathematical models for complex physical processes such as those studied here. The linear stability calculations reported here are based on the plasma conditions of an ion-ITB plasma in the LHD configuration. The plasma parameters and the magnetic geometry differ from previous benchmarks involving these codes. We find excellent agreement between the independently written pre-processors that calculate the geometrical coefficients used in the gyrokinetic equations. Grid convergence tests are used to establish the resolution and domain size needed to obtain converged linear stability results. The agreement of the frequencies, growth rates and eigenfunctions in the benchmarks reported here provides additional verification that the algorithms used by the GS2 and GKV-X codes are correctly finding the linear eigenvalues and eigenfunctions of the gyrokinetic Vlasov-Poisson equations.

  14. Verification of gyrokinetic microstability codes with an LHD configuration

    SciTech Connect

    Mikkelsen, D. R.; Nunami, M.; Sugama, H.; Tanaka, K.; Watanabe, T.-H.

    2014-11-15

    We extend previous benchmarks of the GS2 and GKV-X codes to verify their algorithms for solving the gyrokinetic Vlasov-Poisson equations for plasma microturbulence. Code benchmarks are the most complete way of verifying the correctness of implementations for the solution of mathematical models for complex physical processes such as those studied here. The linear stability calculations reported here are based on the plasma conditions of an ion-ITB plasma in the LHD configuration. The plasma parameters and the magnetic geometry differ from previous benchmarks involving these codes. We find excellent agreement between the independently written pre-processors that calculate the geometrical coefficients used in the gyrokinetic equations. Grid convergence tests are used to establish the resolution and domain size needed to obtain converged linear stability results. The agreement of the frequencies, growth rates, and eigenfunctions in the benchmarks reported here provides additional verification that the algorithms used by the GS2 and GKV-X codes are correctly finding the linear eigenvalues and eigenfunctions of the gyrokinetic Vlasov-Poisson equations.

  15. ColDICE: A parallel Vlasov-Poisson solver using moving adaptive simplicial tessellation

    NASA Astrophysics Data System (ADS)

    Sousbie, Thierry; Colombi, Stéphane

    2016-09-01

    Resolving numerically Vlasov-Poisson equations for initially cold systems can be reduced to following the evolution of a three-dimensional sheet evolving in six-dimensional phase-space. We describe a public parallel numerical algorithm consisting in representing the phase-space sheet with a conforming, self-adaptive simplicial tessellation of which the vertices follow the Lagrangian equations of motion. The algorithm is implemented both in six- and four-dimensional phase-space. Refinement of the tessellation mesh is performed using the bisection method and a local representation of the phase-space sheet at second order relying on additional tracers created when needed at runtime. In order to preserve in the best way the Hamiltonian nature of the system, refinement is anisotropic and constrained by measurements of local Poincaré invariants. Resolution of Poisson equation is performed using the fast Fourier method on a regular rectangular grid, similarly to particle in cells codes. To compute the density projected onto this grid, the intersection of the tessellation and the grid is calculated using the method of Franklin and Kankanhalli [65-67] generalised to linear order. As preliminary tests of the code, we study in four dimensional phase-space the evolution of an initially small patch in a chaotic potential and the cosmological collapse of a fluctuation composed of two sinusoidal waves. We also perform a "warm" dark matter simulation in six-dimensional phase-space that we use to check the parallel scaling of the code.

  16. A Fully-Implicit, Ion-Electron, Vlasov-Poisson Algorithm

    NASA Astrophysics Data System (ADS)

    Taitano, William; Knoll, Dana; Chacon, Luis

    2010-11-01

    The Jacobian-Free-Newton-Krylov method (JFNK) is an advanced non- linear alogorithm that allows solution to a coupled systems of non-linear equations [1]. We put forth a new JFNK-based implicit plasma simulation algorithm. We have studied this algorithm within the context of a two-species Vlasov-Poisson system where the Vlasov equations are solved in an Eulerian frame [2]. We have investigated the route of non-linear-elimination/kinetic- enslavement to reduce the size of block Jacobian matrix in order to solve the field-kinetic system implicitly. The non-linear- elimination/kinetic-enslavement technique allows reduction in the size of non-linear system but still retains high order temporal accuracy and strong non-linear coupling. Our new algorithm make implicit time-dependent, coupled, field-kinetic systems more attractive. As will be shown, a fully implicit run was able to achieve 22 times speed-up compared to the explicit run for our ion-acoustic-showckwave simulation [2].[4pt] [1] D.A. Knoll and D.E. Keyes, J. Comput. Phys. vol. 193 (2004)[0pt] [2] W.T. Taitano, Masters Thesis, Nuclear Engineering, University of Idaho (2010)

  17. Highly efficient numerical algorithm based on random trees for accelerating parallel Vlasov-Poisson simulations

    NASA Astrophysics Data System (ADS)

    Acebrón, Juan A.; Rodríguez-Rozas, Ángel

    2013-10-01

    An efficient numerical method based on a probabilistic representation for the Vlasov-Poisson system of equations in the Fourier space has been derived. This has been done theoretically for arbitrary dimensional problems, and particularized to unidimensional problems for numerical purposes. Such a representation has been validated theoretically in the linear regime comparing the solution obtained with the classical results of the linear Landau damping. The numerical strategy followed requires generating suitable random trees combined with a Padé approximant for approximating accurately a given divergent series. Such series are obtained by summing the partial contributions to the solution coming from trees with arbitrary number of branches. These contributions, coming in general from multi-dimensional definite integrals, are efficiently computed by a quasi-Monte Carlo method. It is shown how the accuracy of the method can be effectively increased by considering more terms of the series. The new representation was used successfully to develop a Probabilistic Domain Decomposition method suited for massively parallel computers, which improves the scalability found in classical methods. Finally, a few numerical examples based on classical phenomena such as the non-linear Landau damping, and the two streaming instability are given, illustrating the remarkable performance of the algorithm, when compared the results with those obtained using a classical method.

  18. Linear stability of stationary solutions of the Vlasov-Poisson system in three dimensions

    SciTech Connect

    Batt, J.; Rein, G. . Mathematisches Inst.); Morrison, P.J. )

    1993-03-01

    Rigorous results on the stability of stationary solutions of the Vlasov-Poisson system are obtained in both the plasma physics and stellar dynamics contexts. It is proven that stationary solutions in the plasma physics (stellar dynamics) case are linearly stable if they are decreasing (increasing) functions of the local, i.e. particle, energy. The main tool in the analysis is the free energy of the system, a conserved quantity. In addition, an appropriate global existence result is proven for the linearized Vlasov-Poisson system and the existence of stationary solutions that satisfy the above stability condition is established.

  19. A Legendre-Fourier spectral method with exact conservation laws for the Vlasov-Poisson system

    NASA Astrophysics Data System (ADS)

    Manzini, G.; Delzanno, G. L.; Vencels, J.; Markidis, S.

    2016-07-01

    We present the design and implementation of an L2-stable spectral method for the discretization of the Vlasov-Poisson model of a collisionless plasma in one space and velocity dimension. The velocity and space dependence of the Vlasov equation are resolved through a truncated spectral expansion based on Legendre and Fourier basis functions, respectively. The Poisson equation, which is coupled to the Vlasov equation, is also resolved through a Fourier expansion. The resulting system of ordinary differential equation is discretized by the implicit second-order accurate Crank-Nicolson time discretization. The non-linear dependence between the Vlasov and Poisson equations is iteratively solved at any time cycle by a Jacobian-Free Newton-Krylov method. In this work we analyze the structure of the main conservation laws of the resulting Legendre-Fourier model, e.g., mass, momentum, and energy, and prove that they are exactly satisfied in the semi-discrete and discrete setting. The L2-stability of the method is ensured by discretizing the boundary conditions of the distribution function at the boundaries of the velocity domain by a suitable penalty term. The impact of the penalty term on the conservation properties is investigated theoretically and numerically. An implementation of the penalty term that does not affect the conservation of mass, momentum and energy, is also proposed and studied. A collisional term is introduced in the discrete model to control the filamentation effect, but does not affect the conservation properties of the system. Numerical results on a set of standard test problems illustrate the performance of the method.

  20. Instability conditions for some periodic BGK waves in the Vlasov-Poisson system

    NASA Astrophysics Data System (ADS)

    Pankavich, Stephen; Allen, Robert

    2014-12-01

    A one-dimensional, collisionless plasma given by the Vlasov-Poisson system is considered and the stability properties of periodic steady state solutions known as Bernstein-Greene-Kruskal (BGK) waves are investigated. Sufficient conditions are determined under which BGK waves are linearly unstable under perturbations that share the same period as the equilibria. It is also shown that such solutions cannot support a monotonically decreasing particle distribution function.

  1. A Uniqueness Criterion for Unbounded Solutions to the Vlasov-Poisson System

    NASA Astrophysics Data System (ADS)

    Miot, Evelyne

    2016-07-01

    We prove uniqueness for the Vlasov-Poisson system in two and three dimensions under the condition that the L p norms of the macroscopic density grow at most linearly with respect to p. This allows for solutions with logarithmic singularities. We provide explicit examples of initial data that fulfill the uniqueness condition and that exhibit a logarithmic blow-up. In the gravitational two-dimensional case, such states are intimately related to radially symmetric steady solutions of the system. Our method relies on the Lagrangian formulation for the solutions, exploiting the second-order structure of the corresponding ODE.

  2. Optimal Gradient Estimates and Asymptotic Behaviour for the Vlasov-Poisson System with Small Initial Data

    NASA Astrophysics Data System (ADS)

    Hwang, Hyung Ju; Rendall, Alan D.; Velázquez, Juan J. L.

    2011-04-01

    The Vlasov-Poisson system describes interacting systems of collisionless particles. For solutions with small initial data in three dimensions it is known that the spatial density of particles decays as t -3 at late times. In this paper this statement is refined to show that each derivative of the density which is taken leads to an extra power of decay, so that in N dimensions for {N ≥q 3} the derivative of the density of order k decays as t - N- k . An asymptotic formula for the solution at late times is also obtained.

  3. Nonlinear scattering term in the gyrokinetic Vlasov equation

    SciTech Connect

    Wang, Shaojie

    2013-08-15

    Nonlinear scattering term is found from the nonlinear gyrokinetic equation by decoupling the perturbed gyrocenter motion from the unperturbed motion. The gyro-center distribution function is determined by the well-understood unperturbed motion, with the effects of fields perturbation included in the nonlinear scattering term, which explicitly reveals the nonlinear stochastic dissipation on the time scale longer than the wave correlation time.

  4. Nonlinear electromagnetic gyrokinetic equations for rotating axisymmetric plasmas

    SciTech Connect

    Artun, M.; Tang, W.M.

    1994-03-01

    The influence of sheared equilibrium flows on the confinement properties of tokamak plasmas is a topic of much current interest. A proper theoretical foundation for the systematic kinetic analysis of this important problem has been provided here by presented the derivation of a set of nonlinear electromagnetic gyrokinetic equations applicable to low frequency microinstabilities in a rotating axisymmetric plasma. The subsonic rotation velocity considered is in the direction of symmetry with the angular rotation frequency being a function of the equilibrium magnetic flux surface. In accordance with experimental observations, the rotation profile is chosen to scale with the ion temperature. The results obtained represent the shear flow generalization of the earlier analysis by Frieman and Chen where such flows were not taken into account. In order to make it readily applicable to gyrokinetic particle simulations, this set of equations is cast in a phase-space-conserving continuity equation form.

  5. Global Existence to the Vlasov-Poisson System and Propagation of Moments Without Assumption of Finite Kinetic Energy

    NASA Astrophysics Data System (ADS)

    Chen, Zili; Zhang, Xianwen

    2016-05-01

    We consider classical as well as weak solutions to the three dimensional Vlasov-Poisson system. Without assuming finiteness of kinetic energy, we prove global existence of classical solutions by assuming the initial datum is smooth enough and has a compact velocity-spatial support, which will be specified in Theorem 1.1. We also establish some propagation results for low moments of weak solutions.

  6. Numerical Solution of the Gyrokinetic Poisson Equation in TEMPEST

    NASA Astrophysics Data System (ADS)

    Dorr, Milo; Cohen, Bruce; Cohen, Ronald; Dimits, Andris; Hittinger, Jeffrey; Kerbel, Gary; Nevins, William; Rognlien, Thomas; Umansky, Maxim; Xiong, Andrew; Xu, Xueqiao

    2006-10-01

    The gyrokinetic Poisson (GKP) model in the TEMPEST continuum gyrokinetic edge plasma code yields the electrostatic potential due to the charge density of electrons and an arbitrary number of ion species including the effects of gyroaveraging in the limit kρ1. The TEMPEST equations are integrated as a differential algebraic system involving a nonlinear system solve via Newton-Krylov iteration. The GKP preconditioner block is inverted using a multigrid preconditioned conjugate gradient (CG) algorithm. Electrons are treated as kinetic or adiabatic. The Boltzmann relation in the adiabatic option employs flux surface averaging to maintain neutrality within field lines and is solved self-consistently with the GKP equation. A decomposition procedure circumvents the near singularity of the GKP Jacobian block that otherwise degrades CG convergence.

  7. Optimal ?-Control for the Global Cauchy Problem of The Relativistic Vlasov-Poisson System

    NASA Astrophysics Data System (ADS)

    Young, Brent

    2011-12-01

    Recently, M.K.-H. Kiessling and A.S. Tahvildar-Zadeh proved that a unique global classical solution to the relativistic Vlasov-Poisson system exists whenever the positive, integrable initial datum is spherically symmetric, compactly supported in momentum space, vanishes on characteristics with vanishing angular momentum, and for β⩾3/2 has ?-norm strictly below a positive, critical value ?. Everything else being equal, data leading to finite time blow-up can be found with ?-norm surpassing ? for any β>1, with ? if and only if β⩾3/2. In their paper, the critical value for β=3/2 is calculated explicitly while the value for all other β is merely characterized as the infimum of a functional over an appropriate function space. In this work, the existence of minimizers is established, and the exact expression of ? is calculated in terms of the famous Lane-Emden functions. Numerical computations of the ? are presented along with some elementary asymptotics near the critical exponent 3/2.

  8. Fully Electromagnetic Nonlinear Gyrokinetic Equations for Tokamak Edge Turbulence

    SciTech Connect

    Hahm, T. S.; Wang, Lu; Madsen, J.

    2008-08-01

    An energy conserving set of the fully electromagnetic nonlinear gyrokinetic Vlasov equation and Maxwell's equations, which is applicable to both L-mode turbulence with large amplitude and H-mode turbulence in the presence of high E Χ B shear has been derived. The phase-space action variational Lie perturbation method ensures the preservation of the conservation laws of the underlying Vlasov-Maxwell system. Our generalized ordering takes ρi<< ρθ¡ ~ LE ~ Lp << R (here ρi is the thermal ion Larmor radius and ρθ¡ = B/Bθ] ρi), as typically observed in the tokamak H-mode edge, with LE and Lp being the radial electric field and pressure gradient lengths. We take κ perpendicular to ρi ~ 1 for generality, and keep the relative fluctuation amplitudes eδφ /Τi ~ δΒ / Β up to the second order. Extending the electrostatic theory in the presence of high E Χ B shear [Hahm, Phys. Plasmas 3, 4658 (1996)], contributions of electromagnetic fluctuations to the particle charge density and current are explicitly evaluated via pull-back transformation from the gyrocenter distribution function in the gyrokinetic Maxwell's equation.

  9. Gyrokinetic equations and full f solution method based on Dirac's constrained Hamiltonian and inverse Kruskal iteration

    SciTech Connect

    Heikkinen, J. A.; Nora, M.

    2011-02-15

    Gyrokinetic equations of motion, Poisson equation, and energy and momentum conservation laws are derived based on the reduced-phase-space Lagrangian and inverse Kruskal iteration introduced by Pfirsch and Correa-Restrepo [J. Plasma Phys. 70, 719 (2004)]. This formalism, together with the choice of the adiabatic invariant J= as one of the averaging coordinates in phase space, provides an alternative to the standard gyrokinetics. Within second order in gyrokinetic parameter, the new equations do not show explicit ponderomotivelike or polarizationlike terms. Pullback of particle information with an iterated gyrophase and field dependent gyroradius function from the gyrocenter position defined by gyroaveraged coordinates allows direct numerical integration of the gyrokinetic equations in particle simulation of the field and particles with full distribution function. As an example, gyrokinetic systems with polarization drift either present or absent in the equations of motion are considered.

  10. Linearized model collision operators for multiple ion species plasmas and gyrokinetic entropy balance equations

    SciTech Connect

    Sugama, H.; Watanabe, T.-H.; Nunami, M.

    2009-11-15

    Linearized model collision operators for multiple ion species plasmas are presented that conserve particles, momentum, and energy and satisfy adjointness relations and Boltzmann's H-theorem even for collisions between different particle species with unequal temperatures. The model collision operators are also written in the gyrophase-averaged form that can be applied to the gyrokinetic equation. Balance equations for the turbulent entropy density, the energy of electromagnetic fluctuations, the turbulent transport fluxes of particle and heat, and the collisional dissipation are derived from the gyrokinetic equation including the collision term and Maxwell equations. It is shown that, in the steady turbulence, the entropy produced by the turbulent transport fluxes is dissipated in part by collisions in the nonzonal-mode region and in part by those in the zonal-mode region after the nonlinear entropy transfer from nonzonal to zonal modes.

  11. Geodesic acoustic mode in anisotropic plasmas using double adiabatic model and gyro-kinetic equation

    SciTech Connect

    Ren, Haijun; Cao, Jintao

    2014-12-15

    Geodesic acoustic mode in anisotropic tokamak plasmas is theoretically analyzed by using double adiabatic model and gyro-kinetic equation. The bi-Maxwellian distribution function for guiding-center ions is assumed to obtain a self-consistent form, yielding pressures satisfying the magnetohydrodynamic (MHD) anisotropic equilibrium condition. The double adiabatic model gives the dispersion relation of geodesic acoustic mode (GAM), which agrees well with the one derived from gyro-kinetic equation. The GAM frequency increases with the ratio of pressures, p{sub ⊥}/p{sub ∥}, and the Landau damping rate is dramatically decreased by p{sub ⊥}/p{sub ∥}. MHD result shows a low-frequency zonal flow existing for all p{sub ⊥}/p{sub ∥}, while according to the kinetic dispersion relation, no low-frequency branch exists for p{sub ⊥}/p{sub ∥}≳ 2.

  12. On the velocity space discretization for the Vlasov-Poisson system: Comparison between implicit Hermite spectral and Particle-in-Cell methods

    NASA Astrophysics Data System (ADS)

    Camporeale, E.; Delzanno, G. L.; Bergen, B. K.; Moulton, J. D.

    2016-01-01

    We describe a spectral method for the numerical solution of the Vlasov-Poisson system where the velocity space is decomposed by means of an Hermite basis, and the configuration space is discretized via a Fourier decomposition. The novelty of our approach is an implicit time discretization that allows exact conservation of charge, momentum and energy. The computational efficiency and the cost-effectiveness of this method are compared to the fully-implicit PIC method recently introduced by Markidis and Lapenta (2011) and Chen et al. (2011). The following examples are discussed: Langmuir wave, Landau damping, ion-acoustic wave, two-stream instability. The Fourier-Hermite spectral method can achieve solutions that are several orders of magnitude more accurate at a fraction of the cost with respect to PIC.

  13. Vlasov-Poisson calculations of electron confinement times in Polywell(TM) devices using a steady-state particle-in-cell method

    NASA Astrophysics Data System (ADS)

    Kollasch, Jeffrey; Sovinec, Carl; Santarius, John

    2013-10-01

    Collisionless electron confinement times in polyhedral magnetic cusp configurations are investigated numerically with a particle-in-cell technique designed for steady-state conditions of the Vlasov-Poisson system. This method is based on iteratively solving particle trajectories in the time-independent electrostatic field produced by trajectories from a previous iteration. A new code based on this technique, SSUBPIC (steady-state unstructured-boundary particle-in-cell), is presented. It is found to converge rapidly for the cases investigated. The implementation is verified on computations of space-charge limited current in 1D and 2D configurations. Here, it is applied to study the effects of an ejecting virtual cathode potential well on a single electron species injected by guns into a Polywell(TM). Adverse effects of non-magnetically shielded structural members on confinement time are also calculated. Work supported by the Department of Defense (DoD) through the National Defense Science and Engineering Graduate Fellowship (NDSEG) Program.

  14. From charge motion in general magnetic fields to the non perturbative gyrokinetic equation

    NASA Astrophysics Data System (ADS)

    Di Troia, C.

    2015-04-01

    The exact analytical description of non relativistic charge motion in general magnetic fields is, apparently, a simple problem, even if it has not been solved until now, apart for rare cases. The key feature of the present derivation is to adopt a non perturbative magnetic field description to find new solutions of motion. Among all solutions, two are particularly important: guiding particle and gyro-particle solutions. The guiding particle has been characterized to be minimally coupled to the magnetic field; the gyro-particle has been defined to be maximally coupled to the magnetic field and, also, to move on a closed orbit. The generic charged particle motion is shown to be expressed as the sum of such particular solutions. This non perturbative approach corresponds to the description of the particle motion in the gyro-center and/or guiding center reference frame obtained at all the orders of the modern gyro-center transformation. The Boltzmann equation is analyzed with the described exact guiding center coordinates. The obtained gyrokinetic equation is solved for the Boltzmann equation at marginal stability conditions.

  15. From charge motion in general magnetic fields to the non perturbative gyrokinetic equation

    SciTech Connect

    Di Troia, C.

    2015-04-15

    The exact analytical description of non relativistic charge motion in general magnetic fields is, apparently, a simple problem, even if it has not been solved until now, apart for rare cases. The key feature of the present derivation is to adopt a non perturbative magnetic field description to find new solutions of motion. Among all solutions, two are particularly important: guiding particle and gyro-particle solutions. The guiding particle has been characterized to be minimally coupled to the magnetic field; the gyro-particle has been defined to be maximally coupled to the magnetic field and, also, to move on a closed orbit. The generic charged particle motion is shown to be expressed as the sum of such particular solutions. This non perturbative approach corresponds to the description of the particle motion in the gyro-center and/or guiding center reference frame obtained at all the orders of the modern gyro-center transformation. The Boltzmann equation is analyzed with the described exact guiding center coordinates. The obtained gyrokinetic equation is solved for the Boltzmann equation at marginal stability conditions.

  16. Asymptotic behaviour of the Boltzmann equation as a cosmological model

    NASA Astrophysics Data System (ADS)

    Lee, Ho

    2016-08-01

    As a Newtonian cosmological model the Vlasov-Poisson-Boltzmann system is considered, and a slightly modified Boltzmann equation, which describes the stability of an expanding universe, is derived. Asymptotic behaviour of solutions turns out to depend on the expansion of the universe, and in this paper we consider the soft potential case and will obtain asymptotic behaviour.

  17. Monte Carlo particle-in-cell methods for the simulation of the Vlasov-Maxwell gyrokinetic equations

    NASA Astrophysics Data System (ADS)

    Bottino, A.; Sonnendrücker, E.

    2015-10-01

    > The particle-in-cell (PIC) algorithm is the most popular method for the discretisation of the general 6D Vlasov-Maxwell problem and it is widely used also for the simulation of the 5D gyrokinetic equations. The method consists of coupling a particle-based algorithm for the Vlasov equation with a grid-based method for the computation of the self-consistent electromagnetic fields. In this review we derive a Monte Carlo PIC finite-element model starting from a gyrokinetic discrete Lagrangian. The variations of the Lagrangian are used to obtain the time-continuous equations of motion for the particles and the finite-element approximation of the field equations. The Noether theorem for the semi-discretised system implies a certain number of conservation properties for the final set of equations. Moreover, the PIC method can be interpreted as a probabilistic Monte Carlo like method, consisting of calculating integrals of the continuous distribution function using a finite set of discrete markers. The nonlinear interactions along with numerical errors introduce random effects after some time. Therefore, the same tools for error analysis and error reduction used in Monte Carlo numerical methods can be applied to PIC simulations.

  18. Conservation of energy and momentum in nonrelativistic plasmas

    SciTech Connect

    Sugama, H.; Watanabe, T.-H.; Nunami, M.

    2013-02-15

    Conservation laws of energy and momentum for nonrelativistic plasmas are derived from applying Noether's theorem to the action integral for the Vlasov-Poisson-Ampere system [Sugama, Phys. Plasmas 7, 466 (2000)]. The symmetric pressure tensor is obtained from modifying the asymmetric canonical pressure tensor with using the rotational symmetry of the action integral. Differences between the resultant conservation laws and those for the Vlasov-Maxwell system including the Maxwell displacement current are clarified. These results provide a useful basis for gyrokinetic conservation laws because gyrokinetic equations are derived as an approximation of the Vlasov-Poisson-Ampere system.

  19. Energetically consistent collisional gyrokinetics

    SciTech Connect

    Burby, J. W.; Brizard, A. J.; Qin, H.

    2015-10-01

    We present a formulation of collisional gyrokinetic theory with exact conservation laws for energy and canonical toroidal momentum. Collisions are accounted for by a nonlinear gyrokinetic Landau operator. Gyroaveraging and linearization do not destroy the operator's conservation properties. Just as in ordinary kinetic theory, the conservation laws for collisional gyrokinetic theory are selected by the limiting collisionless gyrokinetic theory. (C) 2015 AIP Publishing LLC.

  20. Energetically consistent collisional gyrokinetics

    SciTech Connect

    Burby, J. W.; Brizard, A. J.; Qin, H.

    2015-10-15

    We present a formulation of collisional gyrokinetic theory with exact conservation laws for energy and canonical toroidal momentum. Collisions are accounted for by a nonlinear gyrokinetic Landau operator. Gyroaveraging and linearization do not destroy the operator's conservation properties. Just as in ordinary kinetic theory, the conservation laws for collisional gyrokinetic theory are selected by the limiting collisionless gyrokinetic theory.

  1. Intrinsic rotation with gyrokinetic models

    SciTech Connect

    Parra, Felix I.; Barnes, Michael; Catto, Peter J.; Calvo, Ivan

    2012-05-15

    The generation of intrinsic rotation by turbulence and neoclassical effects in tokamaks is considered. To obtain the complex dependences observed in experiments, it is necessary to have a model of the radial flux of momentum that redistributes the momentum within the tokamak in the absence of a preexisting velocity. When the lowest order gyrokinetic formulation is used, a symmetry of the model precludes this possibility, making small effects in the gyroradius over scale length expansion necessary. These effects that are usually small become important for momentum transport because the symmetry of the lowest order gyrokinetic formulation leads to the cancellation of the lowest order momentum flux. The accuracy to which the gyrokinetic equation needs to be obtained to retain all the physically relevant effects is discussed.

  2. A Short Introduction to General Gyrokinetic Theory

    SciTech Connect

    H. Qin

    2005-02-14

    Interesting plasmas in the laboratory and space are magnetized. General gyrokinetic theory is about a symmetry, gyro-symmetry, in the Vlasov-Maxwell system for magnetized plasmas. The most general gyrokinetic theory can be geometrically formulated. First, the coordinate-free, geometric Vlasov-Maxwell equations are developed in the 7-D phase space, which is defined as a fiber bundle over the space-time. The Poincar{copyright}-Cartan-Einstein 1-form pullbacked onto the 7-D phase space determines particles' worldlines in the phase space, and realizes the momentum integrals in kinetic theory as fiber integrals. The infinite small generator of the gyro-symmetry is then asymptotically constructed as the base for the gyrophase coordinate of the gyrocenter coordinate system. This is accomplished by applying the Lie coordinate perturbation method to the Poincar{copyright}-Cartan-Einstein 1-form, which also generates the most relaxed condition under which the gyro-symmetry still exists. General gyrokinetic Vlasov-Maxwell equations are then developed as the Vlasov-Maxwell equations in the gyrocenter coordinate system, rather than a set of new equations. Since the general gyrokinetic system-developed is geometrically the same as the Vlasov-Maxwell equations, all the coordinate independent properties of the Vlasov-Maxwell equations, such as energy conservation, momentum conservation, and Liouville volume conservation, are automatically carried over to the general gyrokinetic system. The pullback transformation associated with the coordinate transformation is shown to be an indispensable part of the general gyrokinetic Vlasov-Maxwell equations. Without this vital element, a number of prominent physics features, such as the presence of the compressional Alfven wave and a proper description of the gyrokinetic equilibrium, cannot be readily recovered. Three examples of applications of the general gyrokinetic theory developed in the areas of plasma equilibrium and plasma waves are

  3. Energetically consistent collisional gyrokinetics

    DOE PAGESBeta

    Burby, J. W.; Brizard, A. J.; Qin, H.

    2015-10-30

    Here, we present a formulation of collisional gyrokinetic theory with exact conservation laws for energy and canonical toroidal momentum. Collisions are accounted for by a nonlinear gyrokinetic Landau operator. Gyroaveraging and linearization do not destroy the operator's conservation properties. Just as in ordinary kinetic theory, the conservation laws for collisional gyrokinetic theory are selected by the limiting collisionless gyrokinetic theory. (C) 2015 AIP Publishing LLC.

  4. Electromagnetic nonlinear gyrokinetics with polarization drift

    SciTech Connect

    Duthoit, F.-X.; Hahm, T. S.; Wang, Lu

    2014-08-15

    A set of new nonlinear electromagnetic gyrokinetic Vlasov equation with polarization drift and gyrokinetic Maxwell equations is systematically derived by using the Lie-transform perturbation method in toroidal geometry. For the first time, we recover the drift-kinetic expression for parallel acceleration [R. M. Kulsrud, in Basic Plasma Physics, edited by A. A. Galeev and R. N. Sudan (North-Holland, Amsterdam, 1983)] from the nonlinear gyrokinetic equations, thereby bridging a gap between the two formulations. This formalism should be useful in addressing nonlinear ion Compton scattering of intermediate-mode-number toroidal Alfvén eigenmodes for which the polarization current nonlinearity [T. S. Hahm and L. Chen, Phys. Rev. Lett. 74, 266 (1995)] and the usual finite Larmor radius effects should compete.

  5. Electromagnetic nonlinear gyrokinetics with polarization drift

    NASA Astrophysics Data System (ADS)

    Duthoit, F.-X.; Hahm, T. S.; Wang, Lu

    2014-08-01

    A set of new nonlinear electromagnetic gyrokinetic Vlasov equation with polarization drift and gyrokinetic Maxwell equations is systematically derived by using the Lie-transform perturbation method in toroidal geometry. For the first time, we recover the drift-kinetic expression for parallel acceleration [R. M. Kulsrud, in Basic Plasma Physics, edited by A. A. Galeev and R. N. Sudan (North-Holland, Amsterdam, 1983)] from the nonlinear gyrokinetic equations, thereby bridging a gap between the two formulations. This formalism should be useful in addressing nonlinear ion Compton scattering of intermediate-mode-number toroidal Alfvén eigenmodes for which the polarization current nonlinearity [T. S. Hahm and L. Chen, Phys. Rev. Lett. 74, 266 (1995)] and the usual finite Larmor radius effects should compete.

  6. Gyrokinetic particle simulation model

    SciTech Connect

    Lee, W.W.

    1986-07-01

    A new type of particle simulation model based on the gyrophase-averaged Vlasov and Poisson equations is presented. The reduced system, in which particle gyrations are removed from the equations of motion while the finite Larmor radius effects are still preserved, is most suitable for studying low frequency microinstabilities in magnetized plasmas. It is feasible to simulate an elongated system (L/sub parallel/ >> L/sub perpendicular/) with a three-dimensional grid using the present model without resorting to the usual mode expansion technique, since there is essentially no restriction on the size of ..delta..x/sub parallel/ in a gyrokinetic plasma. The new approach also enables us to further separate the time and spatial scales of the simulation from those associated with global transport through the use of multiple spatial scale expansion. Thus, the model can be a very efficient tool for studying anomalous transport problems related to steady-state drift-wave turbulence in magnetic confinement devices. It can also be applied to other areas of plasma physics.

  7. Intercode comparison of gyrokinetic global electromagnetic modes

    NASA Astrophysics Data System (ADS)

    Görler, T.; Tronko, N.; Hornsby, W. A.; Bottino, A.; Kleiber, R.; Norscini, C.; Grandgirard, V.; Jenko, F.; Sonnendrücker, E.

    2016-07-01

    Aiming to fill a corresponding lack of sophisticated test cases for global electromagnetic gyrokinetic codes, a new hierarchical benchmark is proposed. Starting from established test sets with adiabatic electrons, fully gyrokinetic electrons, and electrostatic fluctuations are taken into account before finally studying the global electromagnetic micro-instabilities. Results from up to five codes involving representatives from different numerical approaches as particle-in-cell methods, Eulerian and Semi-Lagrangian are shown. By means of spectrally resolved growth rates and frequencies and mode structure comparisons, agreement can be confirmed on ion-gyro-radius scales, thus providing confidence in the correct implementation of the underlying equations.

  8. Second order gyrokinetic theory for particle-in-cell codes

    NASA Astrophysics Data System (ADS)

    Tronko, Natalia; Bottino, Alberto; Sonnendrücker, Eric

    2016-08-01

    The main idea of the gyrokinetic dynamical reduction consists in a systematical removal of the fast scale motion (the gyromotion) from the dynamics of the plasma, resulting in a considerable simplification and a significant gain of computational time. The gyrokinetic Maxwell-Vlasov equations are nowadays implemented in for modeling (both laboratory and astrophysical) strongly magnetized plasmas. Different versions of the reduced set of equations exist, depending on the construction of the gyrokinetic reduction procedure and the approximations performed in the derivation. The purpose of this article is to explicitly show the connection between the general second order gyrokinetic Maxwell-Vlasov system issued from the modern gyrokinetic theory and the model currently implemented in the global electromagnetic Particle-in-Cell code ORB5. Necessary information about the modern gyrokinetic formalism is given together with the consistent derivation of the gyrokinetic Maxwell-Vlasov equations from first principles. The variational formulation of the dynamics is used to obtain the corresponding energy conservation law, which in turn is used for the verification of energy conservation diagnostics currently implemented in ORB5. This work fits within the context of the code verification project VeriGyro currently run at IPP Max-Planck Institut in collaboration with others European institutions.

  9. Geometric Gyrokinetic Theory for Edge Plasma

    SciTech Connect

    Qin, H; Cohen, R H; Nevins, W M; Xu, X Q

    2007-01-18

    It turns out that gyrokinetic theory can be geometrically formulated as special cases of a geometrically generalized Vlasov-Maxwell system. It is proposed that the phase space of the spacetime is a 7-dimensional fiber bundle P over the 4-dimensional spacetime M, and that a Poincare-Cartan-Einstein 1-form {gamma} on the 7-dimensional phase space determines particles worldlines in the phase space. Through Liouville 6-form {Omega} and fiber integral, the 1-form {gamma} also uniquely defines a geometrically generalized Vlasov-Maxwell system as a field theory for the collective electromagnetic field. The geometric gyrokinetic theory is then developed as a special case of the geometrically generalized Vlasov-Maxwell system. In its most general form, gyrokinetic theory is about a symmetry, called gyro-symmetry, for magnetized plasmas, and the 1-form {gamma} again uniquely defines the gyro-symmetry. The objective is to decouple the gyro-phase dynamics from the rest of particle dynamics by finding the gyro-symmetry in {gamma}. Compared with other methods of deriving the gyrokinetic equations, the advantage of the geometric approach is that it allows any approximation based on mathematical simplification or physical intuition to be made at the 1-form level, and yet the field theories still have the desirable exact conservation properties such as phase space volume conservation and energy-momentum conservation if the 1-form does not depend on the spacetime coordinate explicitly. A set of generalized gyrokinetic equations valid for the edge plasmas is then derived using this geometric method. This formalism allows large-amplitude, time-dependent background electromagnetic fields to be developed fully nonlinearly in addition to small-amplitude, short-wavelength electromagnetic perturbations. The fact that we adopted the geometric method in the present study does not necessarily imply that the major results reported here can not be achieved using classical methods. What the

  10. Nonlinear Gyrokinetic Theory With Polarization Drift

    SciTech Connect

    L. Wang and T.S. Hahm

    2010-03-25

    A set of the electrostatic toroidal gyrokinetic Vlasov equation and the Poisson equation, which explicitly includes the polarization drift, is derived systematically by using Lie-transform method. The polarization drift is introduced in the gyrocenter equations of motion, and the corresponding polarization density is derived. Contrary to the wide-spread expectation, the inclusion of the polarization drift in the gyrocenter equations of motion does not affect the expression for the polarization density significantly. This is due to modification of the gyrocenter phase-space volume caused by the electrostatic potential [T. S. Hahm, Phys. Plasmas 3, 4658 (1996)] .

  11. Magnetohydrodynamics for collisionless plasmas from the gyrokinetic perspective

    NASA Astrophysics Data System (ADS)

    Lee, W. W.

    2016-07-01

    The effort to obtain a set of MagnetoHydroDynamic (MHD) equations for a magnetized collisionless plasma was started nearly 60 years ago by Chew et al. [Proc. R. Soc. London, Ser. A 236(1204), 112-118 (1956)]. Many attempts have been made ever since. Here, we will show the derivation of a set of these equations from the gyrokinetic perspective, which we call it gyrokinetic MHD, and it is different from the conventional ideal MHD. However, this new set of equations still has conservation properties and, in the absence of fluctuations, recovers the usual MHD equilibrium. Furthermore, the resulting equations allow for the plasma pressure balance to be further modified by finite-Larmor-radius effects in regions with steep pressure gradients. The present work is an outgrowth of the paper on "Alfven Waves in Gyrokinetic Plasmas" by Lee and Qin [Phys. Plasmas 10, 3196 (2003)].

  12. Free energy balance in gyrokinetic turbulence

    SciTech Connect

    Banon Navarro, A.; Morel, P.; Albrecht-Marc, M.; Carati, D.; Merz, F.; Goerler, T.; Jenko, F.

    2011-09-15

    Free energy plays an important role in gyrokinetic theory, since it is known to be a nonlinear invariant. Its evolution equations are derived and analyzed for the case of ion temperature gradient driven turbulence, using the formalism adopted in the Gene code. In particular, the ion temperature gradient drive, the collisional dissipation as well as entropy/electrostatic energy transfer channels represented by linear curvature and parallel terms are analyzed in detail.

  13. Neoclassical equilibrium in gyrokinetic simulations

    SciTech Connect

    Garbet, X.; Dif-Pradalier, G.; Nguyen, C.; Sarazin, Y.; Grandgirard, V.; Ghendrih, Ph.

    2009-06-15

    This paper presents a set of model collision operators, which reproduce the neoclassical equilibrium and comply with the constraints of a full-f global gyrokinetic code. The assessment of these operators is based on an entropy variational principle, which allows one to perform a fast calculation of the neoclassical diffusivity and poloidal velocity. It is shown that the force balance equation is recovered at lowest order in the expansion parameter, the normalized gyroradius, hence allowing one to calculate correctly the radial electric field. Also, the conventional neoclassical transport and the poloidal velocity are reproduced in the plateau and banana regimes. The advantages and drawbacks of the various model operators are discussed in view of the requirements for neoclassical and turbulent transport.

  14. Noiseless Vlasov-Poisson simulations with linearly transformed particles

    SciTech Connect

    Pinto, Martin C.; Sonnendrucker, Eric; Friedman, Alex; Grote, David P.; Lund, Steve M.

    2014-06-25

    We introduce a deterministic discrete-particle simulation approach, the Linearly-Transformed Particle-In-Cell (LTPIC) method, that employs linear deformations of the particles to reduce the noise traditionally associated with particle schemes. Formally, transforming the particles is justified by local first order expansions of the characteristic flow in phase space. In practice the method amounts of using deformation matrices within the particle shape functions; these matrices are updated via local evaluations of the forward numerical flow. Because it is necessary to periodically remap the particles on a regular grid to avoid excessively deforming their shapes, the method can be seen as a development of Denavit's Forward Semi-Lagrangian (FSL) scheme (Denavit, 1972 [8]). However, it has recently been established (Campos Pinto, 2012 [20]) that the underlying Linearly-Transformed Particle scheme converges for abstract transport problems, with no need to remap the particles; deforming the particles can thus be seen as a way to significantly lower the remapping frequency needed in the FSL schemes, and hence the associated numerical diffusion. To couple the method with electrostatic field solvers, two specific charge deposition schemes are examined, and their performance compared with that of the standard deposition method. Finally, numerical 1d1v simulations involving benchmark test cases and halo formation in an initially mismatched thermal sheet beam demonstrate some advantages of our LTPIC scheme over the classical PIC and FSL methods. Lastly, benchmarked test cases also indicate that, for numerical choices involving similar computational effort, the LTPIC method is capable of accuracy comparable to or exceeding that of state-of-the-art, high-resolution Vlasov schemes.

  15. Noiseless Vlasov-Poisson simulations with linearly transformed particles

    DOE PAGESBeta

    Pinto, Martin C.; Sonnendrucker, Eric; Friedman, Alex; Grote, David P.; Lund, Steve M.

    2014-06-25

    We introduce a deterministic discrete-particle simulation approach, the Linearly-Transformed Particle-In-Cell (LTPIC) method, that employs linear deformations of the particles to reduce the noise traditionally associated with particle schemes. Formally, transforming the particles is justified by local first order expansions of the characteristic flow in phase space. In practice the method amounts of using deformation matrices within the particle shape functions; these matrices are updated via local evaluations of the forward numerical flow. Because it is necessary to periodically remap the particles on a regular grid to avoid excessively deforming their shapes, the method can be seen as a development ofmore » Denavit's Forward Semi-Lagrangian (FSL) scheme (Denavit, 1972 [8]). However, it has recently been established (Campos Pinto, 2012 [20]) that the underlying Linearly-Transformed Particle scheme converges for abstract transport problems, with no need to remap the particles; deforming the particles can thus be seen as a way to significantly lower the remapping frequency needed in the FSL schemes, and hence the associated numerical diffusion. To couple the method with electrostatic field solvers, two specific charge deposition schemes are examined, and their performance compared with that of the standard deposition method. Finally, numerical 1d1v simulations involving benchmark test cases and halo formation in an initially mismatched thermal sheet beam demonstrate some advantages of our LTPIC scheme over the classical PIC and FSL methods. Lastly, benchmarked test cases also indicate that, for numerical choices involving similar computational effort, the LTPIC method is capable of accuracy comparable to or exceeding that of state-of-the-art, high-resolution Vlasov schemes.« less

  16. Noiseless Vlasov-Poisson simulations with linearly transformed particles

    NASA Astrophysics Data System (ADS)

    Campos Pinto, Martin; Sonnendrücker, Eric; Friedman, Alex; Grote, David P.; Lund, Steve M.

    2014-10-01

    We introduce a deterministic discrete-particle simulation approach, the Linearly-Transformed Particle-In-Cell (LTPIC) method, that employs linear deformations of the particles to reduce the noise traditionally associated with particle schemes. Formally, transforming the particles is justified by local first order expansions of the characteristic flow in phase space. In practice the method amounts of using deformation matrices within the particle shape functions; these matrices are updated via local evaluations of the forward numerical flow. Because it is necessary to periodically remap the particles on a regular grid to avoid excessively deforming their shapes, the method can be seen as a development of Denavit's Forward Semi-Lagrangian (FSL) scheme (Denavit, 1972 [8]). However, it has recently been established (Campos Pinto, 2012 [20]) that the underlying Linearly-Transformed Particle scheme converges for abstract transport problems, with no need to remap the particles; deforming the particles can thus be seen as a way to significantly lower the remapping frequency needed in the FSL schemes, and hence the associated numerical diffusion. To couple the method with electrostatic field solvers, two specific charge deposition schemes are examined, and their performance compared with that of the standard deposition method. Finally, numerical 1d1v simulations involving benchmark test cases and halo formation in an initially mismatched thermal sheet beam demonstrate some advantages of our LTPIC scheme over the classical PIC and FSL methods. Benchmarked test cases also indicate that, for numerical choices involving similar computational effort, the LTPIC method is capable of accuracy comparable to or exceeding that of state-of-the-art, high-resolution Vlasov schemes.

  17. Transport of momentum in full f gyrokinetics

    SciTech Connect

    Parra, Felix I.; Catto, Peter J.

    2010-05-15

    Full f electrostatic gyrokinetic formulations employ two gyrokinetic equations, one for ions and the other for electrons, and quasineutrality to obtain the ion and electron distribution functions and the electrostatic potential. We demonstrate with several examples that the long wavelength radial electric field obtained with full f approaches is extremely sensitive to errors in the ion and electron density since small deviations in density give rise to large, nonphysical deviations in the conservation of toroidal angular momentum. For typical tokamak values, a relative error of 10{sup -7} in the ion or electron densities is enough to obtain the incorrect toroidal rotation. Based on the insights gained with the examples considered, three simple tests to check transport of toroidal angular momentum in full f simulations are proposed.

  18. Incorrectness of the usual gyrokinetic treatment in cylindrically symmetric systems

    SciTech Connect

    Linsker, R.

    1980-07-01

    It is shown that the usual gyrokinetic theory does not consistently retain all terms of leading order in the expansion parameter epsilon = gyroradius/equilibrium scale length. This is illustrated for cylindrically symmetric systems by comparing the perturbed distribution function calculated by the gyrokinetic method with that obtained by explicitly integrating the Vlasov equation over the unperturbed orbit. The integral equation used in some recent treatments of drift waves in sheared-slab geometry is shown to be incorrect. The correct calculation of the ion density perturbation for a collisionless ..beta.. = 0 plasma with cylindrical symmetry is presented.

  19. Gyrokinetic Statistical Absolute Equilibrium and Turbulence

    SciTech Connect

    Jian-Zhou Zhu and Gregory W. Hammett

    2011-01-10

    A paradigm based on the absolute equilibrium of Galerkin-truncated inviscid systems to aid in understanding turbulence [T.-D. Lee, "On some statistical properties of hydrodynamical and magnetohydrodynamical fields," Q. Appl. Math. 10, 69 (1952)] is taken to study gyrokinetic plasma turbulence: A finite set of Fourier modes of the collisionless gyrokinetic equations are kept and the statistical equilibria are calculated; possible implications for plasma turbulence in various situations are discussed. For the case of two spatial and one velocity dimension, in the calculation with discretization also of velocity v with N grid points (where N + 1 quantities are conserved, corresponding to an energy invariant and N entropy-related invariants), the negative temperature states, corresponding to the condensation of the generalized energy into the lowest modes, are found. This indicates a generic feature of inverse energy cascade. Comparisons are made with some classical results, such as those of Charney-Hasegawa-Mima in the cold-ion limit. There is a universal shape for statistical equilibrium of gyrokinetics in three spatial and two velocity dimensions with just one conserved quantity. Possible physical relevance to turbulence, such as ITG zonal flows, and to a critical balance hypothesis are also discussed.

  20. Continuum Edge Gyrokinetic Theory and Simulations

    SciTech Connect

    Xu, X Q; Xiong, Z; Dorr, M R; Hittinger, J A; Bodi, K; Candy, J; Cohen, B I; Cohen, R H; Colella, P; Kerbel, G D; Krasheninnikov, S; Nevins, W M; Qin, H; Rognlien, T D; Snyder, P B; Umansky, M V

    2007-01-09

    The following results are presented from the development and application of TEMPEST, a fully nonlinear (full-f) five dimensional (3d2v) gyrokinetic continuum edge-plasma code. (1) As a test of the interaction of collisions and parallel streaming, TEMPEST is compared with published analytic and numerical results for endloss of particles confined by combined electrostatic and magnetic wells. Good agreement is found over a wide range of collisionality, confining potential, and mirror ratio; and the required velocity space resolution is modest. (2) In a large-aspect-ratio circular geometry, excellent agreement is found for a neoclassical equilibrium with parallel ion flow in the banana regime with zero temperature gradient and radial electric field. (3) The four-dimensional (2d2v) version of the code produces the first self-consistent simulation results of collisionless damping of geodesic acoustic modes and zonal flow (Rosenbluth-Hinton residual) with Boltzmann electrons using a full-f code. The electric field is also found to agree with the standard neoclassical expression for steep density and ion temperature gradients in the banana regime. In divertor geometry, it is found that the endloss of particles and energy induces parallel flow stronger than the core neoclassical predictions in the SOL. (5) Our 5D gyrokinetic formulation yields a set of nonlinear electrostatic gyrokinetic equations that are for both neoclassical and turbulence simulations.

  1. Gyrofluid-Gyrokinetic Hybrid Turbulence Model

    NASA Astrophysics Data System (ADS)

    Dorland, William; Mandell, Noah

    2015-11-01

    Gyrofluid models of tokamak turbulence are efficient compared to gyrokinetic models, in three senses. First, it is typically easier to develop one's intuition from fluid equations than kinetic equations. Second, because gyrofluid equations are only three-dimensional (instead of 5D or 6D), simulations with gyrofluid models require less memory than kinetic simulations and can therefore more easily fit on highly-optimized computing hardware, such as graphics processors. The third advantage is a result of the first two: one can develop and test ideas quickly with gyrofluid models. The disadvantage of gyrofluid models is their potential lack of physics fidelity. In this poster, we present our attempt to take full advantage of gyrofluid models, without sacrificing physics fidelity. Our approach is encapsulated in the Gryf-X code, which is an implementation of hybrid gyrofluid/gyrokinetic equations. The key improvements that we have brought to bear are: an improved understanding of the cascade of free energy simultaneously in k⊥ and v⊥ an improved model of zonal flow physics; and an implementation of the equations on modern heterogeneous computing platforms, both as a standalone simulation tool and as a component of TRINITY (a transport modeling code for tokamaks).

  2. Gyrokinetic large eddy simulations

    SciTech Connect

    Morel, P.; Navarro, A. Banon; Albrecht-Marc, M.; Carati, D.; Merz, F.; Goerler, T.; Jenko, F.

    2011-07-15

    The large eddy simulation approach is adapted to the study of plasma microturbulence in a fully three-dimensional gyrokinetic system. Ion temperature gradient driven turbulence is studied with the GENE code for both a standard resolution and a reduced resolution with a model for the sub-grid scale turbulence. A simple dissipative model for representing the effect of the sub-grid scales on the resolved scales is proposed and tested. Once calibrated, the model appears to be able to reproduce most of the features of the free energy spectra for various values of the ion temperature gradient.

  3. Gyrokinetic microtearing turbulence.

    PubMed

    Doerk, H; Jenko, F; Pueschel, M J; Hatch, D R

    2011-04-15

    The nonlinear dynamics of microtearing modes in standard tokamak plasmas are investigated by means of ab initio gyrokinetic simulations. The saturation levels of the magnetic field fluctuations can be understood in the framework of a balance between (small poloidal wave number) linear drive and small-scale dissipation. The resulting heat transport is dominated by the electron magnetic component, and the transport levels are found to be experimentally relevant. Microtearing modes thus constitute another candidate for explaining turbulent transport in such toroidal systems. PMID:21568567

  4. Integral equation for electrostatic waves generated by a point source in a spatially homogeneous magnetized plasma

    SciTech Connect

    Podesta, John J.

    2012-08-15

    The electric field generated by a time varying point charge in a three-dimensional, unbounded, spatially homogeneous plasma with a uniform background magnetic field and a uniform (static) flow velocity is studied in the electrostatic approximation which is often valid in the near field. For plasmas characterized by Maxwell distribution functions with isotropic temperatures, the linearized Vlasov-Poisson equations may be formulated in terms of an equivalent integral equation in the time domain. The kernel of the integral equation has a relatively simple mathematical form consisting of elementary functions such as exponential and trigonometric functions (sines and cosines), and contains no infinite sums of Bessel functions. Consequently, the integral equation is amenable to numerical solutions and may be useful for the study of the impulse response of magnetized plasmas and, more generally, the response to arbitrary waveforms.

  5. Gyrokinetic Formalism in Plasmas with Flows

    NASA Astrophysics Data System (ADS)

    Chiu, S. C.; Chan, V. S.; Lin-Liu, Y. R.; Chu, M. S.

    1998-11-01

    There is considerable theoretical and experimental evidence that plasma rotations in tokamaks can improve stability and confinement. Understanding the physics of rotating tokamak plasmas is thus of great importance for attaining high performance in reactors. Since present and future tokamaks are likely to have auxiliary heating such as radio frequency (RF) waves or neutral beam injection, it is of interest to understand the interaction of auxiliary heating with plasma rotation, and its consequences to transport and stability. In the past, rotation has not been self-consistently treated in auxiliary heating in that it is either ignored or ambipolarity is not self-consistently maintained. In this work, we describe a gyrokinetic formalism for RF in rotating plasmas which takes ambipolarity and equilibrium into account. A non-canonical guiding center Lagrangian for rotating plasma in a fluctuating field is obtained, and from which the gyrokinetic equation is deduced. Limiting the wave equations to the cyclotron frequency or below, quasilinear equations using an eikonal approximation are derived. Conservation laws for the system shall be discussed.

  6. The theory of gyrokinetic turbulence: A multiple-scales approach

    NASA Astrophysics Data System (ADS)

    Plunk, Gabriel Galad

    Gyrokinetics is a rich and rewarding playground to study some of the mysteries of modern physics -- such as turbulence, universality, self-organization and dynamic criticality -- which are found in physical systems that are driven far from thermodynamic equilibrium. One such system is of particular importance, as it is central in the development of fusion energy -- this system is the turbulent plasma found in magnetically confined fusion device. In this thesis I present work, motivated by the quest for fusion energy, which seeks to uncover some of the inner workings of turbulence in magnetized plasmas. I present three projects, based on the work of me and my collaborators, which take a tour of different aspects and approaches to the gyrokinetic turbulence problem. I begin with the fundamental theory of gyrokinetics, and a novel formulation of its extension to the equations for mean-scale transport -- the equations which must be solved to determine the performance of Magnetically confined fusion devices. The results of this work include (1) the equations of evolution for the mean scale (equilibrium) density, temperature and magnetic field of the plasma, (2) a detailed Poynting's theorem for the energy balance and (3) the entropy balance equations. The second project presents gyrokinetic secondary instability theory as a mechanism to bring about saturation of the basic instabilities that drive gyrokinetic turbulence. Emphasis is put on the ability for this analytic theory to predict basic properties of the nonlinear state, which can be applied to a mixing length phenomenology of transport. The results of this work include (1) an integral equation for the calculation of the growth rate of the fully gyrokinetic secondary instability with finite Larmor radius (FLR) affects included exactly, (2) the demonstration of the robustness of the secondary instability at fine scales (krhoi for ion temperature gradient (ITG) turbulence and krhoe ≪ 1 for electron temperature

  7. Nonequilibrium Gyrokinetic Fluctuation Theory and Sampling Noise in Gyrokinetic Particle-in-cell Simulations

    SciTech Connect

    John A. Krommes

    2007-10-09

    The present state of the theory of fluctuations in gyrokinetic GK plasmas and especially its application to sampling noise in GK particle-in-cell PIC simulations is reviewed. Topics addressed include the Δf method, the fluctuation-dissipation theorem for both classical and GK many-body plasmas, the Klimontovich formalism, sampling noise in PIC simulations, statistical closure for partial differential equations, the theoretical foundations of spectral balance in the presence of arbitrary noise sources, and the derivation of Kadomtsev-type equations from the general formalism.

  8. Transport in gyrokinetic tokamaks

    SciTech Connect

    Mynick, H.E.; Parker, S.E.

    1995-01-01

    A comprehensive study of transport in full-volume gyrokinetic (gk) simulations of ion temperature gradient driven turbulence in core tokamak plasmas is presented. Though this ``gyrokinetic tokamak`` is much simpler than experimental tokamaks, such simplicity is an asset, because a dependable nonlinear transport theory for such systems should be more attainable. Toward this end, we pursue two related lines of inquiry. (1) We study the scalings of gk tokamaks with respect to important system parameters. In contrast to real machines, the scalings of larger gk systems (a/{rho}{sub s} {approx_gt} 64) with minor radius, with current, and with a/{rho}{sub s} are roughly consistent with the approximate theoretical expectations for electrostatic turbulent transport which exist as yet. Smaller systems manifest quite different scalings, which aids in interpreting differing mass-scaling results in other work. (2) With the goal of developing a first-principles theory of gk transport, we use the gk data to infer the underlying transport physics. The data indicate that, of the many modes k present in the simulation, only a modest number (N{sub k} {approximately} 10) of k dominate the transport, and for each, only a handful (N{sub p} {approximately} 5) of couplings to other modes p appear to be significant, implying that the essential transport physics may be described by a far simpler system than would have been expected on the basis of earlier nonlinear theory alone. Part of this analysis is the inference of the coupling coefficients M{sub kpq} governing the nonlinear mode interactions, whose measurement from tokamak simulation data is presented here for the first time.

  9. Final Technical Report: Global Field Aligned Mesh and Gyrokinetic Field Solver in a Tokamak Edge Geometry

    SciTech Connect

    Cummings, Julian C.

    2013-05-15

    This project was a collaboration between researchers at the California Institute of Technology and the University of California, Irvine to investigate the utility of a global field-aligned mesh and gyrokinetic field solver for simulations of the tokamak plasma edge region. Mesh generation software from UC Irvine was tested with specific tokamak edge magnetic geometry scenarios and the quality of the meshes and the solutions to the gyrokinetic Poisson equation were evaluated.

  10. Nonlinear gyrokinetic theory for finite-BETA plasmas

    SciTech Connect

    Hahm, T.S.; Lee, W.W.; Brizard, A.

    1988-02-01

    A self-consistent and energy-conserving set of nonlinear gyrokinetic equations, consisting of the averaged Vlasov and Maxwell's equations for finite-..beta.. plasmas, is derived. The method utilized in the present investigation is based on the Hamiltonian formalism and Lie transformation. The resulting formation is valid for arbitrary values of k/perpendicular//rho//sub i/ and, therefore, is most suitable for studying linear and nonlinear evolution of microinstabilities in tokamak plasmas as well as other areas of plasma physics where the finite Larmor radius effects are important. Because the underlying Hamiltonian structure is preserved in the present formalism, these equations are directly applicable to numerical studies based on the existing gyrokinetic particle simulation techniques. 31 refs.

  11. Gyrokinetic particle simulation of microturbulence for general magnetic geometry and experimental profiles

    SciTech Connect

    Xiao, Yong; Holod, Ihor; Wang, Zhixuan; Lin, Zhihong; Zhang, Taige

    2015-02-15

    Developments in gyrokinetic particle simulation enable the gyrokinetic toroidal code (GTC) to simulate turbulent transport in tokamaks with realistic equilibrium profiles and plasma geometry, which is a critical step in the code–experiment validation process. These new developments include numerical equilibrium representation using B-splines, a new Poisson solver based on finite difference using field-aligned mesh and magnetic flux coordinates, a new zonal flow solver for general geometry, and improvements on the conventional four-point gyroaverage with nonuniform background marker loading. The gyrokinetic Poisson equation is solved in the perpendicular plane instead of the poloidal plane. Exploiting these new features, GTC is able to simulate a typical DIII-D discharge with experimental magnetic geometry and profiles. The simulated turbulent heat diffusivity and its radial profile show good agreement with other gyrokinetic codes. The newly developed nonuniform loading method provides a modified radial transport profile to that of the conventional uniform loading method.

  12. Simulation of neoclassical transport with the continuum gyrokinetic code COGENT

    SciTech Connect

    Dorf, M. A.; Cohen, R. H.; Dorr, M.; Rognlien, T.; Hittinger, J.; Compton, J.; Colella, P.; Martin, D.; McCorquodale, P.

    2013-01-15

    The development of the continuum gyrokinetic code COGENT for edge plasma simulations is reported. The present version of the code models a nonlinear axisymmetric 4D (R, v{sub Parallel-To }, {mu}) gyrokinetic equation coupled to the long-wavelength limit of the gyro-Poisson equation. Here, R is the particle gyrocenter coordinate in the poloidal plane, and v{sub Parallel-To} and {mu} are the guiding center velocity parallel to the magnetic field and the magnetic moment, respectively. The COGENT code utilizes a fourth-order finite-volume (conservative) discretization combined with arbitrary mapped multiblock grid technology (nearly field-aligned on blocks) to handle the complexity of tokamak divertor geometry with high accuracy. Topics presented are the implementation of increasingly detailed model collision operators, and the results of neoclassical transport simulations including the effects of a strong radial electric field characteristic of a tokamak pedestal under H-mode conditions.

  13. Simulation of neoclassical transport with the continuum gyrokinetic code COGENT

    DOE PAGESBeta

    Dorf, M. A.; Cohen, R. H.; Dorr, M.; Rognlien, T.; Hittinger, J.; Compton, J.; Colella, P.; Martin, D.; McCorquodale, P.

    2013-01-25

    The development of the continuum gyrokinetic code COGENT for edge plasma simulations is reported. The present version of the code models a nonlinear axisymmetric 4D (R, v∥, μ) gyrokinetic equation coupled to the long-wavelength limit of the gyro-Poisson equation. Here, R is the particle gyrocenter coordinate in the poloidal plane, and v∥ and μ are the guiding center velocity parallel to the magnetic field and the magnetic moment, respectively. The COGENT code utilizes a fourth-order finite-volume (conservative) discretization combined with arbitrary mapped multiblock grid technology (nearly field-aligned on blocks) to handle the complexity of tokamak divertor geometry with high accuracy.more » Furthermore, topics presented are the implementation of increasingly detailed model collision operators, and the results of neoclassical transport simulations including the effects of a strong radial electric field characteristic of a tokamak pedestal under H-mode conditions.« less

  14. Simulation of neoclassical transport with the continuum gyrokinetic code COGENT

    SciTech Connect

    Dorf, M. A.; Cohen, R. H.; Dorr, M.; Rognlien, T.; Hittinger, J.; Compton, J.; Colella, P.; Martin, D.; McCorquodale, P.

    2013-01-25

    The development of the continuum gyrokinetic code COGENT for edge plasma simulations is reported. The present version of the code models a nonlinear axisymmetric 4D (R, v∥, μ) gyrokinetic equation coupled to the long-wavelength limit of the gyro-Poisson equation. Here, R is the particle gyrocenter coordinate in the poloidal plane, and v∥ and μ are the guiding center velocity parallel to the magnetic field and the magnetic moment, respectively. The COGENT code utilizes a fourth-order finite-volume (conservative) discretization combined with arbitrary mapped multiblock grid technology (nearly field-aligned on blocks) to handle the complexity of tokamak divertor geometry with high accuracy. Furthermore, topics presented are the implementation of increasingly detailed model collision operators, and the results of neoclassical transport simulations including the effects of a strong radial electric field characteristic of a tokamak pedestal under H-mode conditions.

  15. Effects of collisions on conservation laws in gyrokinetic field theory

    SciTech Connect

    Sugama, H.; Nunami, M.; Watanabe, T.-H.

    2015-08-15

    Effects of collisions on conservation laws for toroidal plasmas are investigated based on the gyrokinetic field theory. Associating the collisional system with a corresponding collisionless system at a given time such that the two systems have the same distribution functions and electromagnetic fields instantaneously, it is shown how the collisionless conservation laws derived from Noether's theorem are modified by the collision term. Effects of the external source term added into the gyrokinetic equation can be formulated similarly with the collisional effects. Particle, energy, and toroidal momentum balance equations including collisional and turbulent transport fluxes are systematically derived using a novel gyrokinetic collision operator, by which the collisional change rates of energy and canonical toroidal angular momentum per unit volume in the gyrocenter space can be given in the conservative forms. The ensemble-averaged transport equations of particles, energy, and toroidal momentum given in the present work are shown to include classical, neoclassical, and turbulent transport fluxes which agree with those derived from conventional recursive formulations.

  16. Effects of collisions on conservation laws in gyrokinetic field theory

    NASA Astrophysics Data System (ADS)

    Sugama, H.; Watanabe, T.-H.; Nunami, M.

    2015-08-01

    Effects of collisions on conservation laws for toroidal plasmas are investigated based on the gyrokinetic field theory. Associating the collisional system with a corresponding collisionless system at a given time such that the two systems have the same distribution functions and electromagnetic fields instantaneously, it is shown how the collisionless conservation laws derived from Noether's theorem are modified by the collision term. Effects of the external source term added into the gyrokinetic equation can be formulated similarly with the collisional effects. Particle, energy, and toroidal momentum balance equations including collisional and turbulent transport fluxes are systematically derived using a novel gyrokinetic collision operator, by which the collisional change rates of energy and canonical toroidal angular momentum per unit volume in the gyrocenter space can be given in the conservative forms. The ensemble-averaged transport equations of particles, energy, and toroidal momentum given in the present work are shown to include classical, neoclassical, and turbulent transport fluxes which agree with those derived from conventional recursive formulations.

  17. Gyrokinetic simulations of the tearing instability

    SciTech Connect

    Numata, Ryusuke; Dorland, William; Howes, Gregory G.; Loureiro, Nuno F.; Tatsuno, Tomoya

    2011-11-15

    Linear gyrokinetic simulations covering the collisional-collisionless transitional regime of the tearing instability are performed. It is shown that the growth rate scaling with collisionality agrees well with that predicted by a two-fluid theory for a low plasma beta case in which ion kinetic dynamics are negligible. Electron wave-particle interactions (Landau damping), finite Larmor radius, and other kinetic effects invalidate the fluid theory in the collisionless regime, in which a general non-polytropic equation of state for pressure (temperature) perturbations should be considered. We also vary the ratio of the background ion to electron temperatures and show that the scalings expected from existing calculations can be recovered, but only in the limit of very low beta.

  18. On push-forward representations in the standard gyrokinetic model

    SciTech Connect

    Miyato, N. Yagi, M.; Scott, B. D.

    2015-01-15

    Two representations of fluid moments in terms of a gyro-center distribution function and gyro-center coordinates, which are called push-forward representations, are compared in the standard electrostatic gyrokinetic model. In the representation conventionally used to derive the gyrokinetic Poisson equation, the pull-back transformation of the gyro-center distribution function contains effects of the gyro-center transformation and therefore electrostatic potential fluctuations, which is described by the Poisson brackets between the distribution function and scalar functions generating the gyro-center transformation. Usually, only the lowest order solution of the generating function at first order is considered to explicitly derive the gyrokinetic Poisson equation. This is true in explicitly deriving representations of scalar fluid moments with polarization terms. One also recovers the particle diamagnetic flux at this order because it is associated with the guiding-center transformation. However, higher-order solutions are needed to derive finite Larmor radius terms of particle flux including the polarization drift flux from the conventional representation. On the other hand, the lowest order solution is sufficient for the other representation, in which the gyro-center transformation part is combined with the guiding-center one and the pull-back transformation of the distribution function does not appear.

  19. Discoveries from the exploration of gyrokinetic momentum transport

    SciTech Connect

    Staebler, G.M.; Waltz, R. E.; Kinsey, J. E.

    2011-05-15

    The momentum transport due to gyroradius scale turbulence in tokamak plasmas is very complex. In general, some type of breaking of the parity of the gyrokinetic equation under simultaneous reflection of the poloidal angle and the sign of the parallel velocity phase space coordinate (poloidal parity) is always involved. There are three distinct types of poloidal parity breaking effects. In this paper, all three types of poloidal parity breaking are explored using the quasi-linear trapped gyro-Landau fluid [G. M. Staebler et al., Phys. Plasmas 12, 102508 (2005)] transport code. Selected results are verified with full nonlinear turbulence simulations using the gyro [J. Candy et al., J. Comput. Phys. 186, 545 (2003)] gyrokinetic code. The observable properties like an energy pinch driven by a parallel velocity shear and a dependence of momentum transport on the direction of the ion grad-B drift relative to the X-point location in single null divertor geometry have been discovered.

  20. Discoveries from the exploration of gyrokinetic momentum transporta)

    NASA Astrophysics Data System (ADS)

    Staebler, G. M.; Waltz, R. E.; Kinsey, J. E.

    2011-05-01

    The momentum transport due to gyroradius scale turbulence in tokamak plasmas is very complex. In general, some type of breaking of the parity of the gyrokinetic equation under simultaneous reflection of the poloidal angle and the sign of the parallel velocity phase space coordinate (poloidal parity) is always involved. There are three distinct types of poloidal parity breaking effects. In this paper, all three types of poloidal parity breaking are explored using the quasi-linear trapped gyro-Landau fluid [G. M. Staebler et al., Phys. Plasmas 12, 102508 (2005)] transport code. Selected results are verified with full nonlinear turbulence simulations using the gyro [J. Candy et al., J. Comput. Phys. 186, 545 (2003)] gyrokinetic code. The observable properties like an energy pinch driven by a parallel velocity shear and a dependence of momentum transport on the direction of the ion grad-B drift relative to the X-point location in single null divertor geometry have been discovered.

  1. High frequency gyrokinetic particle simulation

    SciTech Connect

    Kolesnikov, R. A.; Lee, W. W.; Qin, H.; Startsev, E.

    2007-07-15

    The gyrokinetic approach for arbitrary frequency dynamics in magnetized plasmas is explored, using the gyrocenter-gauge kinetic theory. Contrary to low-frequency gyrokinetics, which views each particle as a rigid charged ring, arbitrary frequency response of a particle is described by a quickly changing Kruskal ring. This approach allows the separation of gyrocenter and gyrophase responses and thus allows for, in many situations, larger time steps for the gyrocenter push than for the gyrophase push. The gyrophase response which determines the shape of Kruskal rings can be described by a Fourier series in gyrophase for some problems, thus allowing control over the cyclotron harmonics at which the plasma responds. A computational algorithm for particle-in-cell simulation based on this concept has been developed. An example of the ion Bernstein wave is used to illustrate its numerical properties, and comparison with a direct Lorentz-force approach is presented.

  2. A new hybrid kinetic electron model for full-f gyrokinetic simulations

    NASA Astrophysics Data System (ADS)

    Idomura, Y.

    2016-05-01

    A new hybrid kinetic electron model is developed for electrostatic full-f gyrokinetic simulations of the ion temperature gradient driven trapped electron mode (ITG-TEM) turbulence at the ion scale. In the model, a full kinetic electron model is applied to the full-f gyrokinetic equation, the multi-species linear Fokker-Planck collision operator, and an axisymmetric part of the gyrokinetic Poisson equation, while in a non-axisymmetric part of the gyrokinetic Poisson equation, turbulent fluctuations are determined only by kinetic trapped electrons responses. By using this approach, the so-called ωH mode is avoided with keeping important physics such as the ITG-TEM, the neoclassical transport, the ambipolar condition, and particle trapping and detrapping processes. The model enables full-f gyrokinetic simulations of ITG-TEM turbulence with a reasonable computational cost. Comparisons between flux driven ITG turbulence simulations with kinetic and adiabatic electrons are presented. Although the similar ion temperature gradients with nonlinear upshift from linear critical gradients are sustained in quasi-steady states, parallel flows and radial electric fields are qualitatively different with kinetic electrons.

  3. A generalized gyrokinetic Poisson solver

    SciTech Connect

    Lin, Z.; Lee, W.W.

    1995-03-01

    A generalized gyrokinetic Poisson solver has been developed, which employs local operations in the configuration space to compute the polarization density response. The new technique is based on the actual physical process of gyrophase-averaging. It is useful for nonlocal simulations using general geometry equilibrium. Since it utilizes local operations rather than the global ones such as FFT, the new method is most amenable to massively parallel algorithms.

  4. Gyro-water-bag approach in nonlinear gyrokinetic turbulence

    SciTech Connect

    Besse, Nicolas Bertrand, Pierre

    2009-06-20

    Turbulent transport is a key issue for controlled thermonuclear fusion based on magnetic confinement. The thermal confinement of a magnetized fusion plasma is essentially determined by the turbulent heat conduction across the equilibrium magnetic field. It has long been acknowledged, that the prediction of turbulent transport requires to solve Vlasov-type gyrokinetic equations. Although the kinetic description is more accurate than fluid models (MHD, gyro-fluid), because among other things it takes into account nonlinear resonant wave-particle interaction, kinetic modeling has the drawback of a huge computer resource request. An unifying approach consists in considering water-bag-like weak solutions of kinetic collisionless equations, which allow to reduce the full kinetic Vlasov equation into a set of hydrodynamic equations, while keeping its kinetic behaviour. As a result this exact reduction induces a multi-fluid numerical resolution cost. Therefore finding water-bag-like weak solutions of the gyrokinetic equations leads to the birth of the gyro-water-bag model. This model is suitable for studying linear and nonlinear low-frequency micro-instabilities and the associated anomalous transport in magnetically-confined plasmas. The present paper addresses the derivation of the nonlinear gyro-water-bag model, its quasilinear approximation and their numerical approximations by Runge-Kutta semi-Lagrangian methods and Runge-Kutta discontinuous Galerkin schemes respectively.

  5. Equilibrium fluctuation energy of gyrokinetic plasma

    SciTech Connect

    Krommes, J.A.; Lee, W.W.; Oberman, C.

    1985-11-01

    The thermal equilibrium electric field fluctuation energy of the gyrokinetic model of magnetized plasma is computed, and found to be smaller than the well-known result (k)/8..pi.. = 1/2T/(1 + (klambda/sub D/)/sup 2/) valid for arbitrarily magnetized plasmas. It is shown that, in a certain sense, the equilibrium electric field energy is minimum in the gyrokinetic regime. 13 refs., 2 figs.

  6. Testing the high turbulence level breakdown of low-frequency gyrokinetics against high-frequency cyclokinetic simulations

    SciTech Connect

    Deng, Zhao; Waltz, R. E.

    2015-05-15

    This paper presents numerical simulations of the nonlinear cyclokinetic equations in the cyclotron harmonic representation [R. E. Waltz and Zhao Deng, Phys. Plasmas 20, 012507 (2013)]. Simulations are done with a local flux-tube geometry and with the parallel motion and variation suppressed using a newly developed rCYCLO code. Cyclokinetic simulations dynamically follow the high-frequency ion gyro-phase motion which is nonlinearly coupled into the low-frequency drift-waves possibly interrupting and suppressing gyro-averaging and increasing the transport over gyrokinetic levels. By comparing the more fundamental cyclokinetic simulations with the corresponding gyrokinetic simulations, the breakdown of gyrokinetics at high turbulence levels is quantitatively tested over a range of relative ion cyclotron frequency 10 < Ω*{sup  }< 100 where Ω*{sup  }= 1/ρ*, and ρ* is the relative ion gyroradius. The gyrokinetic linear mode rates closely match the cyclokinetic low-frequency rates for Ω*{sup  }> 5. Gyrokinetic transport recovers cyclokinetic transport at high relative ion cyclotron frequency (Ω*{sup  }≥ 50) and low turbulence level as required. Cyclokinetic transport is found to be lower than gyrokinetic transport at high turbulence levels and low-Ω* values with stable ion cyclotron (IC) modes. The gyrokinetic approximation is found to break down when the density perturbations exceed 20%. For cyclokinetic simulations with sufficiently unstable IC modes and sufficiently low Ω*{sup  }∼ 10, the high-frequency component of cyclokinetic transport level can exceed the gyrokinetic transport level. However, the low-frequency component of the cyclokinetic transport and turbulence level does not exceed that of gyrokinetics. At higher and more physically relevant Ω*{sup  }≥ 50 values and physically realistic IC driving rates, the low-frequency component of the cyclokinetic transport and turbulence level is still smaller than that of

  7. Gyrokinetic modeling: A multi-water-bag approach

    SciTech Connect

    Morel, P.; Gravier, E.; Besse, N.; Klein, R.; Ghizzo, A.; Bertrand, P.; Garbet, X.; Ghendrih, P.; Grandgirard, V.; Sarazin, Y.

    2007-11-15

    Predicting turbulent transport in nearly collisionless fusion plasmas requires one to solve kinetic (or, more precisely, gyrokinetic) equations. In spite of considerable progress, several pending issues remain; although more accurate, the kinetic calculation of turbulent transport is much more demanding in computer resources than fluid simulations. An alternative approach is based on a water-bag representation of the distribution function that is not an approximation but rather a special class of initial conditions, allowing one to reduce the full kinetic Vlasov equation into a set of hydrodynamic equations while keeping its kinetic character. The main result for the water-bag model is a lower cost in the parallel velocity direction since no differential operator associated with some approximate numerical scheme has to be carried out on this variable v{sub parallel}. Indeed, a small bag number is sufficient to correctly describe the ion temperature gradient instability.

  8. A gyrokinetic one-dimensional scrape-off layer model of an edge-localized mode heat pulse

    DOE PAGESBeta

    Shi, E. L.; Hakim, A. H.; Hammett, G. W.

    2015-02-03

    An electrostatic gyrokinetic-based model is applied to simulate parallel plasma transport in the scrape-off layer to a divertor plate. We focus on a test problem that has been studied previously, using parameters chosen to model a heat pulse driven by an edge-localized mode in JET. Previous work has used direct particle-in-cellequations with full dynamics, or Vlasov or fluid equations with only parallel dynamics. With the use of the gyrokinetic quasineutrality equation and logical sheathboundary conditions, spatial and temporal resolution requirements are no longer set by the electron Debye length and plasma frequency, respectively. Finally, this test problem also helps illustratemore » some of the physics contained in the Hamiltonian form of the gyrokineticequations and some of the numerical challenges in developing an edge gyrokinetic code.« less

  9. A gyrokinetic one-dimensional scrape-off layer model of an edge-localized mode heat pulse

    SciTech Connect

    Shi, E. L.; Hakim, A. H.; Hammett, G. W.

    2015-02-03

    An electrostatic gyrokinetic-based model is applied to simulate parallel plasma transport in the scrape-off layer to a divertor plate. We focus on a test problem that has been studied previously, using parameters chosen to model a heat pulse driven by an edge-localized mode in JET. Previous work has used direct particle-in-cellequations with full dynamics, or Vlasov or fluid equations with only parallel dynamics. With the use of the gyrokinetic quasineutrality equation and logical sheathboundary conditions, spatial and temporal resolution requirements are no longer set by the electron Debye length and plasma frequency, respectively. Finally, this test problem also helps illustrate some of the physics contained in the Hamiltonian form of the gyrokineticequations and some of the numerical challenges in developing an edge gyrokinetic code.

  10. Dynamic procedure for filtered gyrokinetic simulations

    SciTech Connect

    Morel, P.; Banon Navarro, A.; Albrecht-Marc, M.; Carati, D.; Merz, F.; Goerler, T.; Jenko, F.

    2012-01-15

    Large eddy simulations (LES) of gyrokinetic plasma turbulence are investigated as interesting candidates to decrease the computational cost. A dynamic procedure is implemented in the gene code, allowing for dynamic optimization of the free parameters of the LES models (setting the amplitudes of dissipative terms). Employing such LES methods, one recovers the free energy and heat flux spectra obtained from highly resolved direct numerical simulations. Systematic comparisons are performed for different values of the temperature gradient and magnetic shear, parameters which are of prime importance in ion temperature gradient driven turbulence. Moreover, the degree of anisotropy of the problem, which can vary with parameters, can be adapted dynamically by the method that shows gyrokinetic large eddy simulation to be a serious candidate to reduce numerical cost of gyrokinetic solvers.

  11. Gyrokinetic theory and simulation of angular momentum transport

    SciTech Connect

    Waltz, R. E.; Staebler, G. M.; Candy, J.; Hinton, F. L.

    2007-12-15

    A gyrokinetic theory of turbulent toroidal angular momentum transport as well as modifications to neoclassical poloidal rotation from turbulence is formulated starting from the fundamental six-dimensional kinetic equation. The gyro-Bohm scaled transport is evaluated from toroidal delta-f gyrokinetic simulations using the GYRO code [Candy and Waltz, J. Comput. Phys. 186, 545 (2003)]. The simulations recover two pinch mechanisms in the radial transport of toroidal angular momentum: The slab geometry ExB shear pinch [Dominguez and Staebler, Phys. Fluids B 5, 387 (1993)] and the toroidal geometry 'Coriolis' pinch [Peeters, Angioni, and Strintzi, Phys. Rev. Lett. 98, 265003 (2007)]. The pinches allow the steady state null stress (or angular momentum transport flow) condition required to understand intrinsic (or spontaneous) toroidal rotation in heated tokamak without an internal source of torque [Staebler, Kinsey, and Waltz, Bull. Am. Phys. Soc. 46, 221 (2001)]. A predicted turbulent shift in the neoclassical poloidal rotation [Staebler, Phys. Plasmas 11, 1064 (2004)] appears to be small at the finite relative gyroradius (rho-star) of current experiments.

  12. Linear multispecies gyrokinetic flux tube benchmarks in shaped tokamak plasmas

    NASA Astrophysics Data System (ADS)

    Merlo, G.; Sauter, O.; Brunner, S.; Burckel, A.; Camenen, Y.; Casson, F. J.; Dorland, W.; Fable, E.; Görler, T.; Jenko, F.; Peeters, A. G.; Told, D.; Villard, L.

    2016-03-01

    Verification is the fundamental step that any turbulence simulation code has to be submitted in order to assess the proper implementation of the underlying equations. We have carried out a cross comparison of three flux tube gyrokinetic codes, GENE [F. Jenko et al., Phys. Plasmas 7, 1904 (2000)], GKW [A. G. Peeters et al., Comput. Phys. Commun. 180, 2650 (2009)], and GS2 [W. Dorland et al., Phys. Rev. Lett. 85, 5579 (2000)], focusing our attention on the effect of realistic geometries described by a series of MHD equilibria with increasing shaping complexity. To simplify the effort, the benchmark has been limited to the electrostatic collisionless linear behaviour of the system. A fully gyrokinetic model has been used to describe the dynamics of both ions and electrons. Several tests have been carried out looking at linear stability at ion and electron scales, where for the assumed profiles Ion Temperature Gradient (ITG)/Trapped Electron Modes and Electron Temperature Gradient modes are unstable. The capability of the codes to handle a non-zero ballooning angle has been successfully benchmarked in the ITG regime. Finally, the standard Rosenbluth-Hinton test has been successfully carried out looking at the effect of shaping on Zonal Flows (ZFs) and Geodesic Acoustic Modes (GAMs). Inter-code comparison as well as validation of simulation results against analytical estimates has been accomplished. All the performed tests confirm that plasma elongation strongly stabilizes plasma instabilities as well as leads to a strong increase in ZF residual and GAM damping.

  13. Free Energy Cascade in Gyrokinetic Turbulence

    SciTech Connect

    Banon Navarro, A.; Morel, P.; Albrecht-Marc, M.; Carati, D.; Merz, F.; Goerler, T.; Jenko, F.

    2011-02-04

    In gyrokinetic theory, the quadratic nonlinearity is known to play an important role in the dynamics by redistributing (in a conservative fashion) the free energy between the various active scales. In the present study, the free energy transfer is analyzed for the case of ion temperature gradient driven turbulence. It is shown that it shares many properties with the energy transfer in fluid turbulence. In particular, one finds a (strongly) local, forward (from large to small scales) cascade of free energy in the plane perpendicular to the background magnetic field. These findings shed light on some fundamental properties of plasma turbulence, and encourage the development of large-eddy-simulation techniques for gyrokinetics.

  14. Gyrokinetic simulation studies on the energetic-particle-induced geodesic acoustic mode

    NASA Astrophysics Data System (ADS)

    Miki, Kazuhiro; Idomura, Yasuhiro

    2014-10-01

    Understanding of the energetic particles physics is of great interest in the future burning plasmas. Particularly, particle loss in the presence of EGAM may be critical for ITER. We thus need to know how EGAM is excited and interacts with turbulence. We here introduce energetic particles in a full-f gyrokinetic code (GT5D). (i) We find linear dynamics of the EGAM driven by bump-on-tail particle distributions. We examine flat-q, homogeneous, axisymmetric, electrostatic gyrokinetic simulations. Above a certain level of the beam intensity, an oscillatory mode grows with about a half of the standard GAM. The observed frequencies are consistent with the eigenmode analyses derived from the perturbed gyrokinetic equations. The theoretical analyses also indicate a bifurcation of the excited modes depending on q-value. Estimation of the finite-orbit-width effects can provide a size dependency of the EGAM growth rate. (ii) We find linear and nonlinear dynamics of the EGAM driven by slowing-down distributions. We examine the axisymmetric gyrokinetic simulations with DIII-D-like parameters. The observed growth rates and frequencies are consistent with results of other hybrid code. Furthermore, we will focus on nonlinear phase space dynamics, namely chirping mode. This work is supported by HPCI Strategic Program Field No.4: Next-Generation Industrial Innovations, funded by the MEXT, Japan.

  15. Gyrokinetic studies of core turbulence features in ASDEX Upgrade: Can gyrokinetic simulations match the fluctuation measurements?

    NASA Astrophysics Data System (ADS)

    Banon Navarro, Alejandro

    2015-11-01

    Worldwide, gyrokinetic codes are used to predict the dominant micro-instabilities as well as the resulting anomalous transport in fusion experiments. A careful verification and validation of these codes is crucial to develop confidence in the model and improving the predictive capabilities of the numerical simulations. To date, the validation of gyrokinetic simulations versus experiments is mainly done at a macroscopic level, namely, by comparing turbulent heat fluxes. This is usually achieved by varying the profile gradients within the experimental error bars until a match with the experimental heat fluxes is obtained. However, since the turbulent fluxes are caused by plasma fluctuations on microscopic scales, it is also necessary to validate gyrokinetic codes on a microscopic level. We will describe a recent step in this direction by presenting simulation results with the gyrokinetic code GENE for an ASDEX Upgrade discharge. In particular, after flux-matched simulations are achieved, density fluctuations measured by means of Doppler reflectometry are compared with results of gyrokinetic simulations. We will also show that density and temperature fluctuation amplitudes and even the fluctuation spectra can be very sensitive to small changes in the profile gradients. This implies that a match of gyrokinetic simulations with experiment measurements for these quantities can be very difficult to achieve. However, it is observed that cross-phases between different quantities are robust to changes in this parameter, indicating that cross-phases could be a better observable for comparisons with experimental measurements.

  16. 3D hybrid simulations with gyrokinetic particle ions and fluid electrons

    SciTech Connect

    Belova, E.V.; Park, W.; Fu, G.Y.; Strauss, H.R.; Sugiyama, L.E.

    1998-12-31

    The previous hybrid MHD/particle model (MH3D-K code) represented energetic ions as gyrokinetic (or drift-kinetic) particles coupled to MHD equations using the pressure or current coupling scheme. A small energetic to bulk ion density ratio was assumed, n{sub h}/n{sub b} {much_lt} 1, allowing the neglect of the energetic ion perpendicular inertia in the momentum equation and the use of MHD Ohm`s law E = {minus}v{sub b} {times} B. A generalization of this model in which all ions are treated as gyrokinetic/drift-kinetic particles and fluid description is used for the electron dynamics is considered in this paper.

  17. Vlasov-Poisson in 1D for initially cold systems: post-collapse Lagrangian perturbation theory

    NASA Astrophysics Data System (ADS)

    Colombi, Stéphane

    2015-01-01

    We study analytically the collapse of an initially smooth, cold, self-gravitating collisionless system in one dimension. The system is described as a central 'S' shape in phase-space surrounded by a nearly stationary halo acting locally like a harmonic background on the S. To resolve the dynamics of the S under its self-gravity and under the influence of the halo, we introduce a novel approach using post-collapse Lagrangian perturbation theory. This approach allows us to follow the evolution of the system between successive crossing times and to describe in an iterative way the interplay between the central S and the halo. Our theoretical predictions are checked against measurements in entropy conserving numerical simulations based on the waterbag method. While our post-collapse Lagrangian approach does not allow us to compute rigorously the long-term behaviour of the system, i.e. after many crossing times, it explains the close to power-law behaviour of the projected density observed in numerical simulations. Pushing the model at late time suggests that the system could build at some point a very small flat core, but this is very speculative. This analysis shows that understanding the dynamics of initially cold systems requires a fine-grained approach for a correct description of their very central part. The analyses performed here can certainly be extended to spherical symmetry.

  18. Status of Continuum Edge Gyrokinetic Code Physics Development

    SciTech Connect

    Xu, X Q; Xiong, Z; Dorr, M R; Hittinger, J A; Kerbel, G D; Nevins, W M; Cohen, B I; Cohen, R H

    2005-05-31

    We are developing an edge gyro-kinetic continuum simulation code to study the boundary plasma over a region extending from inside the H-mode pedestal across the separatrix to the divertor plates. A 4-D ({psi}, {theta}, {epsilon}, {mu}) version of this code is presently being implemented, en route to a full 5-D version. A set of gyrokinetic equations[1] are discretized on computational grid which incorporates X-point divertor geometry. The present implementation is a Method of Lines approach where the phase-space derivatives are discretized with finite differences and implicit backwards differencing formulas are used to advance the system in time. A fourth order upwinding algorithm is used for particle cross-field drifts, parallel streaming, and acceleration. Boundary conditions at conducting material surfaces are implemented on the plasma side of the sheath. The Poisson-like equation is solved using GMRES with multi-grid preconditioner from HYPRE. A nonlinear Fokker-Planck collision operator from STELLA[2] in ({nu}{sub {parallel}},{nu}{sub {perpendicular}}) has been streamlined and integrated into the gyro-kinetic package using the same implicit Newton-Krylov solver and interpolating F and dF/dt|{sub coll} to/from ({epsilon}, {mu}) space. With our 4D code we compute the ion thermal flux, ion parallel velocity, self-consistent electric field, and geo-acoustic oscillations, which we compare with standard neoclassical theory for core plasma parameters; and we study the transition from collisional to collisionless end-loss. In the real X-point geometry, we find that the particles are trapped near outside midplane and in the X-point regions due to the magnetic configurations. The sizes of banana orbits are comparable to the pedestal width and/or the SOL width for energetic trapped particles. The effect of the real X-point geometry and edge plasma conditions on standard neoclassical theory will be evaluated, including a comparison of our 4D code with other kinetic

  19. Extension of Discontinuous Galerkin Algorithms To Preserve the Locality of Parallel Gyrokinetic Dynamics

    NASA Astrophysics Data System (ADS)

    Hammett, G. W.; Hakim, A.

    2012-10-01

    A wide range of physics problems, including gyrokinetics, have an underlying Hamiltonian structure that can be expressed in terms of a Poisson bracket, which leads to two quadratic invariants, such as the energy and enstrophy invariants in 2-D hydrodynamics or Hasegawa-Mima equations. A type of Discontinuous Galerkin (DG) algorithm has been developed in the literature that can preserve both invariants, by coupling the DG algorithm for the advection part of the problem with a continuous Finite Element Method for the elliptic field equations. This algorithm can preserve both invariants if centered fluxes are used, and still preserves energy conservation even if upwind fluxes are used. However, when applied to gyrokinetics, the weak form of the continuous finite-element part of the algorithm causes a coupling along the field line that would require a full 3-D elliptic solver. We show a new type of DG algorithm that allows the potential to be discontinuous along the field line, just as the particle distribution function can be, thus restoring the property that the fields in gyrokinetics can determined by a set of uncoupled 2-D elliptic problems. By accounting for the delta-function electric field as particles cross cell boundaries, energy can still be preserved.

  20. Neoclassical simulation of tokamak plasmas using the continuum gyrokinetic code TEMPEST

    NASA Astrophysics Data System (ADS)

    Xu, X. Q.

    2008-07-01

    We present gyrokinetic neoclassical simulations of tokamak plasmas with a self-consistent electric field using a fully nonlinear (full- f ) continuum code TEMPEST in a circular geometry. A set of gyrokinetic equations are discretized on a five-dimensional computational grid in phase space. The present implementation is a method of lines approach where the phase-space derivatives are discretized with finite differences, and implicit backward differencing formulas are used to advance the system in time. The fully nonlinear Boltzmann model is used for electrons. The neoclassical electric field is obtained by solving the gyrokinetic Poisson equation with self-consistent poloidal variation. With a four-dimensional (ψ,θ,γ,μ) version of the TEMPEST code, we compute the radial particle and heat fluxes, the geodesic-acoustic mode, and the development of the neoclassical electric field, which we compare with neoclassical theory using a Lorentz collision model. The present work provides a numerical scheme for self-consistently studying important dynamical aspects of neoclassical transport and electric field in toroidal magnetic fusion devices.

  1. Global gyrokinetic simulation of tokamak transport

    SciTech Connect

    Furnish, G.; Horton, W.; Kishimoto, Y.; LeBrun, M.J.; Tajima, T. |

    1998-10-01

    A kinetic simulation code based on the gyrokinetic ion dynamics in global general metric (including a tokamak with circular or noncircular cross-section) has been developed. This gyrokinetic simulation is capable of examining the global and semi-global driftwave structures and their associated transport in a tokamak plasma. The authors investigate the property of the ion temperature gradient (ITG) or {eta}{sub i}({eta}{sub i} {equivalent_to} {partial_derivative}{ell}nT{sub i}/{partial_derivative}{ell}n n{sub i}) driven drift waves in a tokamak plasma. The emergent semi-global drift wave modes give rise to thermal transport characterized by the Bohm scaling.

  2. Chasing hamiltonian structure in gyrokinetic theory

    NASA Astrophysics Data System (ADS)

    Burby, J. W.

    Hamiltonian structure is pursued and uncovered in collisional and collisionless gyrokinetic theory. A new Hamiltonian formulation of collisionless electromagnetic theory is presented that is ideally suited to implementation on modern supercomputers. The method used to uncover this structure is described in detail and applied to a number of examples, where several well-known plasma models are endowed with a Hamiltonian structure for the first time. The first energy- and momentum-conserving formulation of full-F collisional gyrokinetics is presented. In an effort to understand the theoretical underpinnings of this result at a deeper level, a emph{stochastic} Hamiltonian modeling approach is presented and applied to pitch angle scattering. Interestingly, the collision operator produced by the Hamiltonian approach is equal to the Lorentz operator plus higher-order terms, but does not exactly conserve energy. Conversely, the classical Lorentz collision operator is provably not Hamiltonian in the stochastic sense.

  3. Parallel magnetic field perturbations in gyrokinetic simulations

    SciTech Connect

    Joiner, N.; Hirose, A.; Dorland, W.

    2010-07-15

    At low beta it is common to neglect parallel magnetic field perturbations on the basis that they are of order beta{sup 2}. This is only true if effects of order beta are canceled by a term in the nablaB drift also of order beta[H. L. Berk and R. R. Dominguez, J. Plasma Phys. 18, 31 (1977)]. To our knowledge this has not been rigorously tested with modern gyrokinetic codes. In this work we use the gyrokinetic code GS2[Kotschenreuther et al., Comput. Phys. Commun. 88, 128 (1995)] to investigate whether the compressional magnetic field perturbation B{sub ||} is required for accurate gyrokinetic simulations at low beta for microinstabilities commonly found in tokamaks. The kinetic ballooning mode (KBM) demonstrates the principle described by Berk and Dominguez strongly, as does the trapped electron mode, in a less dramatic way. The ion and electron temperature gradient (ETG) driven modes do not typically exhibit this behavior; the effects of B{sub ||} are found to depend on the pressure gradients. The terms which are seen to cancel at long wavelength in KBM calculations can be cumulative in the ion temperature gradient case and increase with eta{sub e}. The effect of B{sub ||} on the ETG instability is shown to depend on the normalized pressure gradient beta{sup '} at constant beta.

  4. Generalized Covariant Gyrokinetic Dynamics of Magnetoplasmas

    SciTech Connect

    Cremaschini, C.; Tessarotto, M.; Nicolini, P.; Beklemishev, A.

    2008-12-31

    A basic prerequisite for the investigation of relativistic astrophysical magnetoplasmas, occurring typically in the vicinity of massive stellar objects (black holes, neutron stars, active galactic nuclei, etc.), is the accurate description of single-particle covariant dynamics, based on gyrokinetic theory (Beklemishev et al., 1999-2005). Provided radiation-reaction effects are negligible, this is usually based on the assumption that both the space-time metric and the EM fields (in particular the magnetic field) are suitably prescribed and are considered independent of single-particle dynamics, while allowing for the possible presence of gravitational/EM perturbations driven by plasma collective interactions which may naturally arise in such systems. The purpose of this work is the formulation of a generalized gyrokinetic theory based on the synchronous variational principle recently pointed out (Tessarotto et al., 2007) which permits to satisfy exactly the physical realizability condition for the four-velocity. The theory here developed includes the treatment of nonlinear perturbations (gravitational and/or EM) characterized locally, i.e., in the rest frame of a test particle, by short wavelength and high frequency. Basic feature of the approach is to ensure the validity of the theory both for large and vanishing parallel electric field. It is shown that the correct treatment of EM perturbations occurring in the presence of an intense background magnetic field generally implies the appearance of appropriate four-velocity corrections, which are essential for the description of single-particle gyrokinetic dynamics.

  5. Velocity-space resolution, entropy production, and upwind dissipation in Eulerian gyrokinetic simulations

    SciTech Connect

    Candy, J.; Waltz, R.E.

    2006-03-15

    Equations which describe the evolution of volume-averaged gyrokinetic entropy are derived and added to GYRO [J. Candy and R.E. Waltz, J. Comput. Phys. 186, 545 (2003)], a Eulerian gyrokinetic turbulence simulation code. In particular, the creation of entropy through spatial upwind dissipation (there is zero velocity-space dissipation in GYRO) and the reduction of entropy via the production of fluctuations are monitored in detail. This new diagnostic has yielded several key confirmations of the validity of the GYRO simulations. First, fluctuations balance dissipation in the ensemble-averaged sense, thus demonstrating that turbulent GYRO simulations achieve a true statistical steady state. Second, at the standard spatial grid size, neither entropy nor energy flux is significantly changed by a 16-fold increase (from 32 to 512 grid points per cell) in the number of grid points in the two-dimensional velocity space. Third, the measured flux is invariant to an eightfold increase in the upwind dissipation coefficients. A notable conclusion is that the lack of change in entropy with grid refinement refutes the familiar but incorrect notion that Eulerian gyrokinetic codes miss important velocity-space structure. The issues of density and energy conservation and their relation to negligible second-order effects such as the parallel nonlinearity are also discussed.

  6. Gyrokinetic simulations of electrostatic microinstabilities with bounce-averaged kinetic electrons for shaped tokamak plasmas

    NASA Astrophysics Data System (ADS)

    Qi, Lei; Kwon, Jaemin; Hahm, T. S.; Jo, Gahyung

    2016-06-01

    Nonlinear bounce-averaged kinetic theory [B. H. Fong and T. S. Hahm, Phys. Plasmas 6, 188 (1999)] is used for magnetically trapped electron dynamics for the purpose of achieving efficient gyrokinetic simulations of Trapped Electron Mode (TEM) and Ion Temperature Gradient mode with trapped electrons (ITG-TEM) in shaped tokamak plasmas. The bounce-averaged kinetic equations are explicitly extended to shaped plasma equilibria from the previous ones for concentric circular plasmas, and implemented to a global nonlinear gyrokinetic code, Gyro-Kinetic Plasma Simulation Program (gKPSP) [J. M. Kwon et al., Nucl. Fusion 52, 013004 (2012)]. Verification of gKPSP with the bounce-averaged kinetic trapped electrons in shaped plasmas is successfully carried out for linear properties of the ITG-TEM mode and Rosenbluth-Hinton residual zonal flow [M. N. Rosenbluth and F. L. Hinton, Phys. Rev. Lett. 80, 724 (1998)]. Physics responsible for stabilizing effects of elongation on both ITG mode and TEM is identified using global gKPSP simulations. These can be understood in terms of magnetic flux expansion, leading to the effective temperature gradient R / L T ( 1 - E ') [P. Angelino et al., Phys. Rev. Lett. 102, 195002 (2009)] and poloidal wave length contraction at low field side, resulting in the effective poloidal wave number kθρi/κ.

  7. Gyrokinetic and kinetic particle-in-cell simulations of guide-field reconnection

    NASA Astrophysics Data System (ADS)

    Munoz Sepulveda, Patricio Alejandro; Büchner, Jörg; Kilian, Patrick; Told, Daniel; Jenko, Frank

    2016-07-01

    Fully kinetic Particle-in-Cell (PIC) simulations of (strong) guide-field reconnection can be computationally very demanding, due to the intrinsic stability and accuracy conditions required by this numerical method. One convenient approach to circumvent this issue is using gyrokinetic theory, an approximation of the Vlasov-Maxwell equations for strongly magnetized plasmas that eliminates the fast gyromotion, and thus reduces the computational cost. Although previous works have started to compare the features of reconnection between both approaches, a complete understanding of the differences is far from being complete. This knowledge is essential to discern the limitations of the gyrokinetic simulations of magnetic reconnection when applied to scenarios with moderate guide fields, such as the Solar corona, in contrast to most of the fusion/laboratory plasmas. We extend a previous work by our group, focused in the differences in the macroscopic flows, by analyzing the heating processes and non-thermal features developed by reconnection between both plasma approximations. We relate these processes by identifying some high-frequency cross-streaming instabilities appearing only in the fully kinetic approach. We characterize the effects of these phenonema such as anisotropic electron heating, beam formation and turbulence under different parameter regimes. And finally, we identify the conditions under which these instabilities tends to become negligible in the fully kinetic model, and thus a comparison with gyrokinetic theory becomes more reliable.

  8. Comparison of linear gyrokinetic and two-fluid stability analyses of DIII-D L-mode plasmas

    NASA Astrophysics Data System (ADS)

    Holland, C.; Bass, E. M.

    2014-10-01

    We present results from a linear stability study of the edge and near-edge regions of well-studied DIII-D tokamak L-mode discharges, using both the gyrokinetic-Maxwell equations (as implemented in the GYRO code) and a range of two-fluid models implemented in the BOUT++ code. The goal is to identify instabilities that may help explain the well-known systematic under-prediction of near-edge DIII-D transport and fluctuation levels by some gyrokinetic codes, in particular those driven by edge physics not included within the gyrokinetic models. We first compare local and global gyrokinetic stability results spanning the region of 0 . 7 <ΨN < 0 . 95 to corresponding predictions from Braginskii-like models implemented in BOUT++, focusing on the influence of magnetic shaping and collisionality scalings for a range of low- to moderate-n modes, consistent with the observed discrepancies in fluctuation spectra. The closed-field line results are then compared against equivalent results that extend across the separatrix to the open field line region 0 . 7 <ΨN < 1 . 05 , in order to assess whether inclusion of this region leads to any significant changes in linear stability. Progress on extending the linear analysis to inclusion of rotational and gyrofluid effects will also be reported.

  9. Gyrokinetic Simulations of ETG and ITG Turbulence

    SciTech Connect

    Dimits, A; Nevins, W; Shumaker, D; Hammett, G; Dannert, T; Jenko, F; Dorland, W; Leboeuf, J; Rhodes, T; Candy, J; Estrada-Mila, C

    2006-10-03

    Published gyrokinetic continuum-code simulations indicated levels of the electron thermal conductivity {chi}{sub e} due to electron-temperature-gradient (ETG) turbulence large enough to be significant in some tokamaks, while subsequent global particle-in-cell (PIC) simulations gave significantly lower values. We have carried out an investigation of this discrepancy. We have reproduced the key features of the aforementioned PIC simulations using the flux-tube gyrokinetic PIC code, PG3EQ, thereby eliminating global effects and as the cause of the discrepancy. We show that the late-time low-transport state in both of these sets of PIC simulations is a result of discrete particle noise, which is a numerical artifact. Thus, the low value of {chi}{sub e} along with conclusions about anomalous transport drawn from these particular PIC simulations are unjustified. In our attempts to benchmark PIC and continuum codes for ETG turbulence at the plasma parameters used above, both produce very large intermittent transport. We have therefore undertaken benchmarks at an alternate reference point, magnetic shear s=0.1 instead of s=0.796, and have found that PIC and continuum codes reproduce the same transport levels. Scans in the magnetic shear show an abrupt transition to a high-{chi}{sub e} state as the shear is increased above s=0.4. When nonadiabatic ions are used, this abrupt transition is absent, and {chi}{sub e} increases gradually reaching values consistent with transport analyses of DIII-D, JET, and JT60-U discharges. New results on the balances of zonal-flow driving and damping terms in late-time quasi-steady ITG turbulence and on real-geometry gyrokinetic simulations of shaped DIII-D discharges are also reported.

  10. Gyrokinetic simulation of current-driven instabilities

    NASA Astrophysics Data System (ADS)

    McClenaghan, Joseph

    The gyrokinetic toroidal code(GTC) capability has been extended for simulating current-driven instabilities in magnetized plasmas such as kink and resistive tearing modes with kinetic effects. This new gyrokinetic capability enables first-principles, integrated simulations of macroscopic magnetohydrodynamic(MHD) modes, which limit the performance of burning plasmas and threaten the integrity of fusion devices. The excitation and evolution of macroscopic MHD modes often depend on the kinetic effects at microscopic scales and the nonlinear coupling of multiple physical processes. GTC simulation in the fluid limit of the internal kink modes in cylindrical geometry has been verified by benchmarking with an MHD eigenvalue code. The global simulation domain covers the magnetic axis which is necessary for simulating the macroscopic MHD modes. Gyrokinetic simulations of the internal kink modes in the toroidal geometry find that ion kinetic effects significantly reduce the growth rate even when the banana orbit width is much smaller than the radial width of the perturbed current layer at the mode rational surface. This new GTC capability for current-driven instability has now been extended to simulate fishbone instabilities excited by energetic particles and resistive tearing modes. GTC has also been applied to study the internal kink modes in astrophysical jets that are formed around supermassive black holes. Linear simulations find that the internal kink modes in astrophysical jets are unstable with a broad eigenmode. Nonlinear saturation amplitude of these kink modes is observed to be small, suggesting that the jets can remain collimated even in the presence of the internal kink modes. Generation of a mean parallel electric field by the nonlinear dynamics of internal kink modes and the potential implication of this field on particle acceleration in jets has been examined.

  11. THE LOCAL LIMIT OF GLOBAL GYROKINETIC SIMULATIONS

    SciTech Connect

    CANDY J; WALTZ RE; DORLAND W

    2003-10-01

    OAK-B135 Global gyrokinetic simulations of turbulence include physical effects that are not retained in local flux-tube simulations. nevertheless, in the limit of sufficiently small {rho}* (gyroradius compared to system size) it is expected that a local simulation should agree with a global one (at the local simulation radius) since all effects that are dropped in the local simulations are expected to vanish as {rho}* {yields} 0. In this note, global simulations of a well-established test case are indeed shown to recover the flux-tube limit at each radius.

  12. Gyrokinetic turbulence simulations at high plasma beta

    SciTech Connect

    Pueschel, M. J.; Kammerer, M.; Jenko, F.

    2008-10-15

    Electromagnetic gyrokinetic turbulence simulations employing Cyclone Base Case parameters are presented for {beta} values up to and beyond the kinetic ballooning threshold. The {beta} scaling of the turbulent transport is found to be linked to a complex interplay of linear and nonlinear effects. Linear investigation of the kinetic ballooning mode is performed in detail, while nonlinearly, it is found to dominate the turbulence only in a fairly narrow range of {beta} values just below the respective ideal limit. The magnetic transport scales like {beta}{sup 2} and is well described by a Rechester-Rosenbluth-type ansatz.

  13. Saturation of Gyrokinetic Turbulence through Damped Eigenmodes

    SciTech Connect

    Hatch, D. R.; Terry, P. W.; Jenko, F.; Merz, F.; Nevins, W. M.

    2011-03-18

    In the context of toroidal gyrokinetic simulations, it is shown that a hierarchy of damped modes is excited in the nonlinear turbulent state. These modes exist at the same spatial scales as the unstable eigenmodes that drive the turbulence. The larger amplitude subdominant modes are weakly damped and exhibit smooth, large-scale structure in velocity space and in the direction parallel to the magnetic field. Modes with increasingly fine-scale structure are excited to decreasing amplitudes. In aggregate, damped modes define a potent energy sink. This leads to an overlap of the spatial scales of energy injection and peak dissipation, a feature that is in contrast with more traditional turbulent systems.

  14. Beyond linear gyrocenter polarization in gyrokinetic theory

    SciTech Connect

    Brizard, Alain J.

    2013-09-15

    The concept of polarization in gyrokinetic theory is clarified and generalized to include contributions from the guiding-center (zeroth-order) polarization as well as the nonlinear (second-order) gyrocenter polarization. The guiding-center polarization, which appears as the antecedent (zeroth-order) of the standard linear (first-order) gyrocenter polarization, is obtained from a modified guiding-center transformation. The nonlinear gyrocenter polarization is derived either variationally from the third-order gyrocenter Hamiltonian or directly by gyrocenter push-forward method.

  15. Energy conserving continuum algorithms for kinetic & gyrokinetic simulations of plasmas

    NASA Astrophysics Data System (ADS)

    Hakim, A.; Hammett, G. W.; Shi, E.; Stoltzfus-Dueck, T.

    2015-11-01

    We present high-order, energy conserving, continuum algorithms for the solution of gyrokinetic equations for use in edge turbulence simulations. The distribution function is evolved with a discontinuous Galerkin scheme, while the fields are evolved with a continuous finite-element method. These algorithms work for a general, possibly non-canonical, Poisson bracket operator and conserve energy exactly. Benchmark simulations with ETG turbulence in 3X/2V are shown, as well as initial applications of the algorithms to turbulence in a simplified SOL geometry. Sheath boundary conditions with recycling and secondary electron emission are implemented, and a Lenard-Bernstein collision operator is included. Extension of these algorithms to full Vlasov-Maxwell equations are presented. It is shown that with a particular choice of numerical fluxes the total (particle+field) energy is conserved. Algorithms are implemented in a flexible and open-source framework, Gkeyll, which also includes fluid models, allowing potential hybrid simulations of various plasma problems. Supported by the Max-Planck/Princeton Center for Plasma Physics, and DOE Contract DE-AC02-09CH11466.

  16. Testing gyrokinetic simulations of electron turbulence

    NASA Astrophysics Data System (ADS)

    Holland, C.; DeBoo, J. C.; Rhodes, T. L.; Schmitz, L.; Hillesheim, J. C.; Wang, G.; White, A. E.; Austin, M. E.; Doyle, E. J.; Peebles, W. A.; Petty, C. C.; Zeng, L.; Candy, J.

    2012-06-01

    An extensive set of tests comparing gyrokinetic predictions of temperature-gradient driven electron turbulence to power balance transport analyses and fluctuation measurements are presented. These tests use data from an L-mode validation study on the DIII-D tokamak (Luxon 2002 Nucl. Fusion 42 614) in which the local value of a/L_{T_e } =-(a/T_e )(dT_e /dr) is varied by modulated electron cyclotron heating; the GYRO code (Candy and Waltz 2003 J. Comput. Phys. 186 545) is used to make the gyrokinetic predictions. Using a variety of novel measures, both local and global nonlinear simulations are shown to predict key characteristics of the electron energy flux Qe and long-wavelength (low-k) Te fluctuations, but systematically underpredict (by roughly a factor of two) the ion energy flux Qi. A new synthetic diagnostic for comparison to intermediate wavelength Doppler backscattering measurements is presented, and used to compare simulation predictions against experiment. In contrast to the agreement observed in the low-k Te fluctuation comparisons, little agreement is found between the predicted and measured intermediate-k density fluctuation responses. The results presented in this paper significantly expand upon those previously reported in DeBoo et al (2010 Phys. Plasmas 17 056105), comparing transport and multiple turbulence predictions from numerically converged local and global simulations for all four experimental heating configurations (instead of only fluxes and low-k Te fluctuations for one condition) to measurements and power balance analyses.

  17. Gyrokinetic simulations of ion and impurity transport

    SciTech Connect

    Estrada-Mila, C.; Candy, J.; Waltz, R.E.

    2005-02-01

    A systematic study of turbulent particle and energy transport in both pure and multicomponent plasmas is presented. In this study, gyrokinetic results from the GYRO code [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] are supplemented with those from the GLF23 [R. E. Waltz, G. M. Staebler, W. Dorland et al., Phys. Plasmas 4, 2482 (1997)] transport model, as well as from quasilinear theory. Various results are obtained. The production of a particle pinch driven by temperature gradients (a thermal pinch) is demonstrated, and further shown to be weakened by finite electron collisionality. Helium transport and the effects of helium density gradient and concentration in a deuterium plasma are examined. Interestingly, it is found that the simple D-v (diffusion versus convective velocity) model of impurity flow is consistent with results obtained from nonlinear gyrokinetic simulations. Also studied is the transport in a 50-50 deuterium-tritium plasma, where a symmetry breaking is observed indicating the potential for fuel separation in a burning plasma. Quasilinear theory together with linear simulations shows that the symmetry breaking which enhances the tritium confinement arises largely from finite-Larmor-radius effects. To justify the numerical methods used in the paper, a variety of linear benchmarks and nonlinear grid refinement studies are detailed.

  18. Gyrokinetic simulation of internal kink modes

    SciTech Connect

    Naitou, Hiroshi; Tsuda, Kenji; Lee, W.W.; Sydora, R.D.

    1995-05-01

    Internal disruption in a tokamak has been simulated using a three-dimensional magneto-inductive gyrokinetic particle code. The code operates in both the standard gyrokinetic mode (total-f code) and the fully nonlinear characteristic mode ({delta}f code). The latter, a recent addition, is a quiet low noise algorithm. The computational model represents a straight tokamak with periodic boundary conditions in the toroidal direction. The plasma is initially uniformly distributed in a square cross section with perfectly conducting walls. The linear mode structure of an unstable m = 1 (poloidal) and n = 1 (toroidal) kinetic internal kink mode is clearly observed, especially in the {delta}f code. The width of the current layer around the x-point, where magnetic reconnection occurs, is found to be close to the collisionless electron skin depth. This is consistent with the theory in which electron inertia has a dominant role. The nonlinear behavior of the mode is found to be quite similar for both codes. Full reconnection in the Alfven time scale is observed along with the electrostatic potential structures created during the full reconnection phase. The E x B drift due to this electrostatic potential dominates the nonlinear phase of the development after the full reconnection.

  19. Gyrokinetic particle simulation of fast-electron driven beta-induced Aflvén eigenmode

    NASA Astrophysics Data System (ADS)

    Cheng, Junyi; Zhang, Wenlu; Lin, Zhihong; Holod, Ihor; Li, Ding; Chen, Yang; Cao, Jintao

    2016-05-01

    The fast-electron driven beta-induced Alfvén eigenmode (e-BAE) in toroidal plasmas is investigated for the first time using global gyrokinetic particle simulations, where the fast electron is described by the drift kinetic equation. The simulation shows that the e-BAE propagates in the fast electron diamagnetic direction and its polarization is close to an ideal MHD mode. The phase space structure shows that only the fast electron processional resonance is responsible for the e-BAE excitations while fast-ion driven BAE can be excited through all the channels, including transit, bounce, and processional resonance.

  20. Nonlinear gyrokinetic theory and its application to computation of the gyrocenter motion in ripple field

    NASA Astrophysics Data System (ADS)

    Zhu, Siqiang; Xu, Yingfeng; Wang, Shaojie

    2016-06-01

    The nonlinear gyrokinetic equation with full electromagnetic potential perturbations is derived by using the two-step transform procedure. The second-order transformed Hamiltonian can be simplified as /1 2 δ A∥ 2 , instead of /1 2 δ A 2 in the long-wave-length limit. A numerical code based on the I-transform method is improved to compute the gyrocenter orbit in the TFTR tokamak with a ripple field, and the numerical results indicate that the collisionless stochastic diffusion criterion agrees well with the theoretical prediction.

  1. Benchmarking gyrokinetic simulations in a toroidal flux-tube

    NASA Astrophysics Data System (ADS)

    Chen, Y.; Parker, S. E.; Wan, W.; Bravenec, R.

    2013-09-01

    A flux-tube model is implemented in the global turbulence code GEM [Y. Chen and S. E. Parker, J. Comput. Phys. 220, 839 (2007)] in order to facilitate benchmarking with Eulerian codes. The global GEM assumes the magnetic equilibrium to be completely given. The initial flux-tube implementation simply selects a radial location as the center of the flux-tube and a radial size of the flux-tube, sets all equilibrium quantities (B, ∇B, etc.) to be equal to the values at the center of the flux-tube, and retains only a linear radial profile of the safety factor needed for boundary conditions. This implementation shows disagreement with Eulerian codes in linear simulations. An alternative flux-tube model based on a complete local equilibrium solution of the Grad-Shafranov equation [J. Candy, Plasma Phys. Controlled Fusion 51, 105009 (2009)] is then implemented. This results in better agreement between Eulerian codes and the particle-in-cell (PIC) method. The PIC algorithm based on the v||-formalism [J. Reynders, Ph.D. dissertation, Princeton University, 1992] and the gyrokinetic ion/fluid electron hybrid model with kinetic electron closure [Y. Chan and S. E. Parker, Phys. Plasmas 18, 055703 (2011)] are also implemented in the flux-tube geometry and compared with the direct method for both the ion temperature gradient driven modes and the kinetic ballooning modes.

  2. Gyrokinetic Study of L-H Transition with Profile Evolution

    NASA Astrophysics Data System (ADS)

    Xie, Hua-Sheng; GTC Team

    2015-11-01

    Recent simulations based on gyrokinetic toroidal code (GTC) and theory based on model eigen equation (H. S. Xie and Y. Xiao, arXiv:1503.04440) have found that the eigenstates of mirco-instabilities (trapped electron mode TEM or ion temperature gradient mode ITG) under strong and weak gradients are not the same. Under weak gradient, the most unstable mode is on the ground state, with conventional ballooning mode structure. When the gradient exceed a critical value, the most unstable mode jump to non-ground state. The mode structures of non-ground state are rich and unconventional, and thus can reduced the transport level, which can provide a explanation to the H-mode in the mirco-scale aspect. Nonlinear simulations (H. S. Xie, Y. Xiao and Z. Lin, 9th West Lake International Symposium on Plasma Simulation, May. 18-21, 2015, Hangzhou, China) verified this and have also found a turning point of the gradient. The turbulent transport coefficient would decrease with the gradient increasing when the gradient exceed a critical value. This provide a new route for the L to H transition without invoking shear flow or zonal flow. In the above works, the profiles are fixed. In this work, we will give some preliminary results on self-consistent simulations of L-H transition including the evolution of the radial plasma profiles. Collaboration with GTC team.

  3. Full f gyrokinetic method for particle simulation of tokamak transport

    SciTech Connect

    Heikkinen, J.A. Janhunen, S.J.; Kiviniemi, T.P.; Ogando, F.

    2008-05-10

    A gyrokinetic particle-in-cell approach with direct implicit construction of the coefficient matrix of the Poisson equation from ion polarization and electron parallel nonlinearity is described and applied in global electrostatic toroidal plasma transport simulations. The method is applicable for calculation of the evolution of particle distribution function f including as special cases strong plasma pressure profile evolution by transport and formation of neoclassical flows. This is made feasible by full f formulation and by recording the charge density changes due to the ion polarization drift and electron acceleration along the local magnetic field while particles are advanced. The code has been validated against the linear predictions of the unstable ion temperature gradient mode growth rates and frequencies. Convergence and saturation in both turbulent and neoclassical limit of the ion heat conductivity is obtained with numerical noise well suppressed by a sufficiently large number of simulation particles. A first global full f validation of the neoclassical radial electric field in the presence of turbulence for a heated collisional tokamak plasma is obtained. At high Mach number (M{sub p}{approx}1) of the poloidal flow, the radial electric field is significantly enhanced over the standard neoclassical prediction. The neoclassical radial electric field together with the related GAM oscillations is found to regulate the turbulent heat and particle diffusion levels particularly strongly in a large aspect ratio tokamak at low plasma current.

  4. Benchmarking gyrokinetic simulations in a toroidal flux-tube

    SciTech Connect

    Chen, Y.; Parker, S. E.; Wan, W.; Bravenec, R.

    2013-09-15

    A flux-tube model is implemented in the global turbulence code GEM [Y. Chen and S. E. Parker, J. Comput. Phys. 220, 839 (2007)] in order to facilitate benchmarking with Eulerian codes. The global GEM assumes the magnetic equilibrium to be completely given. The initial flux-tube implementation simply selects a radial location as the center of the flux-tube and a radial size of the flux-tube, sets all equilibrium quantities (B, ∇B, etc.) to be equal to the values at the center of the flux-tube, and retains only a linear radial profile of the safety factor needed for boundary conditions. This implementation shows disagreement with Eulerian codes in linear simulations. An alternative flux-tube model based on a complete local equilibrium solution of the Grad-Shafranov equation [J. Candy, Plasma Phys. Controlled Fusion 51, 105009 (2009)] is then implemented. This results in better agreement between Eulerian codes and the particle-in-cell (PIC) method. The PIC algorithm based on the v{sub ||}-formalism [J. Reynders, Ph.D. dissertation, Princeton University, 1992] and the gyrokinetic ion/fluid electron hybrid model with kinetic electron closure [Y. Chan and S. E. Parker, Phys. Plasmas 18, 055703 (2011)] are also implemented in the flux-tube geometry and compared with the direct method for both the ion temperature gradient driven modes and the kinetic ballooning modes.

  5. The next-generation ESL continuum gyrokinetic edge code

    NASA Astrophysics Data System (ADS)

    Cohen, R.; Dorr, M.; Hittinger, J.; Rognlien, T.; Collela, P.; Martin, D.

    2009-05-01

    The Edge Simulation Laboratory (ESL) project is developing continuum-based approaches to kinetic simulation of edge plasmas. A new code is being developed, based on a conservative formulation and fourth-order discretization of full-f gyrokinetic equations in parallel-velocity, magnetic-moment coordinates. The code exploits mapped multiblock grids to deal with the geometric complexities of the edge region, and utilizes a new flux limiter [P. Colella and M.D. Sekora, JCP 227, 7069 (2008)] to suppress unphysical oscillations about discontinuities while maintaining high-order accuracy elsewhere. The code is just becoming operational; we will report initial tests for neoclassical orbit calculations in closed-flux surface and limiter (closed plus open flux surfaces) geometry. It is anticipated that the algorithmic refinements in the new code will address the slow numerical instability that was observed in some long simulations with the existing TEMPEST code. We will also discuss the status and plans for physics enhancements to the new code.

  6. Comparisons of gyrofluid and gyrokinetic simulations

    SciTech Connect

    Parker, S.E.; Dorland, W.; Santoro, R.A.; Beer, M.A.; Liu, Q.P.; Lee, W.W.; Hammett, G.W.

    1994-03-01

    The gyrokinetic and gyrofluid models show the most promise for large scale simulations of tokamak microturbulence. This paper discusses detailed comparisons of these two complementary approaches. Past comparisons with linear theory have been fairly good, therefore the emphasis here is on nonlinear comparisons. Simulations include simple two dimensional slab test cases, turbulent three dimensional slab cases, and toroidal cases, each modeling the nonlinear evolution of the ion temperature gradient instability. There is good agreement in both turbulent and coherent nonlinear slab comparisons in terms of the ion heat flux, both in magnitude and scaling with magnetic shear. However, the nonlinear saturation level for {vert_bar}{Phi}{vert_bar} in the slab comparisons show differences of approximately 40%. Preliminary toroidal comparisons show agreement within 50%, in terms of ion heat flux and saturation level.

  7. Nonlinear electromagnetic gyrokinetic simulations of tokamak plasmas

    NASA Astrophysics Data System (ADS)

    Jenko, F.; Dorland, W.

    2001-12-01

    One of the central physics issues currently targeted by nonlinear gyrokinetic simulations is the role of finite-β effects. The latter change the MHD equilibrium, introduce new dynamical space and time scales, alter and enlarge the zoo of electrostatic microinstabilities and saturation mechanisms, and lead to turbulent transport along fluctuating magnetic field lines. It is shown that the electromagnetic effects on primarily electrostatic microinstabilities are generally weakly or moderately stabilizing. However, the saturation of these modes and hence the determination of the transport level in the quasi-stationary turbulent state can be dominated by nonlinear electromagnetic effects and yield surprising results. Despite this, the induced transport is generally electrostatic in nature well below the ideal ballooning limit.

  8. Saturation of gyrokinetic turbulence through damped eigenmodes.

    PubMed

    Hatch, D R; Terry, P W; Jenko, F; Merz, F; Nevins, W M

    2011-03-18

    In the context of toroidal gyrokinetic simulations, it is shown that a hierarchy of damped modes is excited in the nonlinear turbulent state. These modes exist at the same spatial scales as the unstable eigenmodes that drive the turbulence. The larger amplitude subdominant modes are weakly damped and exhibit smooth, large-scale structure in velocity space and in the direction parallel to the magnetic field. Modes with increasingly fine-scale structure are excited to decreasing amplitudes. In aggregate, damped modes define a potent energy sink. This leads to an overlap of the spatial scales of energy injection and peak dissipation, a feature that is in contrast with more traditional turbulent systems. PMID:21469869

  9. DIRECT INTEGRATION OF THE COLLISIONLESS BOLTZMANN EQUATION IN SIX-DIMENSIONAL PHASE SPACE: SELF-GRAVITATING SYSTEMS

    SciTech Connect

    Yoshikawa, Kohji; Umemura, Masayuki; Yoshida, Naoki

    2013-01-10

    We present a scheme for numerical simulations of collisionless self-gravitating systems which directly integrates the Vlasov-Poisson equations in six-dimensional phase space. Using the results from a suite of large-scale numerical simulations, we demonstrate that the present scheme can simulate collisionless self-gravitating systems properly. The integration scheme is based on the positive flux conservation method recently developed in plasma physics. We test the accuracy of our code by performing several test calculations, including the stability of King spheres, the gravitational instability, and the Landau damping. We show that the mass and the energy are accurately conserved for all the test cases we study. The results are in good agreement with linear theory predictions and/or analytic solutions. The distribution function keeps the property of positivity and remains non-oscillatory. The largest simulations are run on 64{sup 6} grids. The computation speed scales well with the number of processors, and thus our code performs efficiently on massively parallel supercomputers.

  10. Petascale Parallelization of the Gyrokinetic Toroidal Code

    SciTech Connect

    Ethier, Stephane; Adams, Mark; Carter, Jonathan; Oliker, Leonid

    2010-05-01

    The Gyrokinetic Toroidal Code (GTC) is a global, three-dimensional particle-in-cell application developed to study microturbulence in tokamak fusion devices. The global capability of GTC is unique, allowing researchers to systematically analyze important dynamics such as turbulence spreading. In this work we examine a new radial domain decomposition approach to allow scalability onto the latest generation of petascale systems. Extensive performance evaluation is conducted on three high performance computing systems: the IBM BG/P, the Cray XT4, and an Intel Xeon Cluster. Overall results show that the radial decomposition approach dramatically increases scalability, while reducing the memory footprint - allowing for fusion device simulations at an unprecedented scale. After a decade where high-end computing (HEC) was dominated by the rapid pace of improvements to processor frequencies, the performance of next-generation supercomputers is increasingly differentiated by varying interconnect designs and levels of integration. Understanding the tradeoffs of these system designs is a key step towards making effective petascale computing a reality. In this work, we examine a new parallelization scheme for the Gyrokinetic Toroidal Code (GTC) [?] micro-turbulence fusion application. Extensive scalability results and analysis are presented on three HEC systems: the IBM BlueGene/P (BG/P) at Argonne National Laboratory, the Cray XT4 at Lawrence Berkeley National Laboratory, and an Intel Xeon cluster at Lawrence Livermore National Laboratory. Overall results indicate that the new radial decomposition approach successfully attains unprecedented scalability to 131,072 BG/P cores by overcoming the memory limitations of the previous approach. The new version is well suited to utilize emerging petascale resources to access new regimes of physical phenomena.

  11. Gyrokinetic Simulations of the ITER Pedestal

    NASA Astrophysics Data System (ADS)

    Kotschenreuther, Mike

    2015-11-01

    It has been reported that low collisionality pedestals for JET parameters are strongly stable to Kinetic Ballooning Modes (KBM), and it is, as simulations with GENE show, the drift-tearing modes that produce the pedestal transport. It would seem, then, that gyrokinetic simulations may be a powerful, perhaps, indispensable tool for probing the characteristics of the H-mode pedestal in ITER especially since projected ITER pedestals have the normalized gyroradius ρ* smaller than the range of present experimental investigation; they do lie, however, within the regime of validity of gyrokinetics. Since ExB shear becomes small as ρ* approaches zero, strong drift turbulence will eventually be excited. Finding an answer to the question whether the ITER ρ* is small enough to place it in the high turbulence regime compels serious investigation. We begin with MHD equilibria (including pedestal bootstrap current) constructed using VMEC. Plasma profile shapes, very close to JET experimental profiles, are scaled to values expected on ITER (e.g., a 4 keV pedestal). The equilibrium ExB shear is computed using a neoclassical formula for the radial electric field. As with JET, the ITER pedestal is found to be strongly stable to KBM. Preliminary nonlinear simulations with GENE show that the turbulent drift transport is strong for ITER; the electrostatic transport has a highly unfavorable scaling from JET to ITER, going from being highly sub-dominant to electromagnetic transport on JET, to dominant on ITER. At burning plasma parameters, pedestals in spherical tokamak H-modes may have much stronger velocity shear, and hence more favorable transport; preliminary investigations will be reported. This research supported by U.S. Department of Energy, Office of Fusion Energy Science: Grant No. DE-FG02-04ER-54742.

  12. Gyrokinetic Studies of Turbulence in Steep Gradient Region: Role of Turbulence Spreading and E x B Shear

    SciTech Connect

    T.S. Hahm; Z. Lin; P.H. Diamond; G. Rewoldt; W.X. Wang; S. Ethier; O. Gurcan; W.W. Lee; W.M. Tang

    2004-12-21

    An integrated program of gyrokinetic particle simulation and theory has been developed to investigate several outstanding issues in both turbulence and neoclassical physics. Gyrokinetic particle simulations of toroidal ion temperature gradient (ITG) turbulence spreading using the GTC code and its related dynamical model have been extended to the case with radially increasing ion temperature gradient, to study the inward spreading of edge turbulence toward the core. Due to turbulence spreading from the edge, the turbulence intensity in the core region is significantly enhanced over the value obtained from simulations of the core region only. Even when the core gradient is within the Dimits shift regime (i.e., self-generated zonal flows reduce the transport to a negligible value), a significant level of turbulence and transport is observed in the core due to spreading from the edge. The scaling of the turbulent front propagation speed is closer to the prediction from our nonlinear diffusion model than one based on linear toroidal coupling. A calculation of ion poloidal rotation in the presence of sharp density and toroidal angular rotation frequency gradients from the GTC-Neo particle simulation code shows that the results are significantly different from the conventional neoclassical theory predictions. An energy conserving set of a fully electromagnetic nonlinear gyrokinetic Vlasov equation and Maxwell's equations, which is applicable to edge turbulence, is being derived via the phase-space action variational Lie perturbation method. Our generalized ordering takes the ion poloidal gyroradius to be on the order of the radial electric field gradient length.

  13. Extended gyrokinetic field theory for time-dependent magnetic confinement fields

    SciTech Connect

    Sugama, H.; Watanabe, T.-H.; Nunami, M.

    2014-01-15

    A gyrokinetic system of equations for turbulent toroidal plasmas in time-dependent axisymmetric background magnetic fields is derived from the variational principle. Besides governing equations for gyrocenter distribution functions and turbulent electromagnetic fields, the conditions which self-consistently determine the background magnetic fields varying on a transport time scale are obtained by using the Lagrangian, which includes the constraint on the background fields. Conservation laws for energy and toroidal angular momentum of the whole system in the time-dependent background magnetic fields are naturally derived by applying Noether's theorem. It is shown that the ensemble-averaged transport equations of particles, energy, and toroidal momentum given in the present work agree with the results from the conventional recursive formulation with the WKB representation except that collisional effects are disregarded here.

  14. Experimentally Relevant Benchmarks for Gyrokinetic Codes

    NASA Astrophysics Data System (ADS)

    Bravenec, Ronald

    2010-11-01

    Although benchmarking of gyrokinetic codes has been performed in the past, e.g., The Numerical Tokamak, The Cyclone Project, The Plasma Microturbulence Project, and various informal activities, these efforts have typically employed simple plasma models. For example, the Cyclone ``base case'' assumed shifted-circle flux surfaces, no magnetic transport, adiabatic electrons, no collisions nor impurities, ρi << a (ρi the ion gyroradius and a the minor radius), and no ExB flow shear. This work presents comparisons of linear frequencies and nonlinear fluxes from GYRO and GS2 with none of the above approximations except ρi << a and no ExB flow shear. The comparisons are performed at two radii of a DIII-D plasma, one in the confinement region (r/a = 0.5) and the other closer to the edge (r/a = 0.7). Many of the plasma parameters differ by a factor of two between these two locations. Good agreement between GYRO and GS2 is found when neglecting collisions. However, differences are found when including e-i collisions (Lorentz model). The sources of the discrepancy are unknown as of yet. Nevertheless, two collisionless benchmarks have been formulated with considerably different plasma parameters. Acknowledgements to J. Candy, E. Belli, and M. Barnes.

  15. Gyrokinetic particle simulation of neoclassical transport

    SciTech Connect

    Lin, Z.; Tang, W.M.; Lee, W.W.

    1995-02-01

    A time varying weighting ({delta} f) scheme for gyrokinetic particle simulation is applied to a steady state, multi-species simulation of neoclassical transport. Accurate collision operators conserving momentum and energy are developed and implemented. Simulation results using these operators are found to agree very well with neoclassical theory. For example, it is dynamically demonstrated in these multispecies simulations that like-particle collisions produce no particle flux and that the neoclassical fluxes are ambipolar for an ion-electron plasma. An important physics feature of the present scheme is the introduction of toroidal sheared flow to the simulations. Simulation results are in agreement with the existing analytical neoclassical theory of Hinton and Wong. The poloidal electric field associated with toroidal mass flow is found to enhance density gradient driven electron particle flux and the bootstrap current while reducing temperature gradient driven flux and current. Finally, neoclassical theory in steep gradient profile relevant to the edge regime is examined by taking into account finite banana width effects. In general, the present work demonstrates a valuable new capability for studying important aspects of neoclassical transport inaccessible by conventional analytical calculation processes.

  16. Gyrokinetic particle simulation of neoclassical transport

    SciTech Connect

    Lin, Z.; Tang, W.M.; Lee, W.W.

    1995-08-01

    A time varying weighting ({delta}{ital f} ) scheme for gyrokinetic particle simulation is applied to a steady-state, multispecies simulation of neoclassical transport. Accurate collision operators conserving momentum and energy are developed and implemented. Simulation results using these operators are found to agree very well with neoclassical theory. For example, it is dynamically demonstrated that like-particle collisions produce no particle flux and that the neoclassical fluxes are ambipolar for an ion--electron plasma. An important physics feature of the present scheme is the introduction of toroidal flow to the simulations. Simulation results are in agreement with the existing analytical neoclassical theory. The poloidal electric field associated with toroidal mass flow is found to enhance density gradient-driven electron particle flux and the bootstrap current while reducing temperature gradient-driven flux and current. Finally, neoclassical theory in steep gradient profile relevant to the edge regime is examined by taking into account finite banana width effects. In general, in the present work a valuable new capability for studying important aspects of neoclassical transport inaccessible by conventional analytical calculation processes is demonstrated. {copyright} {ital 1995} {ital American} {ital Institute} {ital of} {ital Physics}.

  17. Visual interrogation of gyrokinetic particle simulations

    NASA Astrophysics Data System (ADS)

    Jones, Chad; Ma, Kwan-Liu; Sanderson, Allen; Myers, Lee Roy, Jr.

    2007-07-01

    Gyrokinetic particle simulations are critical to the study of anomalous energy transport associated with plasma microturbulence in magnetic confinement fusion experiments. The simulations are conducted on massively parallel computers and produce large quantities of particles, variables, and time steps, thus presenting a formidable challenge to data analysis tasks. We present two new visualization techniques for scientists to improve their understanding of the time-varying, multivariate particle data. One technique allows scientists to examine correlations in multivariate particle data with tightly coupled views of the data in both physical space and variable space, and to visually identify and track features of interest. The second technique, built into SCIRun, allows scientists to perform range-based queries over a series of time slices and visualize the resulting particles using glyphs. The ability to navigate the multiple dimensions of the particle data, as well as query individual or a collection of particles, enables scientists to not only validate their simulations but also discover new phenomena in their data.

  18. Edge gyrokinetic theory and continuum simulations

    NASA Astrophysics Data System (ADS)

    Xu, X. Q.; Xiong, Z.; Dorr, M. R.; Hittinger, J. A.; Bodi, K.; Candy, J.; Cohen, B. I.; Cohen, R. H.; Colella, P.; Kerbel, G. D.; Krasheninnikov, S.; Nevins, W. M.; Qin, H.; Rognlien, T. D.; Snyder, P. B.; Umansky, M. V.

    2007-08-01

    The following results are presented from the development and application of TEMPEST, a fully nonlinear (full-f) five-dimensional (3d2v) gyrokinetic continuum edge-plasma code. (1) As a test of the interaction of collisions and parallel streaming, TEMPEST is compared with published analytic and numerical results for endloss of particles confined by combined electrostatic and magnetic wells. Good agreement is found over a wide range of collisionality, confining potential and mirror ratio, and the required velocity space resolution is modest. (2) In a large-aspect-ratio circular geometry, excellent agreement is found for a neoclassical equilibrium with parallel ion flow in the banana regime with zero temperature gradient and radial electric field. (3) The four-dimensional (2d2v) version of the code produces the first self-consistent simulation results of collisionless damping of geodesic acoustic modes and zonal flow (Rosenbluth-Hinton residual) with Boltzmann electrons using a full-f code. The electric field is also found to agree with the standard neoclassical expression for steep density and ion temperature gradients in the plateau regime. In divertor geometry, it is found that the endloss of particles and energy induces parallel flow stronger than the core neoclassical predictions in the SOL.

  19. Evolution of the phase-space density and the Jeans scale for dark matter derived from the Vlasov-Einstein equation

    SciTech Connect

    Piattella, O.F.; Rodrigues, D.C.; Fabris, J.C.; Pacheco, J.A. de Freitas E-mail: davi.rodrigues@ufes.br E-mail: pacheco@oca.eu

    2013-11-01

    We discuss solutions of Vlasov-Einstein equation for collisionless dark matter particles in the context of a flat Friedmann universe. We show that, after decoupling from the primordial plasma, the dark matter phase-space density indicator Q = ρ/(σ{sub 1D}{sup 2}){sup 3/2} remains constant during the expansion of the universe, prior to structure formation. This well known result is valid for non-relativistic particles and is not ''observer dependent'' as in solutions derived from the Vlasov-Poisson system. In the linear regime, the inclusion of velocity dispersion effects permits to define a physical Jeans length for collisionless matter as function of the primordial phase-space density indicator: λ{sub J} = (5π/G){sup 1/2}Q{sup −1/3}ρ{sub dm}{sup −1/6}. The comoving Jeans wavenumber at matter-radiation equality is smaller by a factor of 2-3 than the comoving wavenumber due to free-streaming, contributing to the cut-off of the density fluctuation power spectrum at the lowest scales. We discuss the physical differences between these two scales. For dark matter particles of mass equal to 200 GeV, the derived Jeans mass is 4.3 × 10{sup −6}M{sub ⊙}.

  20. Gyrokinetic theory and simulation of turbulent energy exchange

    SciTech Connect

    Waltz, R. E.; Staebler, G. M.

    2008-01-15

    A previous gyrokinetic theory of turbulent heating [F. L. Hinton and R. E. Waltz, Phys. Plasma 13, 102301 (2006)] is simplified and extended to show that the local radial average of terms in the gyrokinetic turbulent heating (which survive in the drift kinetic limit) are actually closer to a turbulent energy exchange between electrons and ions. The integrated flow for the local exchange is simulated with the GYRO [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] delta-f gyrokinetic code and found to be small in a well studied DIII-D [M. A. Mahdavi and J. L. Luxon, ''DIII-D Tokamak Special Issue'' Fusion Sci. Technol. 48, 2 (2005)] L-mode discharge.

  1. Gyrokinetic particle simulation of beta-induced Alfven eigenmode

    SciTech Connect

    Zhang, H. S.; Lin, Z.; Holod, I.; Xiao, Y.; Wang, X.; Zhang, W. L.

    2010-11-15

    The beta-induced Alfven eigenmode (BAE) in toroidal plasmas is studied using global gyrokinetic particle simulations. The BAE real frequency and damping rate measured in the initial perturbation simulation and in the antenna excitation simulation agree well with each other. The real frequency is slightly higher than the ideal magnetohydrodynamic (MHD) accumulation point frequency due to the kinetic effects of thermal ions. Simulations with energetic particle density gradient show exponential growth of BAE with a growth rate sensitive to the energetic particle temperature and density. The nonperturbative contributions by energetic particles modify the mode structure and reduce the frequency relative to the MHD theory. The finite Larmor radius effects of energetic particles reduce the BAE growth rate. Benchmarks between gyrokinetic particle simulation and hybrid MHD-gyrokinetic simulation show good agreement in BAE real frequency and mode structure.

  2. Gyrokinetic particle simulation of a field reversed configuration

    NASA Astrophysics Data System (ADS)

    Fulton, D. P.; Lau, C. K.; Holod, I.; Lin, Z.; Dettrick, S.

    2016-01-01

    Gyrokinetic particle simulation of the field-reversed configuration (FRC) has been developed using the gyrokinetic toroidal code (GTC). The magnetohydrodynamic equilibrium is mapped from cylindrical coordinates to Boozer coordinates for the FRC core and scrape-off layer (SOL), respectively. A field-aligned mesh is constructed for solving self-consistent electric fields using a semi-spectral solver in a partial torus FRC geometry. This new simulation capability has been successfully verified and driftwave instability in the FRC has been studied using the gyrokinetic simulation for the first time. Initial GTC simulations find that in the FRC core, the ion-scale driftwave is stabilized by the large ion gyroradius. In the SOL, the driftwave is unstable on both ion and electron scales.

  3. Nonlinear Gyrokinetics: A Powerful Tool for the Description of Microturbulence in Magnetized Plasmas

    SciTech Connect

    John E. Krommes

    2010-09-27

    Gyrokinetics is the description of low-frequency dynamics in magnetized plasmas. In magnetic-confinement fusion, it provides the most fundamental basis for numerical simulations of microturbulence; there are astrophysical applications as well. In this tutorial, a sketch of the derivation of the novel dynamical system comprising the nonlinear gyrokinetic (GK) equation (GKE) and the coupled electrostatic GK Poisson equation will be given by using modern Lagrangian and Lie perturbation methods. No background in plasma physics is required in order to appreciate the logical development. The GKE describes the evolution of an ensemble of gyrocenters moving in a weakly inhomogeneous background magnetic field and in the presence of electromagnetic perturbations with wavelength of the order of the ion gyroradius. Gyrocenters move with effective drifts, which may be obtained by an averaging procedure that systematically, order by order, removes gyrophase dependence. To that end, the use of the Lagrangian differential one-form as well as the content and advantages of Lie perturbation theory will be explained. The electromagnetic fields follow via Maxwell's equations from the charge and current density of the particles. Particle and gyrocenter densities differ by an important polarization effect. That is calculated formally by a "pull-back" (a concept from differential geometry) of the gyrocenter distribution to the laboratory coordinate system. A natural truncation then leads to the closed GK dynamical system. Important properties such as GK energy conservation and fluctuation noise will be mentioned briefly, as will the possibility (and diffculties) of deriving nonlinear gyro fluid equations suitable for rapid numerical solution -- although it is probably best to directly simulate the GKE. By the end of the tutorial, students should appreciate the GKE as an extremely powerful tool and will be prepared for later lectures describing its applications to physical problems.

  4. A verification of the gyrokinetic microstability codes GEM, GYRO, and GS2

    NASA Astrophysics Data System (ADS)

    Bravenec, R. V.; Chen, Y.; Candy, J.; Wan, W.; Parker, S.

    2013-10-01

    A previous publication [R. V. Bravenec et al., Phys. Plasmas 18, 122505 (2011)] presented favorable comparisons of linear frequencies and nonlinear fluxes from the Eulerian gyrokinetic codes gyro [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] and gs2 [W. Dorland et al., Phys. Rev. Lett. 85, 5579 (2000)]. The motivation was to verify the codes, i.e., demonstrate that they correctly solve the gyrokinetic-Maxwell equations. The premise was that it is highly unlikely for both codes to yield the same incorrect results. In this work, we add the Lagrangian particle-in-cell code gem [Y. Chen and S. Parker, J. Comput. Phys. 220, 839 (2007)] to the comparisons, not simply to add another code, but also to demonstrate that the codes' algorithms do not matter. We find good agreement of gem with gyro and gs2 for the plasma conditions considered earlier, thus establishing confidence that the codes are verified and that ongoing validation efforts for these plasma parameters are warranted.

  5. A verification of the gyrokinetic microstability codes GEM, GYRO, and GS2

    SciTech Connect

    Bravenec, R. V.; Chen, Y.; Wan, W.; Parker, S.; Candy, J.

    2013-10-15

    A previous publication [R. V. Bravenec et al., Phys. Plasmas 18, 122505 (2011)] presented favorable comparisons of linear frequencies and nonlinear fluxes from the Eulerian gyrokinetic codes gyro[J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] and gs2[W. Dorland et al., Phys. Rev. Lett. 85, 5579 (2000)]. The motivation was to verify the codes, i.e., demonstrate that they correctly solve the gyrokinetic-Maxwell equations. The premise was that it is highly unlikely for both codes to yield the same incorrect results. In this work, we add the Lagrangian particle-in-cell code gem[Y. Chen and S. Parker, J. Comput. Phys. 220, 839 (2007)] to the comparisons, not simply to add another code, but also to demonstrate that the codes' algorithms do not matter. We find good agreement of gem with gyro and gs2 for the plasma conditions considered earlier, thus establishing confidence that the codes are verified and that ongoing validation efforts for these plasma parameters are warranted.

  6. Relevance of the parallel nonlinearity in gyrokinetic simulations of tokamak plasmas

    SciTech Connect

    Candy, J.; Waltz, R. E.; Parker, S. E.; Chen, Y.

    2006-07-15

    The influence of the parallel nonlinearity on transport in gyrokinetic simulations is assessed for values of {rho}{sub *} which are typical of current experiments. Here, {rho}{sub *}={rho}{sub s}/a is the ratio of gyroradius, {rho}{sub s}, to plasma minor radius, a. The conclusion, derived from simulations with both GYRO [J. Candy and R. E. Waltz, J. Comput. Phys., 186, 585 (2003)] and GEM [Y. Chen and S. E. Parker J. Comput. Phys., 189, 463 (2003)] is that no measurable effect of the parallel nonlinearity is apparent for {rho}{sub *}<0.012. This result is consistent with scaling arguments, which suggest that the parallel nonlinearity should be O({rho}{sub *}) smaller than the ExB nonlinearity. Indeed, for the plasma parameters under consideration, the magnitude of the parallel nonlinearity is a factor of 8{rho}{sub *} smaller (for 0.000 75<{rho}{sub *}<0.012) than the other retained terms in the nonlinear gyrokinetic equation.

  7. Benchmarking of the Gyrokinetic Microstability Codes GYRO, GS2, and GEM

    NASA Astrophysics Data System (ADS)

    Bravenec, Ronald; Chen, Yang; Wan, Weigang; Parker, Scott; Candy, Jeff; Barnes, Michael; Howard, Nathan; Holland, Christopher; Wang, Eric

    2012-10-01

    The physics capabilities of modern gyrokinetic microstability codes are now so extensive that they cannot be verified fully for realistic tokamak plasmas using purely analytic approaches. Instead, verification (demonstrating that the codes correctly solve the gyrokinetic-Maxwell equations) must rely on benchmarking (comparing code results for identical plasmas and physics). Benchmarking exercises for a low-power DIII-D discharge at the mid-radius have been presented recently for the Eulerian codes GYRO and GS2 [R.V. Bravenec, J. Candy, M. Barnes, C. Holland, Phys. Plasmas 18, 122505 (2011)]. This work omitted ExB flow shear, but we include it here. We also present GYRO/GS2 comparisons for a high-power Alcator C-Mod discharge. To add further confidence to the verification exercises, we have recently added the particle-in-cell (PIC) code GEM to the efforts. We find good agreement of linear frequencies between GEM and GYRO/GS2 for the DIII-D plasma. We also present preliminary nonlinear comparisons. This benchmarking includes electromagnetic effects, plasma shaping, kinetic electrons and one impurity. In addition, we compare linear results among the three codes for the steep-gradient edge region of a DIII-D plasma between edge-localized modes.

  8. Fully Nonlinear Edge Gyrokinetic Simulations of Kinetic Geodesic-Acoustic Modes and Boundary Flows

    SciTech Connect

    Xu, X Q; Belli, E; Bodi, K; Candy, J; Chang, C S; Cohen, B I; Cohen, R H; Colella, P; Dimits, A M; Dorr, M R; Gao, Z; Hittinger, J A; Ko, S; Krasheninnikov, S; McKee, G R; Nevins, W M; Rognlien, T D; Snyder, P B; Suh, J; Umansky, M V

    2008-09-18

    We present edge gyrokinetic neoclassical simulations of tokamak plasmas using the fully nonlinear (full-f) continuum code TEMPEST. A nonlinear Boltzmann model is used for the electrons. The electric field is obtained by solving the 2D gyrokinetic Poisson Equation. We demonstrate the following: (1) High harmonic resonances (n > 2) significantly enhance geodesic-acoustic mode (GAM) damping at high-q (tokamak safety factor), and are necessary to explain both the damping observed in our TEMPEST q-scans and experimental measurements of the scaling of the GAM amplitude with edge q{sub 95} in the absence of obvious evidence that there is a strong q dependence of the turbulent drive and damping of the GAM. (2) The kinetic GAM exists in the edge for steep density and temperature gradients in the form of outgoing waves, its radial scale is set by the ion temperature profile, and ion temperature inhomogeneity is necessary for GAM radial propagation. (3) The development of the neoclassical electric field evolves through different phases of relaxation, including GAMs, their radial propagation, and their long-time collisional decay. (4) Natural consequences of orbits in the pedestal and scrape-off layer region in divertor geometry are substantial non-Maxwellian ion distributions and flow characteristics qualitatively like those observed in experiments.

  9. Poloidal tilting symmetry of high order tokamak flux surface shaping in gyrokinetics

    NASA Astrophysics Data System (ADS)

    Ball, Justin; Parra, Felix I.; Barnes, Michael

    2016-04-01

    A poloidal tilting symmetry of the local nonlinear δ f gyrokinetic model is demonstrated analytically and verified numerically. This symmetry shows that poloidally rotating all the flux surface shaping effects with large poloidal mode number by a single tilt angle has an exponentially small effect on the transport properties of a tokamak. This is shown using a generalization of the Miller local equilibrium model to specify an arbitrary flux surface geometry. With this geometry specification we find that, when performing an expansion in large flux surface shaping mode number, the governing equations of gyrokinetics are symmetric in the poloidal tilt of the high order shaping effects. This allows us to take the fluxes from a single configuration and calculate the fluxes in any configuration that can be produced by tilting the large mode number shaping effects. This creates a distinction between tokamaks with mirror symmetric flux surfaces and tokamaks without mirror symmetry, which is expected to have important consequences for generating toroidal rotation using up-down asymmetry.

  10. Quasisteady and steady states in global gyrokinetic particle-in-cell simulations

    SciTech Connect

    Jolliet, S.; McMillan, B. F.; Vernay, T.; Villard, L.; Bottino, A.; Angelino, P.

    2009-05-15

    Collisionless delta-f gyrokinetic particle-in-cell simulations suffer from the entropy paradox, in which the entropy grows linearly in time while low-order moments are saturated. As a consequence, these simulations do not reach a steady state and are unsuited to make quantitative predictions. A solution to this issue is the introduction of artificial dissipation. The notion of steady state in gyrokinetic simulations is studied by deriving an evolution equation for the fluctuation entropy and applying it to the global collisionless particle-in-cell code ORB5 [S. Jolliet et al., Comput. Phys. Commun. 177, 409 (2007)]. It is shown that a recently implemented noise-control algorithm [B. F. McMillan et al., Phys. Plasmas 15, 052308 (2008)] based on a W-stat provides the necessary dissipation to reach a steady state. The two interesting situations of decaying and driven turbulence are considered. In addition, it is shown that a separate heating algorithm, not based on a W-stat, does not lead to a statistical steady state.

  11. Introduction to Gyrokinetic Theory with Applications in Magnetic Confinement Research in Plasma Physics

    SciTech Connect

    W.M. Tang

    2005-01-03

    The present lecture provides an introduction to the subject of gyrokinetic theory with applications in the area of magnetic confinement research in plasma physics--the research arena from which this formalism was originally developed. It was presented as a component of the ''Short Course in Kinetic Theory within the Thematic Program in Partial Differential Equations'' held at the Fields Institute for Research in Mathematical Science (24 March 2004). This lecture also discusses the connection between the gyrokinetic formalism and powerful modern numerical simulations. Indeed, simulation, which provides a natural bridge between theory and experiment, is an essential modern tool for understanding complex plasma behavior. Progress has been stimulated in particular by the exponential growth of computer speed along with significant improvements in computer technology. The advances in both particle and fluid simulations of fine-scale turbulence and large-scale dynamics have produced increasingly good agreement between experimental observations and computational modeling. This was enabled by two key factors: (i) innovative advances in analytic and computational methods for developing reduced descriptions of physics phenomena spanning widely disparate temporal and spatial scales and (ii) access to powerful new computational resources.

  12. Gyrokinetic turbulence cascade via predator-prey interactions between different scales

    SciTech Connect

    Kobayashi, Sumire Gurcan, Ozgur D.

    2015-05-15

    Gyrokinetic simulations in a closed fieldline geometry are presented to explore the physics of nonlinear transfer in plasma turbulence. As spontaneously formed zonal flows and small-scale turbulence demonstrate “predator-prey” dynamics, a particular cascade spectrum emerges. The electrostatic potential and the density spectra appear to be in good agreement with the simple theoretical prediction based on Charney-Hasegawa-Mima equation | ϕ{sup ~}{sub k} |{sup 2}∼| n{sup ~}{sub k} |{sup 2}∝k{sup −3}/(1+k{sup 2}){sup 2}, with the spectra becoming anisotropic at small scales. The results indicate that the disparate scale interactions, in particular, the refraction and shearing of larger scale eddies by the self-consistent zonal flows, dominate over local interactions, and contrary to the common wisdom, the comprehensive scaling relation is created even within the energy injection region.

  13. Gyrokinetic turbulence cascade via predator-prey interactions between different scales

    NASA Astrophysics Data System (ADS)

    Kobayashi, Sumire; Gurcan, Ozgur D.

    2015-05-01

    Gyrokinetic simulations in a closed fieldline geometry are presented to explore the physics of nonlinear transfer in plasma turbulence. As spontaneously formed zonal flows and small-scale turbulence demonstrate "predator-prey" dynamics, a particular cascade spectrum emerges. The electrostatic potential and the density spectra appear to be in good agreement with the simple theoretical prediction based on Charney-Hasegawa-Mima equation |ϕ˜ k | 2˜|n˜ k | 2∝k-3/(1+k2 ) 2 , with the spectra becoming anisotropic at small scales. The results indicate that the disparate scale interactions, in particular, the refraction and shearing of larger scale eddies by the self-consistent zonal flows, dominate over local interactions, and contrary to the common wisdom, the comprehensive scaling relation is created even within the energy injection region.

  14. Gyrokinetic determination of the electrostatic potential of rotating magnetic islands in tokamaks

    SciTech Connect

    Siccinio, M.; Poli, E.; Casson, F. J.; Hornsby, W. A.; Peeters, A. G.

    2011-12-15

    The electrostatic potential related to a magnetic island structure with imposed width and rotation frequency is studied by means of gyrokinetic simulations, which allow its self-consistent determination via the Poisson equation. An adiabatic response of the trapped ions at the island separatrix leads to a significant smoothing of the potential with respect to analytic calculations based on a complete flattening of the pressure profile inside the island. As a consequence, the magnitude of the polarization current is drastically reduced. When the island size is comparable to the ion banana width, the adiabatic response covers the whole island region, leading to a reduced density flattening for islands rotating in the electron diamagnetic direction. This confirms previous results based on drift-kinetic simulations.

  15. Gyrokinetic study of the spatial entropy dynamics in turbulent plasmas with zonal flow

    NASA Astrophysics Data System (ADS)

    Imadera, Kenji; Kishimoto, Yasuaki; Li, Jiquan; Utsumi, Takayuki

    2009-11-01

    We have developed a new computational algorithm based on the IDO-CF (Conservative Form of Interpolated Differential Operator) scheme [1], which is efficient in capturing sharp domain structure in long time scale, for solving full-f Gyrokineitc Vlasov-Poisson system. By using the developed code, we have performed the ITG simulation focusing on entropy dynamics and associated zonal flow formation. Here, we have introduced the modified local entropy defined asδSm(x)=<δf^2/ 2f0(-1+v||^2 /T) >yzdv , which retains the spatial information. It is found that the entropy balances with the acoustic coupling driven by ITG mode in the linear stage, and then the zonal flows expel the entropy to outside region via its convection. The spatial structure of the entropy is regulated by the zonal flows, and finally, the quasi-steady state where the entropy and zonal flows have similar structure is established. This indicates that the zonal flows couple with the entropy spatially [1] Y.Imai et al., J. Comput. Phys. 227 (2008) 2263.

  16. Eulerian action principles for linearized reduced dynamical equations

    NASA Astrophysics Data System (ADS)

    Brizard, Alain

    1994-08-01

    New Eulerian action principles for the linearized gyrokinetic Maxwell-Vlasov equations and the linearized kinetic-magnetohydrodynamic (kinetic-MHD) equations are presented. The variational fields for the linearized gyrokinetic Vlasov-Maxwell equations are the perturbed electromagnetic potentials (φ1,A1) and the gyroangle-independent gyrocenter (gy) function Sgy, while the variational fields for the linearized kinetic-MHD equations are the ideal MHD fluid displacement ξ and the gyroangle-independent drift-kinetic (dk) function Sdk (defined as the drift-kinetic limit of Sgy). According to the Lie-transform approach to Vlasov perturbation theory, Sgy generates first-order perturbations in the gyrocenter distribution F1≡{Sgy, F0}gc, where F1 satisfies the linearized gyrokinetic Vlasov equation and {, }gc denotes the unperturbed guiding-center (gc) Poisson bracket. Previous quadratic variational forms were constructed ad hoc from the linearized equations, and required the linearized gyrokinetic (or drift-kinetic) Vlasov equation to be solved a priori (e.g., by integration along an unperturbed guiding-center orbit) through the use of the normal-mode and ballooning-mode representations. The presented action principles ignore these requirements and, thus, apply to more general perturbations.

  17. Simulating Gyrokinetic/fluid hybrid electromagnetic modes in the total-f gyrokinetic code XGC1

    NASA Astrophysics Data System (ADS)

    Lang, Jianying; Hager, Robert; Ku, Seung-Hoe; Chang, Choong-Seock

    2015-11-01

    XGC1 code has been extended to include the electronmagnetic capability using the hybrid model with gyrokinetic ions and fluid electrons. This feature will enable a more complete description of the MHD/fluid type mode activities including ELMs and low-n tearing modes. Their interaction with the kinetic neoclassical and microturbulence dynamics needs to be simulated together. Evolution of the background profile should also be captured self-consistently. We report recent development and verification of this hybrid model in the limit of small delta-B. The code has been verified for Alfven waves and ITG/KBM transition, and low-n resistive tearing modes. The KBM capability of XGC1 has been verified against the published results from Gyro, GEM, GS2, Gene, and GTC. Detailed verification of resistive tearing modes and kink modes in the toroidal geometry will be also presented. An implicit method is implemented in XGC1 to bypass the Courant condition caused by fast Alfven oscillations. Work supported by US DOE OFES and OASCR.

  18. The Implementation of Magnetic Islands in Gyrokinetic Toroidal Code

    NASA Astrophysics Data System (ADS)

    Jiang, Peng; Lin, Zhihong; Ihor, Holod; Xiao, Chijie

    2016-02-01

    The implementation of magnetic islands in gyrokinetic simulation has been verified in the gyrokinetic toroidal code (GTC). The ion and electron density profiles become partially flattened inside the islands. The density profile at the low field side is less flattened than that at the high field side due to toroidally trapped particles in the low field side, which do not move along the perturbed magnetic field lines. When the fraction of trapped particles decreases, the density profile at the low field becomes more flattened. supported by National Special Research Program of China for ITER (Nos. 2013GB111000 and 2014GB107004), China Scholarship Council (No. 2011601098), U.S. DOE Grants DE-SC0010416 and DE-FG02-07ER54916

  19. Gyrokinetics for high-frequency modes in tokamaks

    NASA Astrophysics Data System (ADS)

    Wang, Z. T.; Wang, L.; Long, L. X.; Dong, J. Q.; He, Zhixiong; Liu, Y.; Tang, C. J.

    2012-07-01

    Gyrokinetics for high-frequency modes in tokamaks is developed. It is found that the breakdown of the invariants by perturbed electromagnetic fields drives microinstability. The obtained diamagnetic frequency, ω∗, is proportional to only the toroidal mode number rather than transverse mode numbers. Therefore, there is no nonadiabatic drive for axisymmetrical modes in gyrokinetics. Meanwhile, the conventional eikonal Ansatz breaks down for the axisymmetrical modes. The ion drift-cyclotron instability discovered in a mirror machine is found for the first time in the toroidal system. The growth rates are proportional to ρi/Ln, and the slope changes with magnetic curvature. In spherical torus, where magnetic curvature is greater than that of traditional tokamaks, instability poses a potential danger to such devices.

  20. Gyrokinetics for high-frequency modes in tokamaks

    SciTech Connect

    Wang, Z. T.; Long, L. X.; Dong, J. Q.; He, Zhixiong; Wang, L.; Liu, Y.; Tang, C. J.

    2012-07-15

    Gyrokinetics for high-frequency modes in tokamaks is developed. It is found that the breakdown of the invariants by perturbed electromagnetic fields drives microinstability. The obtained diamagnetic frequency, {omega}{sup *}, is proportional to only the toroidal mode number rather than transverse mode numbers. Therefore, there is no nonadiabatic drive for axisymmetrical modes in gyrokinetics. Meanwhile, the conventional eikonal Ansatz breaks down for the axisymmetrical modes. The ion drift-cyclotron instability discovered in a mirror machine is found for the first time in the toroidal system. The growth rates are proportional to {rho}{sub i}/L{sub n}, and the slope changes with magnetic curvature. In spherical torus, where magnetic curvature is greater than that of traditional tokamaks, instability poses a potential danger to such devices.

  1. A Numerical Instability in an ADI Algorithm for Gyrokinetics

    SciTech Connect

    E.A. Belli; G.W. Hammett

    2004-12-17

    We explore the implementation of an Alternating Direction Implicit (ADI) algorithm for a gyrokinetic plasma problem and its resulting numerical stability properties. This algorithm, which uses a standard ADI scheme to divide the field solve from the particle distribution function advance, has previously been found to work well for certain plasma kinetic problems involving one spatial and two velocity dimensions, including collisions and an electric field. However, for the gyrokinetic problem we find a severe stability restriction on the time step. Furthermore, we find that this numerical instability limitation also affects some other algorithms, such as a partially implicit Adams-Bashforth algorithm, where the parallel motion operator v{sub {parallel}} {partial_derivative}/{partial_derivative}z is treated implicitly and the field terms are treated with an Adams-Bashforth explicit scheme. Fully explicit algorithms applied to all terms can be better at long wavelengths than these ADI or partially implicit algorithms.

  2. The energetic coupling of scales in gyrokinetic plasma turbulence

    SciTech Connect

    Teaca, Bogdan; Jenko, Frank

    2014-07-15

    In magnetized plasma turbulence, the couplings of perpendicular spatial scales that arise due to the nonlinear interactions are analyzed from the perspective of the free-energy exchanges. The plasmas considered here, with appropriate ion or electron adiabatic electro-neutrality responses, are described by the gyrokinetic formalism in a toroidal magnetic geometry. Turbulence develops due to the electrostatic fluctuations driven by temperature gradient instabilities, either ion temperature gradient (ITG) or electron temperature gradient (ETG). The analysis consists in decomposing the system into a series of scale structures, while accounting separately for contributions made by modes possessing special symmetries (e.g., the zonal flow modes). The interaction of these scales is analyzed using the energy transfer functions, including a forward and backward decomposition, scale fluxes, and locality functions. The comparison between the ITG and ETG cases shows that ETG turbulence has a more pronounced classical turbulent behavior, exhibiting a stronger energy cascade, with implications for gyrokinetic turbulence modeling.

  3. Graphics Processing Unit Acceleration of Gyrokinetic Turbulence Simulations

    NASA Astrophysics Data System (ADS)

    Hause, Benjamin; Parker, Scott

    2012-10-01

    We find a substantial increase in on-node performance using Graphics Processing Unit (GPU) acceleration in gyrokinetic delta-f particle-in-cell simulation. Optimization is performed on a two-dimensional slab gyrokinetic particle simulation using the Portland Group Fortran compiler with the GPU accelerator compiler directives. We have implemented the GPU acceleration on a Core I7 gaming PC with a NVIDIA GTX 580 GPU. We find comparable, or better, acceleration relative to the NERSC DIRAC cluster with the NVIDIA Tesla C2050 computing processor. The Tesla C 2050 is about 2.6 times more expensive than the GTX 580 gaming GPU. Optimization strategies and comparisons between DIRAC and the gaming PC will be presented. We will also discuss progress on optimizing the comprehensive three dimensional general geometry GEM code.

  4. Plasma Simulation Using Gyrokinetic-Gyrofluid Hybrid Models

    SciTech Connect

    Scott Parker

    2009-04-09

    We are developing kinetic ion models for the simulation of extended MHD phenomena. The model they have developed uses full Lorentz force ions, and either drift-kinetic or gyro-kinetic electrons. Quasi-neutrality is assumed and the displacement current is neglected. They are also studying alpha particle driven Toroidal Alfven Eigenmodes (TAE) in the GEM gyrokinetic code [Chen 07]. The basic kinetic ion MHD model was recently reported in an invited talk given by Dan Barnes at the 2007 American Physical Society - Division of Plasma Physics (APS-DPP) and it has been published [Jones 04, Barnes 08]. The model uses an Ohm's law that includes the Hall term, pressure term and the electron inertia [Jones 04]. These results focused on the ion physics and assumed an isothermal electron closure. It is found in conventional gyrokinetic turbulence simulations that the timestep cannot be made much greater than the ion cyclotron period. However, the kinetic ion MHD model has the compressional mode, which further limits the timestep. They have developed an implicit scheme to avoid this timestep constraint. They have also added drift kinetic electrons. This model has been benchmarked linearly. Waves investigated where shear and compressional Alfven, whisterl, ion acoustic, and drift waves, including the kinetic damping rates. This work is ongoing and was first reported at the 2008 Sherwood Fusion Theory Conference [Chen 08] and they are working on a publication. They have also formulated an integrated gyrokinetic electron model, which is of interest for studying electron gradient instabilities and weak guide-field magnetic reconnection.

  5. Status of edge gyrokinetic turbulence simulation in XGC1

    NASA Astrophysics Data System (ADS)

    Ku, Seung-Hoe; Chang, C. S.; Zorin, D.; Greengard, L.; Adams, M.; Cummings, J.; Worley, P.; D'Azevedo, E.; Lee, W.; Parker, S.; Chen, Y.; Lin, Z.

    2007-11-01

    Gyrokinetic simulation of a tokamak edge plasma is one of the highest priority research items for ITER and the magnetic fusion program. Due to the complex physical modeling required in the edge plasma (closed and open magnetic field lines with the magnetic separatrix in between, the importance of neoclassical physics, the material wall boundary, steep pressure gradients, a non-Maxwellian distribution function, and the neutral particle physics), most of the gyrokinetic simulation activities have so far been focused on the core plasmas. The status of the gyrokinetic edge turbulence simulation in the XGC1 particle code in the SciDAC Prototype FSP Center for Plasma Edge Simulation (CPES) will be reported. XGC1 includes the above mentioned edge complexities with full- f/delta-f particle technology on an unstructured mesh. Special physics/math/CS features will be discussed. Our current electrostatic turbulence/neoclassical capabilities will be presented and verified. Plans for incorporating full electromagnetic turbulence will also be discussed.

  6. Tokamak profile prediction using direct gyrokinetic and neoclassical simulation

    SciTech Connect

    Candy, Jeff; Holland, Chris; Waltz, R. E.; Fahey, Mark R; Belli, E

    2009-01-01

    okamak transport modeling scenarios, including ITER ITER Physics Basis Editors, Nucl. Fusion 39, 2137 1999 performance predictions, are based exclusively on reduced models for core thermal and particle transport. The reason for this is simple: computational cost. A typical modeling scenario may require the evaluation of thousands of individual transport fluxes local transport models calculate the energy and particle fluxes across a specified flux surface given fixed profiles . Despite continuous advances in direct gyrokinetic simulation, the cost of an individual simulation remains so high that direct gyrokinetic transport calculations have been avoided. By developing a steady-state iteration scheme suitable for direct gyrokinetic and neoclassical simulations, we can now compute steady-state temperature profiles for DIII-D J. L. Luxon, Nucl. Fusion 42, 614 2002 plasmas given known plasma sources. The new code, TGYRO, encapsulates the GYRO J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 2003 code, for turbulent transport, and the NEO E. A. Belli and J. Candy, Plasma Phys. Controlled Fusion 50, 095010 2008 code, for kinetic neoclassical transport. Results for DIII-D L-mode discharge 128913 are given, with computational and experimental results consistent in the region 0 <= r/a <= 0.8.

  7. Tokamak profile prediction using direct gyrokinetic and neoclassical simulation

    SciTech Connect

    Candy, J.; Waltz, R. E.; Belli, E.; Holland, C.; Fahey, M. R.

    2009-06-15

    Tokamak transport modeling scenarios, including ITER [ITER Physics Basis Editors, Nucl. Fusion 39, 2137 (1999)] performance predictions, are based exclusively on reduced models for core thermal and particle transport. The reason for this is simple: computational cost. A typical modeling scenario may require the evaluation of thousands of individual transport fluxes (local transport models calculate the energy and particle fluxes across a specified flux surface given fixed profiles). Despite continuous advances in direct gyrokinetic simulation, the cost of an individual simulation remains so high that direct gyrokinetic transport calculations have been avoided. By developing a steady-state iteration scheme suitable for direct gyrokinetic and neoclassical simulations, we can now compute steady-state temperature profiles for DIII-D [J. L. Luxon, Nucl. Fusion 42, 614 (2002)] plasmas given known plasma sources. The new code, TGYRO, encapsulates the GYRO[J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] code, for turbulent transport, and the NEO[E. A. Belli and J. Candy, Plasma Phys. Controlled Fusion 50, 095010 (2008)] code, for kinetic neoclassical transport. Results for DIII-D L-mode discharge 128913 are given, with computational and experimental results consistent in the region 0{<=}r/a{<=}0.8.

  8. Predictive Gyrokinetic Transport Simulations and Application of Synthetic Diagnostics

    NASA Astrophysics Data System (ADS)

    Candy, J.

    2009-11-01

    In this work we make use of the gyrokinetic transport solver TGYRO [1] to predict kinetic plasma profiles consistent with energy and particle fluxes in the DIII-D tokamak. TGYRO uses direct nonlinear and neoclassical fluxes calculated by the GYRO and NEO codes, respectively, to solve for global, self-consistent temperature and density profiles via Newton iteration. Previous work has shown that gyrokinetic simulation results for DIII-D discharge 128913 match experimental data rather well in the plasma core, but with a discrepancy in both fluxes and fluctuation levels emerging closer to the edge (r/a > 0.8). The present work will expand on previous results by generating model predictions across the entire plasma core, rather than at isolated test radii. We show that TGYRO predicts temperature and density profiles in good agreement with experimental observations which simultaneously yield near-exact (to within experimental uncertainties) agreement with power balance calculations of the particle and energy fluxes for r/a <=0.8. Moreover, we use recently developed synthetic diagnostic algorithms [2] to show that TGYRO also predicts density and electron temperature fluctuation levels in close agreement with experimental measurements across the simulated plasma volume. 8pt [1] J. Candy, C. Holland, R.E. Waltz, M.R. Fahey, and E. Belli, ``Tokamak profile prediction using direct gyrokinetic and neoclassical simulation," Phys. Plasmas 16, 060704 (2009). [2] C. Holland, A.E. White, G.R. McKee, M.W. Shafer, J. Candy, R.E. Waltz, L. Schmitz, and G.R. Tynan, ``Implementation and application of two synthetic diagnostics for validating simulations of core tokamak turbulence," Phys. Plasmas 16, 052301 (2009).

  9. Gyrokinetic simulation of isotope scaling in tokamak plasmas

    SciTech Connect

    Lee, W.W.; Santoro, R.A.

    1995-07-01

    A three-dimensional global gyrokinetic particle code in toroidal geometry has been used for investigating the transport properties of ion temperature gradient (ITG) drift instabilities in tokamak plasmas. Using the isotopes of hydrogen (H{sup +}), deuterium (D{sup +}) and tritium (T{sup +}), we have found that, under otherwise identical conditions, there exists a favorable isotope scaling for the ion thermal diffusivity, i.e., Xi decreases with mass. Such a scaling, which exists both at the saturation of the instability and also at the nonlinear steady state, can be understood from the resulting wavenumber and frequency spectra.

  10. Applications of large eddy simulation methods to gyrokinetic turbulence

    SciTech Connect

    Bañón Navarro, A. Happel, T.; Teaca, B. [Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB; Max-Planck für Sonnensystemforschung, Max-Planck-Str. 2, D-37191 Katlenburg-Lindau; Max-Planck Jenko, F. [Max-Planck-Institut für Plasmaphysik, EURATOM Association, D-85748 Garching; Max-Planck Hammett, G. W. [Max-Planck Collaboration: ASDEX Upgrade Team

    2014-03-15

    The large eddy simulation (LES) approach—solving numerically the large scales of a turbulent system and accounting for the small-scale influence through a model—is applied to nonlinear gyrokinetic systems that are driven by a number of different microinstabilities. Comparisons between modeled, lower resolution, and higher resolution simulations are performed for an experimental measurable quantity, the electron density fluctuation spectrum. Moreover, the validation and applicability of LES is demonstrated through a series of diagnostics based on the free energetics of the system.

  11. Study of high-N modes in tokamaks using a high speed nonlocal gyrokinetic model

    NASA Astrophysics Data System (ADS)

    Elia, Michael Aldo

    Gyrokinetic theory has been used to derive a system of integral equations which nonlocally describe low frequency, short wavelength modes in a plasma of axisymmetrical toroidal geometry with low-beta and circular nonconcentric flux surfaces with small Shafranov shift. The eigenmode equations contain the two potential approximation in ϕ and A ∥ with full finite Larmor radius and trapped electron effects in the collisionless limit. The analysis makes use of the so-called "ballooning formalism" to lowest order in 1/n which yields a radially local calculation for the eigenfrequencies and the eigenfunctions. This representation, in conjunction with an efficient numerical algorithm, allows the eigenfrequencies to be computed with sufficient accuracy and high speed for arbitrary high- n modes in the drift and shear-Alfven branches. This is the main accomplishment of this work. Test cases using artificial and actual tokamak experimental discharge parameters for the collisionless-trapped-electron, ion-temperature-gradient and ballooning modes have been benchmarked with the premium, comprehensive kinetic formulation of Rewoldt exhibiting favourable results.

  12. Gyrokinetic δ f simulation of collisionless and semi-collisional tearing mode instabilities

    NASA Astrophysics Data System (ADS)

    Wan, Weigang; Chen, Yang; Parker, Scott

    2004-11-01

    The evolution of collisionless and semi-collisional tearing mode instabilities is studied using a three-dimensional particle-in-cell simulation model that utilizes the δ f-method with the split-weight scheme to enhance the time step, and a novel algorithm(Y. Chen and S.E. Parker, J. Comput. Phys. 198), 463 (2003) to accurately solve the Ampere's equation for experimentally relevant β values, βfracm_im_e≫ 1. We use the model of drift-kinetic electrons and gyrokinetic ions. Linear simulation results are benchmarked with eigenmode analysis for the case of fixed ions. In small box simulations the ions response can be neglected but for large box simulations the ions response is important because the width of perturbed current is larger than ρ_i.The nonlinear dynamics of magnetic islands will be studied and the results will be compared with previous theoretical studiesfootnote J.F. Drake and Y. C. Lee, Phys. Rev. Lett. 39, 453 (1977) on the saturation level and the electron bounce frequency. A collision operator is included in the electron drift kinetic equation to study the simulation in the semi-collisional regime. The algebraical growth stage has been observed and compared quantitatively with theory. Our progress on three-dimensional simulations of tearing mode instabilities will be reported.

  13. Gyrokinetic Simulation of Residual Stress from Diamagnetic Velocity Shears

    NASA Astrophysics Data System (ADS)

    Waltz, R. E.; Staebler, G. M.; Solomon, W. M.

    2010-11-01

    Residual stress refers to the remaining toroidal angular momentum (TAM) flux (divided by major radius) when the shear in the parallel velocity (and parallel velocity itself) vanishes. Previously [1] we demonstrated with gyrokinetic (GYRO) simulations that TAM pinching from the diamagnetic level shear in the ExB velocity could provide the residual stress needed for spontaneous toroidal rotation. Here we show that the shear in the diamagnetic velocities themselves provide comparable residual stress (and level of stabilization). The sign of the residual stress, quantified by the ratio of TAM flow to ion power flow (M/P), depends on the signs of the various velocity shears as well as ion (ITG) versus electron (TEM) mode directed turbulence. The residual stress from these temperature and density gradient diamagnetic velocity shears is demonstrated in global gyrokinetic simulation of ``null'' rotation DIIID discharges by matching M/P profiles within experimental error. 8pt [1] R.E. Waltz, G.M. Staebler, J. Candy, and F.L. Hinton, Phys. Plasmas 14, 122507 (2007); errata 16, 079902 (2009).

  14. Linear signatures in nonlinear gyrokinetics: interpreting turbulence with pseudospectra

    NASA Astrophysics Data System (ADS)

    Hatch, D. R.; Jenko, F.; Bañón Navarro, A.; Bratanov, V.; Terry, P. W.; Pueschel, M. J.

    2016-07-01

    A notable feature of plasma turbulence is its propensity to retain features of the underlying linear eigenmodes in a strongly turbulent state—a property that can be exploited to predict various aspects of the turbulence using only linear information. In this context, this work examines gradient-driven gyrokinetic plasma turbulence through three lenses—linear eigenvalue spectra, pseudospectra, and singular value decomposition (SVD). We study a reduced gyrokinetic model whose linear eigenvalue spectra include ion temperature gradient driven modes, stable drift waves, and kinetic modes representing Landau damping. The goal is to characterize in which ways, if any, these familiar ingredients are manifest in the nonlinear turbulent state. This pursuit is aided by the use of pseudospectra, which provide a more nuanced view of the linear operator by characterizing its response to perturbations. We introduce a new technique whereby the nonlinearly evolved phase space structures extracted with SVD are linked to the linear operator using concepts motivated by pseudospectra. Using this technique, we identify nonlinear structures that have connections to not only the most unstable eigenmode but also subdominant modes that are nonlinearly excited. The general picture that emerges is a system in which signatures of the linear physics persist in the turbulence, albeit in ways that cannot be fully explained by the linear eigenvalue approach; a non-modal treatment is necessary to understand key features of the turbulence.

  15. Direct identification of predator-prey dynamics in gyrokinetic simulations

    SciTech Connect

    Kobayashi, Sumire Gürcan, Özgür D; Diamond, Patrick H.

    2015-09-15

    The interaction between spontaneously formed zonal flows and small-scale turbulence in nonlinear gyrokinetic simulations is explored in a shearless closed field line geometry. It is found that when clear limit cycle oscillations prevail, the observed turbulent dynamics can be quantitatively captured by a simple Lotka-Volterra type predator-prey model. Fitting the time traces of full gyrokinetic simulations by such a reduced model allows extraction of the model coefficients. Scanning physical plasma parameters, such as collisionality and density gradient, it was observed that the effective growth rates of turbulence (i.e., the prey) remain roughly constant, in spite of the higher and varying level of primary mode linear growth rates. The effective growth rate that was extracted corresponds roughly to the zonal-flow-modified primary mode growth rate. It was also observed that the effective damping of zonal flows (i.e., the predator) in the parameter range, where clear predator-prey dynamics is observed, (i.e., near marginal stability) agrees with the collisional damping expected in these simulations. This implies that the Kelvin-Helmholtz-like instability may be negligible in this range. The results imply that when the tertiary instability plays a role, the dynamics becomes more complex than a simple Lotka-Volterra predator prey.

  16. Multi-code benchmark of global gyrokinetic electromagnetic instabilities

    NASA Astrophysics Data System (ADS)

    Goerler, Tobias; Bottino, Alberto; Hornsby, William A.; Kleiber, Ralf; Tronko, Natalia; Grandgirard, Virginie; Norscini, Claudia; Sonnendruecker, Eric

    2015-11-01

    Considering the recent major extensions of global gyrokinetic codes towards a comprehensive and self-consistent treatment of electromagnetic (EM) effects, corresponding verification tests are obvious and necessary steps to be taken. While a number of (semi-)analytic test cases and benchmarks exist in the axisymmetric limit, microinstabilities and particularly EM turbulence are rarely addressed. In order to remedy this problem, a hierarchical linear gyrokinetic benchmark study is presented starting with electrostatic ion temperature gradient microinstabilities with adiabatic electron response and progressing finally to the characterization of fully EM instabilities as a function of β. The inter-code comparison involves contributions from Eulerian Vlasov, Lagrangian PIC, and Semi-Lagrange codes at least in one level of this verification exercise, thus confirming a high degree of reliability for the implementations that has rarely been achieved before in this context. Additionally, possible extensions of this benchmark into the physically more relevant nonlinear turbulence regime will be discussed, e.g., relaxation problems or gradient-driven setups. This work has been carried out within the framework of the EUROfusion Consortium.

  17. Gyrokinetic stability theory of electron-positron plasmas

    NASA Astrophysics Data System (ADS)

    Helander, P.; Connor, J. W.

    2016-06-01

    > The linear gyrokinetic stability properties of magnetically confined electron-positron plasmas are investigated in the parameter regime most likely to be relevant for the first laboratory experiments involving such plasmas, where the density is small enough that collisions can be ignored and the Debye length substantially exceeds the gyroradius. Although the plasma beta is very small, electromagnetic effects are retained, but magnetic compressibility can be neglected. The work of a previous publication (Helander, Phys. Rev. Lett., vol. 113, 2014a, 135003) is thus extended to include electromagnetic instabilities, which are of importance in closed-field-line configurations, where such instabilities can occur at arbitrarily low pressure. It is found that gyrokinetic instabilities are completely absent if the magnetic field is homogeneous: any instability must involve magnetic curvature or shear. Furthermore, in dipole magnetic fields, the stability threshold for interchange modes with wavelengths exceeding the Debye radius coincides with that in ideal magnetohydrodynamics. Above this threshold, the quasilinear particle flux is directed inward if the temperature gradient is sufficiently large, leading to spontaneous peaking of the density profile.

  18. Gyrokinetic modelling of stationary electron and impurity profiles in tokamaks

    SciTech Connect

    Skyman, A. Tegnered, D. Nordman, H. Strand, P.

    2014-09-15

    Particle transport due to Ion Temperature Gradient (ITG)/Trapped Electron Mode (TEM) turbulence is investigated using the gyrokinetic code GENE. Both a reduced quasilinear treatment and nonlinear simulations are performed for typical tokamak parameters corresponding to ITG dominated turbulence. The gyrokinetic results are compared and contrasted with results from a computationally efficient fluid model. A selfconsistent treatment is used, where the stationary local profiles are calculated corresponding to zero particle flux simultaneously for electrons and trace impurities. The scaling of the stationary profiles with magnetic shear, safety factor, electron-to-ion temperature ratio, collisionality, toroidal sheared rotation, plasma β, triangularity, and elongation is investigated. In addition, the effect of different main ion mass on the zero flux condition is discussed. The electron density gradient can significantly affect the stationary impurity profile scaling. It is therefore expected that a selfconsistent treatment will yield results more comparable to experimental results for parameter scans where the stationary background density profile is sensitive. This is shown to be the case in scans over magnetic shear, collisionality, elongation, and temperature ratio, for which the simultaneous zero flux electron and impurity profiles are calculated. A slight asymmetry between hydrogen, deuterium, and tritium with respect to profile peaking is obtained, in particular, for scans in collisionality and temperature ratio.

  19. Gyrokinetic simulations of microturbulence in DIII-D tokamak pedestal

    NASA Astrophysics Data System (ADS)

    Holod, Ihor; Fulton, Daniel; Taimourzadeh, Sam; Lin, Zhihong; Nazikian, Raffi; Spong, Donald

    2015-11-01

    The characteristics of H-mode pedestal are generally believed to be constrained by current-driven peeling-ballooning modes and pressure-driven instabilities, such as kinetic ballooning mode (KBM). In this work we use global gyrokinetic code (GTC) to identify and study the edge pressure-driven instabilities in the H-mode pedestal using realistic geometry and plasma profiles of DIII-D shot 131997. In our simulations we observe the KBM mode marginally dominant in the steep gradient region (ψN = 0 . 98), in the range of kθ ~ 1 cm-1 which corresponds to the most unstable mode number in the nonlinearly saturated state. For shorter wavelengths the trapped electron mode becomes dominant since its linear growth rate increases with the mode number, while the KBM gets saturated. In the pedestal top region (ψN = 0 . 95) the ITG dominates. Resonant magnetic perturbations (RMP) are widely applied for ELM mitigation. During RMP suppression, the increase of edge turbulence is often observed. To understand this phenomena we use gyrokinetic simulations to address the direct effect of magnetic perturbations on the microturbulence. Simulations with 3D equilibrium reconstructed by VMEC code have been compared with toroidally averaged equilibrium, using identical pressure profiles. Work supported by DOE grant DE-SC0010416 and by General Atomics subcontract.

  20. Graphics Processing Unit Acceleration of Gyrokinetic Turbulence Simulations

    NASA Astrophysics Data System (ADS)

    Hause, Benjamin; Parker, Scott; Chen, Yang

    2013-10-01

    We find a substantial increase in on-node performance using Graphics Processing Unit (GPU) acceleration in gyrokinetic delta-f particle-in-cell simulation. Optimization is performed on a two-dimensional slab gyrokinetic particle simulation using the Portland Group Fortran compiler with the OpenACC compiler directives and Fortran CUDA. Mixed implementation of both Open-ACC and CUDA is demonstrated. CUDA is required for optimizing the particle deposition algorithm. We have implemented the GPU acceleration on a third generation Core I7 gaming PC with two NVIDIA GTX 680 GPUs. We find comparable, or better, acceleration relative to the NERSC DIRAC cluster with the NVIDIA Tesla C2050 computing processor. The Tesla C 2050 is about 2.6 times more expensive than the GTX 580 gaming GPU. We also see enormous speedups (10 or more) on the Titan supercomputer at Oak Ridge with Kepler K20 GPUs. Results show speed-ups comparable or better than that of OpenMP models utilizing multiple cores. The use of hybrid OpenACC, CUDA Fortran, and MPI models across many nodes will also be discussed. Optimization strategies will be presented. We will discuss progress on optimizing the comprehensive three dimensional general geometry GEM code.

  1. Direct identification of predator-prey dynamics in gyrokinetic simulations

    NASA Astrophysics Data System (ADS)

    Kobayashi, Sumire; Gürcan, Özgür D.; Diamond, Patrick H.

    2015-09-01

    The interaction between spontaneously formed zonal flows and small-scale turbulence in nonlinear gyrokinetic simulations is explored in a shearless closed field line geometry. It is found that when clear limit cycle oscillations prevail, the observed turbulent dynamics can be quantitatively captured by a simple Lotka-Volterra type predator-prey model. Fitting the time traces of full gyrokinetic simulations by such a reduced model allows extraction of the model coefficients. Scanning physical plasma parameters, such as collisionality and density gradient, it was observed that the effective growth rates of turbulence (i.e., the prey) remain roughly constant, in spite of the higher and varying level of primary mode linear growth rates. The effective growth rate that was extracted corresponds roughly to the zonal-flow-modified primary mode growth rate. It was also observed that the effective damping of zonal flows (i.e., the predator) in the parameter range, where clear predator-prey dynamics is observed, (i.e., near marginal stability) agrees with the collisional damping expected in these simulations. This implies that the Kelvin-Helmholtz-like instability may be negligible in this range. The results imply that when the tertiary instability plays a role, the dynamics becomes more complex than a simple Lotka-Volterra predator prey.

  2. Transport and discrete particle noise in gyrokinetic simulations

    NASA Astrophysics Data System (ADS)

    Jenkins, Thomas; Lee, W. W.

    2006-10-01

    We present results from our recent investigations regarding the effects of discrete particle noise on the long-time behavior and transport properties of gyrokinetic particle-in-cell simulations. It is found that the amplitude of nonlinearly saturated drift waves is unaffected by discreteness-induced noise in plasmas whose behavior is dominated by a single mode in the saturated state. We further show that the scaling of this noise amplitude with particle count is correctly predicted by the fluctuation-dissipation theorem, even though the drift waves have driven the plasma from thermal equilibrium. As well, we find that the long-term behavior of the saturated system is unaffected by discreteness-induced noise even when multiple modes are included. Additional work utilizing a code with both total-f and δf capabilities is also presented, as part of our efforts to better understand the long- time balance between entropy production, collisional dissipation, and particle/heat flux in gyrokinetic plasmas.

  3. Turbulence spreading in gyro-kinetic theory

    NASA Astrophysics Data System (ADS)

    Migliano, P.; Buchholz, R.; Grosshauser, S. R.; Hornsby, W. A.; Peeters, A. G.; Stauffert, O.

    2016-01-01

    In this letter a new operative definition for the turbulence intensity in connection with magnetized plasmas is given. In contrast to previous definitions the new definition satisfies a Fisher-Kolmogorov-Petrovskii-Piskunov type equation. Furthermore, explicit expressions for the turbulence intensity and the turbulence intensity flux, that allow for the first time direct numerical evaluation, are derived. A carefully designed numerical experiment for the case of a tokamak is performed to study the impact of turbulence spreading. The effective turbulence diffusion coefficient is measured to be smaller than the heat conduction coefficient and the turbulence spreading length is found to be of the order of the turbulence correlation length. The results show that turbulence spreading can play a role in the non-local flux gradient relation, or in the scaling of transport coefficients with the normalized Larmor radius, only over lengths scale of the order of the turbulence correlation length. A new turbulence convection mechanism, due to the drift connected with the magnetic field inhomogeneities, is described. The convective flux integrates to zero under the flux surface average unless there is an up-down asymmetry in the tubulence intensity. The latter asymmetry can be generated through a radial inhomogeneity or plasma rotation. It is shown that the turbulence convection can lead to a spreading of the order of the correlation length.

  4. A new hybrid-Lagrangian numerical scheme for gyrokinetic simulation of tokamak edge plasma

    NASA Astrophysics Data System (ADS)

    Ku, S.; Hager, R.; Chang, C. S.; Kwon, J. M.; Parker, S. E.

    2016-06-01

    In order to enable kinetic simulation of non-thermal edge plasmas at a reduced computational cost, a new hybrid-Lagrangian δf scheme has been developed that utilizes the phase space grid in addition to the usual marker particles, taking advantage of the computational strengths from both sides. The new scheme splits the particle distribution function of a kinetic equation into two parts. Marker particles contain the fast space-time varying, δf, part of the distribution function and the coarse-grained phase-space grid contains the slow space-time varying part. The coarse-grained phase-space grid reduces the memory-requirement and the computing cost, while the marker particles provide scalable computing ability for the fine-grained physics. Weights of the marker particles are determined by a direct weight evolution equation instead of the differential form weight evolution equations that the conventional delta-f schemes use. The particle weight can be slowly transferred to the phase space grid, thereby reducing the growth of the particle weights. The non-Lagrangian part of the kinetic equation - e.g., collision operation, ionization, charge exchange, heat-source, radiative cooling, and others - can be operated directly on the phase space grid. Deviation of the particle distribution function on the velocity grid from a Maxwellian distribution function - driven by ionization, charge exchange and wall loss - is allowed to be arbitrarily large. The numerical scheme is implemented in the gyrokinetic particle code XGC1, which specializes in simulating the tokamak edge plasma that crosses the magnetic separatrix and is in contact with the material wall.

  5. Nonlinear theory of drift-cyclotron kinetics and the possible breakdown of gyro-kinetics

    SciTech Connect

    Waltz, R. E.; Deng Zhao

    2013-01-15

    A nonlinear theory of drift-cyclotron kinetics (termed cyclo-kinetics here) is formulated to test the breakdown of the gyro-kinetic approximations. Six dimensional cyclo-kinetics can be regarded as an extension of five dimensional gyro-kinetics to include high-frequency cyclotron waves, which can interrupt the low-frequency gyro-averaging in the (sixth velocity grid) gyro-phase angle. Nonlinear cyclo-kinetics has no limit on the amplitude of the perturbations. Formally, there is no gyro-averaging when all cyclotron (gyro-phase angle) harmonics of the perturbed distribution function (delta-f) are retained. Retaining only the (low frequency) zeroth cyclotron harmonic in cyclo-kinetics recovers both linear and nonlinear gyro-kinetics. Simple recipes are given for converting continuum nonlinear delta-f gyro-kinetic transport simulation codes to cyclo-kinetics codes by retaining (at least some) higher cyclotron harmonics.

  6. The isotope effect in turbulent transport control by GAMs. Observation and gyrokinetic modeling

    NASA Astrophysics Data System (ADS)

    Gurchenko, A. D.; Gusakov, E. Z.; Niskala, P.; Altukhov, A. B.; Esipov, L. A.; Kiviniemi, T. P.; Korpilo, T.; Kouprienko, D. V.; Lashkul, S. I.; Leerink, S.; Perevalov, A. A.; Irzak, M. A.

    2016-04-01

    A comparative investigation of the isotope effect in multi-scale anomalous transport phenomena is performed both experimentally by highly localized turbulence diagnostics in comparable hydrogen and deuterium FT-2 tokamak discharges and theoretically with the help of global gyrokinetic modeling. Substantial excess of the geodesic acoustic mode (GAM) amplitude, radial wavelength and correlation length in a wide spatial region of deuterium discharge resulting in stronger modulation of drift-wave turbulence level is demonstrated by both approaches. A larger turbulence radial correlation length is found at LFS in D-discharge in experiment and a stronger modulation of gyrokinetic particles and energy fluxes is shown there by the gyrokinetic code. The gyrokinetic modeling demonstrated comparable levels of drift wave density and electric field fluctuations in hydrogen and deuterium discharges. Nevertheless, the mean value of the ion energy and particle anomalous flux provided by modeling shows the systematic isotope effect at all radii.

  7. On the Existence of Canonical Gyrokinetic Variables for Chaotic Magnetic Fields

    SciTech Connect

    Nicolini, Piero; Tessarotto, Massimo

    2008-12-31

    The gyrokinetic description of particle dynamics faces a basic difficulty when a special type of canonical variables is sought, i.e., the so-called gyrokinetic canonical variables. These are defined in such a way that two of them are respectively identified with the gyrophase-angle, describing the fast particle gyration motion around magnetic field lines, and its canonically conjugate momentum. In this paper we intend to discuss the conditions of existence for these variables.

  8. Analysis and gyrokinetic simulation of MHD Alfven wave interactions

    NASA Astrophysics Data System (ADS)

    Nielson, Kevin Derek

    The study of low-frequency turbulence in magnetized plasmas is a difficult problem due to both the enormous range of scales involved and the variety of physics encompassed over this range. Much of the progress that has been made in turbulence theory is based upon a result from incompressible magnetohydrodynamics (MHD), in which energy is only transferred from large scales to small via the collision of Alfven waves propagating oppositely along the mean magnetic field. Improvements in laboratory devices and satellite measurements have demonstrated that, while theories based on this premise are useful over inertial ranges, describing turbulence at scales that approach particle gyroscales requires new theory. In this thesis, we examine the limits of incompressible MHD theory in describing collisions between pairs of Alfven waves. This interaction represents the fundamental unit of plasma turbulence. To study this interaction, we develop an analytic theory describing the nonlinear evolution of interacting Alfven waves and compare this theory to simulations performed using the gyrokinetic code AstroGK. Gyrokinetics captures a much richer set of physics than that described by incompressible MHD, and is well-suited to describing Alfvenic turbulence around the ion gyroscale. We demonstrate that AstroGK is well suited to the study of physical Alfven waves by reproducing laboratory Alfven dispersion data collected using the LAPD. Additionally, we have developed an initialization alogrithm for use with AstroGK that allows exact Alfven eigenmodes to be initialized with user specified amplitudes and phases. We demonstrate that our analytic theory based upon incompressible MHD gives excellent agreement with gyrokinetic simulations for weakly turbulent collisions in the limit that k⊥rho i << 1. In this limit, agreement is observed in the time evolution of nonlinear products, and in the strength of nonlinear interaction with respect to polarization and scale. We also examine the

  9. Full-f gyrokinetic simulations for neoclassical toroidal viscosity in a perturbed tokamak configuration

    NASA Astrophysics Data System (ADS)

    Matsuoka, Seikichi; Idomura, Yasuhiro; Satake, Shinsuke

    2014-10-01

    A magnetic field perturbation in tokamak plasmas plays a key role in determining the intrinsic rotation and velocity shear, since even a small perturbation can break the axisymmetry in the toroidal direction and induces the finite neoclassical toroidal viscosity (NTV). A simulation study for the NTV evaluation in an axisymmetric tokamak with a small resonant magnetic field perturbation using the full-f gyrokinetic Eulerian code GT5D is presented. The magnetic field perturbation is included in the particle orbit of GT5D only through the Hamiltonian by replacing the axisymmetric magnetic field with the sum of the axisymmetric field and the perturbation, which enables us to perform GT5D simulations without changing the symplectic structure of the single-particle Lagrangian constructed for the equilibrium (axisymmetric) magnetic field. Numerical results are benchmarked with those obtained by the neoclassical transport code, FORTEC-3D, which solves the drift kinetic equation by two-weight δf Monte Carlo method. The NTV of GT5D with a single-helicity perturbation is found to have a similar peaked profile around the resonant surface as that of FORTEC-3D.

  10. A gyro-kinetic model for trapped electron and ion modes

    NASA Astrophysics Data System (ADS)

    Drouot, Thomas; Gravier, Etienne; Reveille, Thierry; Ghizzo, Alain; Bertrand, Pierre; Garbet, Xavier; Sarazin, Yanick; Cartier-Michaud, Thomas

    2014-10-01

    In tokamak plasmas, it is recognized that ITG (ion temperature gradient instability) and trapped electron modes (TEM) are held responsible for turbulence giving rise to anomalous transport. The present work focuses on the building of a model including trapped kinetic ions and trapped kinetic electrons. For this purpose, the dimensionality is reduced by averaging the motion over the cyclotron motion and the "banana" orbits, according to the fact that the instabilities are characterized by frequencies of the order of the low trapped particle precession frequency. Moreover, a set of action-angle variables is used. The final model is 4D (two-dimensional phase space parametrized by the two first adiabatic invariants namely the particle energy and the trapping parameter). In this paper, the trapped ion and electron modes (TIM and TEM) are studied by using a linear analysis of the model. This work is currently performed in order to include trapped electrons in an existing semi lagrangian code for which TIM modes are already taken into account. This study can be considered as a first step in order to include kinetic trapped electrons in the 5D gyrokinetic code GYSELA [J. Abiteboul et al., ESAIM Proc. 32, 103 (2011)]. Contribution to the Topical Issue "Theory and Applications of the Vlasov Equation", edited by Francesco Pegoraro, Francesco Califano, Giovanni Manfredi and Philip J. Morrison.

  11. Collisional tests and an extension of the TEMPEST continuum gyrokinetic code

    NASA Astrophysics Data System (ADS)

    Cohen, R. H.; Dorr, M.; Hittinger, J.; Kerbel, G.; Nevins, W. M.; Rognlien, T.; Xiong, Z.; Xu, X. Q.

    2006-04-01

    An important requirement of a kinetic code for edge plasmas is the ability to accurately treat the effect of colllisions over a broad range of collisionalities. To test the interaction of collisions and parallel streaming, TEMPEST has been compared with published analytic and numerical (Monte Carlo, bounce-averaged Fokker-Planck) results for endloss of particles confined by combined electrostatic and magnetic wells. Good agreement is found over a wide range of collisionality, confining potential and mirror ratio, and the required velocity space resolution is modest. We also describe progress toward extension of (4-dimensional) TEMPEST into a ``kinetic edge transport code'' (a kinetic counterpart of UEDGE). The extension includes averaging of the gyrokinetic equations over fast timescales and approximating the averaged quadratic terms by diffusion terms which respect the boundaries of inaccessable regions in phase space. F. Najmabadi, R.W. Conn and R.H. Cohen, Nucl. Fusion 24, 75 (1984); T.D. Rognlien and T.A. Cutler, Nucl. Fusion 20, 1003 (1980).

  12. Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq Equations

    NASA Astrophysics Data System (ADS)

    Brenier, Yann

    2009-10-01

    We establish a connection between optimal transport theory (see Villani in Topics in optimal transportation. Graduate studies in mathematics, vol. 58, AMS, Providence, 2003, for instance) and classical convection theory for geophysical flows (Pedlosky, in Geophysical fluid dynamics, Springer, New York, 1979). Our starting point is the model designed few years ago by Angenent, Haker, and Tannenbaum (SIAM J. Math. Anal. 35:61-97, 2003) to solve some optimal transport problems. This model can be seen as a generalization of the Darcy-Boussinesq equations, which is a degenerate version of the Navier-Stokes-Boussinesq (NSB) equations. In a unified framework, we relate different variants of the NSB equations (in particular what we call the generalized hydrostatic-Boussinesq equations) to various models involving optimal transport (and the related Monge-Ampère equation, Brenier in Commun. Pure Appl. Math. 64:375-417, 1991; Caffarelli in Commun. Pure Appl. Math. 45:1141-1151, 1992). This includes the 2D semi-geostrophic equations (Hoskins in Annual review of fluid mechanics, vol. 14, pp. 131-151, Palo Alto, 1982; Cullen et al. in SIAM J. Appl. Math. 51:20-31, 1991, Arch. Ration. Mech. Anal. 185:341-363, 2007; Benamou and Brenier in SIAM J. Appl. Math. 58:1450-1461, 1998; Loeper in SIAM J. Math. Anal. 38:795-823, 2006) and some fully nonlinear versions of the so-called high-field limit of the Vlasov-Poisson system (Nieto et al. in Arch. Ration. Mech. Anal. 158:29-59, 2001) and of the Keller-Segel for Chemotaxis (Keller and Segel in J. Theor. Biol. 30:225-234, 1971; Jäger and Luckhaus in Trans. Am. Math. Soc. 329:819-824, 1992; Chalub et al. in Mon. Math. 142:123-141, 2004). Mathematically speaking, we establish some existence theorems for local smooth, global smooth or global weak solutions of the different models. We also justify that the inertia terms can be rigorously neglected under appropriate scaling assumptions in the generalized Navier-Stokes-Boussinesq equations

  13. Electron heat transport from stochastic fields in gyrokinetic simulationsa)

    NASA Astrophysics Data System (ADS)

    Wang, E.; Nevins, W. M.; Candy, J.; Hatch, D.; Terry, P.; Guttenfelder, W.

    2011-05-01

    GYRO is used to examine the perturbed magnetic field structure generated by electromagnetic gyrokinetic simulations of the CYCLONE base case as βe is varied from 0.1% to 0.7%, as investigated by J. Candy [Phys. Plasmas 12, 072307 (2005)]. Poincare surface of section plots obtained from integrating the self-consistent magnetic field demonstrates widespread stochasticity for all nonzero values of βe. Despite widespread stochasticity of the perturbed magnetic fields, no significant increase in electron transport is observed. The magnetic diffusion, dm [A. B. Rechester and M. N. Rosenbluth, Phys. Rev. Lett 40, 38 (1978)], is used to quantify the degree of stochasticity and related to the electron heat transport for hundreds of time slices in each simulation.

  14. Electron heat transport from stochastic fields in gyrokinetic simulations

    SciTech Connect

    Wang, E.; Nevins, W. M.; Candy, J.; Hatch, D.; Terry, P.; Guttenfelder, W.

    2011-05-15

    GYRO is used to examine the perturbed magnetic field structure generated by electromagnetic gyrokinetic simulations of the CYCLONE base case as {beta}{sub e} is varied from 0.1% to 0.7%, as investigated by J. Candy [Phys. Plasmas 12, 072307 (2005)]. Poincare surface of section plots obtained from integrating the self-consistent magnetic field demonstrates widespread stochasticity for all nonzero values of {beta}{sub e}. Despite widespread stochasticity of the perturbed magnetic fields, no significant increase in electron transport is observed. The magnetic diffusion, d{sub m}[A. B. Rechester and M. N. Rosenbluth, Phys. Rev. Lett 40, 38 (1978)], is used to quantify the degree of stochasticity and related to the electron heat transport for hundreds of time slices in each simulation.

  15. Gyrokinetic Simulation of Global Turbulent Transport Properties in Tokamak Experiments

    SciTech Connect

    Wang, W.X.; Lin, Z.; Tang, W.M.; Lee, W.W.; Ethier, S.; Lewandowski, J.L.V.; Rewoldt, G.; Hahm, T.S.; Manickam, J.

    2006-01-01

    A general geometry gyro-kinetic model for particle simulation of plasma turbulence in tokamak experiments is described. It incorporates the comprehensive influence of noncircular cross section, realistic plasma profiles, plasma rotation, neoclassical (equilibrium) electric fields, and Coulomb collisions. An interesting result of global turbulence development in a shaped tokamak plasma is presented with regard to nonlinear turbulence spreading into the linearly stable region. The mutual interaction between turbulence and zonal flows in collisionless plasmas is studied with a focus on identifying possible nonlinear saturation mechanisms for zonal flows. A bursting temporal behavior with a period longer than the geodesic acoustic oscillation period is observed even in a collisionless system. Our simulation results suggest that the zonal flows can drive turbulence. However, this process is too weak to be an effective zonal flow saturation mechanism.

  16. Gyrokinetic studies of microinstabilities in the reversed field pinch

    SciTech Connect

    Carmody, D.; Pueschel, M. J.; Terry, P. W.

    2013-05-15

    An analytic equilibrium, the Toroidal Bessel Function Model, is used in conjunction with the gyrokinetic code GYRO to investigate the nature of microinstabilities in a reversed field pinch plasma. The effect of the normalized electron plasma pressure β on the characteristics of the microinstabilities is studied. At a β of 4.5%, a transition between an ion temperature gradient (ITG) and a microtearing mode is observed. Suppression of the ITG mode occurs as in the tokamak, through coupling to shear Alfvén waves, with a critical β for stability higher than its tokamak equivalent due to a shorter parallel connection length. A steep dependence of the microtearing growth rate on the temperature gradient suggests high profile stiffness. There is evidence for a collisionless microtearing mode. The properties of this mode are investigated, and it is found that electron curvature drift plays an important role in the instability.

  17. Gyrokinetic investigation of ITG turbulence in helical RFPs

    NASA Astrophysics Data System (ADS)

    Predebon, I.; Xanthopoulos, P.; Terranova, D.

    2014-10-01

    Micro-instabilities in reversed field pinch (RFP) plasmas have been investigated in the last years from several viewpoints and with various numerical tools. So far, axisymmetry of the magnetic equilibrium has always been postulated. Nevertheless, experimental evidence suggests that the physical conditions mostly favoring the onset of electrostatic/electromagnetic turbulence, e.g., the occurrence of large pressure gradients, emerge when magnetic surfaces become helical, during the single helicity states. In this work, we investigate ion-temperature-gradient driven turbulence focusing on the 3D feature, with the aim to describe its distinct properties compared to the axisymmetric geometry. For this study, we will apply the 3D nonlinear gyrokinetic code GENE to RFP equilibria generated by the VMEC code.

  18. Global spectral investigation of plasma turbulence in gyrokinetic simulations

    SciTech Connect

    Henriksson, S. V.; Janhunen, S. J.; Kiviniemi, T. P.; Heikkinen, J. A.

    2006-07-15

    Gyrokinetic global particle-in-cell simulations for a small torus with a large aspect ratio ({epsilon}{sup -1}>{approx}7) indicate a k{sub perpendicular}{sup -{alpha}} spectrum for electrostatic turbulence. When electrons are treated kinetically, the simulation results fit {alpha} that grows from about 1 at the plasma core to about 3 at the plasma edge for the flux surface component of the wave vector perpendicular to the magnetic field, while for adiabatic electrons {alpha}=4 is found for all radii, in agreement with the Hasegawa-Mima model. The relation between spectra and transport is investigated through the formation of an internal transport barrier. The role of flow shear in suppressing turbulence is illustrated by spectral diagnostics. A strong dependence between the presence of small wavenumbers and transport is explicitly observed. The simulated spectra are compared to recent experimental results.

  19. Gyrokinetic simulation of the collisional micro-tearing mode instability

    NASA Astrophysics Data System (ADS)

    Startsev, Edward; Lee, Wei-Li; Wang, Weixing

    2015-11-01

    An application of recently developed perturbative particle simulation scheme for finite- β plasmas in the presence of background inhomogeneities is presented. Originally, using similar scheme, we were able to simulate shear-Alfven waves, finite- β modified drift waves and ion temperature gradient modes using a simple gyrokinetic particle code based on realistic fusion plasma parameters. Recently, we have successfully used the scheme for simulation of linear tearing and drift-tearing modes, in both collisionless semi-collisional regimes in slab geometry with sheared magnetic field. Here, we present further development of this scheme for the simulation of linear semi-collisional micro-tearing mode driven by electron temperature gradient in high-aspect ratio cylindrical cross-section tokamak using the modified turbulence code GTS. Research supported by the U. S. Department of Energy.

  20. Nonlinear Full-f Edge Gyrokinetic Turbulence Simulations

    NASA Astrophysics Data System (ADS)

    Xu, X. Q.; Dimits, A. M.; Umansky, M. V.

    2008-11-01

    TEMPEST is a nonlinear full-f 5D electrostatic gyrokinetic code for simulations of neoclassical and turbulent transport for tokamak plasmas. Given an initial density perturbation, 4D TEMPEST simulations show that the kinetic GAM exists in the edge in the form of outgoing waves [1], its radial scale is set by plasma profiles, and the ion temperature inhomogeneity is necessary for GAM radial propagation. From an initial Maxwellian distribution with uniform poloidal profiles on flux surfaces, the 5D TEMPEST simulations in a flux coordinates with Boltzmann electron model in a circular geometry show the development of neoclassical equilibrium, the generation of the neoclassical electric field due to neoclassical polarization, and followed by a growth of instability due to the spatial gradients. 5D TEMPEST simulations of kinetic GAM turbulent generation, radial propagation, and its impact on transport will be reported. [1] X. Q. Xu, Phys. Rev. E., 78 (2008).

  1. Overview of gyrokinetic studies of finite-β microturbulence

    NASA Astrophysics Data System (ADS)

    Terry, P. W.; Carmody, D.; Doerk, H.; Guttenfelder, W.; Hatch, D. R.; Hegna, C. C.; Ishizawa, A.; Jenko, F.; Nevins, W. M.; Predebon, I.; Pueschel, M. J.; Sarff, J. S.; Whelan, G. G.

    2015-10-01

    Recent results on electromagnetic turbulence from gyrokinetic studies in different magnetic configurations are overviewed, detailing the physics of electromagnetic turbulence and transport, and the effect of equilibrium magnetic field scale lengths. Ion temperature gradient (ITG) turbulence is shown to produce magnetic stochasticity through nonlinear excitation of linearly stable tearing-parity modes. The excitation, which is catalyzed by the zonal flow, produces an electron heat flux proportional to β2 that deviates markedly from quasilinear theory. Above a critical beta known as the non-zonal transition (NZT), the magnetic fluctuations disable zonal flows by allowing electron streaming that shorts zonal potential between flux surfaces. This leads to a regime of very high transport levels. Kinetic ballooning mode (KBM) saturation is described. For tokamaks saturation involves twisted structures arising from magnetic shear; for helical plasmas oppositely inclined convection cells interact by mutual shearing. Microtearing modes are unstable in the magnetic geometry of tokamaks and the reversed field pinch (RFP). In NSTX instability requires finite collisionality, large beta, and is favored by increasing magnetic shear and decreasing safety factor. In the RFP, a new branch of microtearing with finite growth rate at vanishing collisionality is shown from analytic theory to require the electron grad-B/curvature drift resonance. However, gyrokinetic modeling of experimental MST RFP discharges at finite beta reveals turbulence that is electrostatic, has large zonal flows, and a large Dimits shift. Analysis shows that the shorter equilibrium magnetic field scale lengths increase the critical gradients associated with the instability of trapped electron modes, ITG and microtearing, while increasing beta thresholds for KBM instability and the NZT.

  2. Global and Kinetic MHD Simulation by the Gpic-MHD Code

    NASA Astrophysics Data System (ADS)

    Hiroshi, Naitou; Yusuke, Yamada; Kenji, Kajiwara; Wei-li, Lee; Shinji, Tokuda; Masatoshi, Yagi

    2011-10-01

    In order to implement large-scale and high-beta tokamak simulation, a new algorithm of the electromagnetic gyrokinetic PIC (particle-in-cell) code was proposed and installed on the Gpic-MHD code [Gyrokinetic PIC code for magnetohydrodynamic (MHD) simulation]. In the new algorithm, the vorticity equation and the generalized Ohm's law along the magnetic field are derived from the basic equations of the gyrokinetic Vlasov, Poisson, and Ampere system and are used to describe the spatio-temporal evolution of the field quantities of the electrostatic potential varphi and the longitudinal component of the vector potential Az. The basic algorithm is equivalent to solving the reduced-MHD-type equations with kinetic corrections, in which MHD physics related to Alfven modes are well described. The estimation of perturbed electron pressure from particle dynamics is dominant, while the effects of other moments are negligible. Another advantage of the algorithm is that the longitudinal induced electric field, ETz = -∂Az/∂t, is explicitly estimated by the generalized Ohm's law and used in the equations of motion. Furthermore, the particle velocities along the magnetic field are used (vz-formulation) instead of generalized momentums (pz-formulation), hence there is no problem of ‘cancellation', which would otherwise appear when Az is estimated from the Ampere's law in the pz-formulation. The successful simulation of the collisionless internal kink mode by the new Gpic-MHD with realistic values of the large-scale and high-beta tokamaks revealed the usefulness of the new algorithm.

  3. Magnetic stochasticity in gyrokinetic simulations of plasma microturbulence

    NASA Astrophysics Data System (ADS)

    Wang, Eric

    2010-11-01

    One of the fundamental components of a steady state tokamak or stellerator fusion reactor is the structural integrity of nested magnetic surfaces. The consequences of losing this integrity can have very serious implications, ranging from sawtooth crashes to disruptions. In the present work, we use GYRO to examine the perturbed magnetic field structure generated by electromagnetic gyrokinetic simulations of the CYCLONE base case as β is varied from .1% to .7%, as first investigated in [J. Candy, Phys. Plasmas 12, 072307 (2005)]. By integrating the self-consistent magnetic field lines to produce Poincare surface of section plots, we demonstrate destruction of magnetic surfaces for all nonzero values of β. Despite widespread stochasticity of the perturbed magnetic fields, no significant increase in electron transport is observed. We can quantify the stochastic electron heat transport by using test particles to estimate the magnetic diffusion coefficient Dst [A.B. Rechester and M.N. Rosenbluth, PRL 40, 38 (1978)] for hundreds of time slices in each simulation and find the time-history of Dst to be highly correlated with the electron heat transport due to ``magnetic-flutter'' computed in the simulations. The mechanism that couples electromagnetic turbulence to the linearly damped high-n tearing modes that are responsible for reconnection will be discussed.

  4. ADVANCES IN COMPREHENSIVE GYROKINETIC SIMULATIONS OF TRANSPORT IN TOKAMAKS

    SciTech Connect

    WALTZ RE; CANDY J; HINTON FL; ESTRADA-MILA C; KINSEY JE

    2004-10-01

    A continuum global gyrokinetic code GYRO has been developed to comprehensively simulate core turbulent transport in actual experimental profiles and enable direct quantitative comparisons to the experimental transport flows. GYRO not only treats the now standard ion temperature gradient (ITG) mode turbulence, but also treats trapped and passing electrons with collisions and finite {beta}, equilibrium ExB shear stabilization, and all in real tokamak geometry. Most importantly the code operates at finite relative gyroradius ({rho}{sub *}) so as to treat the profile shear stabilization and nonlocal effects which can break gyroBohm scaling. The code operates in either a cyclic flux-tube limit (which allows only gyroBohm scaling) or a globally with physical profile variation. Rohm scaling of DIII-D L-mode has been simulated with power flows matching experiment within error bars on the ion temperature gradient. Mechanisms for broken gyroBohm scaling, neoclassical ion flows embedded in turbulence, turbulent dynamos and profile corrugations, plasma pinches and impurity flow, and simulations at fixed flow rather than fixed gradient are illustrated and discussed.

  5. ADVANCES IN COMPREHENSIVE GYROKINETIC SIMULATIONS OF TRANSPORT IN TOKAMAKS

    SciTech Connect

    WALTZ,R.E; CANDY,J; HINTON,F.L; ESTRADA-MILA,C; KINSEY,J.E

    2004-10-01

    A continuum global gyrokinetic code GYRO has been developed to comprehensively simulate core turbulent transport in actual experimental profiles and enable direct quantitative comparisons to the experimental transport flows. GYRO not only treats the now standard ion temperature gradient (ITG) mode turbulence, but also treats trapped and passing electrons with collisions and finite {beta}, equilibrium ExB shear stabilization, and all in real tokamak geometry. Most importantly the code operates at finite relative gyroradius ({rho}{sub *}) so as to treat the profile shear stabilization and nonlocal effects which can break gyroBohm scaling. The code operates in either a cyclic flux-tube limit (which allows only gyroBohm scaling) or globally with physical profile variation. Bohm scaling of DIII-D L-mode has been simulated with power flows matching experiment within error bars on the ion temperature gradient. Mechanisms for broken gyroBohm scaling, neoclassical ion flows embedded in turbulence, turbulent dynamos and profile corrugations, are illustrated.

  6. Gyrokinetic simulation of driftwave instability in field-reversed configuration

    NASA Astrophysics Data System (ADS)

    Fulton, D. P.; Lau, C. K.; Schmitz, L.; Holod, I.; Lin, Z.; Tajima, T.; Binderbauer, M. W.

    2016-05-01

    Following the recent remarkable progress in magnetohydrodynamic (MHD) stability control in the C-2U advanced beam driven field-reversed configuration (FRC), turbulent transport has become one of the foremost obstacles on the path towards an FRC-based fusion reactor. Significant effort has been made to expand kinetic simulation capabilities in FRC magnetic geometry. The recently upgraded Gyrokinetic Toroidal Code (GTC) now accommodates realistic magnetic geometry from the C-2U experiment at Tri Alpha Energy, Inc. and is optimized to efficiently handle the FRC's magnetic field line orientation. Initial electrostatic GTC simulations find that ion-scale instabilities are linearly stable in the FRC core for realistic pressure gradient drives. Estimated instability thresholds from linear GTC simulations are qualitatively consistent with critical gradients determined from experimental Doppler backscattering fluctuation data, which also find ion scale modes to be depressed in the FRC core. Beyond GTC, A New Code (ANC) has been developed to accurately resolve the magnetic field separatrix and address the interaction between the core and scrape-off layer regions, which ultimately determines global plasma confinement in the FRC. The current status of ANC and future development targets are discussed.

  7. Effects of Plasma Shaping on Nonlinear Gyrokinetic Turbulence

    SciTech Connect

    E. A. Belli; Hammett, G. W.; Dorland, W.

    2008-08-01

    The effects of flux surface shape on the gyrokinetic stability and transport of tokamak plasmas are studied using the GS2 code [M. Kotschenreuther, G. Rewoldt, and W.M. Tang, Comput. Phys. Commun. 88, 128 (1995); W. Dorland, F. Jenko, M. Kotschenreuther, and B.N. Rogers, Phys. Rev. Lett. 85, 5579 (2000)]. Studies of the scaling of nonlinear turbulence with shaping parameters are performed using analytic equilibria based on interpolations of representative shapes of the Joint European Torus (JET) [P.H. Rebut and B.E. Keen, Fusion Technol. 11, 13 (1987)]. High shaping is found to be a stabilizing influence on both the linear ion-temperature-gradient (ITG) instability and the nonlinear ITG turbulence. For the parameter regime studied here, a scaling of the heat flux with elongation of χ ~ κ-1.5 or κ-2.0, depending on the triangularity, is observed at fixed average temperature gradient. While this is not as strong as empirical elongation scalings, it is also found that high shaping results in a larger Dimits upshift of the nonlinear critical temperature gradient due to an enhancement of the Rosenbluth-Hinton residual zonal flows.

  8. Web Interface Connecting Gyrokinetic Turbulence Simulations with Tokamak Fusion Data

    NASA Astrophysics Data System (ADS)

    Suarez, A.; Ernst, D. R.

    2005-10-01

    We are developing a comprehensive interface to connect plasma microturbulence simulation codes with experimental data in the U.S. and abroad. This website automates the preparation and launch of gyrokinetic simulations utilizing plasma profile and magnetic equilibrium data. The functionality of existing standalone interfaces, such as GS2/PREP [D. R. Ernst et al., Phys. Plasmas 11(5) 2637 (2004)], in use for several years for the GS2 code [W. Dorland et al., Phys. Rev. Lett. 85(26) 5579 (2000)], will be extended to other codes, including GYRO [J. Candy / R.E. Waltz, J. Comput. Phys.186, (2003) 545]. Data is read from mdsplus and TRANSP [\\underline {http://w3.pppl.gov/transp}] and can be viewed using a java plotter, Webgraph, developed for this project by previous students Geoffrey Catto and Bo Feng. User sessions are tracked and saved to allow users to access their previous simulations, which can be used as templates for future work.

  9. Gyrokinetic particle simulations of kinetic ballooning mode in tokamak pedestal

    NASA Astrophysics Data System (ADS)

    Holod, Ihor

    2014-10-01

    The pedestal height and width in tokamak H-mode operation are widely believed to be constrained by mesoscale peeling-ballooning modes and microscopic kinetic ballooning modes (KBM). However, direct evidences of the KBM turbulence in pedestal are very limited. The role of the drift-Alfvenic microturbulence during the pedestal recovery period is not clear. Here we use gyrokinetic toroidal code (GTC) to study the edge instability of a DIII-D discharge #131997 using realistic geometry and plasma profiles and focusing on the pedestal region with steep pressure gradient. First, electrostatic simulations find a reactive trapped electron mode with an unusual eigenmode structure, which peaks at the poloidal angle θ = +/- π /2. The electron collisions decrease the growth rate by about one-half. Next, the plasma pressure is scanned in GTC electromagnetic simulations to identify the boundary for the KBM onset. At the finite electron beta an electromagnetic instability is found with KBM characteristics. The linear growth rate increases with βe and the mode propagation is in the ion diamagnetic direction. Nonlinear simulations of the KBM turbulence will also be presented. Work supported by DOE Grant DE-SC0010416, and in collaborations with GTC team.

  10. Electromagnetic gyrokinetic turbulence in finite-beta helical plasmas

    SciTech Connect

    Ishizawa, A.; Watanabe, T.-H.; Sugama, H.; Nakajima, N.; Maeyama, S.

    2014-05-15

    A saturation mechanism for microturbulence in a regime of weak zonal flow generation is investigated by means of electromagnetic gyrokinetic simulations. The study identifies a new saturation process of the kinetic ballooning mode (KBM) turbulence originating from the spatial structure of the KBM instabilities in a finite-beta Large Helical Device (LHD) plasma. Specifically, the most unstable KBM in LHD has an inclined mode structure with respect to the mid-plane of a torus, i.e., it has a finite radial wave-number in flux tube coordinates, in contrast to KBMs in tokamaks as well as ion-temperature gradient modes in tokamaks and helical systems. The simulations reveal that the growth of KBMs in LHD is saturated by nonlinear interactions of oppositely inclined convection cells through mutual shearing as well as by the zonal flow. The saturation mechanism is quantitatively investigated by analysis of the nonlinear entropy transfer that shows not only the mutual shearing but also a self-interaction with an elongated mode structure along the magnetic field line.

  11. Full-f gyrokinetic simulation over a confinement time

    SciTech Connect

    Idomura, Yasuhiro

    2014-02-15

    A long time ion temperature gradient driven turbulence simulation over a confinement time is performed using the full-f gyrokinetic Eulerian code GT5D. The convergence of steady temperature and rotation profiles is examined, and it is shown that the profile relaxation can be significantly accelerated when the simulation is initialized with linearly unstable temperature profiles. In the steady state, the temperature profile and the ion heat diffusivity are self-consistently determined by the power balance condition, while the intrinsic rotation profile is sustained by complicated momentum transport processes without momentum input. The steady turbulent momentum transport is characterized by bursty non-diffusive fluxes, and the resulting turbulent residual stress is consistent with the profile shear stress theory [Y. Camenen et al., “Consequences of profile shearing on toroidal momentum transport,” Nucl. Fusion 51, 073039 (2011)] in which the residual stress depends not only on the profile shear and the radial electric field shear but also on the radial electric field itself. Based on the toroidal angular momentum conservation, it is found that in the steady null momentum transport state, the turbulent residual stress is cancelled by the neoclassical counterpart, which is greatly enhanced in the presence of turbulent fluctuations.

  12. Gyrokinetic Simulation of Microturbulent Saturation at Finite β

    NASA Astrophysics Data System (ADS)

    Terry, P. W.; Pueschel, M. J.; Carmody, D.; Whelan, G. G.

    2014-10-01

    Saturation and zonal flow physics for microturbulence is investigated for tokamaks and the RFP using gyrokinetic computation to understand scalings with respect magnetic shear and β. Modeling an MST discharge shows that the critical instability gradient for TEM is higher than the tokamak threshold by the aspect ratio (R / a) . This factor is rooted in the shorter magnetic field scale length of the RFP. Nonlinear simulations show strong zonal flows and a large Dimits shift exceeding the tokamak shift by a factor of order (R / a) . The non zonal transition (NZT), a critical β for which zonal flows are disabled by flutter-induced charge loss is also considered. The critical β occurs when the radial displacement of a magnetic field line over a half connection length is equal to the radial correlation length. These quantities scale with the connection length and magnetic drift scale lengths entering the instability threshold and quasilinear diffusivities, making the RFP critical β for NZT higher than the tokamak value by (R / a) 1 . 5 times tokamak q. These results are consistent with magnetic shear and q dependence in the kinetic ballooning threshold, indicating that β effects will only arise at high β relative to typical RFP operation.

  13. Global gyrokinetic ion temperature gradient turbulence simulations of ITER

    NASA Astrophysics Data System (ADS)

    Villard, L.; Angelino, P.; Bottino, A.; Brunner, S.; Jolliet, S.; McMillan, B. F.; Tran, T. M.; Vernay, T.

    2013-07-01

    Global gyrokinetic simulations of ion temperature gradient (ITG) driven turbulence in an ideal MHD ITER equilibrium plasma are performed with the ORB5 code. The noise control and field-aligned Fourier filtering procedures implemented in ORB5 are essential in obtaining numerically healthy results with a reasonable amount of computational effort: typical simulations require 109 grid points, 109 particles and, despite a particle per cell ratio of unity, achieve a signal to noise ratio larger than 50. As compared with a circular concentric configuration with otherwise similar parameters (same ρ* = 1/720), the effective heat diffusivity is considerably reduced for the ITER MHD equilibrium. A self-organized radial structure appears, with long-lived zonal flows (ZF), modulating turbulence heat transport and resulting in a corrugated temperature gradient profile. The ratio of long-lived ZF to the fluctuating ZF is markedly higher for the ITER MHD equilibrium as compared with circular configurations, thereby producing a more effective ITG turbulence suppression, in spite of a higher linear growth rate. As a result, the nonlinear critical temperature gradient, R/LTcrit,NL, is about twice the linear critical temperature gradient, R/LTcrit,lin. Moreover, the heat transport stiffness above the nonlinear threshold is considerably reduced as compared with circular cases. Plasma elongation is probably one of the essential causes of this behaviour: indeed, undamped ZF residual levels and geodesic acoustic mode damping are both increasing with elongation. Other possible causes of the difference, such as magnetic shear profile effects, are also investigated.

  14. Gyrokinetic Studies of Microturbulence in the Madison Symmetric Torus

    NASA Astrophysics Data System (ADS)

    Williams, Zachary; Duff, James; Pueschel, M. J.; Terry, Paul

    2015-11-01

    Reversed-field pinches operating with Pulsed Poloidal Current Drive (PPCD) exhibit microturbulence that contributes to heat and particle transport. This work focuses on the analysis of high-frequency fluctuations in a recent 200 kA PPCD discharge in the Madison Symmetric Torus, for which strong experimental evidence of microturbulence exists. Local gyrokinetic simulations were performed at multiple radial positions outside the reversal surface using the Gene code. Linear analysis identifies the dominant instability at all positions to be a density-gradient-driven trapped electron mode. An accurate description of turbulence requires the inclusion of residual tearing mode fluctuations: though reduced in PPCD, large-scale tearing modes introduce non-negligible levels of magnetic perturbations. In simulations, they can be seen to weaken zonal flows and degrade confinement, increasing transport to experimentally observed levels. Importantly, imposed fluctuations appear to be self-consistently reinforced, contrary to the usual island-healing picture in tokamaks. Simulations also include B∥ fluctuations, which provide finite contributions to transport, particularly when artificially zeroing out tearing modes entirely.

  15. Database-driven web interface automating gyrokinetic simulations for validation

    NASA Astrophysics Data System (ADS)

    Ernst, D. R.

    2010-11-01

    We are developing a web interface to connect plasma microturbulence simulation codes with experimental data. The website automates the preparation of gyrokinetic simulations utilizing plasma profile and magnetic equilibrium data from TRANSP analysis of experiments, read from MDSPLUS over the internet. This database-driven tool saves user sessions, allowing searches of previous simulations, which can be restored to repeat the same analysis for a new discharge. The website includes a multi-tab, multi-frame, publication quality java plotter Webgraph, developed as part of this project. Input files can be uploaded as templates and edited with context-sensitive help. The website creates inputs for GS2 and GYRO using a well-tested and verified back-end, in use for several years for the GS2 code [D. R. Ernst et al., Phys. Plasmas 11(5) 2637 (2004)]. A centralized web site has the advantage that users receive bug fixes instantaneously, while avoiding the duplicated effort of local compilations. Possible extensions to the database to manage run outputs, toward prototyping for the Fusion Simulation Project, are envisioned. Much of the web development utilized support from the DoE National Undergraduate Fellowship program [e.g., A. Suarez and D. R. Ernst, http://meetings.aps.org/link/BAPS.2005.DPP.GP1.57.

  16. Theory, Verification and Validation of Finite-Beta Gyrokinetics

    NASA Astrophysics Data System (ADS)

    Candy, J.

    2012-10-01

    In this tutorial presentation, both historical challenges and present open issues related to finite-beta gyrokinetic simulation will be discussed, with emphasis on the GYRO code but including material from relevant codes worldwide. There will be focus on numerical issues and solutions (such as the Ampere cancellation problem and its resolution), complicating physical effects (such as magnetic surface destruction and stochastic electron transport) and unresolved issues (for example, the finite-beta runaway observed in multiple codes). Successes related to code benchmarking exercises, and the role of finite-beta effects in experimental validation (for example, when finite-beta effects must be retained and when they may be neglected) will also be discussed. Categorization of finite-beta effects as they relate to different modes (ion temperature gradient, trapped electron, kinetic ballooning, microtearing and toroidal Alfv'en) will be clarified, and recent success in both numerically resolving and explaining electron transport in spherical tokamaks via flutter nonlinearity and associated magnetic stochasticity will be summarized. The material will be organized in roughly chronological order. Also covered will be practical aspects of the topic, including simulation strategies and best practices. This is relevant because the overall difficulty and potential pitfalls of electromagnetic turbulence simulation are far greater than for electrostatic turbulence simulations.

  17. A multi-species collisional operator for full-F gyrokinetics

    SciTech Connect

    Estève, D.; Garbet, X.; Sarazin, Y.; Grandgirard, V.; Cartier-Michaud, T.; Dif-Pradalier, G.; Ghendrih, P.; Latu, G.; Norscini, C.

    2015-12-15

    A linearized multi-species collision operator has been developed for an efficient implementation in gyrokinetic codes. This operator satisfies the main expected properties: particle, momentum, and energy conservation, and existence of an H-theorem. A gyrokinetic version is then calculated, which involves derivatives with respect to the gyrocenter position, parallel velocity, and magnetic momentum. An isotropic version in the velocity space can be constructed for the specific problem of trace impurities colliding with a main species. A simpler version that involves derivatives with parallel velocity only has been developed. This reduced version has been implemented in the GYSELA gyrokinetic code, and is shown to comply with particle, momentum, and energy conservation laws. Moreover, the interspecies relaxation rates for momentum and energy agree very well with the theoretical values.

  18. Performance of particle in cell methods on highly concurrent computational architectures

    SciTech Connect

    M.F.Adams; S. Ethier; N. Wichmann

    2009-09-23

    Particle in cell (PIC) methods are effective in computing Vlasov-Poisson system of equations used in simulations of magnetic fusion plasmas. PIC methods use grid based computations, for solving Poisson’s equation or more generally Maxwell’s equations, as well as Monte-Carlo type methods to sample the Vlasov equation. The presence of two types of discretizations, deterministic field solves and Monte-Carlo methods for the Vlasov equation, pose challenges in understanding and optimizing performance on today large scale computers which require high levels of concurrency. These challenges arises from the need to optimize two very different types of processes and the interactions between them. Modern cache based high-end computers have very deep memory hierarchies and high degrees of concurrency which must be utilized effectively to achieve good performance. The effective use of these machines requires maximizing concurrency by eliminating serial or redundant work and minimizing global communication. A related issue is minimizing the memory traffic between levels of the memory hierarchy because performance is often limited by the bandwidths and latencies of the memory system. This paper discusses some of the performance issues, particularly in regard to parallelism, of PIC methods. The gyrokinetic toroidal code (GTC) is used for these studies and a new radial grid decomposition is presented and evaluated. Scaling of the code is demonstrated on ITER sized plasmas with up to 16K Cray XT3/4 cores.

  19. Generalized parallel heat transport equations in collisional to weakly collisional plasmas

    NASA Astrophysics Data System (ADS)

    Zawaideh, Emad; Kim, N. S.; Najmabadi, Farrokh

    1988-11-01

    A new set of two-fluid heat-transport equations for heat conduction in collisional to weakly collisional plasmas was derived on the basis of gyrokinetic equations in flux coordinates. In these equations, no restrictions on the anisotropy of the ion distribution function or the collisionality are imposed. In the highly collisional limit, these equations reduce to the classical heat conduction equation of Spitzer and Haerm (1953), while in the weakly collisional limit, they describe a saturated heat flux. Numerical examples comparing these equations with conventional heat transport equations are presented.

  20. Validation of the gyrokinetic model in ITG and TEM dominated L-mode plasmas

    NASA Astrophysics Data System (ADS)

    Howard, N. T.; White, A. E.; Reinke, M. L.; Greenwald, M.; Holland, C.; Candy, J.; Walk, J. R.

    2013-12-01

    A rigorous validation of the gyrokinetic model was performed in both ion temperature gradient (ITG) and trapped electron mode (TEM) dominated Alcator C-Mod plasmas at (normalized midplane minor radius) r/a = 0.5 and 0.8. Analysis focuses on two L-mode discharges operated with 1.2 and 3.5 MW of ion cyclotron resonance heating. In depth investigation into the experimental uncertainties and simulation sensitivities in these discharges allows for a stringent test of the gyrokinetic model implemented by the GYRO code (Candy and Waltz 2003 J. Comput. Phys. 186 545) in both the centre of the stiff gradient region (r/a = 0.5) and the middle of the region often associated with the transport ‘shortfall’(r/a = 0.8). To identify the nature of the plasma turbulence and to ensure a robust evaluation of the model's ability to reproduce experiment, the sensitivity of the simulation results to experimental uncertainty in turbulence drive and suppression terms were determined at both radial locations. When significant TEM activity is present, nonlinear gyrokinetic simulations are found to reproduce both electron and ion experimental heat fluxes within their diagnosed uncertainties. In contrast, in the absence of TEM, electron heat fluxes are robustly under predicted by low-k, gyrokinetic simulation.

  1. Verification of Gyrokinetic (delta)f Simulations of Electron Temperature Gradient Turbulence

    SciTech Connect

    Nevins, W M; Parker, S E; Chen, Y; Candy, J; Dimits, A; Dorland, W; Hammett, G W; Jenko, F

    2007-05-07

    The GEM gyrokinetic {delta}f simulation code [Chen, 2003] [Chen, 2007] is shown to reproduce electron temperature gradient turbulence at the benchmark operating point established in previous work [Nevins, 2006]. The electron thermal transport is within 10% of the expected value, while the turbulent fluctuation spectrum is shown to have the expected intensity and two-point correlation function.

  2. Theoretical and Numerical Properties of a Gyrokinetic Plasma: Issues Related to Transport Time Scale Simulation

    SciTech Connect

    W.W. Lee

    2003-09-17

    Particle simulation has played an important role for the recent investigations on turbulence in magnetically confined plasmas. In this paper, theoretical and numerical properties of a gyrokinetic plasma as well as its relationship with magnetohydrodynamics (MHD) are discussed with the ultimate aim of simulating microturbulence in transport time scale using massively parallel computers.

  3. Global gyrokinetic particle-in-cell simulations of internal kink instabilities

    SciTech Connect

    Mishchenko, Alexey; Zocco, Alessandro

    2012-12-15

    Internal kink instabilities have been studied in straight tokamak geometry employing an electromagnetic gyrokinetic particle-in-cell (PIC) code. The ideal-MHD internal kink mode and the collisionless m=1 tearing mode have been successfully simulated with the PIC code. Diamagnetic effects on the internal kink modes have also been investigated.

  4. Center for Gyrokinetic/MHD Hybrid Simulation of Energetic Particle Physics in Toroidal Plasmas (CSEPP). Final report

    SciTech Connect

    Chen, Yang

    2012-03-07

    in the hybrid model we have studied a kinetic electron closure scheme for the fluid electron model. The most important element of the closure scheme is a complete Ohm’s law for the parallel electric field E{sub ||}, derived by combining the quasi-neutrality condition, the Ampere’s equation and the v{sub ||} moment of the gyrokinetic equations. A discretization method for the closure scheme is studied in detail for a three-dimensional shear-less slab plasma. It is found that for long-wavelength shear Alfven waves the kinetic closure scheme is both more accurate and robust than the previous GEM algorithm using the split-scheme, whereas for the ion-gradient-driven instability the previous algorithm is more efficient. This kinetic electron closure scheme will be implemented in GEM in the future. We have studied the beam driven Reverse Shear Alfven Eigenmodes (RSAE) observed in DIII-D discharge 142111. For this purpose a new scheme for obtaining the electric potential is implemented, i.e., by solving the gyrokinetic moment (GKM) equation, which is essentially the equation for {partial_derivative}{phi}}/{partial_derivative}t used in GEM’s split-weight scheme, and then integrating in time. Due to charge-neutrality the ExB motions of the equilibrium densities of all species cancel each other and do not cause charge separation if there is no finite Larmor radius effect. The advantage of solving the GKM equation is that this lowest-order cancellation can be made explicit. The GKM approach is found to be more accurate and robust. GEM simulations have reproduced many features of RSAE seen in the experiment, such as frequency chirping and the chirping range. It has been reported by other simulation codes that the shearing direction of the mode structure in the poloidal plane disagrees with observation. We found that the mode structure, including the shearing in the poloidal plane, is in general sensitive to the beam distribution. Using the same beam density profile as in

  5. Linear and nonlinear verification of gyrokinetic microstability codes

    SciTech Connect

    Bravenec, R. V.; Candy, J.; Barnes, M.

    2011-12-15

    Verification of nonlinear microstability codes is a necessary step before comparisons or predictions of turbulent transport in toroidal devices can be justified. By verification we mean demonstrating that a code correctly solves the mathematical model upon which it is based. Some degree of verification can be accomplished indirectly from analytical instability threshold conditions, nonlinear saturation estimates, etc., for relatively simple plasmas. However, verification for experimentally relevant plasma conditions and physics is beyond the realm of analytical treatment and must rely on code-to-code comparisons, i.e., benchmarking. The premise is that the codes are verified for a given problem or set of parameters if they all agree within a specified tolerance. True verification requires comparisons for a number of plasma conditions, e.g., different devices, discharges, times, and radii. Running the codes and keeping track of linear and nonlinear inputs and results for all conditions could be prohibitive unless there was some degree of automation. We have written software to do just this and have formulated a metric for assessing agreement of nonlinear simulations. We present comparisons, both linear and nonlinear, between the gyrokinetic codes GYRO[J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] and GS2[W. Dorland, F. Jenko, M. Kotschenreuther, and B. N. Rogers, Phys. Rev. Lett. 85, 5579 (2000)]. We do so at the mid-radius for the same discharge as in earlier work [C. Holland, A. E. White, G. R. McKee, M. W. Shafer, J. Candy, R. E. Waltz, L. Schmitz, and G. R. Tynan, Phys. Plasmas 16, 052301 (2009)]. The comparisons include electromagnetic fluctuations, passing and trapped electrons, plasma shaping, one kinetic impurity, and finite Debye-length effects. Results neglecting and including electron collisions (Lorentz model) are presented. We find that the linear frequencies with or without collisions agree well between codes, as do the time averages of

  6. Linear and nonlinear verification of gyrokinetic microstability codes

    NASA Astrophysics Data System (ADS)

    Bravenec, R. V.; Candy, J.; Barnes, M.; Holland, C.

    2011-12-01

    Verification of nonlinear microstability codes is a necessary step before comparisons or predictions of turbulent transport in toroidal devices can be justified. By verification we mean demonstrating that a code correctly solves the mathematical model upon which it is based. Some degree of verification can be accomplished indirectly from analytical instability threshold conditions, nonlinear saturation estimates, etc., for relatively simple plasmas. However, verification for experimentally relevant plasma conditions and physics is beyond the realm of analytical treatment and must rely on code-to-code comparisons, i.e., benchmarking. The premise is that the codes are verified for a given problem or set of parameters if they all agree within a specified tolerance. True verification requires comparisons for a number of plasma conditions, e.g., different devices, discharges, times, and radii. Running the codes and keeping track of linear and nonlinear inputs and results for all conditions could be prohibitive unless there was some degree of automation. We have written software to do just this and have formulated a metric for assessing agreement of nonlinear simulations. We present comparisons, both linear and nonlinear, between the gyrokinetic codes GYRO [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] and GS2 [W. Dorland, F. Jenko, M. Kotschenreuther, and B. N. Rogers, Phys. Rev. Lett. 85, 5579 (2000)]. We do so at the mid-radius for the same discharge as in earlier work [C. Holland, A. E. White, G. R. McKee, M. W. Shafer, J. Candy, R. E. Waltz, L. Schmitz, and G. R. Tynan, Phys. Plasmas 16, 052301 (2009)]. The comparisons include electromagnetic fluctuations, passing and trapped electrons, plasma shaping, one kinetic impurity, and finite Debye-length effects. Results neglecting and including electron collisions (Lorentz model) are presented. We find that the linear frequencies with or without collisions agree well between codes, as do the time averages of

  7. Gyrokinetic full f analysis of electric field dynamics and poloidal velocity in the FT2-tokamak configuration

    SciTech Connect

    Leerink, S.; Heikkinen, J. A.; Janhunen, S. J.; Kiviniemi, T. P.; Nora, M.; Ogando, F.

    2008-09-15

    The ELMFIRE gyrokinetic simulation code has been used to perform full f simulations of the FT-2 tokamak. The dynamics of the radial electric field and the creation of poloidal velocity in the presence of turbulence are presented.

  8. Double layers without current

    SciTech Connect

    Perkins, F.W.; Sun, Y.C.

    1980-11-01

    The steady-state solution of the nonlinear Vlasov-Poisson equations is reduced to a nonlinear eigenvalue problem for the case of double-layer (potential drop) boundary conditions. Solutions with no relative electron-ion drifts are found. The kinetic stability is discussed. Suggestions for creating these states in experiments and computer simulations are offered.

  9. Simulating the effects of stellarator geometry on gyrokinetic drift-wave turbulence

    NASA Astrophysics Data System (ADS)

    Baumgaertel, Jessica Ann

    Nuclear fusion is a clean, safe form of energy with abundant fuel. In magnetic fusion energy (MFE) experiments, the plasma fuel is confined by magnetic fields at very high temperatures and densities. One fusion reactor design is the non-axisymmetric, torus-shaped stellarator. Its fully-3D fields have advantages over the simpler, better-understood axisymmetric tokamak, including the ability to optimize magnetic configurations for desired properties, such as lower transport (longer confinement time). Turbulence in the plasma can break MFE confinement. While turbulent transport is known to cause a significant amount of heat loss in tokamaks, it is a new area of research in stellarators. Gyrokinetics is a good mathematical model of the drift-wave instabilities that cause turbulence. Multiple gyrokinetic turbulence codes that had great success comparing to tokamak experiments are being converted for use with stellarator geometry. This thesis describes such adaptations of the gyrokinetic turbulence code, GS2. Herein a new computational grid generator and upgrades to GS2 itself are described, tested, and benchmarked against three other gyrokinetic codes. Using GS2, detailed linear studies using the National Compact Stellarator Experiment (NCSX) geometry were conducted. The first compares stability in two equilibria with different β=(plasma pressure)/(magnetic pressure). Overall, the higher β case was more stable than the lower β case. As high β is important for MFE experiments, this is encouraging. The second compares NCSX linear stability to a tokamak case. NCSX was more stable with a 20% higher critical temperature gradient normalized by the minor radius, suggesting that the fusion power might be enhanced by ˜ 50%. In addition, the first nonlinear, non-axisymmetric GS2 simulations are presented. Finally, linear stability of two locations in a W7-AS plasma were compared. The experimentally-measured parameters used were from a W7-AS shot in which measured heat fluxes

  10. Gyrokinetic study of electromagnetic effects on toroidal momentum transport in tokamak plasmas

    SciTech Connect

    Hein, T.; Angioni, C.; Fable, E.; Candy, J.; Peeters, A. G.

    2011-07-15

    The effect of a finite {beta}{sub e} = 8{pi}n{sub e}T{sub e}/B{sup 2} on the turbulent transport of toroidal momentum in tokamak plasmas is discussed. From an analytical gyrokinetic model as well as local linear gyrokinetic simulations, it is shown that the modification of the parallel mode structure due to the nonadiabatic response of passing electrons, which changes the parallel wave vector k{sub ||} with increasing {beta}{sub e}, leads to a decrease in size of both the diagonal momentum transport as well as the Coriolis pinch under ion temperature gradient turbulence conditions, while for trapped electron modes, practically no modification is found. The decrease is particularly strong close to the onset of the kinetic ballooning modes. There, the Coriolis pinch even reverses its direction.

  11. GYSELA, a full-f global gyrokinetic Semi-Lagrangian code for ITG turbulence simulations

    SciTech Connect

    Grandgirard, V.; Sarazin, Y.; Garbet, X.; Dif-Pradalier, G.; Ghendrih, Ph.; Besse, N.; Bertrand, P.

    2006-11-30

    This work addresses non-linear global gyrokinetic simulations of ion temperature gradient (ITG) driven turbulence with the GYSELA code. The particularity of GYSELA code is to use a fixed grid with a Semi-Lagrangian (SL) scheme and this for the entire distribution function. The 4D non-linear drift-kinetic version of the code already showns the interest of such a SL method which exhibits good properties of energy conservation in non-linear regime as well as an accurate description of fine spatial scales. The code has been upgrated to run 5D simulations of toroidal ITG turbulence. Linear benchmarks and non-linear first results prove that semi-lagrangian codes can be a credible alternative for gyrokinetic simulations.

  12. Numerical comparison between a gyrofluid and gyrokinetic model investigating collisionless magnetic reconnection

    SciTech Connect

    Zacharias, O.; Kleiber, R.; Borchardt, M.; Comisso, L.; Grasso, D.; Hatzky, R.

    2014-06-15

    The first detailed comparison between gyrokinetic and gyrofluid simulations of collisionless magnetic reconnection has been carried out. Both the linear and nonlinear evolution of the collisionless tearing mode have been analyzed. In the linear regime, we have found a good agreement between the two approaches over the whole spectrum of linearly unstable wave numbers, both in the drift kinetic limit and for finite ion temperature. Nonlinearly, focusing on the small-Δ′ regime, with Δ′ indicating the standard tearing stability parameter, we have compared relevant observables such as the evolution and saturation of the island width, as well as the island oscillation frequency in the saturated phase. The results are basically the same, with small discrepancies only in the value of the saturated island width for moderately high values of Δ′. Therefore, in the regimes investigated here, the gyrofluid approach can describe the collisionless reconnection process as well as the more complete gyrokinetic model.

  13. Electromagnetic gyrokinetic simulation of turbulent transport in high ion temperature discharge of Large Helical Device

    NASA Astrophysics Data System (ADS)

    Ishizawa, Akihiro; Watanabe, Tomo-Hiko; Sugama, Hideo; Maeyama, Shinya; Nunami, Masanori; Nakajima, Noriyoshi

    2014-10-01

    Turbulent transport in a high ion temperature discharge of Large Helical Device (LHD) is investigated by means of electromagnetic gyrokinetic simulations including kinetic electrons. A new electromagnetic gyrokinetic simulation code GKV+enables us to examine electron heat and particle fluxes as well as ion heat flux in finite beta heliotron/stellarator plasmas. This problem has not been previously explored because of numerical difficulties associated with complex three-dimensional magnetic structures as well as multiple spatio-temporal scales related to electromagnetic ion and electron dynamics. The turbulent fluxes, which are evaluated through a nonlinear simulation carried out in the K-super computer system, will be reported. This research uses computational resources of K at RIKEN Advanced Institute for Computational Science through the HPCI System Research project (Project ID: hp140044).

  14. Nature of turbulent transport across sheared zonal flows: insights from gyro-kinetic simulations

    SciTech Connect

    Sanchez, Raul; Newman, David E; Leboeuf, Jean-Noel; Decyk, Viktor

    2011-01-01

    The traditional view regarding the reduction of turbulence-induced transport across a stable sheared flow invokes a reduction of the characteristic length scale in the direction perpendicular to the flow as a result of the shearing and stretching of eddies caused by the differential pull exerted in the direction of the flow. A reduced effective transport coefficient then suffices to capture the reduction, that can then be readily incorporated into a transport model. However, recent evidence from gyrokinetic simulations of the toroidal ion-temperature-gradient mode suggests that the dynamics of turbulent transport across sheared flows changes in a more fundamental manner, and that the use of reduced effective transport coefficients fails to capture the full dynamics that may exhibit both subdiffusion and non-Gaussian statistics. In this contribution, after briefly reviewing these results, we propose some candidates for the physical mechanisms responsible for endowing transport with such non-diffusive characteristics, backing these proposals with new numerical gyrokinetic data

  15. Advanced methods in global gyrokinetic full f particle simulation of tokamak transport

    SciTech Connect

    Ogando, F.; Heikkinen, J. A.; Henriksson, S.; Janhunen, S. J.; Kiviniemi, T. P.; Leerink, S.

    2006-11-30

    A new full f nonlinear gyrokinetic simulation code, named ELMFIRE, has been developed for simulating transport phenomena in tokamak plasmas. The code is based on a gyrokinetic particle-in-cell algorithm, which can consider electrons and ions jointly or separately, as well as arbitrary impurities. The implicit treatment of the ion polarization drift and the use of full f methods allow for simulations of strongly perturbed plasmas including wide orbit effects, steep gradients and rapid dynamic changes. This article presents in more detail the algorithms incorporated into ELMFIRE, as well as benchmarking comparisons to both neoclassical theory and other codes.Code ELMFIRE calculates plasma dynamics by following the evolution of a number of sample particles. Because of using an stochastic algorithm its results are influenced by statistical noise. The effect of noise on relevant magnitudes is analyzed.Turbulence spectra of FT-2 plasma has been calculated with ELMFIRE, obtaining results consistent with experimental data.

  16. Gyrokinetic studies of core turbulence features in ASDEX Upgrade H-mode plasmas

    SciTech Connect

    Navarro, A. Bañón Told, D.; Happel, T.; Görler, T.; Abiteboul, J.; Bustos, A.; Doerk, H.; Jenko, F.

    2015-04-15

    Gyrokinetic validation studies are crucial for developing confidence in the model incorporated in numerical simulations and thus improving their predictive capabilities. As one step in this direction, we simulate an ASDEX Upgrade discharge with the GENE code, and analyze various fluctuating quantities and compare them to experimental measurements. The approach taken is the following. First, linear simulations are performed in order to determine the turbulence regime. Second, the heat fluxes in nonlinear simulations are matched to experimental fluxes by varying the logarithmic ion temperature gradient within the expected experimental error bars. Finally, the dependence of various quantities with respect to the ion temperature gradient is analyzed in detail. It is found that density and temperature fluctuations can vary significantly with small changes in this parameter, thus making comparisons with experiments very sensitive to uncertainties in the experimental profiles. However, cross-phases are more robust, indicating that they are better observables for comparisons between gyrokinetic simulations and experimental measurements.

  17. A Gyrokinetic Approach to Low Frequency Anisotropy-Driven Instabilities in the Solar Wind

    NASA Astrophysics Data System (ADS)

    Johnson, J.; Porazik, P.

    2014-12-01

    Observational surveys of temperature anisotropy in the solar wind indicate that anisotropy is bounded over a wide range of plasma beta and the anisotropy bounds appear to be predominately controlled by wave-particle interactions associated with mirror and oblique firehose instabilities. We present a reduced kinetic description that exploits gyrosymmetry (a symmetry associated with the gyromotion), providing an efficient, self-consistent approach that can be utilized in global models of the solar wind. We discuss the underlying physics of the mirror and firehose instabilities that allow for a reduced gyrokinetic description, and we verify the approach through comparisons of theory and simulations using gyrokinetic, hybrid, and fully kinetic descriptions. We present simulations showing the nonlinear development and saturation of the mirror instability and explain the amplitude and structure of the nonlinear state in terms of particle trapping. We also consider the nonlinear development of the oblique firehose instability and the associated wave spectra.

  18. Canonical Lie-transform method in Hamiltonian gyrokinetics: a new approach

    SciTech Connect

    Nicolini, Piero; Tessarotto, Massimo

    2005-05-16

    The well-known gyrokinetic problem regards the perturbative expansion related to the dynamics of a charged particle subject to fast gyration motion due to the presence of a strong magnetic field. Although a variety of approaches have been formulated in the past to this well known problem, surprisingly a purely canonical approach based on Lie transform methods is still missing. This paper aims to fill in this gap and provide at the same time new insight in Lie-transform approaches.

  19. Verification of electromagnetic fluid-kinetic hybrid electron model in global gyrokinetic particle simulation

    SciTech Connect

    Holod, I.; Lin, Z.

    2013-03-15

    The fluid-kinetic hybrid electron model is verified in global gyrokinetic particle simulation of linear electromagnetic drift-Alfvenic instabilities in tokamak. In particular, we have recovered the {beta}-stabilization of the ion temperature gradient mode, transition to collisionless trapped electron mode, and the onset of kinetic ballooning mode as {beta}{sub e} (ratio of electron kinetic pressure to magnetic pressure) increases.

  20. Gyrokinetic simulation on the effect of radio frequency waves on ion-temperature-gradient-driven modes

    NASA Astrophysics Data System (ADS)

    Imadera, K.; Kishimoto, Y.; Sen, S.; Vahala, G.

    2016-02-01

    The ion-temperature-driven modes are studied in the presence of radio frequency waves by the use of the Gyro-Kinetic simulation Code. It is shown that the radio frequency waves through the ponderomotive force can stabilise the ion-temperature-gradient instabilities and contrary to the usual belief no radio frequency wave-induced flow generation hypothesis is required. This might be a major way to create a transport barrier in the fusion energy generation.

  1. Gyrokinetic particle simulations of reversed shear Alfven eigenmode excited by antenna and fast ions

    SciTech Connect

    Deng Wenjun; Holod, Ihor; Xiao Yong; Lin Zhihong; Wang Xin; Zhang Wenlu

    2010-11-15

    Global gyrokinetic particle simulations of reversed shear Alfven eigenmode (RSAE) have been successfully performed and verified. We have excited the RSAE by initial perturbation, by external antenna, and by energetic ions. The RSAE excitation by antenna provides verifications of the mode structure, the frequency, and the damping rate. When the kinetic effects of the background plasma are artificially suppressed, the mode amplitude shows a near-linear growth. With kinetic thermal ions, the mode amplitude eventually saturates due to the thermal ion damping. The damping rates measured from the antenna excitation and from the initial perturbation simulation agree very well. The RSAE excited by fast ions shows an exponential growth. The finite Larmor radius effects of the fast ions are found to significantly reduce the growth rate. With kinetic thermal ions and electron pressure, the mode frequency increases due to the elevation of the Alfven continuum by the geodesic compressibility. The nonperturbative contributions from the fast ions and kinetic thermal ions modify the mode structure relative to the ideal magnetohydrodynamic (MHD) theory. The gyrokinetic simulations have been benchmarked with extended hybrid MHD-gyrokinetic simulations.

  2. The role of zonal flows in the saturation of multi-scale gyrokinetic turbulence

    NASA Astrophysics Data System (ADS)

    Staebler, G. M.; Candy, J.; Howard, N. T.; Holland, C.

    2016-06-01

    The 2D spectrum of the saturated electric potential from gyrokinetic turbulence simulations that include both ion and electron scales (multi-scale) in axisymmetric tokamak geometry is analyzed. The paradigm that the turbulence is saturated when the zonal (axisymmetic) ExB flow shearing rate competes with linear growth is shown to not apply to the electron scale turbulence. Instead, it is the mixing rate by the zonal ExB velocity spectrum with the turbulent distribution function that competes with linear growth. A model of this mechanism is shown to be able to capture the suppression of electron-scale turbulence by ion-scale turbulence and the threshold for the increase in electron scale turbulence when the ion-scale turbulence is reduced. The model computes the strength of the zonal flow velocity and the saturated potential spectrum from the linear growth rate spectrum. The model for the saturated electric potential spectrum is applied to a quasilinear transport model and shown to accurately reproduce the electron and ion energy fluxes of the non-linear gyrokinetic multi-scale simulations. The zonal flow mixing saturation model is also shown to reproduce the non-linear upshift in the critical temperature gradient caused by zonal flows in ion-scale gyrokinetic simulations.

  3. Gyrokinetic simulation of edge blobs and divertor heat-load footprint

    NASA Astrophysics Data System (ADS)

    Chang, C. S.; Ku, S.; Hager, R.; Churchill, M.; D'Azevedo, E.; Worley, P.

    2015-11-01

    Gyrokinetic study of divertor heat-load width Lq has been performed using the edge gyrokinetic code XGC1. Both neoclassical and electrostatic turbulence physics are self-consistently included in the simulation with fully nonlinear Fokker-Planck collision operation and neutral recycling. Gyrokinetic ions and drift kinetic electrons constitute the plasma in realistic magnetic separatrix geometry. The electron density fluctuations from nonlinear turbulence form blobs, as similarly seen in the experiments. DIII-D and NSTX geometries have been used to represent today's conventional and tight aspect ratio tokamaks. XGC1 shows that the ion neoclassical orbit dynamics dominates over the blob physics in setting Lq in the sample DIII-D and NSTX plasmas, re-discovering the experimentally observed 1/Ip type scaling. Magnitude of Lq is in the right ballpark, too, in comparison with experimental data. However, in an ITER standard plasma, XGC1 shows that the negligible neoclassical orbit excursion effect makes the blob dynamics to dominate Lq. Differently from Lq 1mm (when mapped back to outboard midplane) as was predicted by simple-minded extrapolation from the present-day data, XGC1 shows that Lq in ITER is about 1 cm that is somewhat smaller than the average blob size. Supported by US DOE and the INCITE program.

  4. Global gyrokinetic models for energetic particle driven Alfvén instabilities in 3D equilibria

    NASA Astrophysics Data System (ADS)

    Spong, Don; Holod, Ihor

    2015-11-01

    The GTC global gyrokinetic PIC model has been adapted to 3D VMEC equilibria and provides a new method for the analysis of Alfvénic instabilities in stellarators, 3D tokamaks, and helical RFP states. The gyrokinetic orderings (k||/k⊥ << 1, ω/Ωci << 1, ρEP/L << 1) are applicable to a range of energetic particle driven instabilities that have been observed in 3D configurations. Applications of this model to stellarators have indicated that a variety of different Alfvén instabilities can be excited, depending on the toroidal mode number, fast ion average energy and fast ion density profile. Both an LHD discharge where bursting n = 1 Alfvén activity in the TAE gap was observed and a W7-X case have been examined. TAE,/EAE/GAE modes have been found in the simulations, depending on the mode family and fast ion profiles used. The dynamical evolution of the instabilities shows the field period coupling between n and n + Nfp expected for a stellarator. The development of gyrofluid reduced models that can capture relevant physics aspects of the gyrokinetic models will also be discussed. Research sponsored by the U.S. Department of Energy under Contract DE-AC05-00OR22725 with UT-Battelle, LLC and the GSEP SciDAC Center.

  5. A flux-matched gyrokinetic analysis of DIII-D L-mode turbulence

    SciTech Connect

    Görler, T. Told, D.; White, A. E.; Jenko, F.; Holland, C.; Rhodes, T. L.

    2014-12-15

    Previous nonlinear gyrokinetic simulations of specific DIII-D L-mode cases have been found to significantly underpredict the ion heat transport and associated density and temperature fluctuation levels by up to almost one of order of magnitude in the outer-core domain, i.e., roughly in the last third of the minor radius. Since then, this so-called shortfall issue has been subject to various speculations on possible reasons and furthermore motivation for a number of dedicated comparisons for L-mode plasmas in comparable machines. However, only a rather limited number of simulations and gyrokinetic codes has been applied to the original scenario, thus calling for further dedicated investigations in order to broaden the scientific basis. The present work contributes along these lines by employing another well-established gyrokinetic code in a numerically and physically comprehensive manner. Contrary to the previous studies, only a mild underprediction is observed at the outer radial positions which can furthermore be overcome by varying the ion temperature gradient within the error bars associated with the experimental measurement. The significance and reliability of these simulations are demonstrated by benchmarks, numerical convergence tests, and furthermore by extensive validation studies. The latter involve cross-phase and cross-power spectra analyses of various fluctuating quantities and confirm a high degree of realism. The code discrepancies come as a surprise since the involved software packages had been benchmarked repeatedly and very successfully in the past. Further collaborative effort in identifying the underlying difference is hence required.

  6. Verification of gyrokinetic particle simulation of current-driven instability in fusion plasmas. I. Internal kink mode

    NASA Astrophysics Data System (ADS)

    McClenaghan, J.; Lin, Z.; Holod, I.; Deng, W.; Wang, Z.

    2014-12-01

    The gyrokinetic toroidal code (GTC) capability has been extended for simulating internal kink instability with kinetic effects in toroidal geometry. The global simulation domain covers the magnetic axis, which is necessary for simulating current-driven instabilities. GTC simulation in the fluid limit of the kink modes in cylindrical geometry is verified by benchmarking with a magnetohydrodynamic eigenvalue code. Gyrokinetic simulations of the kink modes in the toroidal geometry find that ion kinetic effects significantly reduce the growth rate even when the banana orbit width is much smaller than the radial width of the perturbed current layer at the mode rational surface.

  7. Verification of gyrokinetic particle simulation of current-driven instability in fusion plasmas. I. Internal kink mode

    SciTech Connect

    McClenaghan, J.; Lin, Z.; Holod, I.; Deng, W.; Wang, Z.

    2014-12-15

    The gyrokinetic toroidal code (GTC) capability has been extended for simulating internal kink instability with kinetic effects in toroidal geometry. The global simulation domain covers the magnetic axis, which is necessary for simulating current-driven instabilities. GTC simulation in the fluid limit of the kink modes in cylindrical geometry is verified by benchmarking with a magnetohydrodynamic eigenvalue code. Gyrokinetic simulations of the kink modes in the toroidal geometry find that ion kinetic effects significantly reduce the growth rate even when the banana orbit width is much smaller than the radial width of the perturbed current layer at the mode rational surface.

  8. Fully electromagnetic gyrokinetic eigenmode analysis of high-beta shaped plasmas

    SciTech Connect

    Belli, E. A.; Candy, J.

    2010-11-15

    A new, more efficient method to compute unstable linear gyrokinetic eigenvalues and eigenvectors has been developed for drift-wave analysis of plasmas with arbitrary flux-surface shape, including both transverse and compressional magnetic perturbations. In high-beta, strongly shaped plasmas like in the National Spherical Torus Experiment (NSTX) [M. Ono et al., Nucl. Fusion 40, 557 (2000)], numerous branches of closely spaced unstable eigenmodes exist. These modes are difficult and time-consuming to adequately resolve with the existing linear initial-value solvers, which are further limited to the most unstable eigenmode. The new method is based on an eigenvalue approach and is an extension of the GYRO code [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)], reusing the existing discretization schemes in both real and velocity-space. Unlike recent methods, which use an iterative solver to compute eigenvalues of the relatively large gyrokinetic response matrix, the present scheme computes the zeros of the much smaller Maxwell dispersion matrix using a direct method. In the present work, the new eigensolver is applied to gyrokinetic stability analysis of a high-beta, NSTX-like plasma. We illustrate the smooth transformation from ion-temperature-gradient (ITG)-like to kinetic-ballooning (KBM)-like modes, and the formation of hybrid ITG/KBM modes, and further demonstrate the existence of high-k Alfvenic drift-wave 'cascades' for which the most unstable mode is a higher excited state along the field line. A new compressional electron drift wave, which is driven by a combination of strong beta and pressure gradient, is also identified for the first time. Overall, we find that accurate calculation of stability boundaries and growth rates cannot, in general, ignore the compressional component {delta}B{sub ||} of the perturbation.

  9. Nonlinear gyrokinetic simulations of the I-mode high confinement regime and comparisons with experiment

    SciTech Connect

    White, A. E. Howard, N. T.; Creely, A. J.; Chilenski, M. A.; Greenwald, M.; Hubbard, A. E.; Hughes, J. W.; Marmar, E.; Rice, J. E.; Sierchio, J. M.; Sung, C.; Walk, J. R.; Whyte, D. G.; Mikkelsen, D. R.; Edlund, E. M.; Kung, C.; Holland, C.; Candy, J.; Petty, C. C.; Reinke, M. L.; and others

    2015-05-15

    For the first time, nonlinear gyrokinetic simulations of I-mode plasmas are performed and compared with experiment. I-mode is a high confinement regime, featuring energy confinement similar to H-mode, but without enhanced particle and impurity particle confinement [D. G. Whyte et al., Nucl. Fusion 50, 105005 (2010)]. As a consequence of the separation between heat and particle transport, I-mode exhibits several favorable characteristics compared to H-mode. The nonlinear gyrokinetic code GYRO [J. Candy and R. E. Waltz, J Comput. Phys. 186, 545 (2003)] is used to explore the effects of E × B shear and profile stiffness in I-mode and compare with L-mode. The nonlinear GYRO simulations show that I-mode core ion temperature and electron temperature profiles are more stiff than L-mode core plasmas. Scans of the input E × B shear in GYRO simulations show that E × B shearing of turbulence is a stronger effect in the core of I-mode than L-mode. The nonlinear simulations match the observed reductions in long wavelength density fluctuation levels across the L-I transition but underestimate the reduction of long wavelength electron temperature fluctuation levels. The comparisons between experiment and gyrokinetic simulations for I-mode suggest that increased E × B shearing of turbulence combined with increased profile stiffness are responsible for the reductions in core turbulence observed in the experiment, and that I-mode resembles H-mode plasmas more than L-mode plasmas with regards to marginal stability and temperature profile stiffness.

  10. Fully electromagnetic gyrokinetic eigenmode analysis of high-beta shaped plasmas

    NASA Astrophysics Data System (ADS)

    Belli, E. A.; Candy, J.

    2010-11-01

    A new, more efficient method to compute unstable linear gyrokinetic eigenvalues and eigenvectors has been developed for drift-wave analysis of plasmas with arbitrary flux-surface shape, including both transverse and compressional magnetic perturbations. In high-beta, strongly shaped plasmas like in the National Spherical Torus Experiment (NSTX) [M. Ono et al., Nucl. Fusion 40, 557 (2000)], numerous branches of closely spaced unstable eigenmodes exist. These modes are difficult and time-consuming to adequately resolve with the existing linear initial-value solvers, which are further limited to the most unstable eigenmode. The new method is based on an eigenvalue approach and is an extension of the GYRO code [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)], reusing the existing discretization schemes in both real and velocity-space. Unlike recent methods, which use an iterative solver to compute eigenvalues of the relatively large gyrokinetic response matrix, the present scheme computes the zeros of the much smaller Maxwell dispersion matrix using a direct method. In the present work, the new eigensolver is applied to gyrokinetic stability analysis of a high-beta, NSTX-like plasma. We illustrate the smooth transformation from ion-temperature-gradient (ITG)-like to kinetic-ballooning (KBM)-like modes, and the formation of hybrid ITG/KBM modes, and further demonstrate the existence of high-k Alfvénic drift-wave "cascades" for which the most unstable mode is a higher excited state along the field line. A new compressional electron drift wave, which is driven by a combination of strong beta and pressure gradient, is also identified for the first time. Overall, we find that accurate calculation of stability boundaries and growth rates cannot, in general, ignore the compressional component δB∥ of the perturbation.

  11. Nonlinear gyrokinetic simulations of the I-mode high confinement regime and comparisons with experimenta)

    NASA Astrophysics Data System (ADS)

    White, A. E.; Howard, N. T.; Creely, A. J.; Chilenski, M. A.; Greenwald, M.; Hubbard, A. E.; Hughes, J. W.; Marmar, E.; Rice, J. E.; Sierchio, J. M.; Sung, C.; Walk, J. R.; Whyte, D. G.; Mikkelsen, D. R.; Edlund, E. M.; Kung, C.; Holland, C.; Candy, J.; Petty, C. C.; Reinke, M. L.; Theiler, C.

    2015-05-01

    For the first time, nonlinear gyrokinetic simulations of I-mode plasmas are performed and compared with experiment. I-mode is a high confinement regime, featuring energy confinement similar to H-mode, but without enhanced particle and impurity particle confinement [D. G. Whyte et al., Nucl. Fusion 50, 105005 (2010)]. As a consequence of the separation between heat and particle transport, I-mode exhibits several favorable characteristics compared to H-mode. The nonlinear gyrokinetic code GYRO [J. Candy and R. E. Waltz, J Comput. Phys. 186, 545 (2003)] is used to explore the effects of E × B shear and profile stiffness in I-mode and compare with L-mode. The nonlinear GYRO simulations show that I-mode core ion temperature and electron temperature profiles are more stiff than L-mode core plasmas. Scans of the input E × B shear in GYRO simulations show that E × B shearing of turbulence is a stronger effect in the core of I-mode than L-mode. The nonlinear simulations match the observed reductions in long wavelength density fluctuation levels across the L-I transition but underestimate the reduction of long wavelength electron temperature fluctuation levels. The comparisons between experiment and gyrokinetic simulations for I-mode suggest that increased E × B shearing of turbulence combined with increased profile stiffness are responsible for the reductions in core turbulence observed in the experiment, and that I-mode resembles H-mode plasmas more than L-mode plasmas with regards to marginal stability and temperature profile stiffness.

  12. Fidelity of reduced and realistic electron mass ratio multi-scale gyrokinetic simulations of tokamak discharges

    NASA Astrophysics Data System (ADS)

    Howard, N. T.; Holland, C.; White, A. E.; Greenwald, M.; Candy, J.

    2015-06-01

    The first study using multi-scale (coupled ITG/TEM/ETG) gyrokinetic simulations at both reduced and realistic electron mass ratios, μ = (mD/me).5 = 20.0, 40.0 and 60.0, has been performed on a standard, Alcator C-Mod, L-mode discharge. Ion-scale (kθρs  ∼  1.0) and multi-scale (up to kθρe  ∼  0.8) gyrokinetic simulations are compared at different simulated mass ratios to investigate the fidelity of reduced electron mass ratio, multi-scale simulation through direct comparison with realistic mass ratio, multi-scale simulation. Detailed description of both the numerical setup and the turbulent scales required to obtain meaningful coupled ITG/TEM/ETG simulation is presented. Significant high-k driven (TEM/ETG) heat flux is found to exist at scales of approximately kθρe  ∼  0.1 at all mass ratios but can only be obtained by simulation capturing turbulence up to kθρe  ∼  1.0. At slightly reduced mass ratio, μ = 40.0, qualitative agreement with realistic mass simulation can be obtained in the studied discharge, consistent with intuition obtained from linear stability analysis. However, realistic electron mass is required for any robust quantitative comparison with experimental heat fluxes for the condition studied, as significant differences are observed at even slightly reduced electron mass ratio. The details of this numerical study are presented to provide a basis for future studies utilizing coupled ITG/TEM/ETG gyrokinetic simulation.

  13. Gyrokinetic Particle Simulation of Turbulent Transport in Burning Plasmas (GPS - TTBP) Final Report

    SciTech Connect

    Chame, Jacqueline

    2011-05-27

    The goal of this project is the development of the Gyrokinetic Toroidal Code (GTC) Framework and its applications to problems related to the physics of turbulence and turbulent transport in tokamaks,. The project involves physics studies, code development, noise effect mitigation, supporting computer science efforts, diagnostics and advanced visualizations, verification and validation. Its main scientific themes are mesoscale dynamics and non-locality effects on transport, the physics of secondary structures such as zonal flows, and strongly coherent wave-particle interaction phenomena at magnetic precession resonances. Special emphasis is placed on the implications of these themes for rho-star and current scalings and for the turbulent transport of momentum. GTC-TTBP also explores applications to electron thermal transport, particle transport; ITB formation and cross-cuts such as edge-core coupling, interaction of energetic particles with turbulence and neoclassical tearing mode trigger dynamics. Code development focuses on major initiatives in the development of full-f formulations and the capacity to simulate flux-driven transport. In addition to the full-f -formulation, the project includes the development of numerical collision models and methods for coarse graining in phase space. Verification is pursued by linear stability study comparisons with the FULL and HD7 codes and by benchmarking with the GKV, GYSELA and other gyrokinetic simulation codes. Validation of gyrokinetic models of ion and electron thermal transport is pursed by systematic stressing comparisons with fluctuation and transport data from the DIII-D and NSTX tokamaks. The physics and code development research programs are supported by complementary efforts in computer sciences, high performance computing, and data management.

  14. Gyrokinetic simulations of off-axis minimum-q profile corrugations

    SciTech Connect

    Waltz, R.E.; Austin, M.E.; Burrell, K.H.; Candy, J.

    2006-05-15

    Quasiequilibrium radial 'profile corrugations' in the electron temperature gradient are found at lowest-order singular surfaces in global gyrokinetic code simulations of both monotonic-q and off-axis minimum-q discharges. The profile corrugations in the temperature and density gradients are time-averaged components of zonal flows. The m/n=2/1 electron temperature gradient corrugation is measurably large and appears to trigger an internal transport barrier as the off-axis minimum-q=2 surfaces enter the plasma.

  15. Verification of gyrokinetic {delta}f simulations of electron temperature gradient turbulence

    SciTech Connect

    Nevins, W. M.; Parker, S. E.; Chen, Y.; Candy, J.; Dimits, A.; Dorland, W.; Hammett, G. W.; Jenko, F.

    2007-08-15

    The GEM gyrokinetic {delta}f simulation code [Y. Chen and S. Parker, J. Comput. Phys. 189, 463 (2003); and ibid.220, 839 (2007)] is shown to reproduce electron temperature gradient turbulence at the benchmark operating point established in previous work [W. M. Nevins, J. Candy, S. Cowley, T. Dannert, A. Dimits, W. Dorland, C. Estrada-Mila, G. W. Hammett, F. Jenko, M. J. Pueschel, and D. E. Shumaker, Phys. Plasmas 13, 122306 (2006)]. The electron thermal transport is within 10% of the expected value, while the turbulent fluctuation spectrum is shown to have the expected intensity and two-point correlation function.

  16. The Hamiltonian Structure and Euler-Poincare Formulation of the Valsov-Maxwell and Gyrokinetic System

    SciTech Connect

    J. Squire, H. Qin and W.M. Tang

    2012-09-25

    We present a new variational principle for the gyrokinetic system, similar to the Maxwell-Vlasov action presented in Ref. 1. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid velocity and particle distribution function. Using a Legendre transform, we explicitly derive the field theoretic Hamiltonian structure of the system. This is carried out with the Dirac theory of constraints, which is used to construct meaningful brackets from those obtained directly from Euler-Poincare theory. Possible applications of these formulations include continuum geometric integration techniques, large-eddy simulation models and Casimir type stability methods. __________________________________________________

  17. Gyrokinetic full-torus simulations of ohmic tokamak plasmas in circular limiter configuration

    NASA Astrophysics Data System (ADS)

    Korpilo, T.; Gurchenko, A. D.; Gusakov, E. Z.; Heikkinen, J. A.; Janhunen, S. J.; Kiviniemi, T. P.; Leerink, S.; Niskala, P.; Perevalov, A. A.

    2016-06-01

    The gyrokinetic full 5D particle distribution code ELMFIRE has been extended to simulate circular tokamak plasmas from the magnetic axis to the limiter scrape-off-layer. The predictive power of the code in the full-torus configuration is tested via its ability to reproduce experimental steady-state profiles in FT-2 ohmic L-mode plasmas. The results show that the experimental profile solution is not reproduced numerically due to the difficulty of obtaining global power balance. This is verified by cross-comparison of ELMFIRE code versions, which shows also the impact of boundary conditions and grid resolution on turbulent transport.

  18. Multiscale Nature of the Dissipation Range in Gyrokinetic Simulations of Alfvénic Turbulence.

    PubMed

    Told, D; Jenko, F; TenBarge, J M; Howes, G G; Hammett, G W

    2015-07-10

    Nonlinear energy transfer and dissipation in Alfvén wave turbulence are analyzed in the first gyrokinetic simulation spanning all scales from the tail of the MHD range to the electron gyroradius scale. For typical solar wind parameters at 1 AU, about 30% of the nonlinear energy transfer close to the electron gyroradius scale is mediated by modes in the tail of the MHD cascade. Collisional dissipation occurs across the entire kinetic range k(⊥)ρ(I)≳1. Both mechanisms thus act on multiple coupled scales, which have to be retained for a comprehensive picture of the dissipation range in Alfvénic turbulence. PMID:26207474

  19. On the definition of a kinetic equilibrium in global gyrokinetic simulations

    NASA Astrophysics Data System (ADS)

    Angelino, P.; Bottino, A.; Hatzky, R.; Jolliet, S.; Sauter, O.; Tran, T. M.; Villard, L.

    2006-05-01

    Nonlinear electrostatic global gyrokinetic simulations of collisionless ion temperature gradient (ITG) turbulence and E ×B zonal flows in axisymmetric toroidal plasmas are examined for different choices of the initial distribution function. Using a local Maxwellian leads to the generation of axisymmetric E ×B flows that can be so strong as to prevent ITG mode growth. A method using a canonical Maxwellian is shown to avoid this spurious generation of E ×B flows. In addition, a revised δf scheme is introduced and compared to the standard δf method.

  20. On the definition of a kinetic equilibrium in global gyrokinetic simulations

    SciTech Connect

    Angelino, P.; Bottino, A.; Hatzky, R.; Jolliet, S.; Sauter, O.; Tran, T.M.; Villard, L.

    2006-05-15

    Nonlinear electrostatic global gyrokinetic simulations of collisionless ion temperature gradient (ITG) turbulence and ExB zonal flows in axisymmetric toroidal plasmas are examined for different choices of the initial distribution function. Using a local Maxwellian leads to the generation of axisymmetric ExB flows that can be so strong as to prevent ITG mode growth. A method using a canonical Maxwellian is shown to avoid this spurious generation of ExB flows. In addition, a revised {delta}f scheme is introduced and compared to the standard {delta}f method.

  1. A gyrokinetic continuum code based on the numerical Lie transform (NLT) method

    NASA Astrophysics Data System (ADS)

    Ye, Lei; Xu, Yingfeng; Xiao, Xiaotao; Dai, Zongliang; Wang, Shaojie

    2016-07-01

    In this work, we report a novel gyrokinetic simulation method named numerical Lie transform (NLT), which depends on a new physical model derived from the I-transform theory. In this model, the perturbed motion of a particle is decoupled from the unperturbed motion. Due to this property, the unperturbed orbit can be computed in advance and saved as numerical tables for real-time computation. A 4D tensor B-spline interpolation module is developed and applied with the semi-Lagrangian scheme to avoid operator splitting. The NLT code is verified by the Rosenbluth-Hinton test and the linear ITG Cyclone test.

  2. Verification of a magnetic island in gyro-kinetics by comparison with analytic theory

    SciTech Connect

    Zarzoso, D. Casson, F. J.; Poli, E.; Hornsby, W. A.; Peeters, A. G.

    2015-02-15

    A rotating magnetic island is imposed in the gyrokinetic code GKW, when finite differences are used for the radial direction, in order to develop the predictions of analytic tearing mode theory and understand its limitations. The implementation is verified against analytics in sheared slab geometry with three numerical tests that are suggested as benchmark cases for every code that imposes a magnetic island. The convergence requirements to properly resolve physics around the island separatrix are investigated. In the slab geometry, at low magnetic shear, binormal flows inside the island can drive Kelvin-Helmholtz instabilities which prevent the formation of the steady state for which the analytic theory is formulated.

  3. Nature of Transport across Sheared Zonal Flows in Electrostatic Ion-Temperature-Gradient Gyrokinetic Plasma Turbulence

    SciTech Connect

    Sanchez, R.; Newman, D. E.; Leboeuf, J.-N.; Decyk, V. K.; Carreras, B. A.

    2008-11-14

    It is shown that the usual picture for the suppression of turbulent transport across a stable sheared flow based on a reduction of diffusive transport coefficients is, by itself, incomplete. By means of toroidal gyrokinetic simulations of electrostatic, collisionless ion-temperature-gradient turbulence, it is found that the nature of the transport is altered fundamentally, changing from diffusive to anticorrelated and subdiffusive. Additionally, whenever the flows are self-consistently driven by turbulence, the transport gains an additional non-Gaussian character. These results suggest that a description of transport across sheared flows using effective diffusivities is oversimplified.

  4. Nature of Transport across Sheared Zonal Flows in Electrostatic Ion-Temperature-Gradient Gyrokinetic Plasma Turbulence

    SciTech Connect

    Sanchez, Raul; Newman, David E; Leboeuf, Jean-Noel; Decyk, Viktor; Carreras, Benjamin A

    2008-01-01

    It is shown that the usual picture for the suppression of turbulent transport across a stable sheared flow based on a reduction of diffusive transport coefficients is, by itself, incomplete. By means of toroidal gyrokinetic simulations of electrostatic, collisionless ion-temperature-gradient turbulence, it is found that the nature of the transport is altered fundamentally, changing from diffusive to anticorrelated and subdiffusive. Additionally, whenever the flows are self-consistently driven by turbulence, the transport gains an additional non-Gaussian character. These results suggest that a description of transport across sheared flows using effective diffusivities is oversimplified.

  5. Effects of the magnetic equilibrium on gyrokinetic simulations of tokamak microinstabilities

    SciTech Connect

    Wan, Weigang; Chen, Yang; Parker, Scott E.; Groebner, Richard J.

    2015-06-15

    The general geometry of the experimental tokamak magnetic equilibrium is implemented in the global gyrokinetic simulation code GEM. Compared to the general geometry, the well used Miller parameterization of the magnetic equilibrium is a good approximation in the core region and up to the top of the pedestal. Linear simulations indicate that results with the two geometries agree for r/a ≤ 0.9. However, in the edge region, the instabilities are sensitive to the magnetic equilibrium in both the L-mode and the H-mode plasmas. A small variation of the plasma shaping parameters leads to large changes to the edge instability.

  6. Gyrokinetic simulations of ion temperature gradient modes in the reversed field pinch

    SciTech Connect

    Predebon, I.; Guo, S. C.; Angioni, C.

    2010-01-15

    Linear gyrokinetic calculations are applied to the reversed field pinch configuration to investigate the occurrence of ion temperature gradient instabilities. The analysis shows this type of instability to be only marginally responsible for particle and energy transport. The required gradients could be reached only in correspondence to the temperature slopes arising at the boundary of the helical structure in the quasisingle helicity states. The dependence of the instability threshold on the relevant macroscopic quantities is considered. A discussion on the main differences in the driving mechanisms existing between the reversed field pinch and the tokamak configuration is addressed.

  7. Measurement of plasma current dependent changes in impurity transport and comparison with nonlinear gyrokinetic simulation

    SciTech Connect

    Howard, N. T.; Greenwald, M.; White, A. E.; Reinke, M. L.; Ernst, D.; Podpaly, Y.; Mikkelsen, D. R.; Candy, J.

    2012-05-15

    Measured impurity transport coefficients are found to demonstrate a strong dependence on plasma current in the core of Alcator C-Mod. These measurements are compared directly with linear and nonlinear gyrokinetic simulation in an attempt to both qualitatively and quantitatively reproduce the measured impurity transport. Discharges constituting a scan of plasma current from 0.6 to 1.2 MA were performed during the 2010 run campaign. The impurity transport from these discharges was determined using a novel set of spectroscopic diagnostics available on Alcator C-Mod. This diagnostic suite allowed for the effective constraint of impurity transport coefficient profiles inside of r/a = 0.6. A decrease in the measured impurity diffusivity and inward convection is found with increased plasma current. Global, nonlinear gyrokinetic simulations were performed using the GYRO code [J. Candy and R. E. Waltz, J Comput. Phys. 186, 545 (2003)] for all discharges in the experimental scan and are found to reproduce the experimental trends, while demonstrating good quantitative agreement with measurement. A more comprehensive quantitative comparison was performed on the 0.8 MA discharge of the current scan which demonstrates that simultaneous agreement between experiment and simulation in both the impurity particle transport and ion heat transport channels is attainable within experimental uncertainties.

  8. Gyrokinetic simulations of momentum transport and fluctuation spectra for ICRF-heated L-Mode plasmas

    NASA Astrophysics Data System (ADS)

    Sierchio, J. M.; White, A. E.; Howard, N. T.; Sung, C.; Ennever, P.; Porkolab, M.; Candy, J.

    2014-10-01

    We examine ICRF-heated L-mode plasmas in Alcator C-Mod, with differing momentum transport (hollow vs. peaked radial profiles of intrinsic toroidal rotation) but similar heat and particle transport. Nonlinear gyrokinetic simulations of heat and particle transport with GYRO [Candy and Waltz, J. Comp. Phys. 186, 545 (2003)] have previously been compared with these experiments [White et al., Phys. Plasmas 20, 056106 (2013); Howard et al. PPCF submitted (2014)] as part of an effort to validate the gyrokinetic model for core turbulent transport in C-Mod plasmas. To further test the model for these plasmas, predicted core turbulence characteristics such as fluctuation spectra will be compared with experiment. Using synthetic diagnostics for the CECE, reflectometry, and PCI systems at C-Mod, synthetic spectra and, when applicable, fluctuation amplitudes, are generated. We compare these generated results with fluctuation measurements from the experiment. We also report the momentum transport results from simulations of these plasmas and compare them to experiment. Supported by USDoE award DE-FC02-99ER54512.

  9. Measurement of plasma current dependent changes in impurity transport and comparison with nonlinear gyrokinetic simulationa)

    NASA Astrophysics Data System (ADS)

    Howard, N. T.; Greenwald, M.; Mikkelsen, D. R.; White, A. E.; Reinke, M. L.; Ernst, D.; Podpaly, Y.; Candy, J.

    2012-05-01

    Measured impurity transport coefficients are found to demonstrate a strong dependence on plasma current in the core of Alcator C-Mod. These measurements are compared directly with linear and nonlinear gyrokinetic simulation in an attempt to both qualitatively and quantitatively reproduce the measured impurity transport. Discharges constituting a scan of plasma current from 0.6 to 1.2 MA were performed during the 2010 run campaign. The impurity transport from these discharges was determined using a novel set of spectroscopic diagnostics available on Alcator C-Mod. This diagnostic suite allowed for the effective constraint of impurity transport coefficient profiles inside of r/a = 0.6. A decrease in the measured impurity diffusivity and inward convection is found with increased plasma current. Global, nonlinear gyrokinetic simulations were performed using the GYRO code [J. Candy and R. E. Waltz, J Comput. Phys. 186, 545 (2003)] for all discharges in the experimental scan and are found to reproduce the experimental trends, while demonstrating good quantitative agreement with measurement. A more comprehensive quantitative comparison was performed on the 0.8 MA discharge of the current scan which demonstrates that simultaneous agreement between experiment and simulation in both the impurity particle transport and ion heat transport channels is attainable within experimental uncertainties.

  10. Gyrokinetic Studies of ETG Turbulence in NSTX, and Comparisons of Advection Algorithms

    NASA Astrophysics Data System (ADS)

    Hammett, G. W.; Peterson, J. L.; Mikkelsen, D. R.; Kaye, S. M.; Waltz, R. E.; Candy, J.

    2009-05-01

    Electron temperature gradient (ETG) driven turbulence has been predicted to be important in some parameter regimes. High-k fluctuations have recently been measured in the National Spherical Torus Experiment (NSTX), at kθρe˜0.1-0.4, in the range of ETG turbulence. We are undertaking studies of ETG turbulence for NSTX cases using the GYRO gyrokinetic code. Among other results, we will show the dependence of some of the linear ETG properties on magnetic shear, q, and ZeffTe/Ti. Microtearing may also be important in some cases. On another topic, we will compare various advection algorithms on simple 1-D and 2-D test problems. The edge region in a magnetic fusion device has very steep density variations, which can cause problems for standard advection algorithms. Gibb's phenomena can lead to negative overshoots in the density solution; however, modern high-order upwind methods, developed initially for shock capturing, can preserve positivity of density, making them useful for edge gyrokinetic simulations. W. Dorland, F. Jenko, et al., Phys. Rev. Lett. 85, 5579 (2000). E. Mazzucato, D. R. Smith, et al., Phys. Rev. Lett. 101, 075001 (2008)

  11. Global gyrokinetic stability of collisionless microtearing modes in large aspect ratio tokamaks

    SciTech Connect

    Swamy, Aditya K.; Ganesh, R.; Chowdhury, J.; Brunner, S.; Vaclavik, J.; Villard, L.

    2014-08-15

    Linear full radius gyrokinetic calculations show the existence of unstable microtearing modes (MTMs) in purely collisionless, high temperature, large aspect ratio tokamak plasmas. The present study takes into account fully gyrokinetic highly passing ions and electrons. The global 2-D structures of the collisionless mode with full radius coupling of the poloidal modes is obtained and compared with another electromagnetic mode, namely, the Alfvén Ion Temperature Gradient (AITG) mode (or Kinetic Ballooning Mode, KBM) for the same equilibrium profile. Several important characteristics of the modes are brought out and compared, such as a clear signature in the symmetry properties of the two modes, the plasma–β dependence, and radial and poloidal length scales of the electrostatic and magnetic vector potential fluctuations. Extensive parameter scans for this collisionless microtearing mode reveal the scaling of the growth rate with β and the electron temperature gradient η{sub e}. Scans at different β values show an inverse relationship between the η{sub e} threshold and β, leading to a stability diagram, and implying that the mode might exist at moderate to strong temperature gradients for finite β plasmas in large aspect ratio tokamaks. In contrast to small aspect ratio tokamaks where the trapped electron magnetic drift resonance is found to be important, in large aspect ratio tokamaks, a strong destabilization due to the magnetic drift resonance of passing electrons is observed and is identified as a possible collisionless drive mechanism for the collisionless MTM.

  12. Profile stiffness measurements in the Helically Symmetric experiment and comparison to nonlinear gyrokinetic calculations

    SciTech Connect

    Weir, G. M.; Faber, B. J.; Likin, K. M.; Talmadge, J. N.; Anderson, D. T.; Anderson, F. S. B.

    2015-05-15

    Stiffness measurements are presented in the quasi-helically symmetric experiment (HSX), in which the neoclassical transport is comparable to that in a tokamak and turbulent transport dominates throughout the plasma. Electron cyclotron emission is used to measure the local electron temperature response to modulated electron cyclotron resonant heating. The amplitude and phase of the heat wave through the steep electron temperature gradient (ETG) region of the plasma are used to determine a transient electron thermal diffusivity that is close to the steady-state diffusivity. The low stiffness in the region between 0.2 ≤ r/a ≤ 0.4 agrees with the scaling of the steady-state heat flux with temperature gradient in this region. These experimental results are compared to gyrokinetic calculations in a flux-tube geometry using the gyrokinetic electromagnetic numerical experiment code with two kinetic species. Linear simulations show that the ETG mode may be experimentally relevant within r/a ≤ 0.2, while the Trapped Electron Mode (TEM) is the dominant long-wavelength microturbulence instability across most of the plasma. The TEM is primarily driven by the density gradient. Non-linear calculations of the saturated heat flux driven by the TEM and ETG bracket the experimental heat flux.

  13. The linear tearing instability in three dimensional, toroidal gyro-kinetic simulations

    SciTech Connect

    Hornsby, W. A. Migliano, P.; Buchholz, R.; Kroenert, L.; Weikl, A.; Peeters, A. G.; Zarzoso, D.; Poli, E.; Casson, F. J.

    2015-02-15

    Linear gyro-kinetic simulations of the classical tearing mode in three-dimensional toroidal geometry were performed using the global gyro-kinetic turbulence code, GKW. The results were benchmarked against a cylindrical ideal MHD and analytical theory calculations. The stability, growth rate, and frequency of the mode were investigated by varying the current profile, collisionality, and the pressure gradients. Both collisionless and semi-collisional tearing modes were found with a smooth transition between the two. A residual, finite, rotation frequency of the mode even in the absence of a pressure gradient is observed, which is attributed to toroidal finite Larmor-radius effects. When a pressure gradient is present at low collisionality, the mode rotates at the expected electron diamagnetic frequency. However, the island rotation reverses direction at high collisionality. The growth rate is found to follow a η{sup 1∕7} scaling with collisional resistivity in the semi-collisional regime, closely following the semi-collisional scaling found by Fitzpatrick. The stability of the mode closely follows the stability analysis as performed by Hastie et al. using the same current and safety factor profiles but for cylindrical geometry, however, here a modification due to toroidal coupling and pressure effects is seen.

  14. Quantitative comparison of experimental impurity transport with nonlinear gyrokinetic simulation in an Alcator C-Mod L-mode plasma

    NASA Astrophysics Data System (ADS)

    Howard, N. T.; Greenwald, M.; Mikkelsen, D. R.; Reinke, M. L.; White, A. E.; Ernst, D.; Podpaly, Y.; Candy, J.

    2012-06-01

    Nonlinear gyrokinetic simulations of impurity transport are compared to experimental impurity transport for the first time. The GYRO code (Candy and Waltz 2003 J. Comput. Phys. 186 545) was used to perform global, nonlinear gyrokinetic simulations of impurity transport for a standard Alcator C-Mod, L-mode discharge. The laser blow-off technique was combined with soft x-ray measurements of a single charge state of calcium to provide time-evolving profiles of this non-intrinsic, non-recycling impurity over a radial range of 0.0 ⩽ r/a ⩽ 0.6. Experimental transport coefficient profiles and their uncertainties were extracted from the measurements using the impurity transport code STRAHL and rigorous Monte Carlo error analysis. To best assess the agreement of gyrokinetic simulations with the experimental profiles, the sensitivity of the GYRO predicted impurity transport to a wide range of turbulence-relevant plasma parameters was investigated. A direct comparison of nonlinear gyrokinetic simulation and experiment is presented with an in depth discussion of error sources and a new data analysis methodology.

  15. Static cylindrically symmetric spacetimes

    NASA Astrophysics Data System (ADS)

    Fjällborg, Mikael

    2007-05-01

    We prove the existence of static solutions to the cylindrically symmetric Einstein Vlasov system, and we show that the matter cylinder has finite extension in two of the three spatial dimensions. The same results are also proved for a quite general class of equations of state for perfect fluids coupled to the Einstein equations, extending the class of equations of state considered by Bicak et al (2004 Class. Quantum Grav.21 1583). We also obtain this result for the Vlasov Poisson system.

  16. Using a local gyrokinetic code to study global ion temperature gradient modes in tokamaks

    NASA Astrophysics Data System (ADS)

    Abdoul, P. A.; Dickinson, D.; Roach, C. M.; Wilson, H. R.

    2015-06-01

    In this paper the global eigenmode structures of linear ion temperature gradient (ITG) modes in tokamak plasmas are obtained using a novel technique which combines results from the local gyrokinetic code GS2 with analytical theory to reconstruct global properties. Local gyrokinetic calculations are performed for a range of radial flux surfaces, x, and ballooning phase angles, p, to map out the local complex mode frequency, Ω0(x, p) = ω0(x, p) + iγ0(x, p) for a single toroidal mode number, n. Taylor expanding Ω0 about a reference surface at x = 0, and employing the Fourier-ballooning representation leads to a second order ODE for the amplitude envelope, A(p), which describes how the local results are combined to form the global mode. The equilibrium profiles impact on the variation of Ω0(x, p) and hence influence the global mode structure. The simulations presented here are based upon a global extension to the CYCLONE base case and employ the circular Miller equilibrium model. In an equilibrium with radially varying profiles of a/LT and a/Ln, peaked at x = 0, and with all other equilibrium profiles held constant, including ηi = Ln/LT, Ω0(x, p) is found to have a stationary point. The reconstructed global mode sits at the outboard mid-plane of the tokamak, with global growth rate, γ ∼ Max[γ0]. Including the radial variation of other equilibrium profiles like safety factor and magnetic shear, leads to a mode that peaks away from the outboard mid-plane, with a reduced global growth rate. Finally, the influence of toroidal flow shear has also been investigated through the introduction of a Doppler shift, {ω0}\\to {ω0}-nΩ φ\\prime x , where Ωϕ is the equilibrium toroidal flow, and a prime denotes the radial derivative. The equilibrium profile variations introduce an asymmetry into the global growth rate spectrum with respect to the sign of Ω φ\\prime , such that the maximum growth rate is achieved with non-zero shearing, consistent with recent global

  17. Particle pinch and collisionality in gyrokinetic simulations of tokamak plasma turbulence

    SciTech Connect

    Angioni, C.; Candy, J.; Waltz, R. E.; Fable, E.; Maslov, M.; Weisen, H.; Peeters, A. G.

    2009-06-15

    The generic problem of how, in a turbulent plasma, the experimentally relevant conditions of a particle flux very close to the null are achieved, despite the presence of strong heat fluxes, is addressed. Nonlinear gyrokinetic simulations of plasma turbulence in tokamaks reveal a complex dependence of the particle flux as a function of the turbulent spatial scale and of the velocity space as collisionality is increased. At experimental values of collisionality, the particle flux is found close to the null, in agreement with the experiment, due to the balance between inward and outward contributions at small and large scales, respectively. These simulations provide full theoretical support to the prediction of a peaked density profile in a future nuclear fusion reactor.

  18. Feasibility study for a correlation electron cyclotron emission turbulence diagnostic based on nonlinear gyrokinetic simulations

    NASA Astrophysics Data System (ADS)

    White, A. E.; Howard, N. T.; Mikkelsen, D. R.; Greenwald, M.; Candy, J.; Waltz, R. E.

    2011-11-01

    This paper describes the use of nonlinear gyrokinetic simulations to assess the feasibility of a new correlation electron cyclotron emission (CECE) diagnostic that has been proposed for the Alcator C-Mod tokamak (Marmar et al 2009 Nucl. Fusion 49 104014). This work is based on a series of simulations performed with the GYRO code (Candy and Waltz 2003 J. Comput. Phys. 186 545). The simulations are used to predict ranges of fluctuation level, peak poloidal wavenumber and radial correlation length of electron temperature fluctuations in the core of the plasma. The impact of antenna pattern and poloidal viewing location on measurable turbulence characteristics is addressed using synthetic diagnostics. An upper limit on the CECE sample volume size is determined. The modeling results show that a CECE diagnostic capable of measuring transport-relevant, long-wavelength (kθρs < 0.5) electron temperature fluctuations is feasible at Alcator C-Mod.

  19. Flux- and gradient-driven global gyrokinetic simulation of tokamak turbulence

    SciTech Connect

    Goerler, Tobias; Jenko, Frank; Marcus, Patrick; Merz, Florian; Told, Daniel; Lapillonne, Xavier; Brunner, Stephan; Aghdam, Sohrab Khosh; McMillan, Ben F.; Sauter, Olivier; Villard, Laurent; Dannert, Tilman

    2011-05-15

    The Eulerian gyrokinetic turbulence code gene has recently been extended to a full torus code. Moreover, it now provides Krook-type sources for gradient-driven simulations where the profiles are maintained on average as well as localized heat sources for a flux-driven type of operation. Careful verification studies and benchmarks are performed successfully. This setup is applied to address three related transport issues concerning nonlocal effects. First, it is confirmed that in gradient-driven simulations, the local limit can be reproduced--provided that finite aspect ratio effects in the geometry are treated carefully. In this context, it also becomes clear that the profile widths (not the device width) may constitute a more appropriate measure for finite-size effects. Second, the nature and role of heat flux avalanches are discussed in the framework of both local and global, flux- and gradient-driven simulations. Third, simulations dedicated to discharges with electron internal barriers are addressed.

  20. Gyrokinetic particle simulation of beta-induced Alfven-acoustic eigenmode

    NASA Astrophysics Data System (ADS)

    Zhang, H. S.; Liu, Y. Q.; Lin, Z.; Zhang, W. L.

    2016-04-01

    The beta-induced Alfven-acoustic eigenmode (BAAE) in toroidal plasmas is verified and studied by global gyrokinetic particle simulations. When ion temperature is much lower than electron temperature, the existence of the weakly damped BAAE is verified in the simulations using initial perturbation, antenna excitation, and energetic particle excitation, respectively. When the ion temperature is comparable to the electron temperature, the unstable BAAE can be excited by realistic energetic particle density gradient, even though the stable BAAE (in the absence of energetic particles) is heavily damped by the thermal ions. In the simulations with reversed magnetic shear, BAAE frequency sweeping is observed and poloidal mode structure has a triangle shape with a poloidal direction similar to that observed in tokamak experiments. The triangle shape changes the poloidal direction, and no frequency sweeping is found in the simulations with normal magnetic shear.

  1. Comprehensive comparisons of geodesic acoustic mode characteristics and dynamics between Tore Supra experiments and gyrokinetic simulations

    SciTech Connect

    Storelli, A. Vermare, L.; Hennequin, P.; Gürcan, Ö. D.; Singh, Rameswar; Morel, P.; Dif-Pradalier, G.; Sarazin, Y.; Garbet, X.; Grandgirard, V.; Ghendrih, P.; Görler, T.

    2015-06-15

    In a dedicated collisionality scan in Tore Supra, the geodesic acoustic mode (GAM) is detected and identified with the Doppler backscattering technique. Observations are compared to the results of a simulation with the gyrokinetic code GYSELA. We found that the GAM frequency in experiments is lower than predicted by simulation and theory. Moreover, the disagreement is higher in the low collisionality scenario. Bursts of non harmonic GAM oscillations have been characterized with filtering techniques, such as the Hilbert-Huang transform. When comparing this dynamical behaviour between experiments and simulation, the probability density function of GAM amplitude and the burst autocorrelation time are found to be remarkably similar. In the simulation, where the radial profile of GAM frequency is continuous, we observed a phenomenon of radial phase mixing of the GAM oscillations, which could influence the burst autocorrelation time.

  2. Grid-based Parallel Data Streaming Implemented for the Gyrokinetic Toroidal Code

    SciTech Connect

    S. Klasky; S. Ethier; Z. Lin; K. Martins; D. McCune; R. Samtaney

    2003-09-15

    We have developed a threaded parallel data streaming approach using Globus to transfer multi-terabyte simulation data from a remote supercomputer to the scientist's home analysis/visualization cluster, as the simulation executes, with negligible overhead. Data transfer experiments show that this concurrent data transfer approach is more favorable compared with writing to local disk and then transferring this data to be post-processed. The present approach is conducive to using the grid to pipeline the simulation with post-processing and visualization. We have applied this method to the Gyrokinetic Toroidal Code (GTC), a 3-dimensional particle-in-cell code used to study microturbulence in magnetic confinement fusion from first principles plasma theory.

  3. The Hamiltonian structure and Euler-Poincare formulation of the Vlasov-Maxwell and gyrokinetic systems

    SciTech Connect

    Squire, J.; Tang, W. M.; Qin, H.; Chandre, C.

    2013-02-15

    We present a new variational principle for the gyrokinetic system, similar to the Maxwell-Vlasov action presented in H. Cendra et al., [J. Math. Phys. 39, 3138 (1998)]. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid velocity and particle distribution function. Using a Legendre transform, we explicitly derive the field theoretic Hamiltonian structure of the system. This is carried out with a modified Dirac theory of constraints, which is used to construct meaningful brackets from those obtained directly from Euler-Poincare theory. Possible applications of these formulations include continuum geometric integration techniques, large-eddy simulation models, and Casimir type stability methods.

  4. Gyrokinetic turbulence simulations of the pedestal region at various lithium doses in NSTX

    NASA Astrophysics Data System (ADS)

    Coury, Mireille; Guttenfelder, Walter; Mikkelsen, David R.; Canik, John M.; Diallo, Ahmed; Maingi, Rajesh

    2015-11-01

    It is shown that lithium-coated walls alter the pedestal structure by, for instance, improving the energy confinement and reducing recycling. Recent work shows improved discharge characteristics with increasing lithium doses in highly shaped discharges. Edge-localized modes triggered by large edge pressure and current gradients are altered, even suppressed with increasing lithium doses. In this work, the plasma edge characteristics under increasing lithium doses are investigated with GS2 gyrokinetic code. Using experimental discharges as input parameters, microinstabilities are investigated in the pedestal region and the effect of increasing lithium doses on these microinstabilities is discussed. This work is supported by U.S. Dept. of Energy under contract DE-AC02-09CH11466.

  5. Finite ballooning angle effects on ion temperature gradient driven mode in gyrokinetic flux tube simulations

    SciTech Connect

    Singh, Rameswar; Brunner, S.; Ganesh, R.; Jenko, F.

    2014-03-15

    This paper presents effects of finite ballooning angles on linear ion temperature gradient (ITG) driven mode and associated heat and momentum flux in Gyrokinetic flux tube simulation GENE. It is found that zero ballooning angle is not always the one at which the linear growth rate is maximum. The ITG mode acquires a short wavelength (SW) branch (k{sub ⊥}ρ{sub i} > 1) when growth rates maximized over all ballooning angles are considered. However, the SW branch disappears on reducing temperature gradient showing characteristics of zero ballooning angle SWITG in case of extremely high temperature gradient. Associated heat flux is even with respect to ballooning angle and maximizes at nonzero ballooning angle while the parallel momentum flux is odd with respect to the ballooning angle.

  6. Gyrokinetic Calculations of Microturbulence and Transport for NSTX and Alcator-CMOD H-modes

    SciTech Connect

    M.H. Redi; W. Dorland; R. Bell; P. Bonoli; C. Bourdelle; J. Candy; D. Ernst; C. Fiore; D. Gates; G. Hammett; K. Hill; S. Kaye; B. LeBlanc; J. Menard; D. Mikkelsen; G. Rewoldt; J. Rice; R. Waltz; S. Wukitch

    2003-07-08

    Recent H-mode experiments on NSTX [National Spherical Torus Experiment] and experiments on Alcator-CMOD, which also exhibit internal transport barriers (ITB), have been examined with gyrokinetic simulations with the GS2 and GYRO codes to identify the underlying key plasma parameters for control of plasma performance and, ultimately, the successful operation of future reactors such as ITER [International Thermonuclear Experimental Reactor]. On NSTX the H-mode is characterized by remarkably good ion confinement and electron temperature profiles highly resilient in time. On CMOD, an ITB with a very steep electron density profile develops following off-axis radio-frequency heating and establishment of H-mode. Both experiments exhibit ion thermal confinement at the neoclassical level. Electron confinement is also good in the CMOD core.

  7. Collision-dependent power law scalings in two dimensional gyrokinetic turbulence

    SciTech Connect

    Cerri, S. S. Bañón Navarro, A.; Told, D.; Jenko, F.

    2014-08-15

    Nonlinear gyrokinetics provides a suitable framework to describe short-wavelength turbulence in magnetized laboratory and astrophysical plasmas. In the electrostatic limit, this system is known to exhibit a free energy cascade towards small scales in (perpendicular) real and/or velocity space. The dissipation of free energy is always due to collisions (no matter how weak the collisionality), but may be spread out across a wide range of scales. Here, we focus on freely decaying two dimensional electrostatic turbulence on sub-ion-gyroradius scales. An existing scaling theory for the turbulent cascade in the weakly collisional limit is generalized to the moderately collisional regime. In this context, non-universal power law scalings due to multiscale dissipation are predicted, and this prediction is confirmed by means of direct numerical simulations.

  8. Gyrokinetic studies of trapped electron mode turbulence in the Helically Symmetric eXperiment stellarator

    SciTech Connect

    Faber, B. J.; Pueschel, M. J.; Terry, P. W.; Proll, J. H. E.; Hegna, C. C.; Weir, G. M.; Likin, K. M.; Talmadge, J. N.

    2015-07-15

    Gyrokinetic simulations of plasma microturbulence in the Helically Symmetric eXperiment are presented. Using plasma profiles relevant to experimental operation, four dominant drift wave regimes are observed in the ion wavenumber range, which are identified as different flavors of density-gradient-driven trapped electron modes. For the most part, the heat transport exhibits properties associated with turbulence driven by these types of modes. Additionally, long-wavelength, radially localized, nonlinearly excited coherent structures near the resonant central flux surface, not predicted by linear simulations, can further enhance flux levels. Integrated heat fluxes are compatible with experimental observations in the corresponding density gradient range. Despite low shearing rates, zonal flows are observed to regulate turbulence but can be overwhelmed at higher density gradients by the long-wavelength coherent structures.

  9. Simulations of 4D edge transport and dynamics using the TEMPEST gyro-kinetic code

    NASA Astrophysics Data System (ADS)

    Rognlien, T. D.; Cohen, B. I.; Cohen, R. H.; Dorr, M. R.; Hittinger, J. A. F.; Kerbel, G. D.; Nevins, W. M.; Xiong, Z.; Xu, X. Q.

    2006-10-01

    Simulation results are presented for tokamak edge plasmas with a focus on the 4D (2r,2v) option of the TEMPEST continuum gyro-kinetic code. A detailed description of a variety of kinetic simulations is reported, including neoclassical radial transport from Coulomb collisions, electric field generation, dynamic response to perturbations by geodesic acoustic modes, and parallel transport on open magnetic-field lines. Comparison is made between the characteristics of the plasma solutions on closed and open magnetic-field line regions separated by a magnetic separatrix, and simple physical models are used to qualitatively explain the differences observed in mean flow and electric-field generation. The status of extending the simulations to 5D turbulence will be summarized. The code structure used in this ongoing project is also briefly described, together with future plans.

  10. Multi-scale gyrokinetic simulation of Alcator C-Mod tokamak discharges

    NASA Astrophysics Data System (ADS)

    Howard, N. T.; White, A. E.; Greenwald, M.; Holland, C.; Candy, J.

    2014-03-01

    Alcator C-Mod tokamak discharges have been studied with nonlinear gyrokinetic simulation simultaneously spanning both ion and electron spatiotemporal scales. These multi-scale simulations utilized the gyrokinetic model implemented by GYRO code [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] and the approximation of reduced electron mass (μ = (mD/me).5 = 20.0) to qualitatively study a pair of Alcator C-Mod discharges: a low-power discharge, previously demonstrated (using realistic mass, ion-scale simulation) to display an under-prediction of the electron heat flux and a high-power discharge displaying agreement with both ion and electron heat flux channels [N. T. Howard et al., Nucl. Fusion 53, 123011 (2013)]. These multi-scale simulations demonstrate the importance of electron-scale turbulence in the core of conventional tokamak discharges and suggest it is a viable candidate for explaining the observed under-prediction of electron heat flux. In this paper, we investigate the coupling of turbulence at the ion (kθρs˜O(1.0)) and electron (kθρe˜O(1.0)) scales for experimental plasma conditions both exhibiting strong (high-power) and marginally stable (low-power) low-k (kθρs < 1.0) turbulence. It is found that reduced mass simulation of the plasma exhibiting marginally stable low-k turbulence fails to provide even qualitative insight into the turbulence present in the realistic plasma conditions. In contrast, multi-scale simulation of the plasma condition exhibiting strong turbulence provides valuable insight into the coupling of the ion and electron scales.

  11. Multi-scale gyrokinetic simulation of Alcator C-Mod tokamak discharges

    SciTech Connect

    Howard, N. T. White, A. E.; Greenwald, M.; Holland, C.; Candy, J.

    2014-03-15

    Alcator C-Mod tokamak discharges have been studied with nonlinear gyrokinetic simulation simultaneously spanning both ion and electron spatiotemporal scales. These multi-scale simulations utilized the gyrokinetic model implemented by GYRO code [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] and the approximation of reduced electron mass (μ = (m{sub D}/m{sub e}){sup .5} = 20.0) to qualitatively study a pair of Alcator C-Mod discharges: a low-power discharge, previously demonstrated (using realistic mass, ion-scale simulation) to display an under-prediction of the electron heat flux and a high-power discharge displaying agreement with both ion and electron heat flux channels [N. T. Howard et al., Nucl. Fusion 53, 123011 (2013)]. These multi-scale simulations demonstrate the importance of electron-scale turbulence in the core of conventional tokamak discharges and suggest it is a viable candidate for explaining the observed under-prediction of electron heat flux. In this paper, we investigate the coupling of turbulence at the ion (k{sub θ}ρ{sub s}∼O(1.0)) and electron (k{sub θ}ρ{sub e}∼O(1.0)) scales for experimental plasma conditions both exhibiting strong (high-power) and marginally stable (low-power) low-k (k{sub θ}ρ{sub s} < 1.0) turbulence. It is found that reduced mass simulation of the plasma exhibiting marginally stable low-k turbulence fails to provide even qualitative insight into the turbulence present in the realistic plasma conditions. In contrast, multi-scale simulation of the plasma condition exhibiting strong turbulence provides valuable insight into the coupling of the ion and electron scales.

  12. Direct Comparison of Gyrokinetic Turbulence Simulations with Phase Contrast Imaging Fluctuation Measurements

    NASA Astrophysics Data System (ADS)

    Ernst, Darin; Long, A.; Basse, N.; Lin, L.; Porkolab, M.; Dorland, W.

    2006-04-01

    We have developed a synthetic diagnostic^1 for the GS2 gyrokinetic code for direct comparisons with phase contrast imaging (PCI) measurements of density fluctuations in Alcator C-Mod. The gyrokinetic simulation is carried out in a local, field line following flux-tube, while PCI measures density fluctuations along 32 chords passing vertically through the plasma cross-section.^2 Transforming from Clebsch to cartesian coordinates, and integrating appropriately over portions of the flux tube viewed by the diagnostic, yields a density fluctuation spectrum versus wavenumber kR in the major radius direction. To achieve vertical localization, we examine an ITB case in which the spectrum is dominated by a strong trapped electron mode, localized near the half-radius. The wavelength spectrum from the simulations, using the synthetic diagnostic, closely reproduces the PCI spectrum. Contributions from kψ, where B=∇αx∇ψ, downshift the GS2 kα spectrum to improve upon our previous raw comparison with the PCI kR spectrum.^3 ^1A. Long, D. R. Ernst et al., Bull. Am. Phys. Soc. 50(8) p. 153, GP1.48, also p. 235, LP1.37 http://www.psfc.mit.edu/research/alcator/pubs/APS/APS2005/ernst.pdf. ^2N. P. Basse et al., Phys. Plasmas 12, 052512 (2005). ^3D. R. Ernst et al., 2004 IAEA Fusion Energy Conference, IAEA-CN116/TH/4-1 http://www-naweb.iaea.org/napc/physics/fec/fec2004/datasets/TH4-1.html, see also Phys. Plasmas 11 (2004) 2637.

  13. Understanding rotation profile structures in ECH-heated plasmas using nonlinear gyrokinetic simulations

    NASA Astrophysics Data System (ADS)

    Wang, Weixing; Brian, B.; Ethier, S.; Chen, J.; Startsev, E.; Diamond, P. H.; Lu, Z.

    2015-11-01

    A non-diffusive momentum flux connecting edge momentum sources/sinks and core plasma flow is required to establish the off-axis peaked ion rotation profile typically observed in ECH-heated DIII-D plasmas without explicit external momentum input. The understanding of the formation of such profile structures provides an outstanding opportunity to test the physics of turbulence driving intrinsic rotation, and validate first-principles-based gyrokinetic simulation models. Nonlinear, global gyrokinetic simulations of DIII-D ECH plasmas indicate a substantial ITG fluctuation-induced residual stress generated around the region of peaked toroidal rotation, along with a diffusive momentum flux. The residual stress profile shows an anti-gradient, dipole structure, which is critical for accounting for the formation of the peaked rotation profile. It is showed that both turbulence intensity gradient and zonal flow ExB shear contribute to the generation of k// asymmetry needed for residual stress generation. By balancing the simulated residual stress and the momentum diffusion, a rotation profile is calculated. In general, the radial structure of core rotation profile is largely determined by the residual stress profile, while the amplitude of core rotation depends on the edge toroidal rotation velocity, which is determined by edge physics and used as a boundary condition in our model. The calculated core rotation profile is consistent with the experimental measurements. Also discussed is the modification of turbulence-generated Reynolds stress on poloidal rotation in those plasmas. Work supported by U.S. DOE Contract DE-AC02-09-CH11466.

  14. Gyrokinetic neoclassical study of the bootstrap current in the tokamak edge pedestal with fully non-linear Coulomb collisions

    NASA Astrophysics Data System (ADS)

    Hager, Robert; Chang, C. S.

    2016-04-01

    As a follow-up on the drift-kinetic study of the non-local bootstrap current in the steep edge pedestal of tokamak plasma by Koh et al. [Phys. Plasmas 19, 072505 (2012)], a gyrokinetic neoclassical study is performed with gyrokinetic ions and drift-kinetic electrons. Besides the gyrokinetic improvement of ion physics from the drift-kinetic treatment, a fully non-linear Fokker-Planck collision operator—that conserves mass, momentum, and energy—is used instead of Koh et al.'s linearized collision operator in consideration of the possibility that the ion distribution function is non-Maxwellian in the steep pedestal. An inaccuracy in Koh et al.'s result is found in the steep edge pedestal that originated from a small error in the collisional momentum conservation. The present study concludes that (1) the bootstrap current in the steep edge pedestal is generally smaller than what has been predicted from the small banana-width (local) approximation [e.g., Sauter et al., Phys. Plasmas 6, 2834 (1999) and Belli et al., Plasma Phys. Controlled Fusion 50, 095010 (2008)], (2) the plasma flow evaluated from the local approximation can significantly deviate from the non-local results, and (3) the bootstrap current in the edge pedestal, where the passing particle region is small, can be dominantly carried by the trapped particles in a broad trapped boundary layer. A new analytic formula based on numerous gyrokinetic simulations using various magnetic equilibria and plasma profiles with self-consistent Grad-Shafranov solutions is constructed.

  15. Gyrokinetic neoclassical study of the bootstrap current in the tokamak edge pedestal with fully non-linear Coulomb collisions

    DOE PAGESBeta

    Hager, Robert; Chang, C. S.

    2016-04-08

    As a follow-up on the drift-kinetic study of the non-local bootstrap current in the steep edge pedestal of tokamak plasma by Koh et al. [Phys. Plasmas 19, 072505 (2012)], a gyrokinetic neoclassical study is performed with gyrokinetic ions and drift-kinetic electrons. Besides the gyrokinetic improvement of ion physics from the drift-kinetic treatment, a fully non-linear Fokker-Planck collision operator—that conserves mass, momentum, and energy—is used instead of Koh et al.'s linearized collision operator in consideration of the possibility that the ion distribution function is non-Maxwellian in the steep pedestal. An inaccuracy in Koh et al.'s result is found in the steepmore » edge pedestal that originated from a small error in the collisional momentum conservation. The present study concludes that (1) the bootstrap current in the steep edge pedestal is generally smaller than what has been predicted from the small banana-width (local) approximation [e.g., Sauter et al., Phys. Plasmas 6, 2834 (1999) and Belli et al., Plasma Phys. Controlled Fusion 50, 095010 (2008)], (2) the plasma flow evaluated from the local approximation can significantly deviate from the non-local results, and (3) the bootstrap current in the edge pedestal, where the passing particle region is small, can be dominantly carried by the trapped particles in a broad trapped boundary layer. In conclusion, a new analytic formula based on numerous gyrokinetic simulations using various magnetic equilibria and plasma profiles with self-consistent Grad-Shafranov solutions is constructed.« less

  16. Gyrokinetic particle-in-cell simulations of Alfvén eigenmodes in presence of continuum effects

    SciTech Connect

    Mishchenko, Alexey Könies, Axel; Hatzky, Roman

    2014-05-15

    First-principle gyrokinetic particle-in-cell simulations of a global Toroidal Alfvén Eigenmode (TAE) are undertaken in the presence of a strong coupling with the continuum. Effects of the bulk plasma temperature on the interplay between the TAE and Kinetic Alfvén Waves (KAWs) are investigated. A global TAE-KAW structure is identified which appears to be more unstable with respect to the fast ions than a simple (fluid-like) TAE mode.

  17. Measurements of core electron temperature and density fluctuations in DIII-D and comparison to nonlinear gyrokinetic simulations

    SciTech Connect

    White, A. E.; Schmitz, L.; Peebles, W. A.; Carter, T. A.; Doyle, E. J.; Rhodes, T. L.; Wang, G.; McKee, G. R.; Shafer, M. W.; Holland, C.; Tynan, G. R.; Austin, M. E.; Burrell, K. H.; Candy, J.; DeBoo, J. C.; Prater, R.; Staebler, G. M.; Waltz, R. E.; Makowski, M. A.

    2008-05-15

    For the first time, profiles (0.3<{rho}<0.9) of electron temperature and density fluctuations in a tokamak have been measured simultaneously and the results compared to nonlinear gyrokinetic simulations. Electron temperature and density fluctuations measured in neutral beam-heated, sawtooth-free low confinement mode (L-mode) plasmas in DIII-D [J. L. Luxon, Nucl. Fusion 42, 614 (2002)] are found to be similar in frequency and normalized amplitude, with amplitude increasing with radius. The measured radial profile of two fluctuation fields allows for a new and rigorous comparison with gyrokinetic results. Nonlinear gyrokinetic flux-tube simulations predict that electron temperature and density fluctuations have similar normalized amplitudes in L-mode. At {rho}=0.5, simulation results match experimental heat diffusivities and density fluctuation amplitude, but overestimate electron temperature fluctuation amplitude and particle diffusivity. In contrast, simulations at {rho}=0.75 do not match either the experimentally derived transport properties or the measured fluctuation levels.

  18. Linear gyrokinetic calculations of toroidal momentum transport in the presence of trapped electron modes in tokamak plasmas

    SciTech Connect

    Kluy, N.; Angioni, C.; Camenen, Y.; Peeters, A. G.

    2009-12-15

    The toroidal momentum transport in the presence of trapped electron mode microinstabilities in tokamak plasmas is studied by means of quasilinear gyrokinetic calculations. In particular, the role of the Coriolis drift in producing an inward convection of toroidal momentum is investigated. The Coriolis drift term has been implemented in the gyrokinetic code GS2 [W. Dorland et al., Phys. Rev. Lett. 85, 5579 (2000)] specifically for the completion of this work. A benchmark between the GS2 implementation of the Coriolis drift and the implementations included in two other gyrokinetic codes is presented. The numerical calculations show that in the presence of trapped electron modes, despite of a weaker symmetry breaking of the eigenfunctions with respect to the case of ion temperature gradient modes, a pinch of toroidal momentum is produced in most conditions. The toroidal momentum viscosity is also computed, and found to be small as compared with the electron heat conductivity, but significantly larger than the ion heat conductivity. In addition, interesting differences are found in the dependence of the toroidal momentum pinch as a function of collisionality between trapped electron modes and ion temperature gradient modes. The results identify also parameter domains in which the pinch is predicted to be small, which are also of interest for comparisons with the experiments.

  19. Comparison of electron temperature fluctuations with gyrokinetic sumulations across the ohmic energy confinement transition in Alcator C-Mod

    NASA Astrophysics Data System (ADS)

    Sung, C.; White, A.; Howard, N.; Mikkelsen, D.; Rice, J.; Reinke, M.; Gao, C.; Ennever, P.; Porkolab, M.; Churchill, R.; Theiler, C.; Hubbard, A.; Greenwald, M.

    2013-10-01

    Long wavelength electron temperature fluctuations (kyρs < 0 . 3) near the edge (r / a ~ 0 . 85) are reduced across the ohmic confinement transition from Linear Ohmic Confinement(LOC) regime to Saturated Ohmic Confinement(SOC) regime in Alcator C-Mod. Linear stability analysis shows that the dominant mode of long wavelength turbulence near the edge is changed from Trapped Electron Mode(TEM) to Ion Temperature Gradient(ITG) mode while the dominant mode is not changed deeper in the core (r / a ~ 0 . 5). This indicates that local turbulence changes near the edge might be responsible for the change of global energy confinement in ohmic plasmas. Further study using nonlinear gyrokinetic simulations is being performed to clarify the relation between the change of local turbulence and global ohmic energy confinement. Through nonlinear gyrokinetic simulation (GYRO), we will investigate the change of fluctuating quantities (T~ , ñ , ϕ~) and their phase relations across ohmic confinement transitions, and relate them to the change of energy transport. A synthetic CECE diagnostic for C-Mod has been developed, and it will be used to validate the gyrokinetic simulations. Research supported by USDoE awards DE-SC0006419, DE-FC02-99ER54512.

  20. Gyrokinetic simulations with external resonant magnetic perturbations: Island torque and nonambipolar transport with plasma rotation

    NASA Astrophysics Data System (ADS)

    Waltz, R. E.; Waelbroeck, F. L.

    2012-03-01

    Static external resonant magnetic field perturbations (RMPs) have been added to the gyrokinetic code GYRO [J. Candy and R. E. Waltz, J. Comp. Phys. 186, 545 (2003)]. This allows nonlinear gyrokinetic simulations of the nonambipolar radial current flow jr, and the corresponding j→×B→ plasma torque (density) R[jrBp/c], induced by magnetic islands that break the toroidal symmetry of a tokamak. This extends the previous GYRO formulation for the transport of toroidal angular momentum (TAM) [R. E. Waltz, G. M. Staebler, J. Candy, and F. L. Hinton, Phys. Plasmas 14, 122507 (2007); errata 16, 079902 (2009)]. The focus is on electrostatic full torus radial slice simulations of externally induced q =m/n=6/3 islands with widths 5% of the minor radius or about 20 ion gyroradii. Up to moderately strong E ×B rotation, the island torque scales with the radial electric field at the resonant surface Er, the island width w, and the intensity I of the high-n micro-turbulence, as Erw√I . The radial current inside the island is carried (entirely in the n =3 component) and almost entirely by the ion E ×B flux, since the electron E ×B and magnetic flutter particle fluxes are cancelled. The net island torque is null at zero Er rather than at zero toroidal rotation. This means that while the expected magnetic braking of the toroidal plasma rotation occurs at strong co- and counter-current rotation, at null toroidal rotation, there is a small co-directed magnetic acceleration up to the small diamagnetic (ion pressure gradient driven) co-rotation corresponding to the zero Er and null torque. This could be called the residual stress from an externally induced island. At zero Er, the only effect is the expected partial flattening of the electron temperature gradient within the island. Finite-beta GYRO simulations demonstrate almost complete RMP field screening and n =3 mode unlocking at strong Er.

  1. Search for the Missing L-mode Edge Transport and Possible Breakdown of Gyrokinetics

    NASA Astrophysics Data System (ADS)

    Waltz, R. E.

    2012-10-01

    While GYRO simulations of typical core (0 < r/a < 0.7) DIII-D L-modes seems to be in good agreement with experiment, simulated low-k (kθρs< 1) transport and turbulence intensity is more than 5-fold lower than experimentally inferred levels in the near edge L-mode (r/a=0.7-0.95) DIII-D shot 128913 [1]. Global edge slice GYRO simulations of this and the well-studied discharge 101391 [2] are presented here to document the shortfall. TGLF transport code simulations over a large L-mode database indicate this short fall is not atypical so that L-mode edges transit to H-like pedestal profiles contrary to experiment. High edge e-i collisionality stabilizes the TEM modes so that diffusivities (χ) decrease like T^7/2/n to the cold edge. The very high magnetic shear and density gradients stabilize the ITG despite the very high temperature gradient drive and high q. High-k ETG can make-up for the shortfall in the electron but increases ion transport very little. Near L-edge transport is highly local. Focusing on local simulations at r/a=0.9, the ion channel short fall can exceed 10-fold. An artificial 10-fold increase in collisionality is needed to reach the expected resistive g-mode scaling with χ increasing like nT-1/2. Identical GYRO drift kinetic ion simulations (suppressing the gyroaverage) are close to experiment levels suggesting a possible breakdown of low-frequency gyrokinetics. Formulation of a nonlinear theory of 6D drift-cyclotron kinetics following the fast time scale of the gyrophase to test the breakdown of 5D gyrokinetics with reduced model simulations is presented. 6pt [1] C. Holland, A.E. White, et al., Phys. Plasmas 16, 052301 (2009). [2] R.E. Waltz, J. Candy, C.C. Petty, Phys. Plasmas 13, 072304 (2006).

  2. Center for Gyrokinetic Particle Simulation of Turbulent Transport in Burning Plasma

    SciTech Connect

    Viktor K. Decyk

    2008-04-24

    The UCLA work on this grant was to design and help implement an object-oriented version of the GTC code, which is written in Fortran90. The GTC code is the main global gyrokinetic code used in this project, and over the years multiple, incompatible versions have evolved. The reason for this effort is to allow multiple authors to work together on GTC and to simplify future enhancements to GTC. The effort was designed to proceed incrementally. Initially, an upper layer of classes (derived types and methods) was implemented which called the original GTC code 'under the hood.' The derived types pointed to data in the original GTC code, and the methods called the original GTC subroutines. The original GTC code was modified only very slightly. This allowed one to define (and refine) a set of classes which described the important features of the GTC code in a new, more abstract way, with a minimum of implementation. Furthermore, classes could be added one at a time, and at the end of the each day, the code continued to work correctly. This work was done in close collaboration with Y. Nishimura from UC Irvine and Stefan Ethier from PPPL. Ten classes were ultimately defined and implemented: gyrokinetic and drift kinetic particles, scalar and vector fields, a mesh, jacobian, FLR, equilibrium, interpolation, and particles species descriptors. In the second state of this development, some of the scaffolding was removed. The constructors in the class objects now allocated the data and the array data in the original GTC code was removed. This isolated the components and now allowed multiple instantiations of the objects to be created, in particular, multiple ion species. Again, the work was done incrementally, one class at a time, so that the code was always working properly. This work was done in close collaboration with Y. Nishimura and W. Zhang from UC Irvine and Stefan Ethier from PPPL. The third stage of this work was to integrate the capabilities of the various versions of

  3. The non-linear evolution of the tearing mode in electromagnetic turbulence using gyrokinetic simulations

    NASA Astrophysics Data System (ADS)

    Hornsby, W. A.; Migliano, P.; Buchholz, R.; Grosshauser, S.; Weikl, A.; Zarzoso, D.; Casson, F. J.; Poli, E.; Peeters, A. G.

    2016-01-01

    The non-linear evolution of a magnetic island is studied using the Vlasov gyro-kinetic code GKW. The interaction of electromagnetic turbulence with a self-consistently growing magnetic island, generated by a tearing unstable {{Δ }\\prime}>0 current profile, is considered. The turbulence is able to seed the magnetic island and bypass the linear growth phase by generating structures that are approximately an ion gyro-radius in width. The non-linear evolution of the island width and its rotation frequency, after this seeding phase, is found to be modified and is dependent on the value of the plasma beta and equilibrium pressure gradients. At low values of beta the island evolves largely independent of the turbulence, while at higher values the interaction has a dramatic effect on island growth, causing the island to grow exponentially at the growth rate of its linear phase, even though the island is larger than linear theory validity. The turbulence forces the island to rotate in the ion-diamagnetic direction as opposed to the electron diamagnetic direction in which it rotates when no turbulence is present. In addition, it is found that the mode rotation slows as the island grows in size.

  4. Validation of gyrokinetic modelling of light impurity transport including rotation in ASDEX Upgrade

    NASA Astrophysics Data System (ADS)

    Casson, F. J.; McDermott, R. M.; Angioni, C.; Camenen, Y.; Dux, R.; Fable, E.; Fischer, R.; Geiger, B.; Manas, P.; Menchero, L.; Tardini, G.; the ASDEX Upgrade Team

    2013-06-01

    Upgraded spectroscopic hardware and an improved impurity concentration calculation allow accurate determination of boron density in the ASDEX Upgrade tokamak. A database of boron measurements is compared to quasilinear and nonlinear gyrokinetic simulations including Coriolis and centrifugal rotational effects over a range of H-mode plasma regimes. The peaking of the measured boron profiles shows a strong anti-correlation with the plasma rotation gradient, via a relationship explained and reproduced by the theory. It is demonstrated that the rotodiffusive impurity flux driven by the rotation gradient is required for the modelling to reproduce the hollow boron profiles at higher rotation gradients. The nonlinear simulations validate the quasilinear approach, and, with the addition of perpendicular flow shear, demonstrate that each symmetry breaking mechanism that causes momentum transport also couples to rotodiffusion. At lower rotation gradients, the parallel compressive convection is required to match the most peaked boron profiles. The sensitivities of both datasets to possible errors is investigated, and quantitative agreement is found within the estimated uncertainties. The approach used can be considered a template for mitigating uncertainty in quantitative comparisons between simulation and experiment.

  5. Three-dimensional gyrokinetic simulation of the relaxation of a magnetized temperature filament

    SciTech Connect

    Sydora, R. D.; Morales, G. J.; Maggs, J. E.; Van Compernolle, B.

    2015-10-15

    An electromagnetic, 3D gyrokinetic particle code is used to study the relaxation of a magnetized electron temperature filament embedded in a large, uniform plasma of lower temperature. The study provides insight into the role played by unstable drift-Alfvén waves observed in a basic electron heat transport experiment [D. C. Pace et al., Phys. Plasmas 15, 122304 (2008)] in which anomalous cross-field transport has been documented. The simulation exhibits the early growth of temperature-gradient-driven, drift-Alfvén fluctuations that closely match the eigenmodes predicted by linear theory. At the onset of saturation, the unstable fluctuations display a spiral spatial pattern, similar to that observed in the laboratory, which causes the rearrangement of the temperature profile. After saturation of the linear instability, the system exhibits a markedly different behavior depending on the inclusion in the computation of modes without variation along the magnetic field, i.e., k{sub z} = 0. In their absence, the initial filament evolves into a broadened temperature profile, self-consistent with undamped, finite amplitude drift-Alfvén waves. But the inclusion of k{sub z} = 0 modes causes the destruction of the filament and damping of the drift-Alfvén modes leading to a final state consisting of undamped convective cells and multiple, smaller-scale filaments.

  6. Investigation of the effect of flow shear and the ITG on gyrokinetic MAST turbulence

    NASA Astrophysics Data System (ADS)

    van Wyk, Louis; Highcock, Edmund; Field, Anthony; Schekochihin, Alexander; Roach, Colin

    2015-11-01

    We study the effect of flow shear γE and ion temperature gradient a /LTi on L-mode turbulence in MAST using gyrokinetic simulations. These parameters play a crucial role in regulating and driving turbulence and together with the ratio of the safety factor to the inverse aspect ratio, q / ɛ , define a ``zero-turbulence manifold'' (ZTM) that represents the critical values needed to sustain turbulence. Nonlinear simulations show that by varying γE and a /LTi within experimental errors the turbulence crosses the ZTM, implying that the experiment operates close to marginality. In this parameter regime flow shear is very effective at regulating the turbulence, which is found to be subcritical. Finally the structure of the turbulence was studied: statistical parameters such as radial, perpendicular and parallel correlation lengths and the correlation time were calculated and found to be in reasonable agreement with experimental results obtained using Beam Emission Spectroscopy. Work supported by STFC and CCFE. Computing time provided by IFERC grant MULTEIM, The Hartree Centre, and EPSRC grants EP/ H002081/1 and EP/L000237/1.

  7. Bounce-Averaged Gyrokinetic Simulation of Current-Collection Feedback in a Laboratory Magnetosphere

    NASA Astrophysics Data System (ADS)

    Roberts, T. M.; Garnier, D.; Kesner, J.; Mauel, M. E.

    2014-10-01

    A self-consistent, nonlinear simulation of interchange dynamics including the bounce-averaged gyro-kinetics of trapped electrons was previously used to understand frequency sweeping and the turbulent cascades observed in dipole-confined plasmas. Through adjustment of the particle and heat sources this code reproduces dynamics that resemble the turbulence measured experimentally, both in spectral power-law trends and in the onset of a steepened density profile. Time stepping is performed in an explicit leap-frog manner and a flux-corrected transport algorithm is implemented. In this presentation, we discuss the physics and numerical methods of the simulations as well as plans for including the effects of a biasing electrode which can collect or inject electrons. By varying this source/sink of electrons at the electrode location based on the potential fluctuations occurring elsewhere, we study the effects of current-collection feedback to compare to recent experiments observed to regulate interchange turbulence. Supported by NSF-DOE Partnership for Plasma Science and DOE Grant DE-FG02-00ER54585 and NSF Award PHY-1201896.

  8. Comparative study of gyrokinetic, hybrid-kinetic and fully kinetic wave physics for space plasmas

    NASA Astrophysics Data System (ADS)

    Told, D.; Cookmeyer, J.; Muller, F.; Astfalk, P.; Jenko, F.

    2016-06-01

    A set of numerical solvers for the linear dispersion relations of the gyrokinetic (GK), the hybrid-kinetic (HK), and the fully kinetic (FK) model is employed to study the physics of the KAW and the fast magnetosonic mode in these models. In particular, we focus on parameters that are relevant for solar wind oriented applications (using a homogeneous, isotropic background), which are characterized by wave propagation angles averaging close to 90°. It is found that the GK model, while lacking high-frequency solutions and cyclotron effects, faithfully reproduces the FK {{Alfv\\acute{e}n}} wave physics close to, and sometimes significantly beyond, the boundaries of its range of validity. The HK model, on the other hand, is much more complete in terms of high-frequency waves, but owing to its simple electron model it is found to severely underpredict wave damping rates even on ion spatial scales across a large range of parameters, despite containing full kinetic ion physics.

  9. Multi-scale gyrokinetic simulations: Comparison with experiment and implications for predicting turbulence and transport

    NASA Astrophysics Data System (ADS)

    Howard, N. T.; Holland, C.; White, A. E.; Greenwald, M.; Candy, J.; Creely, A. J.

    2016-05-01

    To better understand the role of cross-scale coupling in experimental conditions, a series of multi-scale gyrokinetic simulations were performed on Alcator C-Mod, L-mode plasmas. These simulations, performed using all experimental inputs and realistic ion to electron mass ratio ((mi/me)1/2 = 60.0), simultaneously capture turbulence at the ion ( kθρs˜O (1.0 ) ) and electron-scales ( kθρe˜O (1.0 ) ). Direct comparison with experimental heat fluxes and electron profile stiffness indicates that Electron Temperature Gradient (ETG) streamers and strong cross-scale turbulence coupling likely exist in both of the experimental conditions studied. The coupling between ion and electron-scales exists in the form of energy cascades, modification of zonal flow dynamics, and the effective shearing of ETG turbulence by long wavelength, Ion Temperature Gradient (ITG) turbulence. The tightly coupled nature of ITG and ETG turbulence in these realistic plasma conditions is shown to have significant implications for the interpretation of experimental transport and fluctuations. Initial attempts are made to develop a "rule of thumb" based on linear physics, to help predict when cross-scale coupling plays an important role and to inform future modeling of experimental discharges. The details of the simulations, comparisons with experimental measurements, and implications for both modeling and experimental interpretation are discussed.

  10. The anisotropic redistribution of free energy for gyrokinetic plasma turbulence in a Z-pinch

    NASA Astrophysics Data System (ADS)

    Navarro, Alejandro Bañón; Teaca, Bogdan; Jenko, Frank

    2016-04-01

    For a Z-pinch geometry, we report on the nonlinear redistribution of free energy across scales perpendicular to the magnetic guide field, for a turbulent plasma described in the framework of gyrokinetics. The analysis is performed using a local flux-surface approximation, in a regime dominated by electrostatic fluctuations driven by the entropy mode, with both ion and electron species being treated kinetically. To explore the anisotropic nature of the free energy redistribution caused by the emergence of zonal flows, we use a polar coordinate representation for the field-perpendicular directions and define an angular density for the scale flux. Positive values for the classically defined (angle integrated) scale flux, which denote a direct energy cascade, are shown to be also composed of negative angular sections, a fact that impacts our understanding of the backscatter of energy and the way in which it enters the modeling of sub-grid scales for turbulence. A definition for the flux of free energy across each perpendicular direction is introduced as well, which shows that the redistribution of energy in the presence of zonal flows is highly anisotropic.

  11. A study of self organized criticality in ion temperature gradient mode driven gyrokinetic turbulence

    SciTech Connect

    Mavridis, M.; Isliker, H.; Vlahos, L.; Görler, T.; Jenko, F.; Told, D.

    2014-10-15

    An investigation on the characteristics of self organized criticality (Soc) in ITG mode driven turbulence is made, with the use of various statistical tools (histograms, power spectra, Hurst exponents estimated with the rescaled range analysis, and the structure function method). For this purpose, local non-linear gyrokinetic simulations of the cyclone base case scenario are performed with the GENE software package. Although most authors concentrate on global simulations, which seem to be a better choice for such an investigation, we use local simulations in an attempt to study the locally underlying mechanisms of Soc. We also study the structural properties of radially extended structures, with several tools (fractal dimension estimate, cluster analysis, and two dimensional autocorrelation function), in order to explore whether they can be characterized as avalanches. We find that, for large enough driving temperature gradients, the local simulations exhibit most of the features of Soc, with the exception of the probability distribution of observables, which show a tail, yet they are not of power-law form. The radial structures have the same radial extent at all temperature gradients examined; radial motion (transport) though appears only at large temperature gradients, in which case the radial structures can be interpreted as avalanches.

  12. Verification and validation of linear gyrokinetic simulation of Alfven eigenmodes in the DIII-D tokamak

    SciTech Connect

    Spong, D. A.; Bass, E. M.; Deng, W.; Heidbrink, W. W.; Lin, Z.; Tobias, B.; Van Zeeland, M. A.; Austin, M. E.; Domier, C. W.; Luhmann, N. C. Jr.

    2012-08-15

    A verification and validation study is carried out for a sequence of reversed shear Alfven instability time slices. The mode frequency increases in time as the minimum (q{sub min}) in the safety factor profile decreases. Profiles and equilibria are based upon reconstructions of DIII-D discharge (no. 142111) in which many such frequency up-sweeping modes were observed. Calculations of the frequency and mode structure evolution from two gyrokinetic codes, GTC and GYRO, and a gyro-Landau fluid code TAEFL are compared. The experimental mode structure of the instability was measured using time-resolved two-dimensional electron cyclotron emission imaging. The three models reproduce the frequency upsweep event within {+-}10% of each other, and the average of the code predictions is within {+-}8% of the measurements; growth rates are predicted that are consistent with the observed spectral line widths. The mode structures qualitatively agree with respect to radial location and width, dominant poloidal mode number, ballooning structure, and the up-down asymmetry, with some remaining differences in the details. Such similarities and differences between the predictions of the different models and the experimental results are a valuable part of the verification/validation process and help to guide future development of the modeling efforts.

  13. Global gyrokinetic simulations of the H-mode tokamak edge pedestal

    SciTech Connect

    Wan, Weigang; Parker, Scott E.; Chen, Yang; Groebner, Richard J.; Yan, Zheng; Pankin, Alexei Y.; Kruger, Scott E.

    2013-05-15

    Global gyrokinetic simulations of DIII-D H-mode edge pedestal show two types of instabilities may exist approaching the onset of edge localized modes: an intermediate-n, high frequency mode which we identify as the “kinetic peeling ballooning mode (KPBM),” and a high-n, low frequency mode. Our previous study [W. Wan et al., Phys. Rev. Lett. 109, 185004 (2012)] has shown that when the safety factor profile is flattened around the steep pressure gradient region, the high-n mode is clearly kinetic ballooning mode and becomes the dominant instability. Otherwise, the KPBM dominates. Here, the properties of the two instabilities are studied by varying the density and temperature profiles. It is found that the KPBM is destabilized by density and ion temperature gradient, and the high-n mode is mostly destabilized by electron temperature gradient. Nonlinear simulations with the KPBM saturate at high levels. The equilibrium radial electric field (E{sub r}) reduces the transport. The effect of the parallel equilibrium current is found to be weak.

  14. Gyrokinetic study of ASDEX Upgrade inter-ELM pedestal profile evolution

    NASA Astrophysics Data System (ADS)

    Hatch, D. R.; Told, D.; Jenko, F.; Doerk, H.; Dunne, M. G.; Wolfrum, E.; Viezzer, E.; The ASDEX Upgrade Team; Pueschel, M. J.

    2015-06-01

    The gyrokinetic GENE code is used to study the inter-ELM H-mode pedestal profile evolution for an ASDEX Upgrade discharge. Density gradient driven trapped electron modes are the dominant pedestal instability during the early density-buildup phase. Nonlinear simulations produce particle transport levels consistent with experimental expectations. Later inter-ELM phases appear to be simultaneously constrained by electron temperature gradient (ETG) and kinetic ballooning mode (KBM) turbulence. The electron temperature gradient achieves a critical value early in the ELM cycle, concurrent with the appearance of both microtearing modes and ETG modes. Nonlinear ETG simulations demonstrate that the profiles lie at a nonlinear critical gradient. The nominal profiles are stable to KBM, but moderate increases in β are sufficient to surpass the KBM threshold. Certain aspects of the dynamics support the premise of KBM-constrained pedestal evolution; the density and temperature profiles separately undergo large changes, but in a manner that keeps the pressure profile constant and near the KBM limit.

  15. Gyrokinetic study of edge blobs and divertor heat-load footprint

    NASA Astrophysics Data System (ADS)

    Chang, C.-S.; Ku, S.-H.; Churchill, M.; Zweben, S.

    2014-10-01

    In an attempt to better understand the complicated physics of the inter-related ``intermittent plasma objects (blobs)'' and divertor heat-load footprint, the full-function gyrokinetic PIC code XGC1 has been used in realistic diverted geometry. Neoclassical and turbulence physics are simulated together self-consistently in the presence of Monte Carlo neutral particles. Blobs are modeled here as electrostatic nonlinear turbulence phenomenon. It is found that the ``blobs'' are generated, together with the ``holes,'' around the steep density gradient region. XGC1 reasserts the previous findings that blobs move out convectively into the scrape-off layer, while the holes move inward toward plasma core. The measured radial width of the divertor heat load, mapped to the outer midplane, is found to be much less than the median radial size of the intermittent plasma objects, but is rather closer to the width of neoclassical orbit excursion from pedestal to divertor, yielding approximately the 1/Ip-type scaling found from our previous pure neoclassical simulation or a heuristic neoclassical argument by Goldston. However, it also shows some spreading by the intermittent turbulence. In ITER plasma edge, where the ion banana width at separatrix becomes negligibly small compared to the meso-scale blob size, blobs may saturate the 1/Ip scaling.

  16. Validation study of gyrokinetic simulation (GYRO) near the edge in Alcator C-Mod ohmic discharges

    NASA Astrophysics Data System (ADS)

    Sung, C.; White, A.; Howard, N.; Mikkelsen, D.; Holland, C.; Rice, J.; Reinke, M.; Gao, C.; Ennever, P.; Porkolab, M.; Churchill, R.; Theiler, C.; Walk, J.; Hughes, J.; Hubbard, A.; Greenwald, M.

    2014-10-01

    A validation study of local gyrokinetic simulations (GYRO) near the edge region (r / a ~ 0 . 85) has been performed for two C-Mod ohmic discharges, namely one that is in the Linear Ohmic Confinement (LOC) regime and the other one in the Saturated Ohmic Confinement (SOC) regime. Comparing the simulated heat fluxes and synthetic Te fluctuations with the experiments, it is found that GYRO can reproduce the ion heat flux and the Te fluctuation level measured by the Correlation ECE (CECE) diagnostic within their uncertainties, while the simulated electron heat flux is under-predicted. Furthermore, the synthetic Te spectral shape is not matched with the measured spectrum in both LOC/SOC discharges. We have also performed global simulations to consider the interaction of turbulence within the sampling volume of the CECE diagnostic, enabling us to evaluate the importance of global simulations in applying a synthetic CECE diagnostic in this study. The LOC/SOC transition physics will be also explored. Research supported by USDoE Awards DE-SC0006419, DE-FC02-99ER54512.

  17. Gyrokinetic Calculations of Microinstabilities and Transport During RF H-Modes on Alcator C-Mod

    SciTech Connect

    M.H. Redi; C. Fiore; P. Bonoli; C. Bourdelle; R. Budny; W.D. Dorland; D. Ernst; G. Hammett; D. Mikkelsen; J. Rice; S. Wukitch

    2002-06-18

    Physics understanding for the experimental improvement of particle and energy confinement is being advanced through massively parallel calculations of microturbulence for simulated plasma conditions. The ultimate goal, an experimentally validated, global, non-local, fully nonlinear calculation of plasma microturbulence is still not within reach, but extraordinary progress has been achieved in understanding microturbulence, driving forces and the plasma response in recent years. In this paper we discuss gyrokinetic simulations of plasma turbulence being carried out to examine a reproducible, H-mode, RF heated experiment on the Alcator CMOD tokamak3, which exhibits an internal transport barrier (ITB). This off axis RF case represents the early phase of a very interesting dual frequency RF experiment, which shows density control with central RF heating later in the discharge. The ITB exhibits steep, spontaneous density peaking: a reduction in particle transport occurring without a central particle source. Since the central temperature is maintained while the central density is increasing, this also suggests a thermal transport barrier exists. TRANSP analysis shows that ceff drops inside the ITB. Sawtooth heat pulse analysis also shows a localized thermal transport barrier. For this ICRF EDA H-mode, the minority resonance is at r/a * 0.5 on the high field side. There is a normal shear profile, with q monotonic.

  18. Gyrokinetic Particle Simulation of Compressible Electromagnetic Turbulence in High-β Plasmas

    SciTech Connect

    Lin, Zhihong

    2014-03-13

    Supported by this award, the PI and his research group at the University of California, Irvine (UCI) have carried out computational and theoretical studies of instability, turbulence, and transport in laboratory and space plasmas. Several massively parallel, gyrokinetic particle simulation codes have been developed to study electromagnetic turbulence in space and laboratory plasmas. In space plasma projects, the simulation codes have been successfully applied to study the spectral cascade and plasma heating in kinetic Alfven wave turbulence, the linear and nonlinear properties of compressible modes including mirror instability and drift compressional mode, and the stability of the current sheet instabilities with finite guide field in the context of collisionless magnetic reconnection. The research results have been published in 25 journal papers and presented at many national and international conferences. Reprints of publications, source codes, and other research-related information are also available to general public on the PI’s webpage (http://phoenix.ps.uci.edu/zlin/). Two PhD theses in space plasma physics are highlighted in this report.

  19. Gyrokinetic verification of the persistence of kinetic ballooning modes in the magnetohydrodynamic second stability regime

    SciTech Connect

    Joiner, N.; Hirose, A.

    2008-08-15

    The kinetic ballooning mode (KBM) has been shown in previous work to be unstable within the magnetohydrodynamic (MHD) region (in s-{alpha} space) of second stability [Hirose et al., Phys. Rev. Lett. 72, 3993 (2004)]. In this work we verify this result using the gyrokinetic code GS2 [Kotschenreuther et al., Comput. Phys. Commun. 88, 128 (1996)] treating both ions and electrons as kinetic species and retaining the magnetosonic perturbation B{sub parallel}. Growth rates calculated using GS2 differ significantly from the previous differential/shooting code analysis. Calculations without B{sub parallel} find the stability region is preserved, while the addition of B{sub parallel} causes the mode to be more unstable than previously calculated within the region of MHD second stability. The inclusion of parallel ion current and B{sub parallel} into the shooting code does not account for the GS2 results. The evidence presented in this paper leads us to the conclusion that the adiabatic electron approximation employed in previous studies is found to be unsuitable for this type of instability. Based on the findings of this work, the KBM becomes an interesting instability in the context of internal transport barriers, where {alpha} is often large and magnetic shear is small (positive or negative)

  20. Comparisons between global and local gyrokinetic simulations of an ASDEX Upgrade H-mode plasma

    NASA Astrophysics Data System (ADS)

    Navarro, Alejandro Bañón; Told, Daniel; Jenko, Frank; Görler, Tobias; Happel, Tim

    2016-04-01

    We investigate by means of local and global nonlinear gyrokinetic GENE simulations an ASDEX Upgrade H-mode plasma. We find that for the outer core positions (i.e., ρ tor ≈ 0.5 - 0.7 ), nonlocal effects are important. For nominal input parameters local simulations over-predict the experimental heat fluxes by a large factor, while a good agreement is found with global simulations. This was a priori not expected, since the values of 1 / ρ ⋆ were large enough that global and local simulations should have been in accordance. Nevertheless, due to the high sensitivity of the heat fluxes with respect to the input parameters, it is still possible to match the heat fluxes in local simulations with the experimental and global results by varying the ion temperature gradient within the experimental uncertainties. In addition to that, once an agreement in the transport quantities between local (flux-matched) and global simulations is achieved, an agreement for other quantities, such as density and temperature fluctuations, is also found. The case presented here clearly shows that even in the presence of global size-effects, the local simulation approach is still a valid and accurate approach.

  1. Four-Dimensional Continuum Gyrokinetic Code: Neoclassical Simulation of Fusion Edge Plasmas

    NASA Astrophysics Data System (ADS)

    Xu, X. Q.

    2005-10-01

    We are developing a continuum gyrokinetic code, TEMPEST, to simulate edge plasmas. Our code represents velocity space via a grid in equilibrium energy and magnetic moment variables, and configuration space via poloidal magnetic flux and poloidal angle. The geometry is that of a fully diverted tokamak (single or double null) and so includes boundary conditions for both closed magnetic flux surfaces and open field lines. The 4-dimensional code includes kinetic electrons and ions, and electrostatic field-solver options, and simulates neoclassical transport. The present implementation is a Method of Lines approach where spatial finite-differences (higher order upwinding) and implicit time advancement are used. We present results of initial verification and validation studies: transition from collisional to collisionless limits of parallel end-loss in the scrape-off layer, self-consistent electric field, and the effect of the real X-point geometry and edge plasma conditions on the standard neoclassical theory, including a comparison of our 4D code with other kinetic neoclassical codes and experiments.

  2. The Mean-Field Limit for a Regularized Vlasov-Maxwell Dynamics

    NASA Astrophysics Data System (ADS)

    Golse, François

    2012-03-01

    The present work establishes the mean-field limit of a N-particle system towards a regularized variant of the relativistic Vlasov-Maxwell system, following the work of Braun-Hepp [Commun Math Phys 56:101-113, 1977] and Dobrushin [Func Anal Appl 13:115-123, 1979] for the Vlasov-Poisson system. The main ingredients in the analysis of this system are (a) a kinetic formulation of the Maxwell equations in terms of a distribution of electromagnetic potential in the momentum variable, (b) a regularization procedure for which an analogue of the total energy—i.e. the kinetic energy of the particles plus the energy of the electromagnetic field—is conserved and (c) an analogue of Dobrushin's stability estimate for the Monge-Kantorovich-Rubinstein distance between two solutions of the regularized Vlasov-Poisson dynamics adapted to retarded potentials.

  3. Complete fluid equations for low-n singular modes in axisymmetric toroidal plasmas

    SciTech Connect

    Glasser, A.H.

    1990-01-01

    The goal of this work is to develop a complete linear theory of the singular region, including all important dynamical effects. The present phase of the work treats the more collision fluid regime. A later phase will treat the less collisional gyrokinetic regime. This paper concerns the derivation and form of the fluid equations for the singular region of low-n modes. Later work will treat high-n ballooning modes. In addition, the ordering in the present work must be amended before it is applicable to the neighborhood of the field reversal surface of the RFP.

  4. Verification of gyrokinetic particle simulation of current-driven instability in fusion plasmas. III. Collisionless tearing mode

    NASA Astrophysics Data System (ADS)

    Liu, Dongjian; Bao, Jian; Han, Tao; Wang, Jiaqi; Lin, Zhihong

    2016-02-01

    A finite-mass electron fluid model for low frequency electromagnetic fluctuations, particularly the collisionless tearing mode, has been implemented in the gyrokinetic toroidal code. Using this fluid model, linear properties of the collisionless tearing mode have been verified. Simulations verify that the linear growth rate of the single collisionless tearing mode is proportional to De2, where De is the electron skin depth. On the other hand, the growth rate of a double tearing mode is proportional to De in the parameter regime of fusion plasmas.

  5. Gyrokinetic study of the impact of the electron to ion heating ratio on the turbulent diffusion of highly charged impurities

    NASA Astrophysics Data System (ADS)

    Angioni, C.

    2015-10-01

    A gyrokinetic study based on numerical and analytical calculations is presented, which computes the dependence of the turbulent diffusion of highly charged impurities on the ratio of the electron to the ion heat flux of the plasma. Nonlinear simulations show that the size of the turbulent diffusion of heavy impurities can vary by one order of magnitude with fixed total heat flux and is an extremely sensitive function of the electron to ion heat flux ratio. Numerical linear calculations are found to reproduce the nonlinear results. Thereby, a quasi-linear analytical approach is used to explain the origin of this dependence.

  6. Gyrokinetic study of the impact of the electron to ion heating ratio on the turbulent diffusion of highly charged impurities

    SciTech Connect

    Angioni, C.

    2015-10-15

    A gyrokinetic study based on numerical and analytical calculations is presented, which computes the dependence of the turbulent diffusion of highly charged impurities on the ratio of the electron to the ion heat flux of the plasma. Nonlinear simulations show that the size of the turbulent diffusion of heavy impurities can vary by one order of magnitude with fixed total heat flux and is an extremely sensitive function of the electron to ion heat flux ratio. Numerical linear calculations are found to reproduce the nonlinear results. Thereby, a quasi-linear analytical approach is used to explain the origin of this dependence.

  7. The physics of the second-order gyrokinetic magnetohydrodynamic Hamiltonian: μ conservation, Galilean invariance, and ponderomotive potential

    SciTech Connect

    Krommes, J. A.

    2013-12-15

    Some physical interpretations are given of the well-known second-order gyrokinetic Hamiltonian in the magnetohydrodynamic limit. Its relations to the conservation of the true (Galilean-invariant) magnetic moment and fluid nonlinearities are described. Subtleties about its derivation as a cold-ion limit are explained; it is important to take that limit in the frame moving with the E×B velocity. The discussion also provides some geometric understanding of certain well-known Lie generating functions, and it makes contact with general discussions of ponderomotive potentials and the thermodynamics of dielectric media.

  8. Multi-channel transport experiments at Alcator C-Mod and comparison with gyrokinetic simulationsa)

    NASA Astrophysics Data System (ADS)

    White, A. E.; Howard, N. T.; Greenwald, M.; Reinke, M. L.; Sung, C.; Baek, S.; Barnes, M.; Candy, J.; Dominguez, A.; Ernst, D.; Gao, C.; Hubbard, A. E.; Hughes, J. W.; Lin, Y.; Mikkelsen, D.; Parra, F.; Porkolab, M.; Rice, J. E.; Walk, J.; Wukitch, S. J.; Team, Alcator C-Mod

    2013-05-01

    Multi-channel transport experiments have been conducted in auxiliary heated (Ion Cyclotron Range of Frequencies) L-mode plasmas at Alcator C-Mod [Marmar and Alcator C-Mod Group, Fusion Sci. Technol. 51(3), 3261 (2007)]. These plasmas provide good diagnostic coverage for measurements of kinetic profiles, impurity transport, and turbulence (electron temperature and density fluctuations). In the experiments, a steady sawtoothing L-mode plasma with 1.2 MW of on-axis RF heating is established and density is scanned by 20%. Measured rotation profiles change from peaked to hollow in shape as density is increased, but electron density and impurity profiles remain peaked. Ion or electron heat fluxes from the two plasmas are the same. The experimental results are compared directly to nonlinear gyrokinetic theory using synthetic diagnostics and the code GYRO [Candy and Waltz, J. Comput. Phys. 186, 545 (2003)]. We find good agreement with experimental ion heat flux, impurity particle transport, and trends in the fluctuation level ratio (T˜e/Te)/(n ˜e/ne), but underprediction of electron heat flux. We find that changes in momentum transport (rotation profiles changing from peaked to hollow) do not correlate with changes in particle transport, and also do not correlate with changes in linear mode dominance, e.g., Ion Temperature Gradient versus Trapped Electron Mode. The new C-Mod results suggest that the drives for momentum transport differ from drives for heat and particle transport. The experimental results are inconsistent with present quasilinear models, and the strong sensitivity of core rotation to density remains unexplained.

  9. Gyrokinetic study of the role of β on electron particle transport in tokamaks

    NASA Astrophysics Data System (ADS)

    Hein, T.; Angioni, C.; Fable, E.; Candy, J.

    2010-10-01

    Electromagnetic effects on the radial transport of electrons in the core of tokamak plasmas are studied by means of linear and nonlinear gyrokinetic simulations with the code GYRO [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] and by an analytical derivation. The impact of a finite β, that is, a finite ratio of the plasma pressure to the magnetic pressure, is considered on the fluctuations of the magnetic field through Ampére's law, as well as on the geometrical modification of the vertical drift produced by the Shafranov shift in the magnetic equilibrium, which, for realistic descriptions, has to be included in both electrostatic and electromagnetic modeling. The condition of turbulent particle flux at the null, which allows the determination of stationary logarithmic density gradients when neoclassical transport and particle sources are negligible, is investigated for increasing values of β, in regimes of ion temperature gradient and trapped electron mode turbulence. The loss of adiabaticity of passing electrons produced by fluctuations in the magnetic vector potential produces an outward convection. When the magnetic equilibrium geometry is kept fixed, this induces a strong reduction of the stationary logarithmic density gradient with increasing β. This effect is partly compensated by the geometrical effect on the vertical drift. This compensation effect, however, is significantly weaker in nonlinear simulations as compared to quasilinear calculations. A detailed comparison between quasilinear and nonlinear results reveals that the predicted value of the logarithmic density gradient is highly sensitive on the assumptions on the wave number spectrum applied in the quasilinear model. The qualitative consistency of the theoretical predictions with the experimental results obtained so far on the dependence of density peaking on β is discussed by considering the additional impact, with increasing β, of a particle source delivered by neutral beam

  10. Gyrokinetic study of the role of {beta} on electron particle transport in tokamaks

    SciTech Connect

    Hein, T.; Angioni, C.; Fable, E.; Candy, J.

    2010-10-15

    Electromagnetic effects on the radial transport of electrons in the core of tokamak plasmas are studied by means of linear and nonlinear gyrokinetic simulations with the code GYRO[J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] and by an analytical derivation. The impact of a finite {beta}, that is, a finite ratio of the plasma pressure to the magnetic pressure, is considered on the fluctuations of the magnetic field through Ampere's law, as well as on the geometrical modification of the vertical drift produced by the Shafranov shift in the magnetic equilibrium, which, for realistic descriptions, has to be included in both electrostatic and electromagnetic modeling. The condition of turbulent particle flux at the null, which allows the determination of stationary logarithmic density gradients when neoclassical transport and particle sources are negligible, is investigated for increasing values of {beta}, in regimes of ion temperature gradient and trapped electron mode turbulence. The loss of adiabaticity of passing electrons produced by fluctuations in the magnetic vector potential produces an outward convection. When the magnetic equilibrium geometry is kept fixed, this induces a strong reduction of the stationary logarithmic density gradient with increasing {beta}. This effect is partly compensated by the geometrical effect on the vertical drift. This compensation effect, however, is significantly weaker in nonlinear simulations as compared to quasilinear calculations. A detailed comparison between quasilinear and nonlinear results reveals that the predicted value of the logarithmic density gradient is highly sensitive on the assumptions on the wave number spectrum applied in the quasilinear model. The qualitative consistency of the theoretical predictions with the experimental results obtained so far on the dependence of density peaking on {beta} is discussed by considering the additional impact, with increasing {beta}, of a particle source delivered

  11. Gyrokinetic simulation of momentum transport with residual stress from diamagnetic level velocity shears

    SciTech Connect

    Waltz, R. E.; Staebler, G. M.; Solomon, W. M.

    2011-04-15

    Residual stress refers to the remaining toroidal angular momentum (TAM) flux (divided by major radius) when the shear in the equilibrium fluid toroidal velocity (and the velocity itself) vanishes. Previously [Waltz et al., Phys. Plasmas 14, 122507 (2007); errata 16, 079902 (2009)], we demonstrated with GYRO [Candy and Waltz, J. Comp. Phys. 186, 545 (2003)] gyrokinetic simulations that TAM pinching from (ion pressure gradient supported or diamagnetic level) equilibrium ExB velocity shear could provide some of the residual stress needed to support spontaneous toroidal rotation against normal diffusive loss. Here we show that diamagnetic level shear in the intrinsic drift wave velocities (or ''profile shear'' in the ion and electron density and temperature gradients) provides a comparable residual stress. The individual signed contributions of these small (rho-star level) ExB and profile velocity shear rates to the turbulence level and (rho-star squared) ion energy transport stabilization are additive if the rates are of the same sign. However because of the additive stabilization effect, the contributions to the small (rho-star cubed) residual stress is not always simply additive. If the rates differ in sign, the residual stress from one can buck out that from the other (and in some cases reduce the stabilization.) The residual stress from these diamagnetic velocity shear rates is quantified by the ratio of TAM flow to ion energy (power) flow (M/P) in a global GYRO core simulation of a ''null'' toroidal rotation DIII-D [Mahdavi and Luxon, Fusion Sci. Technol. 48, 2 (2005)] discharge by matching M/P profiles within experimental uncertainty. Comparison of global GYRO (ion and electron energy as well as particle) transport flow balance simulations of TAM transport flow in a high-rotation DIII-D L-mode quantifies and isolates the ExB shear and parallel velocity (Coriolis force) pinching components from the larger ''diffusive'' parallel velocity shear driven component and

  12. Multispecies density peaking in gyrokinetic turbulence simulations of low collisionality Alcator C-Mod plasmas

    NASA Astrophysics Data System (ADS)

    Mikkelsen, D. R.; Bitter, M.; Delgado-Aparicio, L.; Hill, K. W.; Greenwald, M.; Howard, N. T.; Hughes, J. W.; Rice, J. E.; Reinke, M. L.; Podpaly, Y.; Ma, Y.; Candy, J.; Waltz, R. E.

    2015-06-01

    Peaked density profiles in low-collisionality AUG and JET H-mode plasmas are probably caused by a turbulently driven particle pinch, and Alcator C-Mod experiments confirmed that collisionality is a critical parameter. Density peaking in reactors could produce a number of important effects, some beneficial, such as enhanced fusion power and transport of fuel ions from the edge to the core, while others are undesirable, such as lower beta limits, reduced radiation from the plasma edge, and consequently higher divertor heat loads. Fundamental understanding of the pinch will enable planning to optimize these impacts. We show that density peaking is predicted by nonlinear gyrokinetic turbulence simulations based on measured profile data from low collisionality H-mode plasma in Alcator C-Mod. Multiple ion species are included to determine whether hydrogenic density peaking has an isotope dependence or is influenced by typical levels of low-Z impurities, and whether impurity density peaking depends on the species. We find that the deuterium density profile is slightly more peaked than that of hydrogen, and that experimentally relevant levels of boron have no appreciable effect on hydrogenic density peaking. The ratio of density at r/a = 0.44 to that at r/a = 0.74 is 1.2 for the majority D and minority H ions (and for electrons), and increases with impurity Z: 1.1 for helium, 1.15 for boron, 1.3 for neon, 1.4 for argon, and 1.5 for molybdenum. The ion temperature profile is varied to match better the predicted heat flux with the experimental transport analysis, but the resulting factor of two change in heat transport has only a weak effect on the predicted density peaking.

  13. SciDAC Center for Gyrokinetic Particle Simulation of Turbulent Transport in Burning Plasmas

    SciTech Connect

    Lin, Zhihong

    2013-12-18

    During the first year of the SciDAC gyrokinetic particle simulation (GPS) project, the GPS team (Zhihong Lin, Liu Chen, Yasutaro Nishimura, and Igor Holod) at the University of California, Irvine (UCI) studied the tokamak electron transport driven by electron temperature gradient (ETG) turbulence, and by trapped electron mode (TEM) turbulence and ion temperature gradient (ITG) turbulence with kinetic electron effects, extended our studies of ITG turbulence spreading to core-edge coupling. We have developed and optimized an elliptic solver using finite element method (FEM), which enables the implementation of advanced kinetic electron models (split-weight scheme and hybrid model) in the SciDAC GPS production code GTC. The GTC code has been ported and optimized on both scalar and vector parallel computer architectures, and is being transformed into objected-oriented style to facilitate collaborative code development. During this period, the UCI team members presented 11 invited talks at major national and international conferences, published 22 papers in peer-reviewed journals and 10 papers in conference proceedings. The UCI hosted the annual SciDAC Workshop on Plasma Turbulence sponsored by the GPS Center, 2005-2007. The workshop was attended by about fifties US and foreign researchers and financially sponsored several gradual students from MIT, Princeton University, Germany, Switzerland, and Finland. A new SciDAC postdoc, Igor Holod, has arrived at UCI to initiate global particle simulation of magnetohydrodynamics turbulence driven by energetic particle modes. The PI, Z. Lin, has been promoted to the Associate Professor with tenure at UCI.

  14. Quasi-linear gyrokinetic predictions of the Coriolis momentum pinch in National Spherical Torus Experiment

    DOE PAGESBeta

    Guttenfelder, W.; Kaye, S. M.; Ren, Y.; Solomon, W.; Bell, R. E.; Candy, J.; Gerhardt, S. P.; LeBlanc, B. P.; Yuh, H.

    2016-05-11

    This paper presents quasi-linear gyrokinetic predictions of the Coriolis momentum pinch for low aspect-ratio National Spherical Torus Experiment (NSTX) H-modes where previous experimental measurements were focused. Local, linear calculations predict that in the region of interest (just outside the mid-radius) of these relatively high-beta plasmas, profiles are most unstable to microtearing modes that are only effective in transporting electron energy. However, sub-dominant electromagnetic and electrostaticballooning modes are also unstable, which are effective at transporting energy, particles, and momentum. The quasi-linear prediction of transport from these weaker ballooning modes, assuming they contribute transport in addition to that from microtearing modes inmore » a nonlinear turbulent state, leads to a very small or outward convection of momentum, inconsistent with the experimentally measured inward pinch, and opposite to predictions in conventional aspect ratio tokamaks. Additional predictions of a low beta L-mode plasma, unstable to more traditional electrostatic ion temperature gradient-trapped electron mode instability, show that the Coriolis pinch is inward but remains relatively weak and insensitive to many parameter variations. The weak or outward pinch predicted in NSTX plasmas appears to be at least partially correlated to changes in the parallel mode structure that occur at a finite beta and low aspect ratio, as discussed in previous theories. The only conditions identified where a stronger inward pinch is predicted occur either in the purely electrostatic limit or if the aspect ratio is increased. Lastly, as the Coriolis pinch cannot explain the measured momentum pinch, additional theoretical momentum transport mechanisms are discussed that may be potentially important.« less

  15. Multi-channel transport experiments at Alcator C-Mod and comparison with gyrokinetic simulations

    SciTech Connect

    White, A. E.; Howard, N. T.; Greenwald, M.; Reinke, M. L.; Sung, C.; Baek, S.; Barnes, M.; Dominguez, A.; Ernst, D.; Gao, C.; Hubbard, A. E.; Hughes, J. W.; Lin, Y.; Parra, F.; Porkolab, M.; Rice, J. E.; Walk, J.; Wukitch, S. J.; Team, Alcator C-Mod; Candy, J.; and others

    2013-05-15

    Multi-channel transport experiments have been conducted in auxiliary heated (Ion Cyclotron Range of Frequencies) L-mode plasmas at Alcator C-Mod [Marmar and Alcator C-Mod Group, Fusion Sci. Technol. 51(3), 3261 (2007)]. These plasmas provide good diagnostic coverage for measurements of kinetic profiles, impurity transport, and turbulence (electron temperature and density fluctuations). In the experiments, a steady sawtoothing L-mode plasma with 1.2 MW of on-axis RF heating is established and density is scanned by 20%. Measured rotation profiles change from peaked to hollow in shape as density is increased, but electron density and impurity profiles remain peaked. Ion or electron heat fluxes from the two plasmas are the same. The experimental results are compared directly to nonlinear gyrokinetic theory using synthetic diagnostics and the code GYRO [Candy and Waltz, J. Comput. Phys. 186, 545 (2003)]. We find good agreement with experimental ion heat flux, impurity particle transport, and trends in the fluctuation level ratio (T(tilde sign){sub e}/T{sub e})/(ñ{sub e}/n{sub e}), but underprediction of electron heat flux. We find that changes in momentum transport (rotation profiles changing from peaked to hollow) do not correlate with changes in particle transport, and also do not correlate with changes in linear mode dominance, e.g., Ion Temperature Gradient versus Trapped Electron Mode. The new C-Mod results suggest that the drives for momentum transport differ from drives for heat and particle transport. The experimental results are inconsistent with present quasilinear models, and the strong sensitivity of core rotation to density remains unexplained.

  16. Gyrokinetic Stability Studies of the Microtearing Mode in the National Spherical Torus Experiment H-mode

    SciTech Connect

    Baumgaertel, J. A.; Redi, M. H.; Budny, R. V.; Rewoldt, G.; Dorland, W.

    2005-10-19

    Insight into plasma microturbulence and transport is being sought using linear simulations of drift waves on the National Spherical Torus Experiment (NSTX), following a study of drift wave modes on the Alcator C-Mod Tokamak. Microturbulence is likely generated by instabilities of drift waves, which cause transport of heat and particles. Understanding this transport is important because the containment of heat and particles is required for the achievement of practical nuclear fusion. Microtearing modes may cause high heat transport through high electron thermal conductivity. It is hoped that microtearing will be stable along with good electron transport in the proposed low collisionality International Thermonuclear Experimental Reactor (ITER). Stability of the microtearing mode is investigated for conditions at mid-radius in a high density NSTX high performance (H-mode) plasma, which is compared to the proposed ITER plasmas. The microtearing mode is driven by the electron temperature gradient, and believed to be mediated by ion collisions and magnetic shear. Calculations are based on input files produced by TRXPL following TRANSP (a time-dependent transport analysis code) analysis. The variability of unstable mode growth rates is examined as a function of ion and electron collisionalities using the parallel gyrokinetic computational code GS2. Results show the microtearing mode stability dependence for a range of plasma collisionalities. Computation verifies analytic predictions that higher collisionalities than in the NSTX experiment increase microtearing instability growth rates, but that the modes are stabilized at the highest values. There is a transition of the dominant mode in the collisionality scan to ion temperature gradient character at both high and low collisionalities. The calculations suggest that plasma electron thermal confinement may be greatly improved in the low-collisionality ITER.

  17. Gyrokinetic simulations of mesoscale energetic particle-driven Alfvenic turbulent transport embedded in microturbulence

    SciTech Connect

    Bass, E. M.; Waltz, R. E.

    2010-11-15

    Energetic particle (EP) transport from local high-n toroidal Alfven eigenmodes (TAEs) and energetic particle modes (EPMs) is simulated with a gyrokinetic code. Linear and nonlinear simulations have identified a parameter range where the longwave TAE and EPM are unstable alongside the well-known ion-temperature-gradient (ITG) and trapped-electron-mode (TEM) instabilities. A new eigenvalue solver in GYRO facilitates this mode identification. States of nonlinearly saturated local TAE/EPM turbulent intensity are identified, showing a 'soft' transport threshold for enhanced energetic particle transport against the TAE/EPM drive from the EP pressure gradient. The very long-wavelength (mesoscale) TAE/EPM transport is saturated partially by nonlinear interaction with microturbulent ITG/TEM-driven zonal flows. Fixed-gradient-length, nonlinearly saturated states are accessible over a relatively narrow range of EP pressure gradient. Within this range, and in the local limit employed, TAE/EPM-driven transport more closely resembles drift-wave microturbulent transport than 'stiff' ideal MHD transport with a clamped critical total pressure gradient. At a higher, critical EP pressure gradient, fixed-gradient nonlinear saturation fails: EP transport increases without limit and background transport decreases. Presumably saturation is then obtained by relaxation of the EP pressure gradient to near this critical EP pressure gradient. If the background plasma gradients driving the ITG/TEM turbulence and zonal flows are weakened, the critical gradient collapses to the TAE/EPM linear stability threshold. Even at the critical EP pressure gradient there is no evidence that TAE/EPM instability significantly increases transport in the background plasma channels.

  18. Quasi-linear gyrokinetic predictions of the Coriolis momentum pinch in National Spherical Torus Experiment

    NASA Astrophysics Data System (ADS)

    Guttenfelder, W.; Kaye, S. M.; Ren, Y.; Solomon, W.; Bell, R. E.; Candy, J.; Gerhardt, S. P.; LeBlanc, B. P.; Yuh, H.

    2016-05-01

    This paper presents quasi-linear gyrokinetic predictions of the Coriolis momentum pinch for low aspect-ratio National Spherical Torus Experiment (NSTX) H-modes where previous experimental measurements were focused. Local, linear calculations predict that in the region of interest (just outside the mid-radius) of these relatively high-beta plasmas, profiles are most unstable to microtearing modes that are only effective in transporting electron energy. However, sub-dominant electromagnetic and electrostatic ballooning modes are also unstable, which are effective at transporting energy, particles, and momentum. The quasi-linear prediction of transport from these weaker ballooning modes, assuming they contribute transport in addition to that from microtearing modes in a nonlinear turbulent state, leads to a very small or outward convection of momentum, inconsistent with the experimentally measured inward pinch, and opposite to predictions in conventional aspect ratio tokamaks. Additional predictions of a low beta L-mode plasma, unstable to more traditional electrostatic ion temperature gradient-trapped electron mode instability, show that the Coriolis pinch is inward but remains relatively weak and insensitive to many parameter variations. The weak or outward pinch predicted in NSTX plasmas appears to be at least partially correlated to changes in the parallel mode structure that occur at a finite beta and low aspect ratio, as discussed in previous theories. The only conditions identified where a stronger inward pinch is predicted occur either in the purely electrostatic limit or if the aspect ratio is increased. As the Coriolis pinch cannot explain the measured momentum pinch, additional theoretical momentum transport mechanisms are discussed that may be potentially important.

  19. Multispecies density peaking in gyrokinetic turbulence simulations of low collisionality Alcator C-Mod plasmas

    SciTech Connect

    Mikkelsen, D. R. Bitter, M.; Delgado-Aparicio, L.; Hill, K. W.; Greenwald, M.; Howard, N. T.; Hughes, J. W.; Rice, J. E.; Reinke, M. L.; Podpaly, Y.; Ma, Y.; Candy, J.; Waltz, R. E.

    2015-06-15

    Peaked density profiles in low-collisionality AUG and JET H-mode plasmas are probably caused by a turbulently driven particle pinch, and Alcator C-Mod experiments confirmed that collisionality is a critical parameter. Density peaking in reactors could produce a number of important effects, some beneficial, such as enhanced fusion power and transport of fuel ions from the edge to the core, while others are undesirable, such as lower beta limits, reduced radiation from the plasma edge, and consequently higher divertor heat loads. Fundamental understanding of the pinch will enable planning to optimize these impacts. We show that density peaking is predicted by nonlinear gyrokinetic turbulence simulations based on measured profile data from low collisionality H-mode plasma in Alcator C-Mod. Multiple ion species are included to determine whether hydrogenic density peaking has an isotope dependence or is influenced by typical levels of low-Z impurities, and whether impurity density peaking depends on the species. We find that the deuterium density profile is slightly more peaked than that of hydrogen, and that experimentally relevant levels of boron have no appreciable effect on hydrogenic density peaking. The ratio of density at r/a = 0.44 to that at r/a = 0.74 is 1.2 for the majority D and minority H ions (and for electrons), and increases with impurity Z: 1.1 for helium, 1.15 for boron, 1.3 for neon, 1.4 for argon, and 1.5 for molybdenum. The ion temperature profile is varied to match better the predicted heat flux with the experimental transport analysis, but the resulting factor of two change in heat transport has only a weak effect on the predicted density peaking.

  20. Comparison of gradient and flux driven gyro-kinetic turbulent transport

    NASA Astrophysics Data System (ADS)

    Rath, F.; Peeters, A. G.; Buchholz, R.; Grosshauser, S. R.; Migliano, P.; Weikl, A.; Strintzi, D.

    2016-05-01

    Flux and gradient driven ion temperature gradient turbulence in tokamak geometry and for Cyclone base case parameters are compared in the local limit using the same underlying gyro-kinetic turbulence model. The gradient driven turbulence described using the flux tube model with periodic boundary conditions has a finite ion heat flux Qi≈10 n0T0ρ*2vth , where n0 (T0) is the background density (temperature), ρ*=ρ/R is the normalized Larmor radius, R is the major radius of the device, and vth is the ion thermal velocity at the nonlinear threshold of the temperature gradient length for turbulence generation. Consequently, the gradient driven local transport model is unable to accurately describe heat fluxes below Qi<10 n0T0ρ*2vt h , since no stationary fully developed turbulent state can be obtained. The turbulence in the flux driven case shows intermittent behaviour and avalanches for Qi<10 n0T0ρ*2vth . Isolated avalanches disappear for Qi>10 n0T0ρ*2vt h , and at higher heat fluxes, the statistics of the turbulence is the same for the flux and gradient driven case. The nonlinear upshift of the temperature gradient length threshold for turbulence generation (known as the Dimits shift) is larger in the case of flux driven turbulence. This higher nonlinear upshift is attributed to the generation of structures in the radial temperature profile, known as staircases [Dif-Pradalier, Phys. Rev. E 82, 025401 (2010)]. Avalanches are initiated at specific locations and have roughly the same radial extent of 50-70 ion Larmor radii. The staircases are obtained at low heating rates, and become unstable and break up at higher heating rates. At the heat fluxes for which staircase formation is observed, no stationary gradient driven simulations can be obtained.

  1. A circular equilibrium model for local gyrokinetic simulations of ion temperature gradient fluctuations in reversed field pinches

    NASA Astrophysics Data System (ADS)

    Tangri, Varun; Terry, P. W.; Waltz, R. E.

    2011-05-01

    A simple large-aspect-ratio (R0/r) circular equilibrium model is developed for low-beta reversed field pinch (RFP) geometry. The model is suitable for treating small scale instability and turbulent transport driven by ion temperature gradient (ITG) and related electron drift modes in gyrokinetic simulations. The equilibrium model is an RFP generalization of the common tokamak s-α model to small safety factor (q), where the poloidal field dominates the toroidal field. The model accommodates the RFP toroidal field reversal (where q vanishes) by generalizing the cylindrical force-free Bessel function model (BFM) [J. B. Taylor, Phys. Rev. Lett. 33, 1139 (1974)] to toroidal geometry. The global equilibrium can be described in terms of the RFP field reversal and pinch parameters [F ,Θ]. This new toroidal Bessel function model (TBFM) has been incorporated into the gyrokinetic code GYRO [J. Candy and R. E. Waltz, J.Comput. Phys. 186, 545 (2003)] and used here to explore local electrostatic ITG adiabatic electron instability rates for typical low-q RFP parameters.

  2. A circular equilibrium model for local gyrokinetic simulations of ion temperature gradient fluctuations in reversed field pinches

    SciTech Connect

    Tangri, Varun; Terry, P. W.; Waltz, R. E.

    2011-05-15

    A simple large-aspect-ratio (R{sub 0}/r) circular equilibrium model is developed for low-beta reversed field pinch (RFP) geometry. The model is suitable for treating small scale instability and turbulent transport driven by ion temperature gradient (ITG) and related electron drift modes in gyrokinetic simulations. The equilibrium model is an RFP generalization of the common tokamak s-{alpha} model to small safety factor (q), where the poloidal field dominates the toroidal field. The model accommodates the RFP toroidal field reversal (where q vanishes) by generalizing the cylindrical force-free Bessel function model (BFM) [J. B. Taylor, Phys. Rev. Lett. 33, 1139 (1974)] to toroidal geometry. The global equilibrium can be described in terms of the RFP field reversal and pinch parameters [F,{Theta}]. This new toroidal Bessel function model (TBFM) has been incorporated into the gyrokinetic code GYRO [J. Candy and R. E. Waltz, J.Comput. Phys. 186, 545 (2003)] and used here to explore local electrostatic ITG adiabatic electron instability rates for typical low-q RFP parameters.

  3. Implementation of 2D domain decomposition in the UCAN gyrokinetic PIC code for non-diffusive transport studies in tokamaks

    NASA Astrophysics Data System (ADS)

    Leboeuf, Jean-Noel; Decyk, Viktor; Newman, David; Sanchez, Raul

    2012-03-01

    The massively parallel, nonlinear, 3D, toroidal, electrostatic, gyrokinetic, PIC, Cartesian geometry UCAN code, with particle ions and adiabatic electrons, has been successfully exercised to identify non-diffusive transport characteristics in DIII-D-like tokamak discharges. The limitation in applying UCAN to larger scale discharges is the 1D domain decomposition in the toroidal (or z-) direction for massively parallel implementation using MPI which has restricted the calculations to a few hundred ion Larmor radii per minor radius. To exceed these sizes, we have implemented 2D domain decomposition in UCAN with the addition of the y-direction to the processor mix. This has been facilitated by use of relevant components in the 2D domain decomposed PLIB2 library of field and particle management routines developed for UCLA's UPIC framework of conventional PIC codes. The gyro-averaging in gyrokinetic codes has necessitated the use of replicated arrays for efficient charge accumulation and particle push. The 2D domain-decomposed UCAN2 code reproduces the original 1D domain results within roundoff. Production calculations at large system sizes have been performed with UCAN2 on 131072 processors of the Cray XE6 at NERSC.

  4. Linear gyrokinetic simulations of microinstabilities within the pedestal region of H-mode NSTX discharges in a highly shaped geometry

    DOE PAGESBeta

    Coury, M.; Guttenfelder, W.; Mikkelsen, D. R.; Canik, J. M.; Canal, G. P.; Diallo, A.; Kaye, S.; Kramer, G. J.; Maingi, R.

    2016-06-30

    Linear (local) gyrokinetic predictions of edge microinstabilities in highly shaped, lithiated and non-lithiated NSTX discharges are reported using the gyrokinetic code GS2. Microtearing modes dominate the non-lithiated pedestal top. The stabilization of these modes at the lithiated pedestal top enables the electron temperature pedestal to extend further inwards, as observed experimentally. Kinetic ballooning modes are found to be unstable mainly at the mid-pedestal of both types of discharges, with un- stable trapped electron modes nearer the separatrix region. At electron wavelengths, ETG modes are found to be unstable from mid-pedestal outwards for ηe, exp ~2.2 with higher growth rates formore » the lithiated discharge. Near the separatrix, the critical temperature gradient for driving ETG modes is reduced in the presence of lithium, re ecting the reduction of the lithiated density gradients observed experimentally. A preliminary linear study in the edge of non-lithiated discharges shows that the equilibrium shaping alters the electrostatic modes stability, found more unstable at high plasma shaping.« less

  5. Teaching Equations.

    ERIC Educational Resources Information Center

    Nibbelink, William H.

    1990-01-01

    Proposed is a gradual transition from arithmetic to the idea of an equation with variables in the elementary grades. Vertical and horizontal formats of open sentences, the instructional sequence, vocabulary, and levels of understanding are discussed in this article. (KR)

  6. Steady-State Gyrokinetics Transport Code (SSGKT), A Scientific Application Partnership with the Framework Application for Core-Edge Transport Simulations, Final Report

    SciTech Connect

    Fahey, Mark R.; Candy, Jeff

    2013-11-07

    This project initiated the development of TGYRO ? a steady-state Gyrokinetic transport code (SSGKT) that integrates micro-scale GYRO turbulence simulations into a framework for practical multi-scale simulation of conventional tokamaks as well as future reactors. Using a lightweight master transport code, multiple independent (each massively parallel) gyrokinetic simulations are coordinated. The capability to evolve profiles using the TGLF model was also added to TGYRO and represents a more typical use-case for TGYRO. The goal of the project was to develop a steady-state Gyrokinetic transport code (SSGKT) that integrates micro-scale gyrokinetic turbulence simulations into a framework for practical multi-scale simulation of a burning plasma core ? the International Thermonuclear Experimental Reactor (ITER) in particular. This multi-scale simulation capability will be used to predict the performance (the fusion energy gain, Q) given the H-mode pedestal temperature and density. At present, projections of this type rely on transport models like GLF23, which are based on rather approximate fits to the results of linear and nonlinear simulations. Our goal is to make these performance projections with precise nonlinear gyrokinetic simulations. The method of approach is to use a lightweight master transport code to coordinate multiple independent (each massively parallel) gyrokinetic simulations using the GYRO code. This project targets the practical multi-scale simulation of a reactor core plasma in order to predict the core temperature and density profiles given the H-mode pedestal temperature and density. A master transport code will provide feedback to O(16) independent gyrokinetic simulations (each massively parallel). A successful feedback scheme offers a novel approach to predictive modeling of an important national and international problem. Success in this area of fusion simulations will allow US scientists to direct the research path of ITER over the next two

  7. Gyrokinetic simulation of momentum transport with residual stress from diamagnetic level velocity shears

    NASA Astrophysics Data System (ADS)

    Waltz, R. E.; Staebler, G. M.; Solomon, W. M.

    2011-04-01

    Residual stress refers to the remaining toroidal angular momentum (TAM) flux (divided by major radius) when the shear in the equilibrium fluid toroidal velocity (and the velocity itself) vanishes. Previously [Waltz et al., Phys. Plasmas 14, 122507 (2007); errata 16, 079902 (2009)], we demonstrated with GYRO [Candy and Waltz, J. Comp. Phys. 186, 545 (2003)] gyrokinetic simulations that TAM pinching from (ion pressure gradient supported or diamagnetic level) equilibrium E ×B velocity shear could provide some of the residual stress needed to support spontaneous toroidal rotation against normal diffusive loss. Here we show that diamagnetic level shear in the intrinsic drift wave velocities (or "profile shear" in the ion and electron density and temperature gradients) provides a comparable residual stress. The individual signed contributions of these small (rho-star level) E ×B and profile velocity shear rates to the turbulence level and (rho-star squared) ion energy transport stabilization are additive if the rates are of the same sign. However because of the additive stabilization effect, the contributions to the small (rho-star cubed) residual stress is not always simply additive. If the rates differ in sign, the residual stress from one can buck out that from the other (and in some cases reduce the stabilization.) The residual stress from these diamagnetic velocity shear rates is quantified by the ratio of TAM flow to ion energy (power) flow (M/P) in a global GYRO core simulation of a "null" toroidal rotation DIII-D [Mahdavi and Luxon, Fusion Sci. Technol. 48, 2 (2005)] discharge by matching M/P profiles within experimental uncertainty. Comparison of global GYRO (ion and electron energy as well as particle) transport flow balance simulations of TAM transport flow in a high-rotation DIII-D L-mode quantifies and isolates the E ×B shear and parallel velocity (Coriolis force) pinching components from the larger "diffusive" parallel velocity shear driven component and

  8. Gyrokinetic Turbulence Driven Toroidal Momentum Transport and Comparison to Experimental Observations

    NASA Astrophysics Data System (ADS)

    Wang, Weixing

    2008-11-01

    Global gyrokinetic simulations using the GTS code [1] have found that a large inward flux of toroidal momentum is driven robustly in the post saturation phase of ion temperature gradient (ITG) turbulence. As a consequence, core plasma rotation spins up resulting in δu a few percent of vth in the case with no momentum source at the edge. The underlying physics for the inward flux is identified to be the generation of residual stress due to the k symmetry breaking [2] induced by self-generated zonal flow shear which is quasi-stationary in global simulations. The elatively low level momentum flux in the long- time steady state appears to be approximately diffusive, with effective χφ/χi on the order of unity, in broad agreement with experimental observations and theory predictions for ITG turbulence [3]. Neoclassical simulations using the GTC- NEO code [4] also show that the ion temperature gradient can drive a significant inward nondiffusive momentum flux. However, the overall neoclassical contribution to the momentum transport is negligibly small compared to experimental levels for NSTX and DIII-D plasmas. It is also found that finite residual turbulence can survive strong mean ExB shear flow induced damping. This residual turbulence in the presence of strong ExB shear may drive an insignificant ion heat flux reasonably close to the neoclassical value, and a finite momentum flux significantly higher than the neoclassical level. Moreover, the equilibrium ExB flow shear is found to reduce the turbulence driven transport for energy more efficiently than for momentum. These findings may offer an explanation for rather peculiar observations of near neoclassical ion heat and anomalous momentum transport in experiments, which has been often observed in various machines, but with little theoretical understanding. [1] W.X. Wang et al., Phys. Plasmas 14, 072306 (2007). [2] O.D. Gurcan et al., Phys. Plasmas 14, 042306 (2007). [3] N. Mattor and P.H. Diamond, Phys. Fluids 31

  9. Gyrokinetic Simulations of Enhanced Alpha Transport by De-stabilized Alfvèn Turbulence

    NASA Astrophysics Data System (ADS)

    Bass, E. M.

    2009-11-01

    Alfvèn turbulence, destabilized by fusion-produced α-particles, is expected to greatly enhance transport of these hot fusion products. Previously, the gyrokinetic code GYRO [1] was used to simulate the convective transport of fusion alpha particles by electrostatic (β=0) ITG/TEM turbulence driven at low k (0

  10. Final Technical Report

    SciTech Connect

    Brizard, Alain J

    2009-12-31

    Final Technical Report for U.S. Department of Energy Grant No. DE-FG02-09ER55005 Nonlinear FLR Effects in Reduced Fluid Models Alain J. Brizard, Saint Michael's College The above-mentioned DoE grant was used to support research activities by the PI during a sabbatical leave from Saint Michael's College in 2009. The major focus of the work was the role played by guiding-center and gyrocenter (linear and nonlinear) polarization and magnetization effects in understanding transport processes in turbulent magnetized plasmas. The theoretical tools used for this work include Lie-transform perturbation methods and Lagrangian (variational) methods developed by the PI in previous work. The present final technical report lists (I) the peer-reviewed publications that were written based on work funded by the Grant; (II) invited and contributed conference presentations during the period funded by the Grant; and (III) seminars presented during the period funded by the Grant. I. Peer-reviewed Publications A.J. Brizard and N. Tronko, 2011, Exact momentum conservation for the gyrokinetic Vlasov- Poisson equations, Physics of Plasmas 18 , 082307:1-14 [http://dx.doi.org/10.1063/1.3625554 ]. J. Decker, Y. Peysson, A.J. Brizard, and F.-X. Duthoit, 2010, Orbit-averaged guiding-center Fokker-Planck operator for numerical applications, Physics of Plasmas 17, 112513:1-12 [http://dx.doi.org/10.1063/1.3519514]. A.J. Brizard, 2010, Noether derivation of exact conservation laws for dissipationless reduced fluid models, Physics of Plasmas 17, 112503:1-8 [http://dx.doi.org/10.1063/1.3515303]. F.-X. Duthoit, A.J. Brizard, Y. Peysson, and J. Decker, 2010, Perturbation analysis of trapped particle dynamics in axisymmetric dipole geometry, Physics of Plasmas 17, 102903:1-9 [http://dx.doi.org/10.1063/1.3486554]. A.J. Brizard, 2010, Exact energy conservation laws for full and truncated nonlinear gyrokinetic equations, Physics of Plasmas 17, 042303:1-11 [http://dx.doi.org/10.1063/1.3374428]. A

  11. Beautiful equations

    NASA Astrophysics Data System (ADS)

    Viljamaa, Panu; Jacobs, J. Richard; Chris; JamesHyman; Halma, Matthew; EricNolan; Coxon, Paul

    2014-07-01

    In reply to a Physics World infographic (part of which is given above) about a study showing that Euler's equation was deemed most beautiful by a group of mathematicians who had been hooked up to a functional magnetic-resonance image (fMRI) machine while viewing mathematical expressions (14 May, http://ow.ly/xHUFi).

  12. Projected profile similarity in gyrokinetic simulations of Bohm and gyro-Bohm scaled DIII-D L and H modes

    SciTech Connect

    Waltz, R. E.; Candy, J.; Petty, C. C.

    2006-07-15

    Global gyrokinetic simulations of DIII-D [M. A. Mahdavi and J. L. Luxon, in 'DIII-D Tokamak Special Issue', Fusion Sci. Technol. 48, 2 (2005)] L- and H-mode dimensionally similar discharge pairs are treated in detail. The simulations confirm the Bohm scaling of the well-matched L-mode pair. The paradoxical but experimentally apparent gyro-Bohm scaling of the H-mode pair at larger relative gyroradius (rho-star) and lower transport levels is due to poor profile similarity. Simulations of projected experimental plasma profiles with perfect similarity show both the L- and H-mode pairs to have Bohm scaling. A {rho}{sub *} stabilization rule for predicting the breakdown of gyro-Bohm scaling from simulations of a single discharge is presented.

  13. Comparisons of Measurements and Gyrokinetic Simulations of Turbulence and Transport in Alcator C-Mod EDA H-Mode Discharges.

    NASA Astrophysics Data System (ADS)

    Sampsell, M. B.; Bravenec, R. V.; Candy, J.; Ernst, D. R.; Alcator C-Mod Team Nevins

    2004-11-01

    Beam-emission spectroscopy (BES) on Alcator C-Mod has observed long wavelength broadband fluctuations and a `quasi-coherent mode' (the latter exclusive to enhanced D_α H modes) in the plasma edge. However, it has not observed broadband fluctuations at the top of the H-mode pedestal or farther in. In an attempt to understand this, we have run the GYRO gyrokinetic code [J. Candy, J. Comput. Phys. 186, 545 (2003)] for this region, applied `synthetic BES' to the fluctuating density output, and compared with the data. The synthetic BES is composed of i) an anti-aliasing filter in GYRO itself, ii) a conversion from density fluctuations to emissivity fluctuations, and iii) a spatial filter to model the finite viewing area of the diagnostic. We find significant attenuation of the density fluctuations. Transport results from GYRO are also compared to data to validate the simulations.

  14. Nonlinear gyrokinetic theory based on a new method and computation of the guiding-center orbit in tokamaks

    SciTech Connect

    Xu, Yingfeng Dai, Zongliang; Wang, Shaojie

    2014-04-15

    The nonlinear gyrokinetic theory in the tokamak configuration based on the two-step transform is developed; in the first step, we transform the magnetic potential perturbation to the Hamiltonian part, and in the second step, we transform away the gyroangle-dependent part of the perturbed Hamiltonian. Then the I-transform method is used to decoupled the perturbation part of the motion from the unperturbed motion. The application of the I-transform method to the computation of the guiding-center orbit and the guiding-center distribution function in tokamaks is presented. It is demonstrated that the I-transform method of the orbit computation which involves integrating only along the unperturbed orbit agrees with the conventional method which integrates along the full orbit. A numerical code based on the I-transform method is developed and two numerical examples are given to verify the new method.

  15. On the nature of radial transport across sheared zonal flows in electrostatic ion-temperature-gradient gyrokinetic tokamak turbulence

    SciTech Connect

    Sanchez, Raul; Newman, David E; Leboeuf, Jean-Noel; Carreras, Benjamin A; Decyk, Viktor

    2009-01-01

    It is argued that the usual understanding of the suppression of radial turbulent transport across a sheared zonal flow based on a reduction in effective transport coefficients is, by itself, incomplete. By means of toroidal gyrokinetic simulations of electrostatic, ion-temperature-gradient turbulence, it is found instead that the character of the radial transport is altered fundamentally by the presence of a sheared zonal flow, changing from diffusive to anticorrelated and subdiffusive. Furthermore, if the flows are self-consistently driven by the turbulence via the Reynolds stresses (in contrast to being induced externally), radial transport becomes non-Gaussian as well. These results warrant a reevaluation of the traditional description of radial transport across sheared flows in tokamaks via effective transport coefficients, suggesting that such description is oversimplified and poorly captures the underlying dynamics, which may in turn compromise its predictive capabilities.

  16. On the nature of radial transport across sheared zonal flows in electrostatic ion-temperature-gradient gyrokinetic tokamak plasma turbulence

    SciTech Connect

    Sanchez, R.; Newman, D. E.; Leboeuf, J.-N.; Carreras, B. A.; Decyk, V. K.

    2009-05-15

    It is argued that the usual understanding of the suppression of radial turbulent transport across a sheared zonal flow based on a reduction in effective transport coefficients is, by itself, incomplete. By means of toroidal gyrokinetic simulations of electrostatic, ion-temperature-gradient turbulence, it is found instead that the character of the radial transport is altered fundamentally by the presence of a sheared zonal flow, changing from diffusive to anticorrelated and subdiffusive. Furthermore, if the flows are self-consistently driven by the turbulence via the Reynolds stresses (in contrast to being induced externally), radial transport becomes non-Gaussian as well. These results warrant a reevaluation of the traditional description of radial transport across sheared flows in tokamaks via effective transport coefficients, suggesting that such description is oversimplified and poorly captures the underlying dynamics, which may in turn compromise its predictive capabilities.

  17. Gyrokinetic particle simulation of neoclassical transport in the pedestal/scrape-off region of a tokamak plasma

    NASA Astrophysics Data System (ADS)

    Ku, S.; Chang, C.-S.; Adams, M.; Cummings, J.; Hinton, F.; Keyes, D.; Klasky, S.; Lee, W.; Lin, Z.; Parker, S.; CPES Team

    2006-09-01

    A gyrokinetic neoclassical solution for a diverted tokamak edge plasma has been obtained for the first time using the massively parallel Jaguar XT3 computer at Oak Ridge National Laboratory. The solutions show similar characteristics to the experimental observations: electric potential is positive in the scrape-off layer and negative in the H-mode layer, and the parallel rotation is positive in the scrape-off layer and at the inside boundary of the H-mode layer. However, the solution also makes a new physical discovery that there is a strong ExB convective flow in the scrape-off plasma. A general introduction to the edge simulation problem is also presented.

  18. Gyrokinetic particle simulation of neoclassical transport in the pedestal/scrape-off region of a tokamak plasma

    SciTech Connect

    Adams, Mark; Chang, C. S.; Cummings, J.; Hinton, F.; Keyes, David E; Klasky, Scott A; Ku, S.; Lee, W. W.; Lin, Z.; Parker, Scott; CPES Team, the

    2006-01-01

    A gyrokinetic neoclassical solution for a diverted tokamak edge plasma has been obtained for the first time using the massively parallel Jaguar XT3 computer at Oak Ridge National Laboratory. The solutions show similar characteristics to the experimental observations: electric potential is positive in the scrape-off layer and negative in the H-mode layer, and the parallel rotation is positive in the scrape-off layer and at the inside boundary of the H-mode layer. However, the solution also makes a new physical discovery that there is a strong ExB convective flow in the scrape-off plasma. A general introduction to the edge simulation problem is also presented.

  19. Multi-scale gyrokinetic simulation of tokamak plasmas: enhanced heat loss due to cross-scale coupling of plasma turbulence

    NASA Astrophysics Data System (ADS)

    Howard, N. T.; Holland, C.; White, A. E.; Greenwald, M.; Candy, J.

    2016-01-01

    The transport of heat in laboratory and astrophysical plasmas is dominated by the complex nonlinear dynamics of plasma turbulence. In magnetically confined plasmas used for fusion energy research, turbulence is responsible for cross-field transport that limits the performance of tokamak reactors. We report a set of novel gyrokinetic simulations that capture ion and electron-scale turbulence simultaneously, revealing the dynamics of cross-scale energy transfer and zonal flow modification that give rise to heat losses. Multi-scale simulations are required to match experimental ion and electron heat fluxes and electron profile stiffness, establishing the applicability of the newly discovered physics to experiment. Importantly, these results provide a likely explanation for the loss of electron heat from tokamak plasmas, the ‘great unsolved problem’ (Bachelor et al (2007 Plasma Sci. Technol. 9 312-87)) in plasma turbulence and the projected dominant loss channel in ITER.

  20. A correlation electron cyclotron emission diagnostic and the importance of multifield fluctuation measurements for testing nonlinear gyrokinetic turbulence simulations

    SciTech Connect

    White, A. E.; Schmitz, L.; Peebles, W. A.; Carter, T. A.; Rhodes, T. L.; Doyle, E. J.; Gourdain, P. A.; Hillesheim, J. C.; Wang, G.; Holland, C.; Tynan, G. R.; Austin, M. E.; McKee, G. R.; Shafer, M. W.; Burrell, K. H.; Candy, J.; DeBoo, J. C.; Prater, R.; Staebler, G. M.; Waltz, R. E.

    2008-10-15

    A correlation electron cyclotron emission (CECE) diagnostic has been used to measure local, turbulent fluctuations of the electron temperature in the core of DIII-D plasmas. This paper describes the hardware and testing of the CECE diagnostic and highlights the importance of measurements of multifield fluctuation profiles for the testing and validation of nonlinear gyrokinetic codes. The process of testing and validating such codes is critical for extrapolation to next-step fusion devices. For the first time, the radial profiles of electron temperature and density fluctuations are compared to nonlinear gyrokinetic simulations. The CECE diagnostic at DIII-D uses correlation radiometry to measure the rms amplitude and spectrum of the electron temperature fluctuations. Gaussian optics are used to produce a poloidal spot size with w{sub o}{approx}1.75 cm in the plasma. The intermediate frequency filters and the natural linewidth of the EC emission determine the radial resolution of the CECE diagnostic, which can be less than 1 cm. Wavenumbers resolved by the CECE diagnostic are k{sub {theta}}{<=}1.8 cm{sup -1} and k{sub r}{<=}4 cm{sup -1}, relevant for studies of long-wavelength turbulence associated with the trapped electron mode and the ion temperature gradient mode. In neutral beam heated L-mode plasmas, core electron temperature fluctuations in the region 0.5

  1. A correlation electron cyclotron emission diagnostic and the importance of multifield fluctuation measurements for testing nonlinear gyrokinetic turbulence simulationsa)

    NASA Astrophysics Data System (ADS)

    White, A. E.; Schmitz, L.; Peebles, W. A.; Carter, T. A.; Rhodes, T. L.; Doyle, E. J.; Gourdain, P. A.; Hillesheim, J. C.; Wang, G.; Holland, C.; Tynan, G. R.; Austin, M. E.; McKee, G. R.; Shafer, M. W.; Burrell, K. H.; Candy, J.; DeBoo, J. C.; Prater, R.; Staebler, G. M.; Waltz, R. E.; Makowski, M. A.

    2008-10-01

    A correlation electron cyclotron emission (CECE) diagnostic has been used to measure local, turbulent fluctuations of the electron temperature in the core of DIII-D plasmas. This paper describes the hardware and testing of the CECE diagnostic and highlights the importance of measurements of multifield fluctuation profiles for the testing and validation of nonlinear gyrokinetic codes. The process of testing and validating such codes is critical for extrapolation to next-step fusion devices. For the first time, the radial profiles of electron temperature and density fluctuations are compared to nonlinear gyrokinetic simulations. The CECE diagnostic at DIII-D uses correlation radiometry to measure the rms amplitude and spectrum of the electron temperature fluctuations. Gaussian optics are used to produce a poloidal spot size with wo˜1.75 cm in the plasma. The intermediate frequency filters and the natural linewidth of the EC emission determine the radial resolution of the CECE diagnostic, which can be less than 1 cm. Wavenumbers resolved by the CECE diagnostic are kθ≤1.8 cm-1 and kr≤4 cm-1, relevant for studies of long-wavelength turbulence associated with the trapped electron mode and the ion temperature gradient mode. In neutral beam heated L-mode plasmas, core electron temperature fluctuations in the region 0.5

  2. A correlation electron cyclotron emission diagnostic and the importance of multifield fluctuation measurements for testing nonlinear gyrokinetic turbulence simulations.

    PubMed

    White, A E; Schmitz, L; Peebles, W A; Carter, T A; Rhodes, T L; Doyle, E J; Gourdain, P A; Hillesheim, J C; Wang, G; Holland, C; Tynan, G R; Austin, M E; McKee, G R; Shafer, M W; Burrell, K H; Candy, J; DeBoo, J C; Prater, R; Staebler, G M; Waltz, R E; Makowski, M A

    2008-10-01

    A correlation electron cyclotron emission (CECE) diagnostic has been used to measure local, turbulent fluctuations of the electron temperature in the core of DIII-D plasmas. This paper describes the hardware and testing of the CECE diagnostic and highlights the importance of measurements of multifield fluctuation profiles for the testing and validation of nonlinear gyrokinetic codes. The process of testing and validating such codes is critical for extrapolation to next-step fusion devices. For the first time, the radial profiles of electron temperature and density fluctuations are compared to nonlinear gyrokinetic simulations. The CECE diagnostic at DIII-D uses correlation radiometry to measure the rms amplitude and spectrum of the electron temperature fluctuations. Gaussian optics are used to produce a poloidal spot size with w(o) approximately 1.75 cm in the plasma. The intermediate frequency filters and the natural linewidth of the EC emission determine the radial resolution of the CECE diagnostic, which can be less than 1 cm. Wavenumbers resolved by the CECE diagnostic are k(theta) < or = 1.8 cm(-1) and k(r) < or = 4 cm(-1), relevant for studies of long-wavelength turbulence associated with the trapped electron mode and the ion temperature gradient mode. In neutral beam heated L-mode plasmas, core electron temperature fluctuations in the region 0.5 < r/a < 0.9, increase with radius from approximately 0.5% to approximately 2%, similar to density fluctuations that are measured simultaneously with beam emission spectroscopy. After incorporating "synthetic diagnostics" to effectively filter the code output, the simulations reproduce the characteristics of the turbulence and transport at one radial location r/a = 0.5, but not at a second location, r/a = 0.75. These results illustrate that measurements of the profiles of multiple fluctuating fields can provide a significant constraint on the turbulence models employed by the code. PMID:19044712

  3. Gyrokinetic simulation of global and local Alfven eigenmodes driven by energetic particles in a DIII-D discharge

    SciTech Connect

    Bass, E. M.; Waltz, R. E.

    2013-01-15

    The unstable spectrum of Alfven eigenmodes (AEs) driven by neutral beam-sourced energetic particles (EPs) in a benchmark DIII-D discharge (142111) is calculated in a fully gyrokinetic model using the GYRO code's massively parallel linear eigenvalue solver. One cycle of the slow (equilibrium scale) frequency sweep of the reverse shear Alfven eigenmode (RSAE) at toroidal mode number n=3 is mapped. The RSAE second harmonic and an unstable beta-induced Alfven eigenmode (BAE) are simultaneously tracked alongside the primary RSAE. An observed twist in the eigenmode pattern, caused mostly by shear in the driving EP profile, is shown through artificially varying the E Multiplication-Sign B rotational velocity shear to depend generally on shear in the local wave phase velocity. Coupling to the BAE and to the toroidal Alfven eigenmode limit the RSAE frequency sweeps at the lower and upper end, respectively. While the present fully gyrokinetic model (including thermal ions and electrons) constitutes the best treatment of compressibility physics available, the BAE frequency is overpredicted by about 20% against experiment here and is found to be sensitive to energetic beam ion pressure. The RSAE frequency is more accurately matched except when it is limited by the BAE. Simulations suggest that the experiment is very close to marginal AE stability at points of RSAE-BAE coupling. A recipe for comparing the radial profile of quasilinear transport flux from local modes to that from global modes paves the way for the development of a stiff (critical gradient) local AE transport model based on local mode stability thresholds.

  4. Marcus equation

    DOE R&D Accomplishments Database

    1998-09-21

    In the late 1950s to early 1960s Rudolph A. Marcus developed a theory for treating the rates of outer-sphere electron-transfer reactions. Outer-sphere reactions are reactions in which an electron is transferred from a donor to an acceptor without any chemical bonds being made or broken. (Electron-transfer reactions in which bonds are made or broken are referred to as inner-sphere reactions.) Marcus derived several very useful expressions, one of which has come to be known as the Marcus cross-relation or, more simply, as the Marcus equation. It is widely used for correlating and predicting electron-transfer rates. For his contributions to the understanding of electron-transfer reactions, Marcus received the 1992 Nobel Prize in Chemistry. This paper discusses the development and use of the Marcus equation. Topics include self-exchange reactions; net electron-transfer reactions; Marcus cross-relation; and proton, hydride, atom and group transfers.

  5. Marcus equation

    SciTech Connect

    1998-11-01

    In the late 1950s to early 1960s Rudolph A. Marcus developed a theory for treating the rates of outer-sphere electron-transfer reactions. Outer-sphere reactions are reactions in which an electron is transferred from a donor to an acceptor without any chemical bonds being made or broken. (Electron-transfer reactions in which bonds are made or broken are referred to as inner-sphere reactions.) Marcus derived several very useful expressions, one of which has come to be known as the Marcus cross-relation or, more simply, as the Marcus equation. It is widely used for correlating and predicting electron-transfer rates. For his contributions to the understanding of electron-transfer reactions, Marcus received the 1992 Nobel Prize in Chemistry. This paper discusses the development and use of the Marcus equation. Topics include self-exchange reactions; net electron-transfer reactions; Marcus cross-relation; and proton, hydride, atom and group transfers.

  6. Center for Gyrokinetic Particle Simulations of Turbulent Transport in Burning Plasmas

    SciTech Connect

    Scott, Parker

    2011-05-02

    This is the Final Technical Report for University of Colorado's portion of the SciDAC project 'Center for Gyrokinetic Particle Simulation of Turbulent Transport.' This is funded as a multi-institutional SciDAC Center and W.W. Lee at the Princeton Plasma Physics Laboratory is the lead Principal Investigator. Scott Parker is the local Principal Investigator for University of Colorado and Yang Chen is a Co-Principal Investigator. This is Cooperative Agreement DE-FC02-05ER54816. Research personnel include Yang Chen (Senior Research Associate), Jianying Lang (Graduate Research Associate, Ph.D. Physics Student) and Scott Parker (Associate Professor). Research includes core microturbulence studies of NSTX, simulation of trapped electron modes, development of efficient particle-continuum hybrid methods and particle convergence studies of electron temperature gradient driven turbulence simulations. Recently, the particle-continuum method has been extended to five-dimensions in GEM. We find that actually a simple method works quite well for the Cyclone base case with either fully kinetic or adiabatic electrons. Particles are deposited on a 5D phase-space grid using nearest-grid-point interpolation. Then, the value of delta-f is reset, but not the particle's trajectory. This has the effect of occasionally averaging delta-f of nearby (in the phase space) particles. We are currently trying to estimate the dissipation (or effective collision operator). We have been using GEM to study turbulence and transport in NSTX with realistic equilibrium density and temperature profiles, including impurities, magnetic geometry and ExB shear flow. Greg Rewoldt, PPPL, has developed a TRANSP interface for GEM that specifies the equilibrium profiles and parameters needed to run realistic NSTX cases. Results were reported at the American Physical Society - Division of Plasma Physics, and we are currently running convergence studies to ensure physical results. We are also studying the effect of

  7. Reduced fluid-kinetic equations for low-frequency dynamics, magnetic reconnection, and electron heating in low-beta plasmas

    SciTech Connect

    Zocco, Alessandro; Schekochihin, Alexander A.

    2011-10-15

    A minimal model for magnetic reconnection and, generally, low-frequency dynamics in low-beta plasmas is proposed. The model combines analytical and computational simplicity with physical realizability: it is a rigorous limit of gyrokinetics for plasma beta of order the electron-ion mass ratio. The model contains collisions and can be used both in the collisional and collisionless reconnection regimes. It includes gyrokinetic ions (not assumed cold) and allows for the topological rearrangement of the magnetic field lines by either resistivity or electron inertia, whichever predominates. The two-fluid dynamics are coupled to electron kinetics--electrons are not assumed isothermal and are described by a reduced drift-kinetic equation. The model, therefore allows for irreversibility and conversion of magnetic energy into electron heat via parallel phase mixing in velocity space. An analysis of the exchanges between various forms of free energy and its conversion into electron heat is provided. It is shown how all relevant linear waves and regimes of the tearing instability (collisionless, semicollisional, and fully resistive) are recovered in various limits of our model. An efficient way to simulate our equations numerically is proposed, via the Hermite representation of the velocity space. It is shown that small scales in velocity space will form, giving rise to a shallow Hermite-space spectrum, whence it is inferred that, for steady-state or sufficiently slow dynamics, the electron heating rate will remain finite in the limit of vanishing collisionality.

  8. LIGKA: A linear gyrokinetic code for the description of background kinetic and fast particle effects on the MHD stability in tokamaks

    SciTech Connect

    Lauber, Ph. Guenter, S.; Koenies, A.; Pinches, S.D.

    2007-09-10

    In a plasma with a population of super-thermal particles generated by heating or fusion processes, kinetic effects can lead to the additional destabilisation of MHD modes or even to additional energetic particle modes. In order to describe these modes, a new linear gyrokinetic MHD code has been developed and tested, LIGKA (linear gyrokinetic shear Alfven physics) [Ph. Lauber, Linear gyrokinetic description of fast particle effects on the MHD stability in tokamaks, Ph.D. Thesis, TU Muenchen, 2003; Ph. Lauber, S. Guenter, S.D. Pinches, Phys. Plasmas 12 (2005) 122501], based on a gyrokinetic model [H. Qin, Gyrokinetic theory and computational methods for electromagnetic perturbations in tokamaks, Ph.D. Thesis, Princeton University, 1998]. A finite Larmor radius expansion together with the construction of some fluid moments and specification to the shear Alfven regime results in a self-consistent, electromagnetic, non-perturbative model, that allows not only for growing or damped eigenvalues but also for a change in mode-structure of the magnetic perturbation due to the energetic particles and background kinetic effects. Compared to previous implementations [H. Qin, mentioned above], this model is coded in a more general and comprehensive way. LIGKA uses a Fourier decomposition in the poloidal coordinate and a finite element discretisation in the radial direction. Both analytical and numerical equilibria can be treated. Integration over the unperturbed particle orbits is performed with the drift-kinetic HAGIS code [S.D. Pinches, Ph.D. Thesis, The University of Nottingham, 1996; S.D. Pinches et al., CPC 111 (1998) 131] which accurately describes the particles' trajectories. This allows finite-banana-width effects to be implemented in a rigorous way since the linear formulation of the model allows the exchange of the unperturbed orbit integration and the discretisation of the perturbed potentials in the radial direction. Successful benchmarks for toroidal Alfven

  9. A time-dependent gyro-kinetic model of thermal ion upflows in the high-latitude F region

    NASA Technical Reports Server (NTRS)

    Loranc, M.; St.-Maurice, J.-P.

    1994-01-01

    Ample evidence supports the significance of the high-latitude ionospheric contribution to magnetospheric plasma. Assuming flux conservation along a flux tube, the upward field-aligned ion flows observed in the magnetosphere require high-latitude ionospheric field-aligned ion upflows of the order of 10(exp 8) to 10(exp 9)/sq cm/s. Since radar and satellite observations of high-latitude F region flows at times exceed this flux requirement by an order of magnitude, the thermal ionospheric upflows are not simply the ionospheric response to a magnetospheric flux requirement. Several ionospheric ion upflow mechanisms have been proposed, but simulations based on fluid theory do not reproduce all the observed features of ionospheric ion upflows. Certain asymmetries in the statistical morphology of high-latitude F region ion upflows suggest that the ion upflows may be generated by ion-neutral frictional heating. We developed a single-component (O(+)), time-dependent gyro-kinetic model of the high-latitude F region response to frictional heating in which the neutral exobase is a discontinuous boundary between fully collisional and collisionless plasmas. The concept of a discontinuous neutreal exobase and the assumption of a constant and uniform polarization electric field reduce the ion velocity distribution function, from which we can compute the ion density, parallel velocity, parallel and perpendicular temperature, and parallel flux. Using our model, we simulated the response of a convecting flux tube between 500 km and 2500 km to various frictional heating inputs; the results were both qualitatively and quantitatively different from fluid model results, which may indicate an inadequacy of the fluid theory approach. The gyro-kinetic frictional heating model responses to the various simulations were qualitatively similar: (1) initial perturbations of all the modeled parameters propagated rapidly up the flux tube, (2) transient values of the ion parallel velocity

  10. Comment on 'Nonlinear gyrokinetic theory with polarization drift' [Phys. Plasmas 17, 082304 (2010)

    SciTech Connect

    Leerink, S.; Parra, F. I.; Heikkinen, J. A.

    2010-12-15

    In this comment, we show that by using the discrete particle distribution function the changes of the phase-space volume of gyrocenter coordinates due to the fluctuating ExB velocity do not explicitly appear in the Poisson equation and the [Sosenko et al., Phys. Scr. 64, 264 (2001)] result is recovered. It is demonstrated that there is no contradiction between the work presented by Sosenko et al. and the work presented by [Wang et al., Phys. Plasmas 17, 082304 (2010)].

  11. Influence of the parallel nonlinearity on zonal flows and heat transport in global gyrokinetic particle-in-cell simulations

    SciTech Connect

    Jolliet, S.; McMillan, B. F.; Vernay, T.; Villard, L.; Hatzky, R.; Bottino, A.; Angelino, P.

    2009-07-15

    In this paper, the influence of the parallel nonlinearity on zonal flows and heat transport in global particle-in-cell ion-temperature-gradient simulations is studied. Although this term is in theory orders of magnitude smaller than the others, several authors [L. Villard, P. Angelino, A. Bottino et al., Plasma Phys. Contr. Fusion 46, B51 (2004); L. Villard, S. J. Allfrey, A. Bottino et al., Nucl. Fusion 44, 172 (2004); J. C. Kniep, J. N. G. Leboeuf, and V. C. Decyck, Comput. Phys. Commun. 164, 98 (2004); J. Candy, R. E. Waltz, S. E. Parker et al., Phys. Plasmas 13, 074501 (2006)] found different results on its role. The study is performed using the global gyrokinetic particle-in-cell codes TORB (theta-pinch) [R. Hatzky, T. M. Tran, A. Koenies et al., Phys. Plasmas 9, 898 (2002)] and ORB5 (tokamak geometry) [S. Jolliet, A. Bottino, P. Angelino et al., Comput. Phys. Commun. 177, 409 (2007)]. In particular, it is demonstrated that the parallel nonlinearity, while important for energy conservation, affects the zonal electric field only if the simulation is noise dominated. When a proper convergence is reached, the influence of parallel nonlinearity on the zonal electric field, if any, is shown to be small for both the cases of decaying and driven turbulence.

  12. Gyrokinetic Simulation of Global and Local Alfv'en Eigenmodes Driven by Neutral Beam Injection in DIII-D

    NASA Astrophysics Data System (ADS)

    Bass, E. M.; Waltz, R. E.

    2012-10-01

    In ITER, convection of fusion-produced alpha particles by energetic particle (EP)-driven Alfv'en eigenmodes (AEs) risks wall damage and loss of alpha heating needed for ignition. We examine beam-excited AEs and induced quasilinear transport in a DIII-D AE experiment using the gyrokinetic code GYRO [1]. Global, linear eigenvalue simulations show reverse-shear AEs (RSAEs), toroidal AEs, and beta-induced AEs interacting over one (equilibrium time scale) RSAE frequency sweep. Eigenfunction modifications over MHD, including a poloidal twist and broad AE footprint observed in electron cyclotron emission imaging [2], show the value of a kinetic approach. Under a simple quasilinear saturation assumption, a sequence of comparatively inexpensive local simulations quantitatively recreates some global features, notably the quasilinear transport footprint. Accordingly, we present here a stiff EP transport model where AEs limit the EP density gradient to the local stability threshold, and a TGLF-driven quasilinear model elsewhere. The model gives some``worst case'' predictions of the AE-limited alpha profile in ITER.[4pt] [1] J. Candy and R.E. Waltz, Phys. Rev. Lett. 91, 045001 (2003). [2] B.J. Tobias, et al., Phys. Rev. Lett. 106, 075003 (2011).

  13. Verification and validation of linear gyrokinetic simulation of Alfv n eigenmodes in the DIII-D tokamak

    SciTech Connect

    Spong, Donald A; Bass, Eric; Deng, Wenjun; Heidbrink, W.; Lin, Zhihong; Tobias, Ben; Van Zeeland, Michael; Austin, M. E.; Domier, C. W.; Luhmann, N.C.

    2012-01-01

    A verification and validation study is carried out for a sequence of reversed shear Alfven instability time slices. The mode frequency increases in time as the minimum (q{sub min}) in the safety factor profile decreases. Profiles and equilibria are based upon reconstructions of DIII-D discharge (No.142111) in which many such frequency up-sweeping modes were observed. Calculations of the frequency and mode structure evolution from two gyrokinetic codes, GTC and GYRO, and a gyro-Landau fluid code TAEFL are compared. The experimental mode structure of the instability was measured using time-resolved two-dimensional electron cyclotron emission imaging. The three models reproduce the frequency upsweep event within {+-}10% of each other, and the average of the code predictions is within {+-}8% of the measurements; growth rates are predicted that are consistent with the observed spectral line widths. The mode structures qualitatively agree with respect to radial location and width, dominant poloidal mode number, ballooning structure, and the up-down asymmetry, with some remaining differences in the details. Such similarities and differences between the predictions of the different models and the experimental results are a valuable part of the verification/validation process and help to guide future development of the modeling efforts.

  14. Interaction between neoclassical effects and ion temperature gradient turbulence in gradient- and flux-driven gyrokinetic simulations

    NASA Astrophysics Data System (ADS)

    Oberparleiter, M.; Jenko, F.; Told, D.; Doerk, H.; Görler, T.

    2016-04-01

    Neoclassical and turbulent transport in tokamaks has been studied extensively over the past decades, but their possible interaction remains largely an open question. The two are only truly independent if the length scales governing each of them are sufficiently separate, i.e., if the ratio ρ* between ion gyroradius and the pressure gradient scale length is small. This is not the case in particularly interesting regions such as transport barriers. Global simulations of a collisional ion-temperature-gradient-driven microturbulence performed with the nonlinear global gyrokinetic code Gene are presented. In particular, comparisons are made between systems with and without neoclassical effects. In fixed-gradient simulations, the modified radial electric field is shown to alter the zonal flow pattern such that a significant increase in turbulent transport is observed for ρ*≳1 /300 . Furthermore, the dependency of the flux on the collisionality changes. In simulations with fixed power input, we find that the presence of neoclassical effects decreases the frequency and amplitude of intermittent turbulent transport bursts (avalanches) and thus plays an important role for the self-organisation behaviour.

  15. 3D electrostatic gyrokinetic electron and fully kinetic ion simulation of lower-hybrid drift instability of Harris current sheet

    DOE PAGESBeta

    Wang, Zhenyu; Lin, Yu; Wang, Xueyi; Tummel, Kurt; Chen, Liu

    2016-07-07

    The eigenmode stability properties of three-dimensional lower-hybrid-drift-instabilities (LHDI) in a Harris current sheet with a small but finite guide magnetic field have been systematically studied by employing the gyrokinetic electron and fully kinetic ion (GeFi) particle-in-cell (PIC) simulation model with a realistic ion-to-electron mass ratio mi/me. In contrast to the fully kinetic PIC simulation scheme, the fast electron cyclotron motion and plasma oscillations are systematically removed in the GeFi model, and hence one can employ the realistic mi/me. The GeFi simulations are benchmarked against and show excellent agreement with both the fully kinetic PIC simulation and the analytical eigenmode theory. Our studies indicate that, for small wavenumbers, ky, along the current direction, the most unstable eigenmodes are peaked at the location wheremore » $$\\vec{k}$$• $$\\vec{B}$$ =0, consistent with previous analytical and simulation studies. Here, $$\\vec{B}$$ is the equilibrium magnetic field and $$\\vec{k}$$ is the wavevector perpendicular to the nonuniformity direction. As ky increases, however, the most unstable eigenmodes are found to be peaked at $$\\vec{k}$$ •$$\\vec{B}$$ ≠0. Additionally, the simulation results indicate that varying mi/me, the current sheet width, and the guide magnetic field can affect the stability of LHDI. Simulations with the varying mass ratio confirm the lower hybrid frequency and wave number scalings.« less

  16. 3D electrostatic gyrokinetic electron and fully kinetic ion simulation of lower-hybrid drift instability of Harris current sheet

    NASA Astrophysics Data System (ADS)

    Wang, Zhenyu; Lin, Yu; Wang, Xueyi; Tummel, Kurt; Chen, Liu

    2016-07-01

    The eigenmode stability properties of three-dimensional lower-hybrid-drift-instabilities (LHDI) in a Harris current sheet with a small but finite guide magnetic field have been systematically studied by employing the gyrokinetic electron and fully kinetic ion (GeFi) particle-in-cell (PIC) simulation model with a realistic ion-to-electron mass ratio mi/me . In contrast to the fully kinetic PIC simulation scheme, the fast electron cyclotron motion and plasma oscillations are systematically removed in the GeFi model, and hence one can employ the realistic mi/me . The GeFi simulations are benchmarked against and show excellent agreement with both the fully kinetic PIC simulation and the analytical eigenmode theory. Our studies indicate that, for small wavenumbers, ky, along the current direction, the most unstable eigenmodes are peaked at the location where k →.B → =0 , consistent with previous analytical and simulation studies. Here, B → is the equilibrium magnetic field and k → is the wavevector perpendicular to the nonuniformity direction. As ky increases, however, the most unstable eigenmodes are found to be peaked at k →.B → ≠0 . In addition, the simulation results indicate that varying mi/me , the current sheet width, and the guide magnetic field can affect the stability of LHDI. Simulations with the varying mass ratio confirm the lower hybrid frequency and wave number scalings.

  17. Validation studies of gyrokinetic ITG and TEM turbulence simulations in a JT-60U tokamak using multiple flux matching

    NASA Astrophysics Data System (ADS)

    Nakata, Motoki; Honda, Mitsuru; Yoshida, Maiko; Urano, Hajime; Nunami, Masanori; Maeyama, Shinya; Watanabe, Tomo-Hiko; Sugama, Hideo

    2016-08-01

    Quantitative validation studies of flux-tube gyrokinetic Vlasov simulations on ion and electron heat transport are carried out for the JT-60U tokamak experiment. The ion temperature gradient (ITG) and/or trapped electron modes (TEM) driven turbulent transport and zonal flow generations are investigated for an L-mode plasma in the local turbulence limit with a sufficiently small normalized ion thermal gyroradius and weak mean radial electric fields. Nonlinear turbulence simulations by the GKV code successfully reproduce radial profiles of the ion and electron energy fluxes in the core region. The numerical results show that the TEM-driven zonal flow generation in the outer region is more significant than that in the core region with ITG- and ITG–TEM-dominated turbulence, leading to moderate transport shortfall of the ion energy flux. Error levels in the prediction of the ion and electron temperature gradient profiles in the core region are estimated as less than +/- 30% , based on a multiple flux matching technique, where the simulated ion and electron energy fluxes are simultaneously matched to the experimental values.

  18. Comparative gyrokinetic analysis of JET baseline H-mode core plasmas with carbon wall and ITER-like wall

    NASA Astrophysics Data System (ADS)

    Tegnered, D.; Strand, P.; Nordman, H.; Giroud, C.; Kim, Hyun-Tae; Maddison, G. P.; Romanelli, M.; Szepesi, G.; Contributors, JET

    2016-04-01

    Following the change of plasma facing components at JET from a carbon wall (CW) to a metal ITER-like wall (ILW) a deterioration of global confinement has been observed for H-mode baseline experiments. The deterioration has been correlated with a degradation of pedestal confinement with lower electron temperatures at the top of the edge barrier region. In order to investigate the change in core confinement, heat transport due to Ion Temperature Gradient (ITG)/Trapped Electron Mode (TEM) turbulence is investigated using the gyrokinetic code GENE. Two pairs of CW and ILW discharges that are matched according to several global parameters are simulated at mid radius. The simulations included effects of collisions, finite β, realistic geometries, and impurities. A sensitivity study is performed with respect to the key dimensionless parameters in the matched pairs. The combined effect of the relative change in these parameters is that the ITG mode is destabilized in the ILW discharges compared to the CW discharges. This is also reflected in nonlinear simulations where the ILW discharges show higher normalized ion and electron heat fluxes and larger stiffness. The ion energy confinement time within ρ =0.5 is found to be comparable while the electron confinement time is shorter for the ILW discharges. The core confinement in the ILW discharges is expected to improve if the edge pedestal is recovered since that would favourably change the key plasma parameters that now serve to destabilize them.

  19. Extended rate equations

    SciTech Connect

    Shore, B.W.

    1981-01-30

    The equations of motion are discussed which describe time dependent population flows in an N-level system, reviewing the relationship between incoherent (rate) equations, coherent (Schrodinger) equations, and more general partially coherent (Bloch) equations. Approximations are discussed which replace the elaborate Bloch equations by simpler rate equations whose coefficients incorporate long-time consequences of coherence.

  20. The effects of spatial sampling on random noise for gyrokinetic PIC simulations in real space

    NASA Astrophysics Data System (ADS)

    Kiviniemi, T. P.; Sauerwein, U.

    2016-06-01

    We study the effects of cloud-in-cell sampling and gyroaveraging on random noise in real space (as opposed to the common Fourier space presentation), and show that together, these can reduce the noise by a factor of 3 compared to nearest grid point sampling without gyroaveraging. Hence an order of magnitude less test particles are needed for the given noise level. We derive equations for noise level as a function of Larmor radius and also investigate the effect of gyroaveraging on noise in local gradients. The effect of number of gyropoints on noise is also discussed.

  1. Astrophysical Gyrokinetics: Kinetic and Fluid Turbulent Cascades In Magentized Weakly Collisional Plasmas

    SciTech Connect

    Schekochihin, A. A.; Cowley, S. C.; Dorland, W.; Hammett, G. W.; Howes, G. G.; Quataert, E.; Tatsuno, T.

    2009-04-23

    This paper presents a theoretical framework for understanding plasma turbulence in astrophysical plasmas. It is motivated by observations of electromagnetic and density fluctuations in the solar wind, interstellar medium and galaxy clusters, as well as by models of particle heating in accretion disks. All of these plasmas and many others have turbulentmotions at weakly collisional and collisionless scales. The paper focuses on turbulence in a strong mean magnetic field. The key assumptions are that the turbulent fluctuations are small compared to the mean field, spatially anisotropic with respect to it and that their frequency is low compared to the ion cyclotron frequency. The turbulence is assumed to be forced at some system-specific outer scale. The energy injected at this scale has to be dissipated into heat, which ultimately cannot be accomplished without collisions. A kinetic cascade develops that brings the energy to collisional scales both in space and velocity. The nature of the kinetic cascade in various scale ranges depends on the physics of plasma fluctuations that exist there. There are four special scales that separate physically distinct regimes: the electron and ion gyroscales, the mean free path and the electron diffusion scale. In each of the scale ranges separated by these scales, the fully kinetic problem is systematically reduced to a more physically transparent and computationally tractable system of equations, which are derived in a rigorous way. In the "inertial range" above the ion gyroscale, the kinetic cascade separates into two parts: a cascade of Alfvenic fluctuations and a passive cascade of density and magnetic-fieldstrength fluctuations. The former are governed by the Reduced Magnetohydrodynamic (RMHD) equations at both the collisional and collisionless scales; the latter obey a linear kinetic equation along the (moving) field lines associated with the Alfvenic component (in the collisional limit, these compressive fluctuations

  2. Non-local gyrokinetic model of linear ion-temperature-gradient modes

    SciTech Connect

    Moradi, S.; Anderson, J.

    2012-08-15

    The non-local properties of anomalous transport in fusion plasmas are still an elusive topic. In this work, a theory of non-local linear ion-temperature-gradient (ITG) drift modes while retaining non-adiabatic electrons and finite temperature gradients is presented, extending the previous work [S. Moradi et al., Phys. Plasmas 18, 062106 (2011)]. A dispersion relation is derived to quantify the effects on the eigenvalues of the unstable ion temperature gradient modes and non-adiabatic electrons on the order of the fractional velocity operator in the Fokker-Planck equation. By solving this relation for a given eigenvalue, it is shown that as the linear eigenvalues of the modes increase, the order of the fractional velocity derivative deviates from two and the resulting equilibrium probability density distribution of the plasma, i.e., the solution of the Fokker-Planck equation, deviates from a Maxwellian and becomes Levy distributed. The relative effect of the real frequency of the ITG mode on the deviation of the plasma from Maxwellian is larger than from the growth rate. As was shown previously the resulting Levy distribution of the plasma may in turn significantly alter the transport as well.

  3. Generalized parallel heat transport equations in collisional to weakly collisional plasmas

    SciTech Connect

    Zawaideh, E.; Kim, N.S.; Najmabadi, F.

    1988-11-01

    A new set of two-fluid heat transport equations that is valid from collisional to weakly collisional limits is derived. Starting from gyrokinetic equations in flux coordinates, a set of moment equations describing plasma energy transport along the field lines of a space- and time-dependent magnetic field is derived. No restrictions on the anisotropy of the ion distribution function or collisionality are imposed. In the highly collisional limit, these equations reduce to the classical heat conduction equation (e.g., Spitzer and Haerm or Braginskii), while in the weakly collisional limit, they describe a saturated heat flux (flux limited). Numerical examples comparing these equations with conventional heat transport equations show that in the limit where the ratio of the mean free path lambda to the scale length of the temperature gradient L/sub T/ approaches zero, there is no significant difference between the solutions of the new and conventional heat transport equations. As lambda/L/sub T/..-->..1, the conventional heat conduction equation contains a significantly larger error than (lambda/L/sub T/)/sup 2/. The error is found to be O(lambda/L)/sup 2/, where L is the smallest of the scale lengths of the gradient in the magnetic field, or the macroscopic plasma parameters (e.g., velocity scale length, temperature scale length, and density scale length). The accuracy of the flux-limited model depends significantly on the value of the flux limit parameter which, in general, is not known. The new set of equations shows that the flux-limited parameter is a function of the magnetic field and plasma parameter profiles.

  4. The effect of plasma shaping on turbulent transport and ExB shear quenching in nonlinear gyrokinetic simulations

    SciTech Connect

    Kinsey, J. E.; Waltz, R. E.; Candy, J.

    2007-10-15

    Nonlinear gyrokinetic simulations with kinetic electron dynamics are used to study the effects of plasma shaping on turbulent transport and ExB shear in toroidal geometry including the presence of kinetic electrons using the GYRO code. Over 120 simulations comprised of systematic scans were performed around several reference cases in the local, electrostatic, collisionless limit. Using a parameterized local equilibrium model for shaped geometry, the GYRO simulations show that elongation {kappa} (and its gradient) stabilizes the energy transport from ion temperature gradient (ITG) and trapped electron mode (TEM) instabilities at fixed midplane minor radius. For scans around a reference set of parameters, the GYRO ion energy diffusivity, in gyro-Bohm units, approximately follows a {kappa}{sup -1} scaling which is qualitatively similar to recent experimental energy confinement scalings. Most of the {kappa} scaling is due to the shear in the elongation rather than the local {kappa} itself. The {kappa} scaling for the electrons is found to vary and can be stronger or weaker than {kappa}{sup -1} depending on the wavenumber where the transport peaks. The {kappa} scaling is weaker when the energy diffusivity peaks at low wavenumbers and is stronger when the peak occurs at high wavenumbers. The simulations also demonstrate a nonlinear upshift in the critical temperature gradient as the elongation increases due to an increase in the residual zonal flow amplitude. Triangularity is found to be slightly destabilizing and its effect is strongest for highly elongated plasmas. Finally, we find less ExB shear is needed to quench the transport at high elongation and low aspect ratio. A new linear ExB shear quench rule, valid for shaped tokamak geometry, is presented.

  5. Quantitative comparison of electron temperature fluctuations to nonlinear gyrokinetic simulations in C-Mod Ohmic L-mode discharges

    NASA Astrophysics Data System (ADS)

    Sung, C.; White, A. E.; Mikkelsen, D. R.; Greenwald, M.; Holland, C.; Howard, N. T.; Churchill, R.; Theiler, C.

    2016-04-01

    Long wavelength turbulent electron temperature fluctuations (kyρs < 0.3) are measured in the outer core region (r/a > 0.8) of Ohmic L-mode plasmas at Alcator C-Mod [E. S. Marmar et al., Nucl. Fusion 49, 104014 (2009)] with a correlation electron cyclotron emission diagnostic. The relative amplitude and frequency spectrum of the fluctuations are compared quantitatively with nonlinear gyrokinetic simulations using the GYRO code [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] in two different confinement regimes: linear Ohmic confinement (LOC) regime and saturated Ohmic confinement (SOC) regime. When comparing experiment with nonlinear simulations, it is found that local, electrostatic ion-scale simulations (kyρs ≲ 1.7) performed at r/a ˜ 0.85 reproduce the experimental ion heat flux levels, electron temperature fluctuation levels, and frequency spectra within experimental error bars. In contrast, the electron heat flux is robustly under-predicted and cannot be recovered by using scans of the simulation inputs within error bars or by using global simulations. If both the ion heat flux and the measured temperature fluctuations are attributed predominantly to long-wavelength turbulence, then under-prediction of electron heat flux strongly suggests that electron scale turbulence is important for transport in C-Mod Ohmic L-mode discharges. In addition, no evidence is found from linear or nonlinear simulations for a clear transition from trapped electron mode to ion temperature gradient turbulence across the LOC/SOC transition, and also there is no evidence in these Ohmic L-mode plasmas of the "Transport Shortfall" [C. Holland et al., Phys. Plasmas 16, 052301 (2009)].

  6. Benchmark studies of the gyro-Landau-fluid code and gyro-kinetic codes on kinetic ballooning modes

    NASA Astrophysics Data System (ADS)

    Tang, T. F.; Xu, X. Q.; Ma, C. H.; Bass, E. M.; Holland, C.; Candy, J.

    2016-03-01

    A Gyro-Landau-Fluid (GLF) 3 + 1 model has been recently implemented in BOUT++ framework, which contains full Finite-Larmor-Radius effects, Landau damping, and toroidal resonance [Ma et al., Phys. Plasmas 22, 055903 (2015)]. A linear global beta scan has been conducted using the JET-like circular equilibria (cbm18 series), showing that the unstable modes are kinetic ballooning modes (KBMs). In this work, we use the GYRO code, which is a gyrokinetic continuum code widely used for simulation of the plasma microturbulence, to benchmark with GLF 3 + 1 code on KBMs. To verify our code on the KBM case, we first perform the beta scan based on "Cyclone base case parameter set." We find that the growth rate is almost the same for two codes, and the KBM mode is further destabilized as beta increases. For JET-like global circular equilibria, as the modes localize in peak pressure gradient region, a linear local beta scan using the same set of equilibria has been performed at this position for comparison. With the drift kinetic electron module in the GYRO code by including small electron-electron collision to damp electron modes, GYRO generated mode structures and parity suggest that they are kinetic ballooning modes, and the growth rate is comparable to the GLF results. However, a radial scan of the pedestal for a particular set of cbm18 equilibria, using GYRO code, shows different trends for the low-n and high-n modes. The low-n modes show that the linear growth rate peaks at peak pressure gradient position as GLF results. However, for high-n modes, the growth rate of the most unstable mode shifts outward to the bottom of pedestal and the real frequency of what was originally the KBMs in ion diamagnetic drift direction steadily approaches and crosses over to the electron diamagnetic drift direction.

  7. Gyrokinetic and kinetic particle-in-cell simulations of guide-field reconnection. I. Macroscopic effects of the electron flows

    SciTech Connect

    Muñoz, P. A. Kilian, P.; Büchner, J.; Told, D.; Jenko, F.

    2015-08-15

    In this work, we compare gyrokinetic (GK) with fully kinetic Particle-in-Cell (PIC) simulations of magnetic reconnection in the limit of strong guide field. In particular, we analyze the limits of applicability of the GK plasma model compared to a fully kinetic description of force free current sheets for finite guide fields (b{sub g}). Here, we report the first part of an extended comparison, focusing on the macroscopic effects of the electron flows. For a low beta plasma (β{sub i} = 0.01), it is shown that both plasma models develop magnetic reconnection with similar features in the secondary magnetic islands if a sufficiently high guide field (b{sub g} ≳ 30) is imposed in the kinetic PIC simulations. Outside of these regions, in the separatrices close to the X points, the convergence between both plasma descriptions is less restrictive (b{sub g} ≳ 5). Kinetic PIC simulations using guide fields b{sub g} ≲ 30 reveal secondary magnetic islands with a core magnetic field and less energetic flows inside of them in comparison to the GK or kinetic PIC runs with stronger guide fields. We find that these processes are mostly due to an initial shear flow absent in the GK initialization and negligible in the kinetic PIC high guide field regime, in addition to fast outflows on the order of the ion thermal speed that violate the GK ordering. Since secondary magnetic islands appear after the reconnection peak time, a kinetic PIC/GK comparison is more accurate in the linear phase of magnetic reconnection. For a high beta plasma (β{sub i} = 1.0) where reconnection rates and fluctuations levels are reduced, similar processes happen in the secondary magnetic islands in the fully kinetic description, but requiring much lower guide fields (b{sub g} ≲ 3)

  8. Investigating profile stiffness and critical gradients in shaped TCV discharges using local gyrokinetic simulations of turbulent transport

    NASA Astrophysics Data System (ADS)

    Merlo, G.; Brunner, S.; Sauter, O.; Camenen, Y.; Görler, T.; Jenko, F.; Marinoni, A.; Told, D.; Villard, L.

    2015-05-01

    The experimental observation made on the TCV tokamak of a significant confinement improvement in plasmas with negative triangularity (δ < 0) compared to those with standard positive triangularity has been interpreted in terms of different degrees of profile stiffness (Sauter et al 2014 Phys. Plasmas 21 055906) and/or different critical gradients. Employing the Eulerian gyrokinetic code GENE (Jenko et al 2000 Phys. Plasmas 7 1904), profile stiffness and critical gradients are studied under TCV relevant conditions. For the considered experimental discharges, trapped electron modes (TEMs) and electron temperature gradient (ETG) modes are the dominant microinstabilities, with the latter providing a significant contribution to the non-linear electron heat fluxes near the plasma edge. Two series of simulations with different levels of realism are performed, addressing the question of profile stiffness at various radial locations. Retaining finite collisionality, impurities and electromagnetic effects, as well as the physical electron-to-ion mass ratio are all necessary in order to approach the experimental flux measurements. However, flux-tube simulations are unable to fully reproduce the TCV results, pointing towards the need to carry out radially nonlocal (global) simulations, i.e. retaining finite machine size effects, in a future study. Some conclusions about the effect of triangularity can nevertheless be drawn based on the flux-tube results. In particular, the importance of considering the sensitivity to both temperature and density gradient is shown. The flux tube results show an increase of the critical gradients towards the edge, further enhanced when δ < 0, and they also appear to indicate a reduction of profile stiffness towards plasma edge.

  9. Gyrokinetic and kinetic particle-in-cell simulations of guide-field reconnection. I. Macroscopic effects of the electron flows

    NASA Astrophysics Data System (ADS)

    Muñoz, P. A.; Told, D.; Kilian, P.; Büchner, J.; Jenko, F.

    2015-08-01

    In this work, we compare gyrokinetic (GK) with fully kinetic Particle-in-Cell (PIC) simulations of magnetic reconnection in the limit of strong guide field. In particular, we analyze the limits of applicability of the GK plasma model compared to a fully kinetic description of force free current sheets for finite guide fields (bg). Here, we report the first part of an extended comparison, focusing on the macroscopic effects of the electron flows. For a low beta plasma (βi = 0.01), it is shown that both plasma models develop magnetic reconnection with similar features in the secondary magnetic islands if a sufficiently high guide field (bg ≳ 30) is imposed in the kinetic PIC simulations. Outside of these regions, in the separatrices close to the X points, the convergence between both plasma descriptions is less restrictive (bg ≳ 5). Kinetic PIC simulations using guide fields bg ≲ 30 reveal secondary magnetic islands with a core magnetic field and less energetic flows inside of them in comparison to the GK or kinetic PIC runs with stronger guide fields. We find that these processes are mostly due to an initial shear flow absent in the GK initialization and negligible in the kinetic PIC high guide field regime, in addition to fast outflows on the order of the ion thermal speed that violate the GK ordering. Since secondary magnetic islands appear after the reconnection peak time, a kinetic PIC/GK comparison is more accurate in the linear phase of magnetic reconnection. For a high beta plasma (βi = 1.0) where reconnection rates and fluctuations levels are reduced, similar processes happen in the secondary magnetic islands in the fully kinetic description, but requiring much lower guide fields (bg ≲ 3).

  10. Spin field equations and Heun's equations

    NASA Astrophysics Data System (ADS)

    Jiang, Min; Wang, Xuejing; Li, Zhongheng

    2015-06-01

    The Kerr-Newman-(anti) de Sitter metric is the most general stationary black hole solution to the Einstein-Maxwell equation with a cosmological constant. We study the separability of the equations of the massless scalar (spin s=0), neutrino ( s=1/2), electromagnetic ( s=1), Rarita-Schwinger ( s=3/2), and gravitational ( s=2) fields propagating on this background. We obtain the angular and radial master equations, and show that the master equations are transformed to Heun's equation. Meanwhile, we give the condition of existence of event horizons for Kerr-Newman-(anti) de Sitter spacetime by using Sturm theorem.

  11. Basic lubrication equations

    NASA Technical Reports Server (NTRS)

    Hamrock, B. J.; Dowson, D.

    1981-01-01

    Lubricants, usually Newtonian fluids, are assumed to experience laminar flow. The basic equations used to describe the flow are the Navier-Stokes equation of motion. The study of hydrodynamic lubrication is, from a mathematical standpoint, the application of a reduced form of these Navier-Stokes equations in association with the continuity equation. The Reynolds equation can also be derived from first principles, provided of course that the same basic assumptions are adopted in each case. Both methods are used in deriving the Reynolds equation, and the assumptions inherent in reducing the Navier-Stokes equations are specified. Because the Reynolds equation contains viscosity and density terms and these properties depend on temperature and pressure, it is often necessary to couple the Reynolds with energy equation. The lubricant properties and the energy equation are presented. Film thickness, a parameter of the Reynolds equation, is a function of the elastic behavior of the bearing surface. The governing elasticity equation is therefore presented.

  12. Wavenumber-resolved core turbulence studies in the ASDEX Upgrade tokamak and comparison with non-linear gyrokinetic simulations with the GENE code

    NASA Astrophysics Data System (ADS)

    Happel, Tim; Bañón Navarro, Alejandro; Conway, Garrard; Görler, Tobias; Jenko, Frank; Ryter, Francois; Stroth, Ulrich; ASDEX Upgrade Team

    2014-10-01

    Core plasma turbulence determines transport properties and impacts on the efficiency of a fusion reactor. Gyrokinetic codes are developed to predict dominant instabilities and the turbulence level, which causes the observed particle and heat losses. A careful validation of these codes is mandatory to improve the reliability of predictions. To this end, core turbulence is investigated in ASDEX Upgrade by means of Doppler reflectometry, which provides the perpendicular velocity of turbulent structures and their fluctuation level. H-mode discharges have been performed in which ECRH is used to drive the turbulence from the ITG turbulence regime towards the TEM regime. In general, the turbulence level increases from core towards the edge. With increasing R /LTe , core large scale structures show larger fluctuation amplitudes while their phase velocity is altered with respect to that of small structures. Results are compared with gyrokinetic simulations with the GENE code. Linear results show a transition from ITG towards TEM turbulence close to the radial ECRH deposition location. After matching of heat fluxes to results from power balance analysis, the radial trend in the turbulence level is reproduced. The response to additional heating is opposite to the experimental findings.

  13. Study on longitudinal dispersion relation in one-dimensional relativistic plasma: Linear theory and Vlasov simulation

    SciTech Connect

    Zhang, H.; Wu, S. Z.; Zhou, C. T.; He, X. T.; Key Laboratory of HEDP of the Ministry of Education, CAPT, Peking University, Beijing 100871 ; Zhu, S. P.

    2013-09-15

    The dispersion relation of one-dimensional longitudinal plasma waves in relativistic homogeneous plasmas is investigated with both linear theory and Vlasov simulation in this paper. From the Vlasov-Poisson equations, the linear dispersion relation is derived for the proper one-dimensional Jüttner distribution. Numerically obtained linear dispersion relation as well as an approximate formula for plasma wave frequency in the long wavelength limit is given. The dispersion of longitudinal wave is also simulated with a relativistic Vlasov code. The real and imaginary parts of dispersion relation are well studied by varying wave number and plasma temperature. Simulation results are in agreement with established linear theory.

  14. A splitting algorithm for Vlasov simulation with filamentation filtration

    NASA Technical Reports Server (NTRS)

    Klimas, A. J.; Farrell, W. M.

    1994-01-01

    A Fourier-Fourier transformed version of the splitting algorithm for simulating solutions of the Vlasov-Poisson system of equations is introduced. It is shown that with the inclusion of filamentation filtration in this transformed algorithm it is both faster and more stable than the standard splitting algorithm. It is further shown that in a scalar computer environment this new algorithm is approximately equal in speed and far less noisy than its particle-in-cell counterpart. It is conjectured that in a multiprocessor environment the filtered splitting algorithm would be faster while producing more precise results.

  15. Impact-Z

    Energy Science and Technology Software Center (ESTSC)

    2008-03-18

    IMPACT-Z is a parallel particle-in-cell code whose primary purpose is to model the dynamics of charged particle beams in linear accelerators. The code includes the effects of externally applied fields from magnets and accelerating cavities as well as the effect of self-fields (space charge fields). Mathematically, the code solves the Vlasov/Poisson equations using a particle-based technique. The code, which is written in Fortran90 with MPI, runs on both single-processor and multi-processor systems.

  16. Chemical Equation Balancing.

    ERIC Educational Resources Information Center

    Blakley, G. R.

    1982-01-01

    Reviews mathematical techniques for solving systems of homogeneous linear equations and demonstrates that the algebraic method of balancing chemical equations is a matter of solving a system of homogeneous linear equations. FORTRAN programs using this matrix method to chemical equation balancing are available from the author. (JN)

  17. Equations and closure methods

    NASA Technical Reports Server (NTRS)

    1977-01-01

    Basic differential equations governing compressible turbulent boundary layer flow are reviewed, including conservation of mass and energy, momentum equations derived from Navier-Stokes equations, and equations of state. Closure procedures were broken down into: (1) simple or zeroth-order methods, (2) first-order or mean field closure methods, and (3) second-order or mean turbulence field methods.

  18. Measurements of the cross-phase angle between density and electron temperature fluctuations and comparison with gyrokinetic simulations

    SciTech Connect

    White, A. E.; Peebles, W. A.; Rhodes, T. L.; Schmitz, L.; Carter, T. A.; Hillesheim, J. C.; Doyle, E. J.; Zeng, L.; Holland, C. H.; Wang, G.; McKee, G. R.; Staebler, G. M.; Waltz, R. E.; DeBoo, J. C.; Petty, C. C.; Burrell, K. H.

    2010-05-15

    This paper presents new measurements of the cross-phase angle, alpha{sub n{sub eT{sub e}}}, between long-wavelength (k{sub t}hetarho{sub s}<0.5) density, n-tilde{sub e}, and electron temperature, T-tilde{sub e}, fluctuations in the core of DIII-D [J. L. Luxon, Nucl. Fusion 42, 614 (2002)] tokamak plasmas. The coherency and cross-phase angle between n-tilde{sub e} and T-tilde{sub e} are measured using coupled reflectometer and correlation electron cyclotron emission diagnostics that view the same plasma volume. In addition to the experimental results, two sets of local, nonlinear gyrokinetic turbulence simulations that are performed with the GYRO code [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] are described. One set, called the pre-experiment simulations, was performed prior to the experiment in order to predict a change in alpha{sub n{sub eT{sub e}}} given experimentally realizable increases in the electron temperature, T{sub e}. In the experiment the cross-phase angle was measured at three radial locations (rho=0.55, 0.65, and 0.75) in both a 'Base' case and a 'High T{sub e}' case. The measured cross-phase angle is in good qualitative agreement with the pre-experiment simulations, which predicted that n-tilde{sub e} and T-tilde{sub e} would be out of phase. The pre-experiment simulations also predicted a decrease in cross-phase angle as T{sub e} is increased. Experimentally, this trend is observed at the inner two radial locations only. The second set of simulations, the postexperiment simulations, is carried out using local parameters taken from measured experimental profiles as input to GYRO. These postexperiment simulation results are in good quantitative agreement with the measured cross-phase angle, despite disagreements with transport fluxes. Directions for future modeling and experimental work are discussed.

  19. Single wall penetration equations

    NASA Technical Reports Server (NTRS)

    Hayashida, K. B.; Robinson, J. H.

    1991-01-01

    Five single plate penetration equations are compared for accuracy and effectiveness. These five equations are two well-known equations (Fish-Summers and Schmidt-Holsapple), two equations developed by the Apollo project (Rockwell and Johnson Space Center (JSC), and one recently revised from JSC (Cour-Palais). They were derived from test results, with velocities ranging up to 8 km/s. Microsoft Excel software was used to construct a spreadsheet to calculate the diameters and masses of projectiles for various velocities, varying the material properties of both projectile and target for the five single plate penetration equations. The results were plotted on diameter versus velocity graphs for ballistic and spallation limits using Cricket Graph software, for velocities ranging from 2 to 15 km/s defined for the orbital debris. First, these equations were compared to each other, then each equation was compared with various aluminum projectile densities. Finally, these equations were compared with test results performed at JSC for the Marshall Space Flight Center. These equations predict a wide variety of projectile diameters at a given velocity. Thus, it is very difficult to choose the 'right' prediction equation. The thickness of a single plate could have a large variation by choosing a different penetration equation. Even though all five equations are empirically developed with various materials, especially for aluminum alloys, one cannot be confident in the shield design with the predictions obtained by the penetration equations without verifying by tests.

  20. Interpretation of Bernoulli's Equation.

    ERIC Educational Resources Information Center

    Bauman, Robert P.; Schwaneberg, Rolf

    1994-01-01

    Discusses Bernoulli's equation with regards to: horizontal flow of incompressible fluids, change of height of incompressible fluids, gases, liquids and gases, and viscous fluids. Provides an interpretation, properties, terminology, and applications of Bernoulli's equation. (MVL)

  1. Reflections on Chemical Equations.

    ERIC Educational Resources Information Center

    Gorman, Mel

    1981-01-01

    The issue of how much emphasis balancing chemical equations should have in an introductory chemistry course is discussed. The current heavy emphasis on finishing such equations is viewed as misplaced. (MP)

  2. The Pendulum Equation

    ERIC Educational Resources Information Center

    Fay, Temple H.

    2002-01-01

    We investigate the pendulum equation [theta] + [lambda][squared] sin [theta] = 0 and two approximations for it. On the one hand, we suggest that the third and fifth-order Taylor series approximations for sin [theta] do not yield very good differential equations to approximate the solution of the pendulum equation unless the initial conditions are…

  3. Gyrokinetic simulation of global and local Alfvén eigenmodes driven by energetic particles in a DIII-D discharge

    NASA Astrophysics Data System (ADS)

    Bass, E. M.; Waltz, R. E.

    2013-01-01

    The unstable spectrum of Alfvén eigenmodes (AEs) driven by neutral beam-sourced energetic particles (EPs) in a benchmark DIII-D discharge (142111) is calculated in a fully gyrokinetic model using the GYRO code's massively parallel linear eigenvalue solver. One cycle of the slow (equilibrium scale) frequency sweep of the reverse shear Alfvén eigenmode (RSAE) at toroidal mode number n =3 is mapped. The RSAE second harmonic and an unstable beta-induced Alfvén eigenmode (BAE) are simultaneously tracked alongside the primary RSAE. An observed twist in the eigenmode pattern, caused mostly by shear in the driving EP profile, is shown through artificially varying the E ×B rotational velocity shear to depend generally on shear in the local wave phase velocity. Coupling to the BAE and to the toroidal Alfvén eigenmode limit the RSAE frequency sweeps at the lower and upper end, respectively. While the present fully gyrokinetic model (including thermal ions and electrons) constitutes the best treatment of compressibility physics available, the BAE frequency is overpredicted by about 20% against experiment here and is found to be sensitive to energetic beam ion pressure. The RSAE frequency is more accurately matched except when it is limited by the BAE. Simulations suggest that the experiment is very close to marginal AE stability at points of RSAE-BAE coupling. A recipe for comparing the radial profile of quasilinear transport flux from local modes to that from global modes paves the way for the development of a stiff (critical gradient) local AE transport model based on local mode stability thresholds.

  4. Continuum Kinetic Plasma Modeling Using a Conservative 4th-Order Method with AMR

    NASA Astrophysics Data System (ADS)

    Vogman, Genia; Colella, Phillip

    2012-10-01

    When the number of particles in a Debye sphere is large, a plasma can be accurately represented by a distribution function, which can be treated as a continuous incompressible fluid in phase space. In the most general case the evolution of such a distribution function is described by the 6D Boltzmann-Maxwell partial differential equation system. To address the challenges associated with solving a 6D hyperbolic governing equation, a simpler 3D Vlasov-Poisson system is considered. A 4th-order accurate Vlasov-Poisson model has been developed in one spatial and two velocity dimensions. The governing equation is cast in conservation law form and is solved with a finite volume representation. Adaptive mesh refinement (AMR) is used to allow for efficient use of computational resources while maintaining desired levels of resolution. The model employs a flux limiter to remedy non-physical effects such as numerical dispersion. The model is tested on the two-stream, beam-plasma, and Dory-Guest-Harris instabilities. All results are compared with linear theory.

  5. Electromagnetic Gyrokinetic Simulations

    SciTech Connect

    Wan, W

    2003-11-19

    A new electromagnetic kinetic electron {delta} particle simulation model has been demonstrated to work well at large values of plasma {beta} times the ion-to-electron mass ratio. The simulation is three-dimensional using toroidal flux-tube geometry and includes electron-ion collisions. The model shows accurate shear Alfven wave damping and microtearing physics. Zonal flows with kinetic electrons are found to be turbulent with the spectrum peaking at zero and having a width in the frequency range of the driving turbulence. This is in contrast with adiabatic electron cases where the zonal flows are near stationary, even though the linear behavior of the zonal flow is not significantly affected by kinetic electrons. zonal fields are found to be very weak, consistent with theoretical predictions for {beta} below the kinetic ballooning limit. Detailed spectral analysis of the turbulence data is presented in the various limits.

  6. Solving Ordinary Differential Equations

    NASA Technical Reports Server (NTRS)

    Krogh, F. T.

    1987-01-01

    Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

  7. Einstein equation at singularities

    NASA Astrophysics Data System (ADS)

    Stoica, Ovidiu-Cristinel

    2014-02-01

    Einstein's equation is rewritten in an equivalent form, which remains valid at the singularities in some major cases. These cases include the Schwarzschild singularity, the Friedmann-Lemaître-Robertson-Walker Big Bang singularity, isotropic singularities, and a class of warped product singularities. This equation is constructed in terms of the Ricci part of the Riemann curvature (as the Kulkarni-Nomizu product between Einstein's equation and the metric tensor).

  8. What Makes a Chemical Equation an Equation?

    ERIC Educational Resources Information Center

    Fensham, Peter J.; Lui, Julia

    2001-01-01

    Explores how well chemistry graduates preparing for teaching can recognize the similarities and differences between the uses of the word "equation" in mathematics and in chemistry. Reports that the conservation similarities were much less frequently recognized than those involved in the creation of new entities. (Author/MM)

  9. Octonic Gravitational Field Equations

    NASA Astrophysics Data System (ADS)

    Demir, Süleyman; Tanişli, Murat; Tolan, Tülay

    2013-08-01

    Generalized field equations of linear gravity are formulated on the basis of octons. When compared to the other eight-component noncommutative hypercomplex number systems, it is demonstrated that associative octons with scalar, pseudoscalar, pseudovector and vector values present a convenient and capable tool to describe the Maxwell-Proca-like field equations of gravitoelectromagnetism in a compact and simple way. Introducing massive graviton and gravitomagnetic monopole terms, the generalized gravitational wave equation and Klein-Gordon equation for linear gravity are also developed.

  10. Octonic massless field equations

    NASA Astrophysics Data System (ADS)

    Demir, Süleyman; Tanişli, Murat; Kansu, Mustafa Emre

    2015-05-01

    In this paper, it is proven that the associative octons including scalar, pseudoscalar, pseudovector and vector values are convenient and capable tools to generalize the Maxwell-Dirac like field equations of electromagnetism and linear gravity in a compact and simple way. Although an attempt to describe the massless field equations of electromagnetism and linear gravity needs the sixteen real component mathematical structures, it is proved that these equations can be formulated in terms of eight components of octons. Furthermore, the generalized wave equation in terms of potentials is derived in the presence of electromagnetic and gravitational charges (masses). Finally, conservation of energy concept has also been investigated for massless fields.

  11. Octonic Massive Field Equations

    NASA Astrophysics Data System (ADS)

    Demir, Süleyman; Kekeç, Seray

    2016-03-01

    In the present paper we propose the octonic form of massive field equations based on the analogy with electromagnetism and linear gravity. Using the advantages of octon algebra the Maxwell-Dirac-Proca equations have been reformulated in compact and elegant way. The energy-momentum relations for massive field are discussed.

  12. On the Diophantine equation

    NASA Astrophysics Data System (ADS)

    Zahari, N. M.; Sapar, S. H.; Mohd Atan, K. A.

    2013-04-01

    This paper discusses an integral solution (a, b, c) of the Diophantine equations x3n+y3n = 2z2n for n ≥ 2 and it is found that the integral solution of these equation are of the form a = b = t2, c = t3 for any integers t.

  13. The lens equation revisited

    NASA Astrophysics Data System (ADS)

    Molesini, Giuseppe

    2005-02-01

    Problems in the general validity of the lens equations are reported, requiring an assessment of the conditions for correct use. A discussion is given on critical behaviour of the lens equation, and a sign and meaning scheme is provided so that apparent inconsistencies are avoided.

  14. Octonic Massive Field Equations

    NASA Astrophysics Data System (ADS)

    Demir, Süleyman; Kekeç, Seray

    2016-07-01

    In the present paper we propose the octonic form of massive field equations based on the analogy with electromagnetism and linear gravity. Using the advantages of octon algebra the Maxwell-Dirac-Proca equations have been reformulated in compact and elegant way. The energy-momentum relations for massive field are discussed.

  15. Reduced Braginskii equations

    SciTech Connect

    Yagi, M.; Horton, W. )

    1994-07-01

    A set of reduced Braginskii equations is derived without assuming flute ordering and the Boussinesq approximation. These model equations conserve the physical energy. It is crucial at finite [beta] that the perpendicular component of Ohm's law be solved to ensure [del][center dot][bold j]=0 for energy conservation.

  16. Linear Equations: Equivalence = Success

    ERIC Educational Resources Information Center

    Baratta, Wendy

    2011-01-01

    The ability to solve linear equations sets students up for success in many areas of mathematics and other disciplines requiring formula manipulations. There are many reasons why solving linear equations is a challenging skill for students to master. One major barrier for students is the inability to interpret the equals sign as anything other than…

  17. The Effective Equation Method

    NASA Astrophysics Data System (ADS)

    Kuksin, Sergei; Maiocchi, Alberto

    In this chapter we present a general method of constructing the effective equation which describes the behavior of small-amplitude solutions for a nonlinear PDE in finite volume, provided that the linear part of the equation is a hamiltonian system with a pure imaginary discrete spectrum. The effective equation is obtained by retaining only the resonant terms of the nonlinearity (which may be hamiltonian, or may be not); the assertion that it describes the limiting behavior of small-amplitude solutions is a rigorous mathematical theorem. In particular, the method applies to the three- and four-wave systems. We demonstrate that different possible types of energy transport are covered by this method, depending on whether the set of resonances splits into finite clusters (this happens, e.g. in case of the Charney-Hasegawa-Mima equation), or is connected (this happens, e.g. in the case of the NLS equation if the space-dimension is at least two). For equations of the first type the energy transition to high frequencies does not hold, while for equations of the second type it may take place. Our method applies to various weakly nonlinear wave systems, appearing in plasma, meteorology and oceanography.

  18. Volterra difference equations

    NASA Astrophysics Data System (ADS)

    Sultana, Nasrin

    This dissertation consists of five papers in which discrete Volterra equations of different types and orders are studied and results regarding the behavior of their solutions are established. The first paper presents some fundamental results about subexponential sequences. It also illustrates the subexponential solutions of scalar linear Volterra sum-difference equations are asymptotically stable. The exact value of the rate of convergence of asymptotically stable solutions is found by determining the asymptotic behavior of the transient renewal equations. The study of subexponential solutions is also continued in the second and third articles. The second paper investigates the same equation using the same process as considered in the first paper. The discussion focuses on a positive lower bound of the rate of convergence of the asymptotically stable solutions. The third paper addresses the rate of convergence of the solutions of scalar linear Volterra sum-difference equations with delay. The result is proved by developing the rate of convergence of transient renewal delay difference equations. The fourth paper discusses the existence of bounded solutions on an unbounded domain of more general nonlinear Volterra sum-difference equations using the Schaefer fixed point theorem and the Lyapunov direct method. The fifth paper examines the asymptotic behavior of nonoscillatory solutions of higher-order integro-dynamic equations and establishes some new criteria based on so-called time scales, which unifies and extends both discrete and continuous mathematical analysis. Beside these five research papers that focus on discrete Volterra equations, this dissertation also contains an introduction, a section on difference calculus, a section on time scales calculus, and a conclusion.

  19. Stochastic Gauss equations

    NASA Astrophysics Data System (ADS)

    Pierret, Frédéric

    2016-02-01

    We derived the equations of Celestial Mechanics governing the variation of the orbital elements under a stochastic perturbation, thereby generalizing the classical Gauss equations. Explicit formulas are given for the semimajor axis, the eccentricity, the inclination, the longitude of the ascending node, the pericenter angle, and the mean anomaly, which are expressed in term of the angular momentum vector H per unit of mass and the energy E per unit of mass. Together, these formulas are called the stochastic Gauss equations, and they are illustrated numerically on an example from satellite dynamics.

  20. Nonlinear ordinary difference equations

    NASA Technical Reports Server (NTRS)

    Caughey, T. K.

    1979-01-01

    Future space vehicles will be relatively large and flexible, and active control will be necessary to maintain geometrical configuration. While the stresses and strains in these space vehicles are not expected to be excessively large, their cumulative effects will cause significant geometrical nonlinearities to appear in the equations of motion, in addition to the nonlinearities caused by material properties. Since the only effective tool for the analysis of such large complex structures is the digital computer, it will be necessary to gain a better understanding of the nonlinear ordinary difference equations which result from the time discretization of the semidiscrete equations of motion for such structures.

  1. A Comparison of IRT Equating and Beta 4 Equating.

    ERIC Educational Resources Information Center

    Kim, Dong-In; Brennan, Robert; Kolen, Michael

    Four equating methods were compared using four equating criteria: first-order equity (FOE), second-order equity (SOE), conditional mean squared error (CMSE) difference, and the equipercentile equating property. The four methods were: (1) three parameter logistic (3PL) model true score equating; (2) 3PL observed score equating; (3) beta 4 true…

  2. Diophantine Equations and Computation

    NASA Astrophysics Data System (ADS)

    Davis, Martin

    Unless otherwise stated, we’ll work with the natural numbers: N = \\{0,1,2,3, dots\\}. Consider a Diophantine equation F(a1,a2,...,an,x1,x2,...,xm) = 0 with parameters a1,a2,...,an and unknowns x1,x2,...,xm For such a given equation, it is usual to ask: For which values of the parameters does the equation have a solution in the unknowns? In other words, find the set: \\{ mid exists x_1,ldots,x_m [F(a_1,ldots,x_1,ldots)=0] \\} Inverting this, we think of the equation F = 0 furnishing a definition of this set, and we distinguish three classes: a set is called Diophantine if it has such a definition in which F is a polynomial with integer coefficients. We write \\cal D for the class of Diophantine sets.

  3. Regularized Structural Equation Modeling

    PubMed Central

    Jacobucci, Ross; Grimm, Kevin J.; McArdle, John J.

    2016-01-01

    A new method is proposed that extends the use of regularization in both lasso and ridge regression to structural equation models. The method is termed regularized structural equation modeling (RegSEM). RegSEM penalizes specific parameters in structural equation models, with the goal of creating easier to understand and simpler models. Although regularization has gained wide adoption in regression, very little has transferred to models with latent variables. By adding penalties to specific parameters in a structural equation model, researchers have a high level of flexibility in reducing model complexity, overcoming poor fitting models, and the creation of models that are more likely to generalize to new samples. The proposed method was evaluated through a simulation study, two illustrative examples involving a measurement model, and one empirical example involving the structural part of the model to demonstrate RegSEM’s utility. PMID:27398019

  4. Nonlinear differential equations

    SciTech Connect

    Dresner, L.

    1988-01-01

    This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.

  5. Equations For Rotary Transformers

    NASA Technical Reports Server (NTRS)

    Salomon, Phil M.; Wiktor, Peter J.; Marchetto, Carl A.

    1988-01-01

    Equations derived for input impedance, input power, and ratio of secondary current to primary current of rotary transformer. Used for quick analysis of transformer designs. Circuit model commonly used in textbooks on theory of ac circuits.

  6. Equating Training to Education.

    ERIC Educational Resources Information Center

    Davis, Lansing J.

    1993-01-01

    Distinguishes between education and employer-sponsored training in terms of process, purpose, and providers. Concludes that work-related training and postsecondary education are cognates within the classification education, and equating their learning outcomes is appropriate. (SK)

  7. Relativistic Guiding Center Equations

    SciTech Connect

    White, R. B.; Gobbin, M.

    2014-10-01

    In toroidal fusion devices it is relatively easy that electrons achieve relativistic velocities, so to simulate runaway electrons and other high energy phenomena a nonrelativistic guiding center formalism is not sufficient. Relativistic guiding center equations including flute mode time dependent field perturbations are derived. The same variables as used in a previous nonrelativistic guiding center code are adopted, so that a straightforward modifications of those equations can produce a relativistic version.

  8. SIMULTANEOUS DIFFERENTIAL EQUATION COMPUTER

    DOEpatents

    Collier, D.M.; Meeks, L.A.; Palmer, J.P.

    1960-05-10

    A description is given for an electronic simulator for a system of simultaneous differential equations, including nonlinear equations. As a specific example, a homogeneous nuclear reactor system including a reactor fluid, heat exchanger, and a steam boiler may be simulated, with the nonlinearity resulting from a consideration of temperature effects taken into account. The simulator includes three operational amplifiers, a multiplier, appropriate potential sources, and interconnecting R-C networks.

  9. Comparison of BES measurements of ion-scale turbulence with direct gyro-kinetic simulations of MAST L-mode plasmas

    NASA Astrophysics Data System (ADS)

    Field, A. R.; Dunai, D.; Ghim, Y.-c.; Hill, P.; McMillan, B.; Roach, C. M.; Saarelma, S.; Schekochihin, A. A.; Zoletnik, S.; the MAST Team

    2014-02-01

    Observations of ion-scale (kyρi ⩽ 1) density turbulence of relative amplitude ≳0.2% are available on the Mega Amp Spherical Tokamak (MAST) using a 2D (8 radial × 4 poloidal channel) imaging beam emission spectroscopy diagnostic. Spatial and temporal characteristics of this turbulence, i.e., amplitudes, correlation times, radial and perpendicular correlation lengths and apparent phase velocities of the density contours, are determined by means of correlation analysis. For a low-density, L-mode discharge with strong equilibrium flow shear exhibiting an internal transport barrier in the ion channel, the observed turbulence characteristics are compared with synthetic density turbulence data generated from global, non-linear, gyro-kinetic simulations using the particle-in-cell code NEMORB. This validation exercise highlights the need to include increasingly sophisticated physics, e.g., kinetic treatment of trapped electrons, equilibrium flow shear and collisions, to reproduce most of the characteristics of the observed turbulence. Even so, significant discrepancies remain: an underprediction by the simulations of the turbulence amplitude and heat flux at plasma periphery and the finding that the correlation times of the numerically simulated turbulence are typically two orders of magnitude longer than those measured in MAST. Comparison of these correlation times with various linear timescales suggests that, while the measured turbulence is strong and may be ‘critically balanced’, the simulated turbulence is weak.

  10. Study of the L-mode tokamak plasma “shortfall” with local and global nonlinear gyrokinetic δf particle-in-cell simulation

    SciTech Connect

    Chowdhury, J.; Wan, Weigang; Chen, Yang; Parker, Scott E.; Groebner, Richard J.; Holland, C.; Howard, N. T.

    2014-11-15

    The δ f particle-in-cell code GEM is used to study the transport “shortfall” problem of gyrokinetic simulations. In local simulations, the GEM results confirm the previously reported simulation results of DIII-D [Holland et al., Phys. Plasmas 16, 052301 (2009)] and Alcator C-Mod [Howard et al., Nucl. Fusion 53, 123011 (2013)] tokamaks with the continuum code GYRO. Namely, for DIII-D the simulations closely predict the ion heat flux at the core, while substantially underpredict transport towards the edge; while for Alcator C-Mod, the simulations show agreement with the experimental values of ion heat flux, at least within the range of experimental error. Global simulations are carried out for DIII-D L-mode plasmas to study the effect of edge turbulence on the outer core ion heat transport. The edge turbulence enhances the outer core ion heat transport through turbulence spreading. However, this edge turbulence spreading effect is not enough to explain the transport underprediction.

  11. Performance of the UCAN2 Gyrokinetic Particle In Cell (PIC) Code on Two Massively Parallel Mainframes with Intel ``Sandy Bridge'' Processors

    NASA Astrophysics Data System (ADS)

    Leboeuf, Jean-Noel; Decyk, Viktor; Newman, David; Sanchez, Raul

    2013-10-01

    The massively parallel, 2D domain-decomposed, nonlinear, 3D, toroidal, electrostatic, gyrokinetic, Particle in Cell (PIC), Cartesian geometry UCAN2 code, with particle ions and adiabatic electrons, has been ported to two emerging mainframes. These two computers, one at NERSC in the US built by Cray named Edison and the other at the Barcelona Supercomputer Center (BSC) in Spain built by IBM named MareNostrum III (MNIII) just happen to share the same Intel ``Sandy Bridge'' processors. The successful port of UCAN2 to MNIII which came online first has enabled us to be up and running efficiently in record time on Edison. Overall, the performance of UCAN2 on Edison is superior to that on MNIII, particularly at large numbers of processors (>1024) for the same Intel IFORT compiler. This appears to be due to different MPI modules (OpenMPI on MNIII and MPICH2 on Edison) and different interconnection networks (Infiniband on MNIII and Cray's Aries on Edison) on the two mainframes. Details of these ports and comparative benchmarks are presented. Work supported by OFES, USDOE, under contract no. DE-FG02-04ER54741 with the University of Alaska at Fairbanks.

  12. ETG-dominated transport regimes in near-edge DIII-D L-mode plasmas: Validation of multiscale gyrokinetic simulations

    NASA Astrophysics Data System (ADS)

    Neiser, Tom; Jenko, Frank; Schmitz, Lothar; Told, Daniel; Banon Navarro, Alejandro; Carter, Troy; Yan, Zheng; McKee, George

    2015-11-01

    A prerequisite for the development of a self-consistent theoretical description of the L-H transition is the ability to quantitatively characterize near-edge L-mode plasmas. It is shown here for the first time that regimes exist in the L-mode near-edge that appear to be dominated by sub-ion-scale turbulence driven by electron temperature gradient (ETG) modes. These are results of gyrokinetic simulations of a DIII-D L-mode discharge in the near edge region (r/a = 0.8) with the GENE code (www.genecode.org). Instructed by a linear analysis, we performed nonlinear simulations of ITG and ETG turbulence, pointing to a dominance of ETG turbulence regarding the anomalous radial heat flux. Direct comparison with experimental data is encouraging. Respective multi-scale simulations, covering both ion and electron scales are underway and will be presented. Implications for L-H transition modeling will also be discussed. Work supported by the US Department of Energy (DOE) under DE-FG02-08ER54984 and DE-FC02-04ER54698, as well as NERSC, a DOE Office of Science User Facility supported under Contract No. DE-AC02-05CH11231.

  13. Set Equation Transformation System.

    Energy Science and Technology Software Center (ESTSC)

    2002-03-22

    Version 00 SETS is used for symbolic manipulation of Boolean equations, particularly the reduction of equations by the application of Boolean identities. It is a flexible and efficient tool for performing probabilistic risk analysis (PRA), vital area analysis, and common cause analysis. The equation manipulation capabilities of SETS can also be used to analyze noncoherent fault trees and determine prime implicants of Boolean functions, to verify circuit design implementation, to determine minimum cost fire protectionmore » requirements for nuclear reactor plants, to obtain solutions to combinatorial optimization problems with Boolean constraints, and to determine the susceptibility of a facility to unauthorized access through nullification of sensors in its protection system. Two auxiliary programs, SEP and FTD, are included. SEP performs the quantitative analysis of reduced Boolean equations (minimal cut sets) produced by SETS. The user can manipulate and evaluate the equations to find the probability of occurrence of any desired event and to produce an importance ranking of the terms and events in an equation. FTD is a fault tree drawing program which uses the proprietary ISSCO DISSPLA graphics software to produce an annotated drawing of a fault tree processed by SETS. The DISSPLA routines are not included.« less

  14. The Bernoulli-Poiseuille Equation.

    ERIC Educational Resources Information Center

    Badeer, Henry S.; Synolakis, Costas E.

    1989-01-01

    Describes Bernoulli's equation and Poiseuille's equation for fluid dynamics. Discusses the application of the combined Bernoulli-Poiseuille equation in real flows, such as viscous flows under gravity and acceleration. (YP)

  15. Introducing Chemical Formulae and Equations.

    ERIC Educational Resources Information Center

    Dawson, Chris; Rowell, Jack

    1979-01-01

    Discusses when the writing of chemical formula and equations can be introduced in the school science curriculum. Also presents ways in which formulae and equations learning can be aided and some examples for balancing and interpreting equations. (HM)

  16. Nonlocal electrical diffusion equation

    NASA Astrophysics Data System (ADS)

    Gómez-Aguilar, J. F.; Escobar-Jiménez, R. F.; Olivares-Peregrino, V. H.; Benavides-Cruz, M.; Calderón-Ramón, C.

    2016-07-01

    In this paper, we present an analysis and modeling of the electrical diffusion equation using the fractional calculus approach. This alternative representation for the current density is expressed in terms of the Caputo derivatives, the order for the space domain is 0<β≤1 and for the time domain is 0<γ≤2. We present solutions for the full fractional equation involving space and time fractional derivatives using numerical methods based on Fourier variable separation. The case with spatial fractional derivatives leads to Levy flight type phenomena, while the time fractional equation is related to sub- or super diffusion. We show that the mathematical concept of fractional derivatives can be useful to understand the behavior of semiconductors, the design of solar panels, electrochemical phenomena and the description of anomalous complex processes.

  17. Parallel tridiagonal equation solvers

    NASA Technical Reports Server (NTRS)

    Stone, H. S.

    1974-01-01

    Three parallel algorithms were compared for the direct solution of tridiagonal linear systems of equations. The algorithms are suitable for computers such as ILLIAC 4 and CDC STAR. For array computers similar to ILLIAC 4, cyclic odd-even reduction has the least operation count for highly structured sets of equations, and recursive doubling has the least count for relatively unstructured sets of equations. Since the difference in operation counts for these two algorithms is not substantial, their relative running times may be more related to overhead operations, which are not measured in this paper. The third algorithm, based on Buneman's Poisson solver, has more arithmetic operations than the others, and appears to be the least favorable. For pipeline computers similar to CDC STAR, cyclic odd-even reduction appears to be the most preferable algorithm for all cases.

  18. Kepler Equation solver

    NASA Technical Reports Server (NTRS)

    Markley, F. Landis

    1995-01-01

    Kepler's Equation is solved over the entire range of elliptic motion by a fifth-order refinement of the solution of a cubic equation. This method is not iterative, and requires only four transcendental function evaluations: a square root, a cube root, and two trigonometric functions. The maximum relative error of the algorithm is less than one part in 10(exp 18), exceeding the capability of double-precision computer arithmetic. Roundoff errors in double-precision implementation of the algorithm are addressed, and procedures to avoid them are developed.

  19. Difference equation for superradiance

    NASA Technical Reports Server (NTRS)

    Lee, C. T.

    1974-01-01

    The evolution of a completely excited system of N two-level atoms, distributed over a large region and interacting with all modes of radiation field, is studied. The distinction between r-conserving (RC) and r-nonconserving (RNC) processes is emphasized. Considering the number of photons emitted as the discrete independent variable, the evolution is described by a partial difference equation. Numerical solution of this equation shows the transition from RNC dominance at the beginning to RC dominance later. This is also a transition from incoherent to coherent emission of radiation.

  20. Obtaining Maxwell's equations heuristically

    NASA Astrophysics Data System (ADS)

    Diener, Gerhard; Weissbarth, Jürgen; Grossmann, Frank; Schmidt, Rüdiger

    2013-02-01

    Starting from the experimental fact that a moving charge experiences the Lorentz force and applying the fundamental principles of simplicity (first order derivatives only) and linearity (superposition principle), we show that the structure of the microscopic Maxwell equations for the electromagnetic fields can be deduced heuristically by using the transformation properties of the fields under space inversion and time reversal. Using the experimental facts of charge conservation and that electromagnetic waves propagate with the speed of light, together with Galilean invariance of the Lorentz force, allows us to finalize Maxwell's equations and to introduce arbitrary electrodynamics units naturally.

  1. The halo Boltzmann equation

    NASA Astrophysics Data System (ADS)

    Biagetti, Matteo; Desjacques, Vincent; Kehagias, Alex; Racco, Davide; Riotto, Antonio

    2016-04-01

    Dark matter halos are the building blocks of the universe as they host galaxies and clusters. The knowledge of the clustering properties of halos is therefore essential for the understanding of the galaxy statistical properties. We derive an effective halo Boltzmann equation which can be used to describe the halo clustering statistics. In particular, we show how the halo Boltzmann equation encodes a statistically biased gravitational force which generates a bias in the peculiar velocities of virialized halos with respect to the underlying dark matter, as recently observed in N-body simulations.

  2. The Statistical Drake Equation

    NASA Astrophysics Data System (ADS)

    Maccone, Claudio

    2010-12-01

    We provide the statistical generalization of the Drake equation. From a simple product of seven positive numbers, the Drake equation is now turned into the product of seven positive random variables. We call this "the Statistical Drake Equation". The mathematical consequences of this transformation are then derived. The proof of our results is based on the Central Limit Theorem (CLT) of Statistics. In loose terms, the CLT states that the sum of any number of independent random variables, each of which may be ARBITRARILY distributed, approaches a Gaussian (i.e. normal) random variable. This is called the Lyapunov Form of the CLT, or the Lindeberg Form of the CLT, depending on the mathematical constraints assumed on the third moments of the various probability distributions. In conclusion, we show that: The new random variable N, yielding the number of communicating civilizations in the Galaxy, follows the LOGNORMAL distribution. Then, as a consequence, the mean value of this lognormal distribution is the ordinary N in the Drake equation. The standard deviation, mode, and all the moments of this lognormal N are also found. The seven factors in the ordinary Drake equation now become seven positive random variables. The probability distribution of each random variable may be ARBITRARY. The CLT in the so-called Lyapunov or Lindeberg forms (that both do not assume the factors to be identically distributed) allows for that. In other words, the CLT "translates" into our statistical Drake equation by allowing an arbitrary probability distribution for each factor. This is both physically realistic and practically very useful, of course. An application of our statistical Drake equation then follows. The (average) DISTANCE between any two neighboring and communicating civilizations in the Galaxy may be shown to be inversely proportional to the cubic root of N. Then, in our approach, this distance becomes a new random variable. We derive the relevant probability density

  3. Comparison of Kernel Equating and Item Response Theory Equating Methods

    ERIC Educational Resources Information Center

    Meng, Yu

    2012-01-01

    The kernel method of test equating is a unified approach to test equating with some advantages over traditional equating methods. Therefore, it is important to evaluate in a comprehensive way the usefulness and appropriateness of the Kernel equating (KE) method, as well as its advantages and disadvantages compared with several popular item…

  4. Balancing Chemical Equations.

    ERIC Educational Resources Information Center

    Savoy, L. G.

    1988-01-01

    Describes a study of students' ability to balance equations. Answers to a test on this topic were analyzed to determine the level of understanding and processes used by the students. Presented is a method to teach this skill to high school chemistry students. (CW)

  5. Structural Equation Model Trees

    ERIC Educational Resources Information Center

    Brandmaier, Andreas M.; von Oertzen, Timo; McArdle, John J.; Lindenberger, Ulman

    2013-01-01

    In the behavioral and social sciences, structural equation models (SEMs) have become widely accepted as a modeling tool for the relation between latent and observed variables. SEMs can be seen as a unification of several multivariate analysis techniques. SEM Trees combine the strengths of SEMs and the decision tree paradigm by building tree…

  6. Parallel Multigrid Equation Solver

    Energy Science and Technology Software Center (ESTSC)

    2001-09-07

    Prometheus is a fully parallel multigrid equation solver for matrices that arise in unstructured grid finite element applications. It includes a geometric and an algebraic multigrid method and has solved problems of up to 76 mullion degrees of feedom, problems in linear elasticity on the ASCI blue pacific and ASCI red machines.

  7. A Quadratic Spring Equation

    ERIC Educational Resources Information Center

    Fay, Temple H.

    2010-01-01

    Through numerical investigations, we study examples of the forced quadratic spring equation [image omitted]. By performing trial-and-error numerical experiments, we demonstrate the existence of stability boundaries in the phase plane indicating initial conditions yielding bounded solutions, investigate the resonance boundary in the [omega]…

  8. Generalized reduced magnetohydrodynamic equations

    SciTech Connect

    Kruger, S.E.

    1999-02-01

    A new derivation of reduced magnetohydrodynamic (MHD) equations is presented. A multiple-time-scale expansion is employed. It has the advantage of clearly separating the three time scales of the problem associated with (1) MHD equilibrium, (2) fluctuations whose wave vector is aligned perpendicular to the magnetic field, and (3) those aligned parallel to the magnetic field. The derivation is carried out without relying on a large aspect ratio assumption; therefore this model can be applied to any general configuration. By accounting for the MHD equilibrium and constraints to eliminate the fast perpendicular waves, equations are derived to evolve scalar potential quantities on a time scale associated with the parallel wave vector (shear-Alfven wave time scale), which is the time scale of interest for MHD instability studies. Careful attention is given in the derivation to satisfy energy conservation and to have manifestly divergence-free magnetic fields to all orders in the expansion parameter. Additionally, neoclassical closures and equilibrium shear flow effects are easily accounted for in this model. Equations for the inner resistive layer are derived which reproduce the linear ideal and resistive stability criterion of Glasser, Greene, and Johnson. The equations have been programmed into a spectral initial value code and run with shear flow that is consistent with the equilibrium input into the code. Linear results of tearing modes with shear flow are presented which differentiate the effects of shear flow gradients in the layer with the effects of the shear flow decoupling multiple harmonics.

  9. Modelling by Differential Equations

    ERIC Educational Resources Information Center

    Chaachoua, Hamid; Saglam, Ayse

    2006-01-01

    This paper aims to show the close relation between physics and mathematics taking into account especially the theory of differential equations. By analysing the problems posed by scientists in the seventeenth century, we note that physics is very important for the emergence of this theory. Taking into account this analysis, we show the…

  10. Do Differential Equations Swing?

    ERIC Educational Resources Information Center

    Maruszewski, Richard F., Jr.

    2006-01-01

    One of the units of in a standard differential equations course is a discussion of the oscillatory motion of a spring and the associated material on forcing functions and resonance. During the presentation on practical resonance, the instructor may tell students that it is similar to when they take their siblings to the playground and help them on…

  11. Supersymmetric fifth order evolution equations

    SciTech Connect

    Tian, K.; Liu, Q. P.

    2010-03-08

    This paper considers supersymmetric fifth order evolution equations. Within the framework of symmetry approach, we give a list containing six equations, which are (potentially) integrable systems. Among these equations, the most interesting ones include a supersymmetric Sawada-Kotera equation and a novel supersymmetric fifth order KdV equation. For the latter, we supply some properties such as a Hamiltonian structures and a possible recursion operator.

  12. Brownian motion from Boltzmann's equation.

    NASA Technical Reports Server (NTRS)

    Montgomery, D.

    1971-01-01

    Two apparently disparate lines of inquiry in kinetic theory are shown to be equivalent: (1) Brownian motion as treated by the (stochastic) Langevin equation and Fokker-Planck equation; and (2) Boltzmann's equation. The method is to derive the kinetic equation for Brownian motion from the Boltzmann equation for a two-component neutral gas by a simultaneous expansion in the density and mass ratios.

  13. Causal electromagnetic interaction equations

    SciTech Connect

    Zinoviev, Yury M.

    2011-02-15

    For the electromagnetic interaction of two particles the relativistic causal quantum mechanics equations are proposed. These equations are solved for the case when the second particle moves freely. The initial wave functions are supposed to be smooth and rapidly decreasing at the infinity. This condition is important for the convergence of the integrals similar to the integrals of quantum electrodynamics. We also consider the singular initial wave functions in the particular case when the second particle mass is equal to zero. The discrete energy spectrum of the first particle wave function is defined by the initial wave function of the free-moving second particle. Choosing the initial wave functions of the free-moving second particle it is possible to obtain a practically arbitrary discrete energy spectrum.

  14. Biaxial constitutive equation development

    NASA Technical Reports Server (NTRS)

    Jordan, E. H.; Walker, K. P.

    1984-01-01

    In developing the constitutive equations an interdisciplinary approach is being pursued. Specifically, both metallurgical and continuum mechanics considerations are recognized in the formulation. Experiments will be utilized to both explore general qualitative features of the material behavior that needs to be modeled and to provide a means of assessing the validity of the equations being developed. The model under development explicitly recognizes crystallographic slip on the individual slip systems. This makes possible direct representation of specific slip system phenomena. The present constitutive formulation takes the anisotropic creep theory and incorporates two state variables into the model to account for the effect of prior inelastic deformation history on the current rate-dependent response of the material.

  15. Nikolaevskiy equation with dispersion.

    PubMed

    Simbawa, Eman; Matthews, Paul C; Cox, Stephen M

    2010-03-01

    The Nikolaevskiy equation was originally proposed as a model for seismic waves and is also a model for a wide variety of systems incorporating a neutral "Goldstone" mode, including electroconvection and reaction-diffusion systems. It is known to exhibit chaotic dynamics at the onset of pattern formation, at least when the dispersive terms in the equation are suppressed, as is commonly the practice in previous analyses. In this paper, the effects of reinstating the dispersive terms are examined. It is shown that such terms can stabilize some of the spatially periodic traveling waves; this allows us to study the loss of stability and transition to chaos of the waves. The secondary stability diagram ("Busse balloon") for the traveling waves can be remarkably complicated. PMID:20365845

  16. Singularities for PRANDTL'S Equations

    NASA Astrophysics Data System (ADS)

    Lo Bosco, G.; Sammartino, M.; Sciacca, V.

    2006-03-01

    We use a mixed spectral/finite-difference numerical method to investigate the possibility of a finite time blow-up of the solutions of Prandtl's equations for the case of the impulsively started cylinder. Our tool is the complex singularity tracking method. We show that a cubic root singularity seems to develop, in a time that can be made arbitrarily short, from a class of data uniformly bounded in H1.

  17. Multinomial diffusion equation

    NASA Astrophysics Data System (ADS)

    Balter, Ariel; Tartakovsky, Alexandre M.

    2011-06-01

    We describe a new, microscopic model for diffusion that captures diffusion induced fluctuations at scales where the concept of concentration gives way to discrete particles. We show that in the limit as the number of particles N→∞, our model is equivalent to the classical stochastic diffusion equation (SDE). We test our new model and the SDE against Langevin dynamics in numerical simulations, and show that our model successfully reproduces the correct ensemble statistics, while the classical model fails.

  18. Multinomial diffusion equation

    SciTech Connect

    Balter, Ariel I.; Tartakovsky, Alexandre M.

    2011-06-24

    We describe a new, microscopic model for diffusion that captures diffusion induced uctuations at scales where the concept of concentration gives way to discrete par- ticles. We show that in the limit as the number of particles N ! 1, our model is equivalent to the classical stochastic diffusion equation (SDE). We test our new model and the SDE against Langevin dynamics in numerical simulations, and show that our model successfully reproduces the correct ensemble statistics, while the classical model fails.

  19. Student-Generated Equations

    ERIC Educational Resources Information Center

    Vasile, Daniela

    2012-01-01

    We are frequently told that Hong Kong has a model system for learning mathematics. In this article Daniela Vasile notes one short-coming in that the pupils are not taught to problem-solve. She begins with a new class by asking them to write down the craziest equation they can come up with and bases her whole lesson, and the following homework,…

  20. Generalized reduced MHD equations

    SciTech Connect

    Kruger, S.E.; Hegna, C.C.; Callen, J.D.

    1998-07-01

    A new derivation of reduced magnetohydrodynamic (MHD) equations is presented. A multiple-time-scale expansion is employed. It has the advantage of clearly separating the three time scales of the problem associated with (1) MHD equilibrium, (2) fluctuations whose wave vector is aligned perpendicular to the magnetic field, and (3) those aligned parallel to the magnetic field. The derivation is carried out without relying on a large aspect ratio assumption; therefore this model can be applied to any general toroidal configuration. By accounting for the MHD equilibrium and constraints to eliminate the fast perpendicular waves, equations are derived to evolve scalar potential quantities on a time scale associated with the parallel wave vector (shear-alfven wave time scale), which is the time scale of interest for MHD instability studies. Careful attention is given in the derivation to satisfy energy conservation and to have manifestly divergence-free magnetic fields to all orders in the expansion parameter. Additionally, neoclassical closures and equilibrium shear flow effects are easily accounted for in this model. Equations for the inner resistive layer are derived which reproduce the linear ideal and resistive stability criterion of Glasser, Greene, and Johnson.

  1. The Drake Equation revisited

    NASA Astrophysics Data System (ADS)

    Konesky, Gregory

    2009-08-01

    In the almost half century since the Drake Equation was first conceived, a number of profound discoveries have been made that require each of the seven variables of this equation to be reconsidered. The discovery of hydrothermal vents on the ocean floor, for example, as well as the ever-increasing extreme conditions in which life is found on Earth, suggest a much wider range of possible extraterrestrial habitats. The growing consensus that life originated very early in Earth's history also supports this suggestion. The discovery of exoplanets with a wide range of host star types, and attendant habitable zones, suggests that life may be possible in planetary systems with stars quite unlike our Sun. Stellar evolution also plays an important part in that habitable zones are mobile. The increasing brightness of our Sun over the next few billion years, will place the Earth well outside the present habitable zone, but will then encompass Mars, giving rise to the notion that some Drake Equation variables, such as the fraction of planets on which life emerges, may have multiple values.

  2. Differential Equations Compatible with Boundary Rational qKZ Equation

    NASA Astrophysics Data System (ADS)

    Takeyama, Yoshihiro

    2011-10-01

    We give diffierential equations compatible with the rational qKZ equation with boundary reflection. The total system contains the trigonometric degeneration of the bispectral qKZ equation of type (Cěen, Cn) which in the case of type GLn was studied by van Meer and Stokman. We construct an integral formula for solutions to our compatible system in a special case.

  3. The compressible adjoint equations in geodynamics: equations and numerical assessment

    NASA Astrophysics Data System (ADS)

    Ghelichkhan, Siavash; Bunge, Hans-Peter

    2016-04-01

    The adjoint method is a powerful means to obtain gradient information in a mantle convection model relative to past flow structure. While the adjoint equations in geodynamics have been derived for the conservation equations of mantle flow in their incompressible form, the applicability of this approximation to Earth is limited, because density increases by almost a factor of two from the surface to the Core Mantle Boundary. Here we introduce the compressible adjoint equations for the conservation equations in the anelastic-liquid approximation. Our derivation applies an operator formulation in Hilbert spaces, to connect to recent work in seismology (Fichtner et al (2006)) and geodynamics (Horbach et al (2014)), where the approach was used to derive the adjoint equations for the wave equation and incompressible mantle flow. We present numerical tests of the newly derived equations based on twin experiments, focusing on three simulations. A first, termed Compressible, assumes the compressible forward and adjoint equations, and represents the consistent means of including compressibility effects. A second, termed Mixed, applies the compressible forward equation, but ignores compressibility effects in the adjoint equations, where the incompressible equations are used instead. A third simulation, termed Incompressible, neglects compressibility effects entirely in the forward and adjoint equations relative to the reference twin. The compressible and mixed formulations successfully restore earlier mantle flow structure, while the incompressible formulation yields noticeable artifacts. Our results suggest the use of a compressible formulation, when applying the adjoint method to seismically derived mantle heterogeneity structure.

  4. Estimating Equating Error in Observed-Score Equating. Research Report.

    ERIC Educational Resources Information Center

    van der Linden, Wim J.

    Traditionally, error in equating observed scores on two versions of a test is defined as the difference between the transformations that equate the quantiles of their distributions in the sample and in the population of examinees. This definition underlies, for example, the well-known approximation to the standard error of equating by Lord (1982).…

  5. Young's equation revisited.

    PubMed

    Makkonen, Lasse

    2016-04-01

    Young's construction for a contact angle at a three-phase intersection forms the basis of all fields of science that involve wetting and capillary action. We find compelling evidence from recent experimental results on the deformation of a soft solid at the contact line, and displacement of an elastic wire immersed in a liquid, that Young's equation can only be interpreted by surface energies, and not as a balance of surface tensions. It follows that the a priori variable in finding equilibrium is not the position of the contact line, but the contact angle. This finding provides the explanation for the pinning of a contact line. PMID:26940644

  6. Noncommutativity and the Friedmann Equations

    NASA Astrophysics Data System (ADS)

    Sabido, M.; Guzmán, W.; Socorro, J.

    2010-07-01

    In this paper we study noncommutative scalar field cosmology, we find the noncommutative Friedmann equations as well as the noncommutative Klein-Gordon equation, interestingly the noncommutative contributions are only present up to second order in the noncommutitive parameter.

  7. Solitons and nonlinear wave equations

    SciTech Connect

    Dodd, Roger K.; Eilbeck, J. Chris; Gibbon, John D.; Morris, Hedley C.

    1982-01-01

    A discussion of the theory and applications of classical solitons is presented with a brief treatment of quantum mechanical effects which occur in particle physics and quantum field theory. The subjects addressed include: solitary waves and solitons, scattering transforms, the Schroedinger equation and the Korteweg-de Vries equation, and the inverse method for the isospectral Schroedinger equation and the general solution of the solvable nonlinear equations. Also considered are: isolation of the Korteweg-de Vries equation in some physical examples, the Zakharov-Shabat/AKNS inverse method, kinks and the sine-Gordon equation, the nonlinear Schroedinger equation and wave resonance interactions, amplitude equations in unstable systems, and numerical studies of solitons. 45 references.

  8. Conservational PDF Equations of Turbulence

    NASA Technical Reports Server (NTRS)

    Shih, Tsan-Hsing; Liu, Nan-Suey

    2010-01-01

    Recently we have revisited the traditional probability density function (PDF) equations for the velocity and species in turbulent incompressible flows. They are all unclosed due to the appearance of various conditional means which are modeled empirically. However, we have observed that it is possible to establish a closed velocity PDF equation and a closed joint velocity and species PDF equation through conditions derived from the integral form of the Navier-Stokes equations. Although, in theory, the resulted PDF equations are neither general nor unique, they nevertheless lead to the exact transport equations for the first moment as well as all higher order moments. We refer these PDF equations as the conservational PDF equations. This observation is worth further exploration for its validity and CFD application

  9. The Dirac equation

    SciTech Connect

    Thaller, B.

    1992-01-01

    This monograph treats most of the usual material to be found in texts on the Dirac equation such as the basic formalism of quantum mechanics, representations of Dirac matrices, covariant realization of the Dirac equation, interpretation of negative energies, Foldy-Wouthuysen transformation, Klein's paradox, spherically symmetric interactions and a treatment of the relativistic hydrogen atom, etc., and also provides excellent additional treatments of a variety of other relevant topics. The monograph contains an extensive treatment of the Lorentz and Poincare groups and their representations. The author discusses in depth Lie algebaic and projective representations, covering groups, and Mackey's theory and Wigner's realization of induced representations. A careful classification of external fields with respect to their behavior under Poincare transformations is supplemented by a basic account of self-adjointness and spectral properties of Dirac operators. A state-of-the-art treatment of relativistic scattering theory based on a time-dependent approach originally due to Enss is presented. An excellent introduction to quantum electrodynamics in external fields is provided. Various appendices containing further details, notes on each chapter commenting on the history involved and referring to original research papers and further developments in the literature, and a bibliography covering all relevant monographs and over 500 articles on the subject, complete this text. This book should satisfy the needs of a wide audience, ranging from graduate students in theoretical physics and mathematics to researchers interested in mathematical physics.

  10. Inequivalence between the Schroedinger equation and the Madelung hydrodynamic equations

    SciTech Connect

    Wallstrom, T.C.

    1994-03-01

    By differentiating the Schroedinger equation and separating the real amd imaginary parts, one obtains the Madelung hydrodynamic equations, which have inspired numerous classical interpretations of quantum mechanics. Such interpretations frequently assume that these equations are equivalent to the Schroedinger equation, and thus provide an alternative basis for quantum mechanics. This paper proves that this is incorrect: to recover the Schroedinger equation, one must add by hand a quantization condition, as in the old quantum theory. The implications for various alternative interpretations of quantum mechanics are discussed.

  11. ``Riemann equations'' in bidifferential calculus

    NASA Astrophysics Data System (ADS)

    Chvartatskyi, O.; Müller-Hoissen, F.; Stoilov, N.

    2015-10-01

    We consider equations that formally resemble a matrix Riemann (or Hopf) equation in the framework of bidifferential calculus. With different choices of a first-order bidifferential calculus, we obtain a variety of equations, including a semi-discrete and a fully discrete version of the matrix Riemann equation. A corresponding universal solution-generating method then either yields a (continuous or discrete) Cole-Hopf transformation, or leaves us with the problem of solving Riemann equations (hence an application of the hodograph method). If the bidifferential calculus extends to second order, solutions of a system of "Riemann equations" are also solutions of an equation that arises, on the universal level of bidifferential calculus, as an integrability condition. Depending on the choice of bidifferential calculus, the latter can represent a number of prominent integrable equations, like self-dual Yang-Mills, as well as matrix versions of the two-dimensional Toda lattice, Hirota's bilinear difference equation, (2+1)-dimensional Nonlinear Schrödinger (NLS), Kadomtsev-Petviashvili (KP) equation, and Davey-Stewartson equations. For all of them, a recent (non-isospectral) binary Darboux transformation result in bidifferential calculus applies, which can be specialized to generate solutions of the associated "Riemann equations." For the latter, we clarify the relation between these specialized binary Darboux transformations and the aforementioned solution-generating method. From (arbitrary size) matrix versions of the "Riemann equations" associated with an integrable equation, possessing a bidifferential calculus formulation, multi-soliton-type solutions of the latter can be generated. This includes "breaking" multi-soliton-type solutions of the self-dual Yang-Mills and the (2+1)-dimensional NLS equation, which are parametrized by solutions of Riemann equations.

  12. The Forced Hard Spring Equation

    ERIC Educational Resources Information Center

    Fay, Temple H.

    2006-01-01

    Through numerical investigations, various examples of the Duffing type forced spring equation with epsilon positive, are studied. Since [epsilon] is positive, all solutions to the associated homogeneous equation are periodic and the same is true with the forcing applied. The damped equation exhibits steady state trajectories with the interesting…

  13. Equating with Miditests Using IRT

    ERIC Educational Resources Information Center

    Fitzpatrick, Joseph; Skorupski, William P.

    2016-01-01

    The equating performance of two internal anchor test structures--miditests and minitests--is studied for four IRT equating methods using simulated data. Originally proposed by Sinharay and Holland, miditests are anchors that have the same mean difficulty as the overall test but less variance in item difficulties. Four popular IRT equating methods…

  14. Successfully Transitioning to Linear Equations

    ERIC Educational Resources Information Center

    Colton, Connie; Smith, Wendy M.

    2014-01-01

    The Common Core State Standards for Mathematics (CCSSI 2010) asks students in as early as fourth grade to solve word problems using equations with variables. Equations studied at this level generate a single solution, such as the equation x + 10 = 25. For students in fifth grade, the Common Core standard for algebraic thinking expects them to…

  15. Evaluating Cross-Lingual Equating.

    ERIC Educational Resources Information Center

    Rapp, Joel; Allalouf, Avi

    This study examined the cross-lingual equating process adopted by a large scale testing system in which target language (TL) forms are equated to the source language (SL) forms using a set of translated items. The focus was on evaluating the degree of error inherent in the routine cross-lingual equating of the Verbal Reasoning subtest of the…

  16. Solving Nonlinear Coupled Differential Equations

    NASA Technical Reports Server (NTRS)

    Mitchell, L.; David, J.

    1986-01-01

    Harmonic balance method developed to obtain approximate steady-state solutions for nonlinear coupled ordinary differential equations. Method usable with transfer matrices commonly used to analyze shaft systems. Solution to nonlinear equation, with periodic forcing function represented as sum of series similar to Fourier series but with form of terms suggested by equation itself.

  17. Generalized Klein-Kramers equations

    NASA Astrophysics Data System (ADS)

    Fa, Kwok Sau

    2012-12-01

    A generalized Klein-Kramers equation for a particle interacting with an external field is proposed. The equation generalizes the fractional Klein-Kramers equation introduced by Barkai and Silbey [J. Phys. Chem. B 104, 3866 (2000), 10.1021/jp993491m]. Besides, the generalized Klein-Kramers equation can also recover the integro-differential Klein-Kramers equation for continuous-time random walk; this means that it can describe the subdiffusive and superdiffusive regimes in the long-time limit. Moreover, analytic solutions for first two moments both in velocity and displacement (for force-free case) are obtained, and their dynamic behaviors are investigated.

  18. Multinomial Diffusion Equation

    SciTech Connect

    Balter, Ariel I.; Tartakovsky, Alexandre M.

    2011-06-01

    We have developed a novel stochastic, space/time discrete representation of particle diffusion (e.g. Brownian motion) based on discrete probability distributions. We show that in the limit of both very small time step and large concentration, our description is equivalent to the space/time continuous stochastic diffusion equation. Being discrete in both time and space, our model can be used as an extremely accurate, efficient, and stable stochastic finite-difference diffusion algorithm when concentrations are so small that computationally expensive particle-based methods are usually needed. Through numerical simulations, we show that our method can generate realizations that capture the statistical properties of particle simulations. While our method converges converges to both the correct ensemble mean and ensemble variance very quickly with decreasing time step, but for small concentration, the stochastic diffusion PDE does not, even for very small time steps.

  19. Structural Equation Model Trees

    PubMed Central

    Brandmaier, Andreas M.; von Oertzen, Timo; McArdle, John J.; Lindenberger, Ulman

    2015-01-01

    In the behavioral and social sciences, structural equation models (SEMs) have become widely accepted as a modeling tool for the relation between latent and observed variables. SEMs can be seen as a unification of several multivariate analysis techniques. SEM Trees combine the strengths of SEMs and the decision tree paradigm by building tree structures that separate a data set recursively into subsets with significantly different parameter estimates in a SEM. SEM Trees provide means for finding covariates and covariate interactions that predict differences in structural parameters in observed as well as in latent space and facilitate theory-guided exploration of empirical data. We describe the methodology, discuss theoretical and practical implications, and demonstrate applications to a factor model and a linear growth curve model. PMID:22984789

  20. Elliptic scattering equations

    NASA Astrophysics Data System (ADS)

    Cardona, Carlos; Gomez, Humberto

    2016-06-01

    Recently the CHY approach has been extended to one loop level using elliptic functions and modular forms over a Jacobian variety. Due to the difficulty in manipulating these kind of functions, we propose an alternative prescription that is totally algebraic. This new proposal is based on an elliptic algebraic curve embedded in a mathbb{C}{P}^2 space. We show that for the simplest integrand, namely the n - gon, our proposal indeed reproduces the expected result. By using the recently formulated Λ-algorithm, we found a novel recurrence relation expansion in terms of tree level off-shell amplitudes. Our results connect nicely with recent results on the one-loop formulation of the scattering equations. In addition, this new proposal can be easily stretched out to hyperelliptic curves in order to compute higher genus.

  1. Λ scattering equations

    NASA Astrophysics Data System (ADS)

    Gomez, Humberto

    2016-06-01

    The CHY representation of scattering amplitudes is based on integrals over the moduli space of a punctured sphere. We replace the punctured sphere by a double-cover version. The resulting scattering equations depend on a parameter Λ controlling the opening of a branch cut. The new representation of scattering amplitudes possesses an enhanced redundancy which can be used to fix, modulo branches, the location of four punctures while promoting Λ to a variable. Via residue theorems we show how CHY formulas break up into sums of products of smaller (off-shell) ones times a propagator. This leads to a powerful way of evaluating CHY integrals of generic rational functions, which we call the Λ algorithm.

  2. On nonautonomous Dirac equation

    SciTech Connect

    Hovhannisyan, Gro; Liu Wen

    2009-12-15

    We construct the fundamental solution of time dependent linear ordinary Dirac system in terms of unknown phase functions. This construction gives approximate representation of solutions which is useful for the study of asymptotic behavior. Introducing analog of Rayleigh quotient for differential equations we generalize Hartman-Wintner asymptotic integration theorems with the error estimates for applications to the Dirac system. We also introduce the adiabatic invariants for the Dirac system, which are similar to the adiabatic invariant of Lorentz's pendulum. Using a small parameter method it is shown that the change in the adiabatic invariants approaches zero with the power speed as a small parameter approaches zero. As another application we calculate the transition probabilities for the Dirac system. We show that for the special choice of electromagnetic field, the only transition of an electron to the positron with the opposite spin orientation is possible.

  3. Parabolized stability equations

    NASA Astrophysics Data System (ADS)

    Herbert, Thorwald

    1994-04-01

    The parabolized stability equations (PSE) are a new approach to analyze the streamwise evolution of single or interacting Fourier modes in weakly nonparallel flows such as boundary layers. The concept rests on the decomposition of every mode into a slowly varying amplitude function and a wave function with slowly varying wave number. The neglect of the small second derivatives of the slowly varying functions with respect to the streamwise variable leads to an initial boundary-value problem that can be solved by numerical marching procedures. The PSE approach is valid in convectively unstable flows. The equations for a single mode are closely related to those of the traditional eigenvalue problems for linear stability analysis. However, the PSE approach does not exploit the homogeneity of the problem and, therefore, can be utilized to analyze forced modes and the nonlinear growth and interaction of an initial disturbance field. In contrast to the traditional patching of local solutions, the PSE provide the spatial evolution of modes with proper account for their history. The PSE approach allows studies of secondary instabilities without the constraints of the Floquet analysis and reproduces the established experimental, theoretical, and computational benchmark results on transition up to the breakdown stage. The method matches or exceeds the demonstrated capabilities of current spatial Navier-Stokes solvers at a small fraction of their computational cost. Recent applications include studies on localized or distributed receptivity and prediction of transition in model environments for realistic engineering problems. This report describes the basis, intricacies, and some applications of the PSE methodology.

  4. A New Class of Non-Linear, Finite-Volume Methods for Vlasov Simulation

    SciTech Connect

    Banks, J W; Hittinger, J A

    2009-11-24

    Methods for the numerical discretization of the Vlasov equation should efficiently use the phase space discretization and should introduce only enough numerical dissipation to promote stability and control oscillations. A new high-order, non-linear, finite-volume algorithm for the Vlasov equation that discretely conserves particle number and controls oscillations is presented. The method is fourth-order in space and time in well-resolved regions, but smoothly reduces to a third-order upwind scheme as features become poorly resolved. The new scheme is applied to several standard problems for the Vlasov-Poisson system, and the results are compared with those from other finite-volume approaches, including an artificial viscosity scheme and the Piecewise Parabolic Method. It is shown that the new scheme is able to control oscillations while preserving a higher degree of fidelity of the solution than the other approaches.

  5. Potential distribution around a test charge in a positive dust-electron plasma

    NASA Astrophysics Data System (ADS)

    Ali, S.

    2016-06-01

    The electrostatic potential caused by a test-charge particle in a positive dust-electron plasma is studied, accounting for the dust-charge fluctuations associated with ultraviolet photoelectron and thermionic emissions. For this purpose, the set of Vlasov-Poisson equations coupled with the dust charging equation is solved by using the space-time Fourier transform technique. As a consequence, a modified dielectric response function is obtained for dust-acoustic waves in a positive dust-electron plasma. By imposing certain conditions on the velocity of the test charge, the electrostatic potential is decomposed into the Debye-H¨uckel (DH), wake-field (WF), and far-field (FF) potentials that are significantly modified in the limit of a large dust-charge relaxation rate both analytically and numerically. The results can be helpful for understanding dust crystallization/coagulation in twocomponent plasmas, where positively charged dust grains are present.

  6. Simulations of plasma sheaths using continuum kinetic models

    NASA Astrophysics Data System (ADS)

    Srinivasan, Bhuvana; Hakim, Ammar

    2015-11-01

    Understanding plasma sheath physics is important for the performance of devices such as Hall thrusters due to the effect of energetic particles on electrode erosion. Plasma sheath physics is studied using kinetic and multi-fluid models with relevance to secondary electron emissions and plasma-surface interactions. Continuum kinetic models are developed to directly solve the Vlasov-Poisson equation using the discontinuous Galerkin method for each of the ion and electron species. A steady-state sheath is simulated by including a simple model for a neutral fluid. Multi-fluid simulations for the plasma sheath are also performed using the discontinuous Galerkin method to solve a complete set of fluid equations for each of the ion and electron species. The kinetic plasma sheath is compared to a multi-fluid plasma sheath. Supported by Air Force Office of Scientific Research.

  7. Mode decomposition evolution equations

    PubMed Central

    Wang, Yang; Wei, Guo-Wei; Yang, Siyang

    2011-01-01

    Partial differential equation (PDE) based methods have become some of the most powerful tools for exploring the fundamental problems in signal processing, image processing, computer vision, machine vision and artificial intelligence in the past two decades. The advantages of PDE based approaches are that they can be made fully automatic, robust for the analysis of images, videos and high dimensional data. A fundamental question is whether one can use PDEs to perform all the basic tasks in the image processing. If one can devise PDEs to perform full-scale mode decomposition for signals and images, the modes thus generated would be very useful for secondary processing to meet the needs in various types of signal and image processing. Despite of great progress in PDE based image analysis in the past two decades, the basic roles of PDEs in image/signal analysis are only limited to PDE based low-pass filters, and their applications to noise removal, edge detection, segmentation, etc. At present, it is not clear how to construct PDE based methods for full-scale mode decomposition. The above-mentioned limitation of most current PDE based image/signal processing methods is addressed in the proposed work, in which we introduce a family of mode decomposition evolution equations (MoDEEs) for a vast variety of applications. The MoDEEs are constructed as an extension of a PDE based high-pass filter (Europhys. Lett., 59(6): 814, 2002) by using arbitrarily high order PDE based low-pass filters introduced by Wei (IEEE Signal Process. Lett., 6(7): 165, 1999). The use of arbitrarily high order PDEs is essential to the frequency localization in the mode decomposition. Similar to the wavelet transform, the present MoDEEs have a controllable time-frequency localization and allow a perfect reconstruction of the original function. Therefore, the MoDEE operation is also called a PDE transform. However, modes generated from the present approach are in the spatial or time domain and can be

  8. Langevin equation approach to reactor noise analysis: stochastic transport equation

    SciTech Connect

    Akcasu, A.Z. ); Stolle, A.M. )

    1993-01-01

    The application of the Langevin equation method to the study of fluctuations in the space- and velocity-dependent neutron density as well as in the detector outputs in nuclear reactors is presented. In this case, the Langevin equation is the stochastic linear neutron transport equation with a space- and velocity-dependent random neutron source, often referred to as the noise equivalent source (NES). The power spectral densities (PSDs) of the NESs in the transport equation, as well as in the accompanying detection rate equations, are obtained, and the cross- and auto-power spectral densities of the outputs of pairs of detectors are explicitly calculated. The transport-level expression for the R([omega]) ratio measured in the [sup 252]Cf source-driven noise analysis method is also derived. Finally, the implementation of the Langevin equation approach at different levels of approximation is discussed, and the stochastic one-speed transport and one-group P[sub 1] equations are derived by first integrating the stochastic transport equation over speed and then eliminating the angular dependence by a spherical harmonics expansion. By taking the large transport rate limit in the P[sub 1] description, the stochastic diffusion equation is obtained as well as the PSD of the NES in it. This procedure also leads directly to the stochastic Fick's law.

  9. A spinor representation of Maxwell equations and Dirac equation

    SciTech Connect

    Vaz, J. Jr.; Rodrigues, W.A. Jr.

    1993-02-01

    Using the Clifford bundle formalism and starting from the free Maxwell equations dF = {delta}F = 0 we show by writing F = b{psi}{gamma}{sup 1}{gamma}{sup 2}{psi}{sup *}, where {psi} is a Dirac-Hestenes spinor field, that the Dirac-Hestenes equation (which is the representative of the standard Dirac equation in the Clifford bundle over Minkowski spacetime) is equivalent under general assumptions to those free Maxwell equations. We briefly discuss the implications of our findings for the interpretation of quantum mechanics. 15 refs.

  10. JWL Equation of State

    SciTech Connect

    Menikoff, Ralph

    2015-12-15

    The JWL equation of state (EOS) is frequently used for the products (and sometimes reactants) of a high explosive (HE). Here we review and systematically derive important properties. The JWL EOS is of the Mie-Grueneisen form with a constant Grueneisen coefficient and a constants specific heat. It is thermodynamically consistent to specify the temperature at a reference state. However, increasing the reference state temperature restricts the EOS domain in the (V, e)-plane of phase space. The restrictions are due to the conditions that P ≥ 0, T ≥ 0, and the isothermal bulk modulus is positive. Typically, this limits the low temperature regime in expansion. The domain restrictions can result in the P-T equilibrium EOS of a partly burned HE failing to have a solution in some cases. For application to HE, the heat of detonation is discussed. Example JWL parameters for an HE, both products and reactions, are used to illustrate the restrictions on the domain of the EOS.

  11. A note on "Kepler's equation".

    NASA Astrophysics Data System (ADS)

    Dutka, J.

    1997-07-01

    This note briefly points out the formal similarity between Kepler's equation and equations developed in Hindu and Islamic astronomy for describing the lunar parallax. Specifically, an iterative method for calculating the lunar parallax has been developed by the astronomer Habash al-Hasib al-Marwazi (about 850 A.D., Turkestan), which is surprisingly similar to the iterative method for solving Kepler's equation invented by Leonhard Euler (1707 - 1783).

  12. Electronic representation of wave equation

    NASA Astrophysics Data System (ADS)

    Veigend, Petr; Kunovský, Jiří; Kocina, Filip; Nečasová, Gabriela; Šátek, Václav; Valenta, Václav

    2016-06-01

    The Taylor series method for solving differential equations represents a non-traditional way of a numerical solution. Even though this method is not much preferred in the literature, experimental calculations done at the Department of Intelligent Systems of the Faculty of Information Technology of TU Brno have verified that the accuracy and stability of the Taylor series method exceeds the currently used algorithms for numerically solving differential equations. This paper deals with solution of Telegraph equation using modelling of a series small pieces of the wire. Corresponding differential equations are solved by the Modern Taylor Series Method.

  13. Transient growth in stable linearized Vlasov-Maxwell plasmas

    NASA Astrophysics Data System (ADS)

    Podesta, J. J.

    2010-12-01

    Large amplitude transient growth of kinetic scale perturbations in stable collisionless magnetized plasmas has recently been demonstrated using a linearized Landau fluid model. Initial perturbations with lengthscales of the order of the ion gyroradius were shown to have transient timescales that in some cases were long compared to the ion gyroperiod, Ωit≫1. Moreover, it was suggested that such perturbations are not rare but instead form a large class within the set of all possible initial conditions. For collisionless plasmas, the Vlasov-Maxwell equations provide a more complete description of kinetic physics and the existence of transient growth of solutions for the linearized Vlasov-Maxwell system is an interesting question. The existence of transient growth of solutions is demonstrated here for a special case of the Vlasov-Maxwell equations, namely, the one dimensional Vlasov-Poisson system. The analysis is different from the standard approach of nonmodal analysis since the initial value problem is described by a Volterra integral equation of the second kind, reflecting the fact that the time evolution of the system depends on the memory of the state from time zero through time t. For the case of a thermal equilibrium plasma, it is shown how initial conditions may be constructed to obtain solutions that grow linearly in time; the duration of this growth is the time required for a thermal electron to traverse the wavelength of the initial perturbation, a timescale that can last for many plasma periods 2π/ωpe, thus demonstrating the existence of transient growth of solutions for the linearized Vlasov-Poisson system. The results suggest that the phenomenon of transient growth may be a common feature of the linearized Vlasov-Maxwell system as well as for Landau fluid models.

  14. Uncertainty of empirical correlation equations

    NASA Astrophysics Data System (ADS)

    Feistel, R.; Lovell-Smith, J. W.; Saunders, P.; Seitz, S.

    2016-08-01

    The International Association for the Properties of Water and Steam (IAPWS) has published a set of empirical reference equations of state, forming the basis of the 2010 Thermodynamic Equation of Seawater (TEOS-10), from which all thermodynamic properties of seawater, ice, and humid air can be derived in a thermodynamically consistent manner. For each of the equations of state, the parameters have been found by simultaneously fitting equations for a range of different derived quantities using large sets of measurements of these quantities. In some cases, uncertainties in these fitted equations have been assigned based on the uncertainties of the measurement results. However, because uncertainties in the parameter values have not been determined, it is not possible to estimate the uncertainty in many of the useful quantities that can be calculated using the parameters. In this paper we demonstrate how the method of generalised least squares (GLS), in which the covariance of the input data is propagated into the values calculated by the fitted equation, and in particular into the covariance matrix of the fitted parameters, can be applied to one of the TEOS-10 equations of state, namely IAPWS-95 for fluid pure water. Using the calculated parameter covariance matrix, we provide some preliminary estimates of the uncertainties in derived quantities, namely the second and third virial coefficients for water. We recommend further investigation of the GLS method for use as a standard method for calculating and propagating the uncertainties of values computed from empirical equations.

  15. Graphical Solution of Polynomial Equations

    ERIC Educational Resources Information Center

    Grishin, Anatole

    2009-01-01

    Graphing utilities, such as the ubiquitous graphing calculator, are often used in finding the approximate real roots of polynomial equations. In this paper the author offers a simple graphing technique that allows one to find all solutions of a polynomial equation (1) of arbitrary degree; (2) with real or complex coefficients; and (3) possessing…

  16. Drug Levels and Difference Equations

    ERIC Educational Resources Information Center

    Acker, Kathleen A.

    2004-01-01

    American university offers a course in finite mathematics whose focus is difference equation with emphasis on real world applications. The conclusion states that students learned to look for growth and decay patterns in raw data, to recognize both arithmetic and geometric growth, and to model both scenarios with graphs and difference equations.

  17. Generalized Multilevel Structural Equation Modeling

    ERIC Educational Resources Information Center

    Rabe-Hesketh, Sophia; Skrondal, Anders; Pickles, Andrew

    2004-01-01

    A unifying framework for generalized multilevel structural equation modeling is introduced. The models in the framework, called generalized linear latent and mixed models (GLLAMM), combine features of generalized linear mixed models (GLMM) and structural equation models (SEM) and consist of a response model and a structural model for the latent…

  18. Complete solution of Boolean equations

    NASA Technical Reports Server (NTRS)

    Tapia, M. A.; Tucker, J. H.

    1980-01-01

    A method is presented for generating a single formula involving arbitary Boolean parameters, which includes in it each and every possible solution of a system of Boolean equations. An alternate condition equivalent to a known necessary and sufficient condition for solving a system of Boolean equations is given.

  19. Students' Understanding of Quadratic Equations

    ERIC Educational Resources Information Center

    López, Jonathan; Robles, Izraim; Martínez-Planell, Rafael

    2016-01-01

    Action-Process-Object-Schema theory (APOS) was applied to study student understanding of quadratic equations in one variable. This required proposing a detailed conjecture (called a genetic decomposition) of mental constructions students may do to understand quadratic equations. The genetic decomposition which was proposed can contribute to help…

  20. The Equations of Oceanic Motions

    NASA Astrophysics Data System (ADS)

    Müller, Peter

    2006-10-01

    Modeling and prediction of oceanographic phenomena and climate is based on the integration of dynamic equations. The Equations of Oceanic Motions derives and systematically classifies the most common dynamic equations used in physical oceanography, from large scale thermohaline circulations to those governing small scale motions and turbulence. After establishing the basic dynamical equations that describe all oceanic motions, M|ller then derives approximate equations, emphasizing the assumptions made and physical processes eliminated. He distinguishes between geometric, thermodynamic and dynamic approximations and between the acoustic, gravity, vortical and temperature-salinity modes of motion. Basic concepts and formulae of equilibrium thermodynamics, vector and tensor calculus, curvilinear coordinate systems, and the kinematics of fluid motion and wave propagation are covered in appendices. Providing the basic theoretical background for graduate students and researchers of physical oceanography and climate science, this book will serve as both a comprehensive text and an essential reference.