Exact momentum conservation laws for the gyrokinetic Vlasov-Poisson equations
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.
On Higher-order Corrections to Gyrokinetic Vlasov-Poisson Equations in the Long Wavelength Limit
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.
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.
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).
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.
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.
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.
Nonlinear gyrokinetic equations
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.
On the singularity of the Vlasov-Poisson system
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.
On the Singularity of the Vlasov-Poisson System
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.
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.
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.
Verification of gyrokinetic microstability codes with an LHD configuration
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.
Verification of gyrokinetic microstability codes with an LHD configuration
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.
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.
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)
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.
Linear stability of stationary solutions of the Vlasov-Poisson system in three dimensions
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.
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.
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.
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.
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.
Nonlinear scattering term in the gyrokinetic Vlasov equation
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.
Nonlinear electromagnetic gyrokinetic equations for rotating axisymmetric plasmas
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.
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.
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.
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.
Fully Electromagnetic Nonlinear Gyrokinetic Equations for Tokamak Edge Turbulence
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 ρ_{i}<< ρ_{θ¡} ~ L_{E} ~ L_{p} << R (here ρ_{i} is the thermal ion Larmor radius and ρ_{θ¡} = B/B_{θ}] ρ_{i}), 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 ρ_{i} ~ 1 for generality, and keep the relative fluctuation amplitudes eδφ /Τ_{i} ~ 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.
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=
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.
Geodesic acoustic mode in anisotropic plasmas using double adiabatic model and gyro-kinetic equation
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.
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.
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.
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.
From charge motion in general magnetic fields to the non perturbative gyrokinetic equation
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.
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.
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.
Conservation of energy and momentum in nonrelativistic plasmas
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.
Energetically consistent collisional gyrokinetics
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.
Energetically consistent collisional gyrokinetics
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.
Intrinsic rotation with gyrokinetic models
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.
A Short Introduction to General Gyrokinetic Theory
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
Energetically consistent collisional gyrokinetics
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.
Electromagnetic nonlinear gyrokinetics with polarization drift
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.
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.
Gyrokinetic particle simulation model
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.
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.
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.
Geometric Gyrokinetic Theory for Edge Plasma
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
Nonlinear Gyrokinetic Theory With Polarization Drift
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)] .
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)].
Free energy balance in gyrokinetic turbulence
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.
Neoclassical equilibrium in gyrokinetic simulations
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.
Noiseless Vlasov-Poisson simulations with linearly transformed particles
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.
Noiseless Vlasov-Poisson simulations with linearly transformed particles
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
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.
Transport of momentum in full f gyrokinetics
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.
Incorrectness of the usual gyrokinetic treatment in cylindrically symmetric systems
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.
Gyrokinetic Statistical Absolute Equilibrium and Turbulence
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.
Continuum Edge Gyrokinetic Theory and Simulations
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.
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).
Gyrokinetic large eddy simulations
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.
Gyrokinetic microtearing turbulence.
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
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.
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.
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
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.
Transport in gyrokinetic tokamaks
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.
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.
Nonlinear gyrokinetic theory for finite-BETA plasmas
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.
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.
Simulation of neoclassical transport with the continuum gyrokinetic code COGENT
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.
Simulation of neoclassical transport with the continuum gyrokinetic code COGENT
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
Simulation of neoclassical transport with the continuum gyrokinetic code COGENT
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.
Effects of collisions on conservation laws in gyrokinetic field theory
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.
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.
Gyrokinetic simulations of the tearing instability
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.
On push-forward representations in the standard gyrokinetic model
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.
Discoveries from the exploration of gyrokinetic momentum transport
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.
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.
High frequency gyrokinetic particle simulation
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.
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.
A generalized gyrokinetic Poisson solver
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.
Gyro-water-bag approach in nonlinear gyrokinetic turbulence
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.
Equilibrium fluctuation energy of gyrokinetic plasma
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
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
Gyrokinetic modeling: A multi-water-bag approach
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.
A gyrokinetic one-dimensional scrape-off layer model of an edge-localized mode heat pulse
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
A gyrokinetic one-dimensional scrape-off layer model of an edge-localized mode heat pulse
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.
Dynamic procedure for filtered gyrokinetic simulations
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.
Gyrokinetic theory and simulation of angular momentum transport
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.
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.
Free Energy Cascade in Gyrokinetic Turbulence
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.
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.
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.
3D hybrid simulations with gyrokinetic particle ions and fluid electrons
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.
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.
Status of Continuum Edge Gyrokinetic Code Physics Development
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
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.
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.
Global gyrokinetic simulation of tokamak transport
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.
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.
Parallel magnetic field perturbations in gyrokinetic simulations
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.
Generalized Covariant Gyrokinetic Dynamics of Magnetoplasmas
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.
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.
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.
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.
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.
Gyrokinetic Simulations of ETG and ITG Turbulence
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.
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.
THE LOCAL LIMIT OF GLOBAL GYROKINETIC SIMULATIONS
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.
Gyrokinetic turbulence simulations at high plasma beta
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.
Saturation of Gyrokinetic Turbulence through Damped Eigenmodes
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.
Beyond linear gyrocenter polarization in gyrokinetic theory
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.
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.
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.
Gyrokinetic simulations of ion and impurity transport
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.
Gyrokinetic simulation of internal kink modes
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.
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.
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.
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.
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.
Full f gyrokinetic method for particle simulation of tokamak transport
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.
Benchmarking gyrokinetic simulations in a toroidal flux-tube
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.
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.
Comparisons of gyrofluid and gyrokinetic simulations
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.
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.
Saturation of gyrokinetic turbulence through damped eigenmodes.
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
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.
Petascale Parallelization of the Gyrokinetic Toroidal Code
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.
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.
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.
Extended gyrokinetic field theory for time-dependent magnetic confinement fields
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.
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.
Gyrokinetic particle simulation of neoclassical transport
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.
Gyrokinetic particle simulation of neoclassical transport
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}.
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.
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.
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 ⊙}.
Gyrokinetic theory and simulation of turbulent energy exchange
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.
Gyrokinetic particle simulation of beta-induced Alfven eigenmode
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.
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.
Nonlinear Gyrokinetics: A Powerful Tool for the Description of Microturbulence in Magnetized Plasmas
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.
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.
A verification of the gyrokinetic microstability codes GEM, GYRO, and GS2
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.
Relevance of the parallel nonlinearity in gyrokinetic simulations of tokamak plasmas
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.
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.
Fully Nonlinear Edge Gyrokinetic Simulations of Kinetic Geodesic-Acoustic Modes and Boundary Flows
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.
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.
Quasisteady and steady states in global gyrokinetic particle-in-cell simulations
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.
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.
Gyrokinetic turbulence cascade via predator-prey interactions between different scales
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.
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.
Gyrokinetic determination of the electrostatic potential of rotating magnetic islands in tokamaks
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.
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.
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.
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.
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
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.
Gyrokinetics for high-frequency modes in tokamaks
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.
A Numerical Instability in an ADI Algorithm for Gyrokinetics
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.
The energetic coupling of scales in gyrokinetic plasma turbulence
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.
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.
Plasma Simulation Using Gyrokinetic-Gyrofluid Hybrid Models
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.
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.
Tokamak profile prediction using direct gyrokinetic and neoclassical simulation
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.
Tokamak profile prediction using direct gyrokinetic and neoclassical simulation
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.
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).
Gyrokinetic simulation of isotope scaling in tokamak plasmas
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.
Applications of large eddy simulation methods to gyrokinetic turbulence
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.
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.
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.
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).
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.
Direct identification of predator-prey dynamics in gyrokinetic simulations
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.
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.
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.
Gyrokinetic modelling of stationary electron and impurity profiles in tokamaks
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.
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.
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.
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.
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.
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.
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.
Nonlinear theory of drift-cyclotron kinetics and the possible breakdown of gyro-kinetics
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.
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.
On the Existence of Canonical Gyrokinetic Variables for Chaotic Magnetic Fields
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.
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
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.
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.
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).
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
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.
Electron heat transport from stochastic fields in gyrokinetic simulations
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.
Gyrokinetic Simulation of Global Turbulent Transport Properties in Tokamak Experiments
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.
Gyrokinetic studies of microinstabilities in the reversed field pinch
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.
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.
Global spectral investigation of plasma turbulence in gyrokinetic simulations
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.
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.
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).
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.
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.
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.
ADVANCES IN COMPREHENSIVE GYROKINETIC SIMULATIONS OF TRANSPORT IN TOKAMAKS
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.
ADVANCES IN COMPREHENSIVE GYROKINETIC SIMULATIONS OF TRANSPORT IN TOKAMAKS
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.
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.
Effects of Plasma Shaping on Nonlinear Gyrokinetic Turbulence
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.
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.
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.
Electromagnetic gyrokinetic turbulence in finite-beta helical plasmas
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.
Full-f gyrokinetic simulation over a confinement time
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.
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.
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.
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
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.
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.
A multi-species collisional operator for full-F gyrokinetics
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.
Performance of particle in cell methods on highly concurrent computational architectures
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.
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.
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.
Verification of Gyrokinetic (delta)f Simulations of Electron Temperature Gradient Turbulence
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.
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.
Global gyrokinetic particle-in-cell simulations of internal kink instabilities
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.
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
Linear and nonlinear verification of gyrokinetic microstability codes
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
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
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.
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.
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
Gyrokinetic study of electromagnetic effects on toroidal momentum transport in tokamak plasmas
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.
GYSELA, a full-f global gyrokinetic Semi-Lagrangian code for ITG turbulence simulations
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.
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.
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).
Nature of turbulent transport across sheared zonal flows: insights from gyro-kinetic simulations
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
Advanced methods in global gyrokinetic full f particle simulation of tokamak transport
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.
Gyrokinetic studies of core turbulence features in ASDEX Upgrade H-mode plasmas
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.
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.
Canonical Lie-transform method in Hamiltonian gyrokinetics: a new approach
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.
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.
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.
Gyrokinetic particle simulations of reversed shear Alfven eigenmode excited by antenna and fast ions
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.
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.
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.
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.
A flux-matched gyrokinetic analysis of DIII-D L-mode turbulence
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.
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.
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.
Fully electromagnetic gyrokinetic eigenmode analysis of high-beta shaped plasmas
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.
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.
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.