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.
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).
A critical comparison of the numerical solution of the 1D filtered Vlasov-Poisson equation
NASA Astrophysics Data System (ADS)
Viñas, A. F.; Klimas, A. J.
2003-04-01
We present a comparison of the numerical solution of the filtered Vlasov-Poisson system of equations using the Fourier-Fourier and the Flux-Balance-MacCormack methods in the electrostatic, non-relativistic case. We show that the splitting method combined with the Flux-Balance-MacCormack scheme provides an efficient and accurate scheme for integrating the filtered Vlasov-Poisson system in their self-consistent field. Finally we present various typical problems of interest in plasma physics research which can be studied with the scheme presented here.
A numerical method for solving the Vlasov-Poisson equation based on the conservative IDO scheme
NASA Astrophysics Data System (ADS)
Imadera, Kenji; Kishimoto, Yasuaki; Saito, Daisuke; Li, Jiquan; Utsumi, Takayuki
2009-12-01
We have applied the conservative form of the Interpolated Differential Operator (IDO-CF) scheme in order to solve the Vlasov-Poisson equation, which is one of the multi-moment schemes. Through numerical tests of the nonlinear Landau damping and two-stream instability, we compared the present scheme with other schemes such as the Spline and CIP ones. We mainly investigated the conservation property of the L1-norm, energy, entropy and phase space area for each scheme, and demonstrated that the IDO-CF scheme is capable of performing stable long time scale simulation while maintaining high accuracy. The scheme is based on an Eulerian approach, and it can thus be directly used for Fokker-Planck, high dimensional Vlasov-Poisson and also guiding-center drift simulations, aiming at particular problems of plasma physics. The benchmark tests for such simulations have shown that the IDO-CF scheme is superior in keeping the conservation properties without causing serious phase error.
Vlasov-Maxwell and Vlasov-Poisson equations as models of a one-dimensional electron plasma
NASA Technical Reports Server (NTRS)
Klimas, A. J.; Cooper, J.
1983-01-01
The Vlasov-Maxwell and Vlasov-Poisson systems of equations for a one-dimensional electron plasma are defined and discussed. A method for transforming a solution of one system which is periodic over a bounded or unbounded spatial interval to a similar solution of the other is constructed.
Nonlinear gyrokinetic equations for tokamak microturbulence
Hahm, T.S.
1988-05-01
A nonlinear electrostatic gyrokinetic Vlasov equation, as well as Poisson equation, has been derived in a form suitable for particle simulation studies of tokamak microturbulence and associated anomalous transport. This work differs from the existing nonlinear gyrokinetic theories in toroidal geometry, since the present equations conserve energy while retaining the crucial linear and nonlinear polarization physics. In the derivation, the action-variational Lie perturbation method is utilized in order to preserve the Hamiltonian structure of the original Vlasov-Poisson system. Emphasis is placed on the dominant physics of the collective fluctuations in toroidal geometry, rather than on details of particle orbits. 13 refs.
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.
Vlasov-Poisson in 1D: waterbags
NASA Astrophysics Data System (ADS)
Colombi, Stéphane; Touma, Jihad
2014-07-01
We revisit in one dimension the waterbag method to solve numerically Vlasov-Poisson equations. In this approach, the phase-space distribution function f (x, v) is initially sampled by an ensemble of patches, the waterbags, where f is assumed to be constant. As a consequence of Liouville theorem, it is only needed to follow the evolution of the border of these waterbags, which can be done by employing an orientated, self-adaptive polygon tracing isocontours of f. This method, which is entropy conserving in essence, is very accurate and can trace very well non-linear instabilities as illustrated by specific examples. As an application of the method, we generate an ensemble of single-waterbag simulations with decreasing thickness to perform a convergence study to the cold case. Our measurements show that the system relaxes to a steady state where the gravitational potential profile is a power law of slowly varying index β, with β close to 3/2 as found in the literature. However, detailed analysis of the properties of the gravitational potential shows that at the centre, β > 1.54. Moreover, our measurements are consistent with the value β = 8/5 = 1.6 that can be analytically derived by assuming that the average of the phase-space density per energy level obtained at crossing times is conserved during the mixing phase. These results are incompatible with the logarithmic slope of the projected density profile β - 2 ≃ -0.47 obtained recently by Schulz et al. using an N-body technique. This sheds again strong doubts on the capability of N-body techniques to converge to the correct steady state expected in the continuous limit.
The Vlasov-Poisson System for Stellar Dynamics in Spaces of Constant Curvature
NASA Astrophysics Data System (ADS)
Diacu, Florin; Ibrahim, Slim; Lind, Crystal; Shen, Shengyi
2016-09-01
We obtain a natural extension of the Vlasov-Poisson system for stellar dynamics to spaces of constant Gaussian curvature {κ ≠ 0}: the unit sphere {S^2}, for {κ > 0}, and the unit hyperbolic sphere {H^2}, for {κ < 0}. These equations can be easily generalized to higher dimensions. When the particles move on a geodesic, the system reduces to a 1-dimensional problem that is more singular than the classical analogue of the Vlasov-Poisson system. In the analysis of this reduced model, we study the well-posedness of the problem and derive Penrose-type conditions for linear stability around homogeneous solutions in the sense of Landau damping.
The Vlasov-Poisson Dynamics as the Mean Field Limit of Extended Charges
NASA Astrophysics Data System (ADS)
Lazarovici, Dustin
2016-10-01
The paper treats the validity problem of the nonrelativistic Vlasov-Poisson equation in {d ≥ 2} dimensions. It is shown that the Vlasov-Poisson dynamics can be derived as a combined mean field and point-particle limit of an N-particle Coulomb system of extended charges. This requires a sufficiently fast convergence of the initial empirical distributions. If the electron radius decreases slower than {N^{-{1/d(d+2)}}}, the corresponding initial configurations are typical. This result entails propagation of molecular chaos for the respective dynamics.
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.
Variational Principle and Stability of Nonmonotonic Vlasov-Poisson Equilibria
NASA Astrophysics Data System (ADS)
Morrison, P. J.
1987-10-01
The stability of nonmonotonic equilibria of the Vlasov-Poisson equation is assessed by using nonlinear constants of motion . The constants of motion make up the free energy of the system , which upon variation yields nonmonotonic equilibria. Such equilibria have not previously been obtainable from a variation principle, but here this is accomplished by the inclusion of a passively advected tracer field. Definiteness of the second variation of the free energy gives a sufficient condition for stability in agreement with Gardner's theorem [5], Previously, we have argued that indefiniteness implies either spectral in stability or negative energy modes, which are generically unstable when one adds dissipation or nonlinearity [6]. Such is the case for the nonmonotonic equilibria considered.
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.
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.
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.
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.
The parallel implementation of the one-dimensional Fourier transformed Vlasov Poisson system
NASA Astrophysics Data System (ADS)
Eliasson, Bengt
2005-08-01
A parallel implementation of an algorithm for solving the one-dimensional, Fourier transformed Vlasov-Poisson system of equations is documented, together with the code structure, file formats and settings to run the code. The properties of the Fourier transformed Vlasov-Poisson system is discussed in connection with the numerical solution of the system. The Fourier method in velocity space is used to treat numerical problems arising due the filamentation of the solution in velocity space. Outflow boundary conditions in the Fourier transformed velocity space removes the highest oscillations in velocity space. A fourth-order compact Padé scheme is used to calculate derivatives in the Fourier transformed velocity space, and spatial derivatives are calculated with a pseudo-spectral method. The parallel algorithms used are described in more detail, in particular the parallel solver of the tri-diagonal systems occurring in the Padé scheme. Program summaryTitle of program:vlasov Catalogue identifier:ADVQ Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADVQ Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Operating system under which the program has been tested: Sun Solaris; HP-UX; Read Hat Linux Programming language used: FORTRAN 90 with Message Passing Interface (MPI) Computers: Sun Ultra Sparc; HP 9000/785; HP IPF (Itanium Processor Family) ia64 Cluster; PCs cluster Number of lines in distributed program, including test data, etc.:3737 Number of bytes in distributed program, including test data, etc.:18 772 Distribution format: tar.gz Nature of physical problem: Kinetic simulations of collisionless electron-ion plasmas. Method of solution: A Fourier method in velocity space, a pseudo-spectral method in space and a fourth-order Runge-Kutta scheme in time. Memory required to execute with typical data: Uses typically of the order 10 5-10 6 double precision numbers. Restriction on the complexity of the problem: The program uses
High-order continuum kinetic Vlasov-Poisson simulations of magnetized plasmas
NASA Astrophysics Data System (ADS)
Vogman, G. V.; Colella, P.; Shumlak, U.
2014-10-01
Continuum methods offer a high-fidelity means of simulating plasma kinetics as modeled by the Boltzmann-Maxwell equation system. These methods are advantageous because they can be cast in conservation law form, are not susceptible to noise, and can be implemented using high-order numerical methods. Thereby the methods can conserve mass, momentum, and energy to a high degree. A fourth-order accurate finite volume method has been developed to solve the continuum kinetic Vlasov-Poisson equation system in one spatial and two velocity dimensions. The method is validated in cartesian coordinates using the Dory-Guest-Harris instability, which is a special case of a perpendicularly-propagating kinetic electrostatic wave in a warm uniformly magnetized plasma. The instability dispersion relation, and its generalization to arbitrary distribution functions, are demonstrated to be well-suited benchmarks for continuum algorithms in higher-dimensional phase space. The numerical method has also been extended to two spatial dimensions, and has been implemented in cylindrical coordinates to simulate axisymmetric configurations such as a Z-pinch. This work was supported by the DOE SCGF fellowship, and grants from DOE ASCR and AFOSR.
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)
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.
Collisional effects on the numerical recurrence in Vlasov-Poisson simulations
NASA Astrophysics Data System (ADS)
Pezzi, Oreste; Camporeale, Enrico; Valentini, Francesco
2016-02-01
The initial state recurrence in numerical simulations of the Vlasov-Poisson system is a well-known phenomenon. Here, we study the effect on recurrence of artificial collisions modeled through the Lenard-Bernstein operator [A. Lenard and I. B. Bernstein, Phys. Rev. 112, 1456-1459 (1958)]. By decomposing the linear Vlasov-Poisson system in the Fourier-Hermite space, the recurrence problem is investigated in the linear regime of the damping of a Langmuir wave and of the onset of the bump-on-tail instability. The analysis is then confirmed and extended to the nonlinear regime through an Eulerian collisional Vlasov-Poisson code. It is found that, despite being routinely used, an artificial collisionality is not a viable way of preventing recurrence in numerical simulations without compromising the kinetic nature of the solution. Moreover, it is shown how numerical effects associated to the generation of fine velocity scales can modify the physical features of the system evolution even in nonlinear regime. This means that filamentation-like phenomena, usually associated with low amplitude fluctuations contexts, can play a role even in nonlinear regime.
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.
A Uniqueness Criterion for Unbounded Solutions to the Vlasov-Poisson System
NASA Astrophysics Data System (ADS)
Miot, Evelyne
2016-09-01
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.
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.
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.
Conservation equations and calculation of mean flows in gyrokinetics
Abiteboul, J.; Garbet, X.; Grandgirard, V.; Allfrey, S. J.; Ghendrih, Ph.; Latu, G.; Sarazin, Y.; Strugarek, A.
2011-08-15
Conservation equations are derived for the gyrocenter toroidal momentum density and the polarization field. These equations are derived from the gyrokinetic model as it is implemented in simulation codes. In view of predicting the toroidal rotation in future fusion devices such as ITER, where external momentum input will be small, accurate simulations of momentum transport are crucial. The evolution equation for gyrocenter toroidal momentum density involves the divergence of the off-diagonal components of the Reynolds and generalized Maxwell stress, while the source term is the radial current of gyrocenters. The time evolution of the polarization field is the opposite of the gyrocenter current. Hence, an evolution equation for the total momentum density, i.e., the sum of gyrocenter and polarization field toroidal momentum density can be written. The force balance equation and the toroidal momentum conservation equations have been numerically tested with the gysela code. They are satisfied with a high level of accuracy.
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.
Nonlinear canonical gyrokinetic Vlasov equation and computation of the gyrocenter motion in tokamaks
Xu Yingfeng; Wang Shaojie
2013-01-15
The nonlinear canonical gyrokinetic Vlasov equation is obtained from the nonlinear noncanonical gyrokinetic theory using the property of the coordinate transform. In the linear approximation, it exactly recovers the previous linear canonical gyrokinetic equations derived by the Lie-transform perturbation method. The computation of the test particle gyrocenter motion in tokamaks with a large magnetic perturbation is presented and discussed. The numerical results indicate that the second-order gyrocenter Hamiltonian is important for the gyrocenter motion of the trapped electron in tokamaks with a large magnetic perturbation.
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)
Besse, Nicolas; Coulette, David
2016-08-01
Achieving plasmas with good stability and confinement properties is a key research goal for magnetic fusion devices. The underlying equations are the Vlasov-Poisson and Vlasov-Maxwell (VPM) equations in three space variables, three velocity variables, and one time variable. Even in those somewhat academic cases where global equilibrium solutions are known, studying their stability requires the analysis of the spectral properties of the linearized operator, a daunting task. We have identified a model, for which not only equilibrium solutions can be constructed, but many of their stability properties are amenable to rigorous analysis. It uses a class of solution to the VPM equations (or to their gyrokinetic approximations) known as waterbag solutions which, in particular, are piecewise constant in phase-space. It also uses, not only the gyrokinetic approximation of fast cyclotronic motion around magnetic field lines, but also an asymptotic approximation regarding the magnetic-field-induced anisotropy: the spatial variation along the field lines is taken much slower than across them. Together, these assumptions result in a drastic reduction in the dimensionality of the linearized problem, which becomes a set of two nested one-dimensional problems: an integral equation in the poloidal variable, followed by a one-dimensional complex Schrödinger equation in the radial variable. We show here that the operator associated to the poloidal variable is meromorphic in the eigenparameter, the pulsation frequency. We also prove that, for all but a countable set of real pulsation frequencies, the operator is compact and thus behaves mostly as a finite-dimensional one. The numerical algorithms based on such ideas have been implemented in a companion paper [D. Coulette and N. Besse, "Numerical resolution of the global eigenvalue problem for gyrokinetic-waterbag model in toroidal geometry" (submitted)] and were found to be surprisingly close to those for the original gyrokinetic
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.
Partially linearized algorithms in gyrokinetic particle simulation
Dimits, A.M.; Lee, W.W.
1990-10-01
In this paper, particle simulation algorithms with time-varying weights for the gyrokinetic Vlasov-Poisson system have been developed. The primary purpose is to use them for the removal of the selected nonlinearities in the simulation of gradient-driven microturbulence so that the relative importance of the various nonlinear effects can be assessed. It is hoped that the use of these procedures will result in a better understanding of the transport mechanisms and scaling in tokamaks. Another application of these algorithms is for the improvement of the numerical properties of the simulation plasma. For instance, implementations of such algorithms (1) enable us to suppress the intrinsic numerical noise in the simulation, and (2) also make it possible to regulate the weights of the fast-moving particles and, in turn, to eliminate the associated high frequency oscillations. Examples of their application to drift-type instabilities in slab geometry are given. We note that the work reported here represents the first successful use of the weighted algorithms in particle codes for the nonlinear simulation of plasmas.
Bernstein, I.B.; Catto, P.J.
1984-05-01
A nonlinear gyrokinetic formalism is developed which permits mean velocities comparable to thermal speeds in arbitrary magnetic field geometry. The theory is fully electromagnetic and does not employ an eikonal ansatz. The freedom in the theory is exploited to display simply the connection with ideal magnetohydrodynamics.
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.
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.
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.
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.
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.
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)] .
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.
A reanalysis of a strong-flow gyrokinetic formalism
Sharma, A. Y.; McMillan, B. F.
2015-03-15
We reanalyse an arbitrary-wavelength gyrokinetic formalism [A. M. Dimits, Phys. Plasmas 17, 055901 (2010)], which orders only the vorticity to be small and allows strong, time-varying flows on medium and long wavelengths. We obtain a simpler gyrocentre Lagrangian up to second order. In addition, the gyrokinetic Poisson equation, derived either via variation of the system Lagrangian or explicit density calculation, is consistent with that of the weak-flow gyrokinetic formalism [T. S. Hahm, Phys. Fluids 31, 2670 (1988)] at all wavelengths in the weak flow limit. The reanalysed formalism has been numerically implemented as a particle-in-cell code. An iterative scheme is described which allows for numerical solution of this system of equations, given the implicit dependence of the Euler-Lagrange equations on the time derivative of the potential.
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.
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
A finite mass based method for Vlasov-Poisson simulations
NASA Astrophysics Data System (ADS)
Larson, David; Young, Christopher
2014-10-01
A method for the numerical simulation of plasma dynamics using discrete particles is introduced. The shape function kinetics (SFK) method is based on decomposing the mass into discrete particles using shape functions of compact support. The particle positions and shape evolve in response to internal velocity spread and external forces. Remapping is necessary in order to maintain accuracy and two strategies for remapping the particles are discussed. Numerical simulations of standard test problems illustrate the advantages of the method which include very low noise compared to the standard particle-in-cell technique, inherent positivity, large dynamic range, and ease of implementation. This work was performed under the auspices of the U.S. Department of Energy by LLNL under Contract DE-AC52-07NA27344. C. V. Young acknowledges the support of the DOE NNSA Stewardship Science Graduate Fellowship under Contract DE-FC52-08NA28752.
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.
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
A numerical method for solving the Vlasov equation
NASA Technical Reports Server (NTRS)
Satofuka, N.
1982-01-01
A numerical procedure is derived for the solution of the Vlasov-Poisson system of equations in two phase-space variables. Derivatives with respect to the phase-space variables are approximated by a weighted sum of the values of the distribution function at property chosen neighboring points. The resulting set of ordinary differential equations is then solved by using an appropriate time intergration scheme. The accuracy of the proposed method is tested with some simple model problems. The results for the free streaming case, linear Landau damping, and nonlinear Landau damping are investigated and compared with those of the splitting scheme. The proposed method is found to be very accurate and efficient.
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.
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.
Improvements in the gyrokinetic simulation method
Matsuda, Y.; Cohen, B.I.; Williams, T.J.
1991-01-01
Gyrokinetic particle-in-cell (PIC) simulations have been proven to be an important and useful tool for studying low frequency waves and instabilities below ion cyclotron frequency. The gyrokinetic formalism eliminates the cyclotron motion by analytically averaging the equation of motion in time, while keeping finite-Larmor radius effects, and therefore allows a time step of integration to be significantly longer than the cyclotron period. At the same time the thermal fluctuation level is reduced well below that of a conventional PIC simulation code. Recent simulations have been performed over a number of wave periods to study nonlinear evolution of drift waves and ion-temperature-gradient modes and the associated transport. With about a quarter million particles and a 64 {times} 128 {times} 32 grid in three dimensions, it takes about 100 hours on the Cray-2 single processor to follow the modes to a nonlinear quasi-steady state for relatively strong gradients and strong growth rates. Much more efficient simulations are needed in order to understand these low-frequency waves and the transport associated with them by the use of this tool, and to facilitate the simulation of more weakly unstable plasmas with parameters more relevant to experimental conditions. We have set a goal of achieving an efficiency gain of a factor of 100 on a present-day computer over what has been achieved on the Cray-2 for gyrokinetic simulations. To reach this goal we have begun a project with two components; one is the use of new PIC techniques such as subcycling, orbit-averaging, and semi-implicit algorithms, and the other is the use of massively parallel computers such as the BBN TC200 and the Thinking Machines CM-2. 6 refs.
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.
Adaptive Wavelet Techniques, Wigner Distributions and the Direct Simulation of the Vlasov Equation
NASA Astrophysics Data System (ADS)
Afeyan, Bedros; Douglas, Melissa; Spielman, Rick
2000-10-01
The formal analogy between the quantum Liouville equation satisfied by the Wigner function in Quantum Mechanics, and the Vlasov equation satisfied by the single particle distribution function in plasma physics is exploited in order to study the long term evolution of nonlinear electrostatic wave phenomena dictated by the Vlasov-Poisson equations. Adaptive wavelet techniques are used to tile phase space in an optimal manner so as to minimize computational domain sizes and simultaneously to retain accuracy over disparate scales. Traditional MHD calculations will also be analyzed with our wavelet techniques to show the favorable data compression and feature extraction capabilities of multiresolution analysis. Specifically Z51 and Z179 will be compared to show the nature of the improvements in double wire array (Z179) implosions on Z to those obtained with a single wire array (Z51).
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.
2013-01-25
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.
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.
A high-accuracy Eulerian gyrokinetic solver for collisional plasmas
NASA Astrophysics Data System (ADS)
Candy, J.; Belli, E. A.; Bravenec, R. V.
2016-11-01
We describe a new approach to solve the electromagnetic gyrokinetic equations which is optimized for accurate treatment of multispecies Fokker-Planck collisions including both pitch-angle and energy diffusion. The new algorithm is spectral/pseudospectral in four of the five phase space dimensions, and in the fieldline direction a novel 5th-order conservative upwind scheme is used to permit high-accuracy electromagnetic simulation even in the limit of very high plasma β and vanishingly small perpendicular wavenumber, k⊥ → 0. To our knowledge, this is the first pseudospectral implementation of the collision operator in a gyrokinetic code. We show that the new solver agrees closely with GYRO in the limit of weak Lorentz collisions, but gives a significantly more realistic description of collisions at high collision frequency. The numerical methods are also designed to be efficient and scalable for multiscale simulations that treat ion-scale and electron-scale turbulence simultaneously.
Kinetic electrons in global electromagnetic gyrokinetic particle simulations
NASA Astrophysics Data System (ADS)
Nishimura, Y.; Wang, W.
2005-10-01
Employing an electromagnetic gyrokinetic simulation model,ootnotetextZ. Lin and L. Chen, Phys. Plasmas 8, 1447 (2001). kinetic electron dynamics in global tokamak geometry is investigated. The massless fluid electron model is developed as a base. We further evolve gyrokinetic equations for non-adiabatic kinetic electrons. To obtain the magnetic perturbation, the fluid-kinetic hybrid electron model^1 employs the inverse of the Faraday's law. Instead, the Ampere's law is used as a closure relation to avoid uncertainties in estimating ue|, the moment of the electron velocities. The physics goal is to investigate the finite beta effects on the turbulent transport, as well as α particle driven turbulence.ootnotetextI. Holod, Z. Lin, et al., this conference. This work is supported by Department of Energy (DOE) Cooperative Agreement No. DE-FC02-03ER54695 (UCI), DOE Contract No. DE-AC02-76CH03073 (PPPL).
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.
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.
New variables for gyrokinetic electromagnetic simulations
Mishchenko, Alexey Cole, Michael; Kleiber, Ralf; Könies, Axel
2014-05-15
A new approach to electromagnetic gyrokinetic simulations based on modified gyrokinetic theory is described. The method is validated using a particle-in-cell code. The Toroidal Alfvén Eigenmode at low perpendicular mode numbers, the so-called “magnetohydrodynamical limit,” has been successfully simulated using this method.
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
Parallel filtering in global gyrokinetic simulations
NASA Astrophysics Data System (ADS)
Jolliet, S.; McMillan, B. F.; Villard, L.; Vernay, T.; Angelino, P.; Tran, T. M.; Brunner, S.; Bottino, A.; Idomura, Y.
2012-02-01
In this work, a Fourier solver [B.F. McMillan, S. Jolliet, A. Bottino, P. Angelino, T.M. Tran, L. Villard, Comp. Phys. Commun. 181 (2010) 715] is implemented in the global Eulerian gyrokinetic code GT5D [Y. Idomura, H. Urano, N. Aiba, S. Tokuda, Nucl. Fusion 49 (2009) 065029] and in the global Particle-In-Cell code ORB5 [S. Jolliet, A. Bottino, P. Angelino, R. Hatzky, T.M. Tran, B.F. McMillan, O. Sauter, K. Appert, Y. Idomura, L. Villard, Comp. Phys. Commun. 177 (2007) 409] in order to reduce the memory of the matrix associated with the field equation. This scheme is verified with linear and nonlinear simulations of turbulence. It is demonstrated that the straight-field-line angle is the coordinate that optimizes the Fourier solver, that both linear and nonlinear turbulent states are unaffected by the parallel filtering, and that the k∥ spectrum is independent of plasma size at fixed normalized poloidal wave number.
Gyrokinetic simulation of the tearing mode instability
NASA Astrophysics Data System (ADS)
Startsev, Edward; Wang, Weixing; Lee, Wei-Li
2014-10-01
A recently developed split-weight perturbative particle simulation scheme for finite- β plasmas in the presence of background inhomogeneities which analytically separates the additional adiabatic response of the particles associated with the quasi-static bending of the magnetic field lines has been generalized to the sheared magnetic field geometry. The new scheme has been implemented in a 2D particle-in-cell code in slab geometry with drift-kinetic electrons and gyrokinetic ions. The electrons pitch-scattering collision operator has also been implemented to study collisionless as well as collisional tearing, and drift-tearing instabilities. In this paper the results of linear simulations of tearing and drift-tearing modes for realistic mass ratio mi /me = 1837 and different values of plasma β, electron-ion collision frequency, density and temperature gradients are presented and compared to the solution of the eigenvalue equation. We will also present preliminary results of collisionless tearing mode simulations in cylindrical geometry using tokamak turbulence code GTS. Research supported by the U.S. Department of Energy.
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.
Studies of numerical algorithms for gyrokinetics and the effects of shaping on plasma turbulence
NASA Astrophysics Data System (ADS)
Belli, Emily Ann
Advanced numerical algorithms for gyrokinetic simulations are explored for more effective studies of plasma turbulent transport. The gyrokinetic equations describe the dynamics of particles in 5-dimensional phase space, averaging over the fast gyromotion, and provide a foundation for studying plasma microturbulence in fusion devices and in astrophysical plasmas. Several algorithms for Eulerian/continuum gyrokinetic solvers are compared. An iterative implicit scheme based on numerical approximations of the plasma response is developed. This method reduces the long time needed to set-up implicit arrays, yet still has larger time step advantages similar to a fully implicit method. Various model preconditioners and iteration schemes, including Krylov-based solvers, are explored. An Alternating Direction Implicit algorithm is also studied and is surprisingly found to yield a severe stability restriction on the time step. Overall, an iterative Krylov algorithm might be the best approach for extensions of core tokamak gyrokinetic simulations to edge kinetic formulations and may be particularly useful for studies of large-scale ExB shear effects. The effects of flux surface shape on the gyrokinetic stability and transport of tokamak plasmas are studied using the nonlinear GS2 gyrokinetic code with analytic equilibria based on interpolations of representative JET-like shapes. High shaping is found to be a stabilizing influence on both the linear ITG instability and nonlinear ITG turbulence. A scaling of the heat flux with elongation of chi ˜ kappa-1.5 or kappa-2 (depending on the triangularity) is observed, which is consistent with previous gyrofluid simulations. Thus, the GS2 turbulence simulations are explaining a significant fraction, but not all, of the empirical elongation scaling. The remainder of the scaling may come from (1) the edge boundary conditions for core turbulence, and (2) the larger Dimits nonlinear critical temperature gradient shift due to the
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.
Gyrokinetic Transport Stiffness Calculations on Stellarator Geometries
NASA Astrophysics Data System (ADS)
Faber, B. J.; Mynick, H.; Weir, G. M.; Likin, K. M.; Talmadge, J. N.
2012-10-01
A significant, unanswered question in plasma physics is the difference in transport ``stiffness'' between tokamaks and stellarators. In an effort to shed light on this issue, presented are nonlinear gyrokinetic calculations on various machine geometries: the Helically Symmetric Experiment, the National Compact Stellarator Experiment and an equivalent tokamak configuration. Nonlinear gyrokinetic fluxes have been compared directly to experimental fluxes observed in HSX power modulation experiments. Linear calculations on HSX reveal large growth rates due to both ion temperature gradient and trapped electron turbulence, necessitating a kinetic treatment of electrons; one of the first calculations of its kind for stellarators. A comparison of transport stiffness profiles computed through nonlinear gyrokinetic calculations of ion temperature gradient turbulence for the different machine configurations will be presented.
Shear-Alfven Waves in Gyrokinetic Plasmas
W.W.Lee; J.L.V.Lewandowski; T.S. Hahm; Z. Lin
2000-10-18
It is found that the thermal fluctuation level of the shear-Alfven waves in a gyrokinetic plasma decreases with plasma b(* cs2/uA2), where cs is the ion acoustic speed and uA is the Alfven velocity. This unique thermodynamic property based on the fluctuation-dissipation theorem is verified in this paper using a new gyrokinetic particle simulation scheme, which splits the particle distribution function into the equilibrium part as well as the adiabatic and nonadiabatic parts.
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.
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.
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
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.
A conservative semi-Lagrangian HWENO method for the Vlasov equation
NASA Astrophysics Data System (ADS)
Cai, Xiaofeng; Qiu, Jianxian; Qiu, Jing-Mei
2016-10-01
In this paper, we propose a high order conservative semi-Lagrangian (SL) finite difference Hermite weighted essentially non-oscillatory (HWENO) method for the Vlasov equation based on dimensional splitting. HWENO was first proposed for solving nonlinear hyperbolic problems by evolving both function values and its first derivative values (Qiu and Shu (2004) [23]). The major advantage of HWENO, compared with the original WENO, lies in its compactness in reconstruction stencils. There are several new ingredients in this paper. Firstly we propose a mass-conservative SL HWENO scheme for a 1-D equation by working with a flux-difference form, following the work of Qiu and Christlieb (2010) [25]. Secondly, we propose a proper splitting for equations of partial derivatives in HWENO framework to ensure local mass conservation. The proposed fifth order SL HWENO scheme with dimensional splitting has been tested to work well in capturing filamentation structures without oscillations when the time step size is within the Eulerian CFL constraint. However, when the time stepping size becomes larger, numerical oscillations are observed for the 'mass conservative' dimensional splitting HWENO scheme, as there are extra source terms in equations of partial derivatives. In this case, we introduce WENO limiters to control oscillations. Classical numerical examples on linear passive transport problems, as well as the nonlinear Vlasov-Poisson system, have been tested to demonstrate the performance of the proposed scheme.
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.
Discoveries From the Exploration of Gyrokinetic Momentum Transport
NASA Astrophysics Data System (ADS)
Staebler, G. M.
2010-11-01
Gyrokinetic momentum transport can be driven by a variety of mechanisms that break the parity along the magnetic field: parallel and ExB velocity shear, parallel velocity, up/down flux surface asymmetry. In this work, the discovery of interesting properties of these mechanisms and a new mechanism will be reported. The first result is that the Kelvin Helmholtz (KH) mode driven by parallel velocity shear can drive a net negative energy flux when the temperature and density gradients are below the threshold for drift-wave instabilities. The signature of a negative ion energy flow from turbulence would be a power balance effective diffusivity that is below the neoclassical ion thermal diffusivity. The second result is the prediction that the effective momentum transport should depend on the relative sign between the toroidal magnetic field and the toroidal rotation. This follows from the relative sign between the ExB velocity shear in the Doppler shift of the gyro-kinetic equation and the parallel velocity shear term. This is a corollary effect to the property that the toroidal viscous stress can be zero (e.g. for no external torque) even when both the velocity shears are not zero. The two terms try and break the linear mode parity and can cancel each other out giving a net zero stress. A practical solution to the longstanding problem of including ExB velocity shear in linear driftwave eigenmodes in toroidal geometry has recently been developed for the TGLF gyro-fluid transport model. Simulations of momentum transport with TGLF will be compared with DIII-D data. Finally, when the ExB velocity is balance by the ion diamagnetic velocity, as in the H-mode edge, it has been discovered that the net stabilizing effect of the ExB shear is far stronger. The shear in the diamagnetic velocity is yet another symmetry breaking mechanism driving momentum transport.
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.
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.
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.
A very general electromagnetic gyrokinetic formalism
NASA Astrophysics Data System (ADS)
McMillan, B. F.; Sharma, A.
2016-09-01
We derive a gyrokinetic formalism which is very generally valid: the ordering allows both large inhomogeneities in plasma flow and magnetic field at long wavelength, such as typical drift-kinetic theories, as well as fluctuations at the gyro-scale. The underlying approach is to order the vorticity to be small, and to assert that the timescales in the local plasma frame are long compared to the gyrofrequency. Unlike most other derivations, we do not treat the long and short wavelength components of the fluctuating fields separately; the single-field description defines the particle motion and their interaction with the electromagnetic field at small-scale, the system-scale, and intermediate length scales in a unified fashion. As in earlier literature, the work consists of identifying a coordinate system where the gyroangle-dependent terms are small, and using a near-unity transform to systematically find a set of coordinates where the gyroangle dependence vanishes. We derive a gyrokinetic Lagrangian which is valid where the vorticity | ∇ × ( E × B / B ) | is small compared to the gyrofrequency Ω, and the magnetic field scale length is long compared to the gyroradius; we also require that time variation be slow in an appropriately chosen reference frame. This appears to be a minimum set of constraints on a gyrokinetic theory and is substantially more general than earlier approaches. It is the general-geometry electromagnetic extension of Dimits, Phys. Plasmas 17, 055901 (2010) (which is an electrostatic formalism with a homogeneous background magnetic field). This approach also does not require a separate treatment of fluctuating and background components of the magnetic field, unlike much of the previous literature. As a consequence, the "cross terms" due to a combination of long- and short-wavelength variation, which were ignored in the earlier work (but derived in a more restrictive ordering in Parra and Calvo, Plasma Phys. Controlled Fusion 53, 045001 (2011
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.
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.
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 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.
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 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.
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.
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.
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.
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.
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.
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.
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.
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.
Accuracy of momentum and gyrodensity transport in global gyrokinetic particle-in-cell simulations
NASA Astrophysics Data System (ADS)
McMillan, B. F.; Villard, L.
2014-05-01
Gyrokinetic Particle-In-Cell (PIC) simulations based on conservative Lagrangian formalisms admit transport equations for conserved quantities such as gyrodensity and toroidal momentum, and these can be derived for arbitrary wavelength, even though previous applications have used the long-wavelength approximation. In control-variate PIC simulations, a consequence of the different treatment of the background (f0) and perturbed parts (δf), when a splitting f = f0 + δf is performed, is that analytical transport relations for the relevant fluxes and moments are only reproduced in the large marker number limit. The transport equations for f can be used to write the inconsistency in the perturbed quantities explicitly in terms of the sampling of the background distribution f0. This immediately allows estimates of the error in consistency of momentum transport in control-variate PIC simulations. This inconsistency tends to accumulate secularly and is not directly affected by the sources and noise control in the system. Although physical tokamaks often rotate quite strongly, the standard gyrokinetic formalism assumes weak perpendicular flows, comparable to the drift speed. For systems with such weak flows, maintaining acceptably small relative errors requires that a number of markers scale with the fourth power of the linear system size to consistently resolve long-wavelength evolution. To avoid this unfavourable scaling, an algorithm for exact gyrodensity transport has been developed, and this is shown to allow accurate simulations with an order of magnitude fewer markers.
Accuracy of momentum and gyrodensity transport in global gyrokinetic particle-in-cell simulations
McMillan, B. F.; Villard, L.
2014-05-15
Gyrokinetic Particle-In-Cell (PIC) simulations based on conservative Lagrangian formalisms admit transport equations for conserved quantities such as gyrodensity and toroidal momentum, and these can be derived for arbitrary wavelength, even though previous applications have used the long-wavelength approximation. In control-variate PIC simulations, a consequence of the different treatment of the background (f{sub 0}) and perturbed parts (δf), when a splitting f = f{sub 0} + δf is performed, is that analytical transport relations for the relevant fluxes and moments are only reproduced in the large marker number limit. The transport equations for f can be used to write the inconsistency in the perturbed quantities explicitly in terms of the sampling of the background distribution f{sub 0}. This immediately allows estimates of the error in consistency of momentum transport in control-variate PIC simulations. This inconsistency tends to accumulate secularly and is not directly affected by the sources and noise control in the system. Although physical tokamaks often rotate quite strongly, the standard gyrokinetic formalism assumes weak perpendicular flows, comparable to the drift speed. For systems with such weak flows, maintaining acceptably small relative errors requires that a number of markers scale with the fourth power of the linear system size to consistently resolve long-wavelength evolution. To avoid this unfavourable scaling, an algorithm for exact gyrodensity transport has been developed, and this is shown to allow accurate simulations with an order of magnitude fewer markers.
Gyrokinetic simulations of reverse shear Alfven eigenmodes in DIII-D plasmas
Chen, Y.; Munsat, T.; Parker, S. E.; Heidbrink, W. W.; Van Zeeland, M. A.; Tobias, B. J.; Domier, C. W.
2013-01-15
A gyrokinetic ion/mass-less fluid electron hybrid model as implemented in the GEM code [Y. Chen and S. E. Parker, J. Comput. Phys. 220, 837 (2007)] is used to study the reverse shear Alfven eigenmodes (RSAE) observed in DIII-D, discharge no. 142111. This is a well diagnosed case with measurement of the core-localized RSAE mode structures and the mode frequency, which can be used to compare with simulations. Simulations reproduce many features of the observation, including the mode frequency up-sweeping in time and the sweeping range. A new algorithmic feature is added to the GEM code for this study. Instead of the gyrokinetic Poisson equation itself, its time derivative, or the vorticity equation, is solved to obtain the electric potential. This permits a numerical scheme that ensures the E Multiplication-Sign B convection of the equilibrium density profiles of each species cancel each other in the absence of any finite-Larmor-radius effects. These nonlinear simulations generally result in an electron temperature fluctuation level that is comparable to measurements, and a mode frequency spectrum broader than the experimental spectrum. The spectral width from simulations can be reduced if less steep beam density profiles are used, but then the experimental fluctuation level can be reproduced only if a collision rate above the classical level is assumed.
Gyrokinetic simulations of reverse shear Alfvén eigenmodes in DIII-D plasmas
NASA Astrophysics Data System (ADS)
Chen, Y.; Munsat, T.; Parker, S. E.; Heidbrink, W. W.; Van Zeeland, M. A.; Tobias, B. J.; Domier, C. W.
2013-01-01
A gyrokinetic ion/mass-less fluid electron hybrid model as implemented in the GEM code [Y. Chen and S. E. Parker, J. Comput. Phys. 220, 837 (2007)] is used to study the reverse shear Alfvén eigenmodes (RSAE) observed in DIII-D, discharge #142111. This is a well diagnosed case with measurement of the core-localized RSAE mode structures and the mode frequency, which can be used to compare with simulations. Simulations reproduce many features of the observation, including the mode frequency up-sweeping in time and the sweeping range. A new algorithmic feature is added to the GEM code for this study. Instead of the gyrokinetic Poisson equation itself, its time derivative, or the vorticity equation, is solved to obtain the electric potential. This permits a numerical scheme that ensures the E × B convection of the equilibrium density profiles of each species cancel each other in the absence of any finite-Larmor-radius effects. These nonlinear simulations generally result in an electron temperature fluctuation level that is comparable to measurements, and a mode frequency spectrum broader than the experimental spectrum. The spectral width from simulations can be reduced if less steep beam density profiles are used, but then the experimental fluctuation level can be reproduced only if a collision rate above the classical level is assumed.
NASA Astrophysics Data System (ADS)
Barnes, Michael
2009-11-01
To faithfully simulate ITER and other modern fusion devices, one must resolve electron and ion fluctuation scales in a five-dimensional phase space and time. Simultaneously, one must account for the interaction of this turbulence with the slow evolution of the large-scale plasma profiles. Because of the enormous range of scales involved and the high dimensionality of the problem, resolved first-principles simulations of the full core volume over the confinement time are very challenging using conventional (brute force) techniques. In order to address this problem, we have developed a new approach in which turbulence calculations from multiple gyrokinetic flux tube simulations are coupled together using gyrokinetic transport equations to obtain self-consistent equilibrium profiles and corresponding turbulent fluxes. This multi-scale approach is embodied in a new code, Trinity, which is capable of evolving equilibrium profiles for multiple species, including electromagnetic effects and realistic magnetic geometry, at a fraction of the cost of conventional direct numerical simulations. Key components in the cost reduction are the extreme parallelism enabled by the use of coupled flux tubes and the use of a nonlinear implicit algorithm to take large time steps when evolving the equilibrium. In this talk, we describe the multi-scale model employed in Trinity and present simulation results using nonlinear fluxes calculated with the gyrokinetic turbulence codes GS2 and GENE. We compare the numerical predictions from Trinity simulations with experimental results from a number of fusion devices, including JET and MAST.
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}.
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.
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.
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.
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.
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.
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.
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.
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.
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 calculations of ITG turbulence in general toroidal geometry within the Summit Framework
NASA Astrophysics Data System (ADS)
Leboeuf, Jean-Noel; Decyk, Viktor; Dimits, Andris; Shumaker, Dan
2003-10-01
The Summit Framework is a gyrokinetic particle-in-cell turbulence simulation environment written in Fortran90 providing a unified object-based facility for sharing common components in a massively parallel setting [http://www.nersc.gov/scidac/summit/]. The Summit Framework is part of the US Department of Energy SciDAC Plasma Microturbulence Project. Work is under way to include kinetic electron models and electromagnetic effects, realistic magnetic geometry and global effects under one software environment. General geometry, realistic equilibria capabilities are being incorporated in the Summit Framework through the pg3eq_nc module, itself an extension of the circular geometry pg3eq module [Dimits et al. Phys. Rev. Letts 77, 71 (1996)]. These modules use quasi-ballooning coordinates to solve the three-dimensional, toroidal, delta-f, gyrokinetic equations for ions in order to model ITG turbulence. Realistic geometry is introduced through an interface to data from the EFIT equilibrium code[ http://fusion.gat.com/efit/]. Massively parallel implementation has been effected using MPI. Successful nonlinear comparisons for a sample shaped and finite beta equilibrium have yielded equivalent results between serial, one-processor and multi-processor parallel implementations. Linear and nonlinear tests are currently under way between the general geometry and circular geometry modules with a circular equilibrium which can be accommodated in both modules. Results from all of these tests will be reported, along with strategies for the global extension of both the circular and general geometry modules.
SUMMIT Framework: Gyrokinetic calculations of ITG turbulence in general toroidal geometry
NASA Astrophysics Data System (ADS)
Leboeuf, Jean-Noel; Decyk, Viktor; Dimits, Andris; Shumaker, Dan
2004-11-01
The SUMMIT Framework [http://www.nersc.gov/scidac/summit/] is a gyrokinetic particle-in-cell turbulence simulation environment written in Fortran90 providing a unified object-based facility for sharing common components in a massively parallel setting. The SUMMIT Framework was part of the US Department of Energy SciDAC Plasma Microturbulence Project. General geometry, realistic equilibria capabilities are being incorporated in the SUMMIT Framework through the pg3eq_nc module, itself an extension of the circular geometry pg3eq module [Dimits et al., Phys. Rev. Letts 77, 71 (1996)]. These modules use quasi-ballooning coordinates to solve the three-dimensional, toroidal, delta-f, gyrokinetic equations for ions in order to model ITG turbulence. Realistic geometry is introduced through an interface to data from the EFIT equilibrium code [http://fusion.gat.com/efit/] which is currently being upgraded. Massively parallel implementation of the pg3eq_nc module has been effected using MPI. Successful nonlinear comparisons for a sample shaped and finite beta equilibrium have yielded equivalent results between serial, one-processor and multi-processor parallel implementations. Linear and nonlinear tests have also been successfully performed between the general geometry and circular geometry modules with a circular equilibrium which can be accommodated in both modules. Results from all of these upgrades and tests will be reported.
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.
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.
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.
Phase space structures in gyrokinetic simulations of fusion plasma turbulence
NASA Astrophysics Data System (ADS)
Ghendrih, Philippe; Norscini, Claudia; Cartier-Michaud, Thomas; Dif-Pradalier, Guilhem; Abiteboul, Jérémie; Dong, Yue; Garbet, Xavier; Gürcan, Ozgür; Hennequin, Pascale; Grandgirard, Virginie; Latu, Guillaume; Morel, Pierre; Sarazin, Yanick; Storelli, Alexandre; Vermare, Laure
2014-10-01
Gyrokinetic simulations of fusion plasmas give extensive information in 5D on turbulence and transport. This paper highlights a few of these challenging physics in global, flux driven simulations using experimental inputs from Tore Supra shot TS45511. The electrostatic gyrokinetic code GYSELA is used for these simulations. The 3D structure of avalanches indicates that these structures propagate radially at localised toroidal angles and then expand along the field line at sound speed to form the filaments. Analysing the poloidal mode structure of the potential fluctuations (at a given toroidal location), one finds that the low modes m = 0 and m = 1 exhibit a global structure; the magnitude of the m = 0 mode is much larger than that of the m = 1 mode. The shear layers of the corrugation structures are thus found to be dominated by the m = 0 contribution, that are comparable to that of the zonal flows. This global mode seems to localise the m = 2 mode but has little effect on the localisation of the higher mode numbers. However when analysing the pulsation of the latter modes one finds that all modes exhibit a similar phase velocity, comparable to the local zonal flow velocity. The consequent dispersion like relation between the modes pulsation and the mode numbers provides a means to measure the zonal flow. Temperature fluctuations and the turbulent heat flux are localised between the corrugation structures. Temperature fluctuations are found to exhibit two scales, small fluctuations that are localised by the corrugation shear layers, and appear to bounce back and forth radially, and large fluctuations, also readily observed on the flux, which are associated to the disruption of the corrugations. The radial ballistic velocity of both avalanche events if of the order of 0.5ρ∗c0 where ρ∗ = ρ0/a, a being the tokamak minor radius and ρ0 being the characteristic Larmor radius, ρ0 = c0/Ω0. c0 is the reference ion thermal velocity and Ω0 = qiB0/mi the reference
Multiscale gyrokinetics for rotating tokamak plasmas: fluctuations, transport and energy flows.
Abel, I G; Plunk, G G; Wang, E; Barnes, M; Cowley, S C; Dorland, W; Schekochihin, A A
2013-11-01
This paper presents a complete theoretical framework for studying turbulence and transport in rapidly rotating tokamak plasmas. The fundamental scale separations present in plasma turbulence are codified as an asymptotic expansion in the ratio ε = ρi/α of the gyroradius to the equilibrium scale length. Proceeding order by order in this expansion, a set of coupled multiscale equations is developed. They describe an instantaneous equilibrium, the fluctuations driven by gradients in the equilibrium quantities, and the transport-timescale evolution of mean profiles of these quantities driven by the interplay between the equilibrium and the fluctuations. The equilibrium distribution functions are local Maxwellians with each flux surface rotating toroidally as a rigid body. The magnetic equilibrium is obtained from the generalized Grad-Shafranov equation for a rotating plasma, determining the magnetic flux function from the mean pressure and velocity profiles of the plasma. The slow (resistive-timescale) evolution of the magnetic field is given by an evolution equation for the safety factor q. Large-scale deviations of the distribution function from a Maxwellian are given by neoclassical theory. The fluctuations are determined by the 'high-flow' gyrokinetic equation, from which we derive the governing principle for gyrokinetic turbulence in tokamaks: the conservation and local (in space) cascade of the free energy of the fluctuations (i.e. there is no turbulence spreading). Transport equations for the evolution of the mean density, temperature and flow velocity profiles are derived. These transport equations show how the neoclassical and fluctuating corrections to the equilibrium Maxwellian act back upon the mean profiles through fluxes and heating. The energy and entropy conservation laws for the mean profiles are derived from the transport equations. Total energy, thermal, kinetic and magnetic, is conserved and there is no net turbulent heating. Entropy is produced
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 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 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.
Morrison, P.J.; Pfirsch, D.
1992-04-01
Expressions for the energy content of one-dimensional electrostatic perturbations about homogeneous equilibria are revisited. The well-known dielectric energy, {var_epsilon}{sub D}, is compared with the exact plasma free energy expression, {delta}{sup 2}F, that is conserved by the Vlasov-Poisson system. The former is an expression in terms of the perturbed electric field amplitude, while the latter is determined by a generating function, which describes perturbations of the distribution function that respect the important constraint of dynamical accessibility of the system. Thus the comparison requires solving the Vlasov equation for such a perturbations of the distribution function in terms of the electric field. This is done for neutral modes of oscillation that occur for equilibria with stationary inflection points, and it is seen that for these special modes {delta}{sup 2}F = {var_epsilon}{sub D}. In the case of unstable and corresponding damped modes it is seen that {delta}{sup 2}F {ne} {var_epsilon}{sub D}; in fact {delta}{sup 2}F {equivalent_to} 0. This failure of the dielectric energy expression persists even for arbitrarily small growth and damping rates since {var_epsilon}{sub D} is nonzero in this limit, whereas {delta}{sup 2}F remains zero. The connection between the new exact energy expression and the at-best approximate {var_epsilon}{sub D} is described. The new expression motivates natural definitions of Hamiltonian action variables and signature. A general linear integral transform is introduced that maps the linear version of the noncanonical Hamiltonian structure, which describes the Vlasov equation, to action-angle (diagonal) form.
Morrison, P.J. . Inst. for Fusion Studies); Pfirsch, D. )
1992-04-01
Expressions for the energy content of one-dimensional electrostatic perturbations about homogeneous equilibria are revisited. The well-known dielectric energy, {var epsilon}{sub D}, is compared with the exact plasma free energy expression, {delta}{sup 2}F, that is conserved by the Vlasov-Poisson system. The former is an expression in terms of the perturbed electric field amplitude, while the latter is determined by a generating function, which describes perturbations of the distribution function that respect the important constraint of dynamical accessibility of the system. Thus the comparison requires solving the Vlasov equation for such a perturbations of the distribution function in terms of the electric field. This is done for neutral modes of oscillation that occur for equilibria with stationary inflection points, and it is seen that for these special modes {delta}{sup 2}F = {var epsilon}{sub D}. In the case of unstable and corresponding damped modes it is seen that {delta}{sup 2}F {ne} {var epsilon}{sub D}; in fact {delta}{sup 2}F {equivalent to} 0. This failure of the dielectric energy expression persists even for arbitrarily small growth and damping rates since {var epsilon}{sub D} is nonzero in this limit, whereas {delta}{sup 2}F remains zero. The connection between the new exact energy expression and the at-best approximate {var epsilon}{sub D} is described. The new expression motivates natural definitions of Hamiltonian action variables and signature. A general linear integral transform is introduced that maps the linear version of the noncanonical Hamiltonian structure, which describes the Vlasov equation, to action-angle (diagonal) form.
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.
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
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.
Nonlinear electromagnetic gyrokinetic particle simulations with the electron hybrid model
NASA Astrophysics Data System (ADS)
Nishimura, Y.; Lin, Z.; Chen, L.; Hahm, T.; Wang, W.; Lee, W.
2006-10-01
The electromagnetic model with fluid electrons is successfully implemented into the global gyrokinetic code GTC. In the ideal MHD limit, shear Alfven wave oscillation and continuum damping is demonstrated. Nonlinear electromagnetic simulation is further pursued in the presence of finite ηi. Turbulence transport in the AITG unstable β regime is studied. This work is supported by Department of Energy (DOE) Grant DE-FG02-03ER54724, Cooperative Agreement No. DE-FC02-04ER54796 (UCI), DOE Contract No. DE-AC02-76CH03073 (PPPL), and in part by SciDAC Center for Gyrokinetic Particle Simulation of Turbulent Transport in Burning Plasmas. Z. Lin, et al., Science 281, 1835 (1998). F. Zonca and L. Chen, Plasma Phys. Controlled Fusion 30, 2240 (1998); G. Zhao and L. Chen, Phys. Plasmas 9, 861 (2002).
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.
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.
Gyrokinetic analysis of shear flow instability in torodial geometry
NASA Astrophysics Data System (ADS)
Yoon, Eisung; Hahm, T. S.
2008-11-01
Motivated by recent observation of intrinsic rotation in tokamak plasmas, we study linear stability of ion gyroradius scale short wavelength fluctuations in the presence of sheared parallel flow, ion temperature gradient, and toroidal mode coupling. Our gyrokinetic approach in toroidal geometry is an extension of previous studies including those by Catto et al., [Phys. Fluids 16 1719 (1973)] Mattor and Diamond [Phys. Fluids 31 1180 (1988)], and Artun and Tang [Phys. Fluids B 4 1102 (1992)].
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.
Testing Gyrokinetics on C-Mod and NSTX
M.H. Redi; W. Dorland; C.L. Fiore; D. Stutman; J.A. Baumgaertel; B. Davis; S.M. Kaye; D.C. McCune; J. Menard; G. Rewoldt
2005-06-20
Quantitative benchmarks of computational physics codes against experiment are essential for the credible application of such codes. Fluctuation measurements can provide necessary critical tests of nonlinear gyrokinetic simulations, but such require extraordinary computational resources. Linear micro-stability calculations with the GS2 [1] gyrokinetic code have been carried out for tokamak and ST experiments which exhibit internal transport barriers (ITB) and good plasma confinement. Qualitative correlation is found for improved confinement before and during ITB plasmas on Alcator C-Mod [2] and NSTX [3], with weaker long wavelength micro-instabilities in the plasma core regions. Mixing length transport models are discussed. The NSTX L-mode is found to be near marginal stability for kinetic ballooning modes. Fully electromagnetic, linear, gyrokinetic calculations of the Alcator C-Mod ITB during off-axis rf heating, following four plasma species and including the complete electron response show ITG/TEM microturbulence is suppressed in the plasma core and in the barrier region before barrier formation, without recourse to the usual requirements of velocity shear or reversed magnetic shear [4-5]. No strongly growing long or short wavelength drift modes are found in the plasma core but strong ITG/TEM and ETG drift wave turbulence is found outside the barrier region. Linear microstability analysis is qualitatively consistent with the experimental transport analysis, showing low transport inside and high transport outside the ITB region before barrier formation, without consideration of ExB shear stabilization.
COMPREHENSIVE GYROKINETIC SIMULATION OF TOKAMAK TURBULENCE AT FINITE RELATIVE GYRORADIUS
WALTZ,RE; CANDY,J; ROSENBLUTH,MN
2002-09-01
OAK B202 COMPREHENSIVE GYROKINETIC SIMULATION OF TOKAMAK TURBULENCE AT FINITE RELATIVE GYRORADIUS. A continuum global gyrokinetic code GYRO has been developed to comprehensively simulate turbulent transport in actual experimental profiles and allow 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, and all in real tokamak geometry. Most importantly the code operates at finite relative gyroradius ({rho}*) so as to treat the profile shear stabilization effects which break gyroBohm scaling. The code operates in a cyclic flux tube limit which allows only gyroBohm scaling and a noncyclic radial annulus with physical profile variation. The later requires an adaptive source to maintain equilibrium profiles. Simple ITG simulations demonstrate the broken gyroBohm scaling depends on the actual rotational velocity shear rates competing with mode growth rates, direct comprehensive simulations of the DIII-D {rho}*-scaled L-mode experiments are presented as a quantitative test of gyrokinetics and the paradigm.
Peeters, A.G.; Angioni, C.
2005-07-15
It is shown from a symmetry in the gyrokinetic equation that for up-down symmetric tokamak equilibria and for u{sub {phi}}>>{rho}{upsilon}{sub thi}/r (where u{sub {phi}} is the toroidal velocity, {upsilon}{sub thi} is the thermal ion velocity, {rho} is the Larmor radius, and r is the radius of the flux surface), the transport of parallel momentum can be written as the sum of a diffusive and a pinch contribution with no off-diagonal terms due to temperature and pressure gradients. The measured parallel velocity gradient in ASDEX Upgrade [O. Gruber, H.-S. Bosch, S. Guenter et al., Nucl. Fusion 39, 1321 (1999)] is insufficient to drive the parallel velocity shear instability. The parallel velocity is then transported by the ion temperature gradient mode. The diffusive contribution to the transport flux is investigated using a linear gyrokinetic approach, and it is found that the diffusion coefficient for parallel velocity transport divided by the ion heat conductivity coefficient is close to 1, and only weakly dependent on plasma parameters.
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 particle simulation of ion temperature gradient drift instabilities
Lee, W.W.; Tang, W.M.
1987-04-01
Ion temperature gradient drift instabilities have been investigated using gyrokinetic particle simulation techniques for the purpose of identifying the mechanisms responsible for their nonlinear saturation as well as the associated anomalous transport. For simplicity, the simulation has been carried out in a shear-free slab geometry, where the background pressure gradient is held fixed in time to represent quasistatic profiles typical of tokamak discharges. It is found that the nonlinearly generated zero-frequency responses for the ion parallel momentum and pressure are the dominant mechanisms giving rise to saturation. This is supported by the excellent agreement between the simulation results and those obtained from mode coupling calculations.
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.
Global Gyrokinetic Simulation of Tokamak Edge Pedestal Instabilities
NASA Astrophysics Data System (ADS)
Wan, Weigang; Parker, Scott E.; Chen, Yang; Yan, Zheng; Groebner, Richard J.; Snyder, Philip B.
2012-11-01
Global electromagnetic gyrokinetic simulations show the existence of near threshold conditions for both a high-n kinetic ballooning mode (KBM) and an intermediate-n kinetic version of peeling-ballooning mode (KPBM) in the edge pedestal of two DIII-D H-mode discharges. When the magnetic shear is reduced in a narrow region of steep pressure gradient, the KPBM is significantly stabilized, while the KBM is weakly destabilized and hence becomes the most-unstable mode. Collisions decrease the KBM’s critical β and increase the growth rate.
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.
NASA Astrophysics Data System (ADS)
Chen, Yang
2000-10-01
The physics of kinetic electrons and electromagnetic fluctuations are key challenges in microturbulence simulation research. Recently, we have made progress in this area by developing a drift-kinetic electron model using both the ``split-weight scheme"(I. Manuilskiy and W. W. Lee, Phys. Plasmas 7 1381 (2000)) and the canonical parallel momemtum formulation of gyrokinetics(T. S. Hahm, W. W. Lee and A. Brizard, Phys. Fluids 31(1988) 1940) in a fully nonlinear three-dimensional toroidal field-line-following simulation. This model includes magnetic field perturbations perpendicular to the equilibrium magnetic field. Numerical issues arising from the resolution of the magnetic skin depth(J. Cummings, Ph.D. Thesis, Princeton Univ. (1994)) currently limit these simulations to small <≈ β, β m_i/me <≈ O(1) and progress in this area will be reported. A complementary hybrid simulation with fully gyrokinetic ions and a zero-inertia electron fluid has been developed as well. The electron fluid equations are derived from moments of the drift kinetic equation and a predictor-corrector scheme for the fluid-hybrid model has been implemented in three-dimensional toroidal field-line-following geometry. This is a much simpler electron model and works well at high β. We are currently using both models to study the effects of electron dynamics on turbulence, including particle transport (which is zero in simulations using adiabatic response), kinetic Alfvén modes and modification to zonal flows due to kinetic electrons and the generation of zonal fields through including A_allel(A. Das and P. H. Diamond, "Kinetic theory of the zonal flow instability in electromagnetic drift-wave turbulence", to appear in Phys. Plasmas). Both hybrid and the fully kinetic simulations have been carefully benchmarked with linear theory in the slab limit. Simulation results for turbulence with both trapped-electron drive and ion-temperature-gradient drive will be presented. We will report results
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.
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.
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.
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.
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.
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.
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.
Properties of Discontinuous Galerkin Algorithms and Implications for Edge Gyrokinetics
NASA Astrophysics Data System (ADS)
Hammett, G. W.; Hakim, A.; Shi, E. L.; Abel, I. G.; Stoltzfus-Dueck, T.
2015-11-01
The continuum gyrokinetic code Gkeyll uses Discontinuous Galerkin (DG) algorithms, which have a lot of flexibility in the choice of basis functions and inner product norm that can be useful in designing algorithms for particular problems. Rather than use regular polynomial basis functions, we consider here Maxwellian-weighted basis functions (which have similarities to Gaussian radial basis functions). The standard Galerkin approach loses particle and energy conservation, but this can be restored with a particular weight for the inner product (this is equivalent to a Petrov-Galerkin method). This allows a full- F code to have some benefits similar to the Gaussian quadrature used in gyrokinetic δf codes to integrate Gaussians times some polynomials exactly. In tests of Gkeyll for electromagnetic fluctuations, we found it is important to use consistent basis functions where the potential is in a higher-order continuity subspace of the space for the vector potential A| |. A regular projection method to this subspace is a non-local operation, while we show a self-adjoint averaging operator that can preserve locality and energy conservation. This does not introduce damping, but like gyro-averaging involves only the reactive part of the dynamics. Supported by the Max-Planck/Princeton Center for Plasma Physics, the SciDAC Center for the Study of Plasma Microturbulence, and DOE Contract DE-AC02-09CH11466.
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).
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.
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
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).
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.
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.
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).
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.
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.
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.
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
Gyrokinetic simulation of driftwave instability in field-reversed configuration
NASA Astrophysics Data System (ADS)
Fulton, Daniel
2015-11-01
Following the recent remarkable progress in MHD stability control in the C-2U advanced beam driven field-reversed configuration (FRC)[M. Binderbauer et al 2015], turbulent transport has become the foremost obstacle on the path towards an FRC-based fusion reactor. Significant effort has been put into expanding kinetic simulation capabilities in FRC magnetic geometry. The Gyrokinetic Toroidal Code (GTC) has been upgraded to accommodate realistic magnetic geometry from the C-2U experiment and to optimize the field solver for the FRC's field line orientation. Initial linear electrostatic GTC simulations find ion-scale instabilities are not present in the FRC core due to the large gyroradius of thermal ions, while electron drift-interchange modes are driven by the electron temperature gradient and bad magnetic curvature. Simulation in the FRC scrape-off layer finds density gradient driven ion scale fluctuations. 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 kinetic code has been developed to accurately resolve the magnetic field separatrix and address the interaction between the core and scrape-off layer regions, which ultimately provide boundary conditions for the plasma confinement. Initial results 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.
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
Freely decaying turbulence in two-dimensional electrostatic gyrokinetics
Tatsuno, T.; Plunk, G. G.; Barnes, M.; Dorland, W.; Howes, G. G.; Numata, R.
2012-12-15
In magnetized plasmas, a turbulent cascade occurs in phase space at scales smaller than the thermal Larmor radius ('sub-Larmor scales') [Tatsuno et al., Phys. Rev. Lett. 103, 015003 (2009)]. When the turbulence is restricted to two spatial dimensions perpendicular to the background magnetic field, two independent cascades may take place simultaneously because of the presence of two collisionless invariants. In the present work, freely decaying turbulence of two-dimensional electrostatic gyrokinetics is investigated by means of phenomenological theory and direct numerical simulations. A dual cascade (forward and inverse cascades) is observed in velocity space as well as in position space, which we diagnose by means of nonlinear transfer functions for the collisionless invariants. We find that the turbulence tends to a time-asymptotic state, dominated by a single scale that grows in time. A theory of this asymptotic state is derived in the form of decay laws. Each case that we study falls into one of three regimes (weakly collisional, marginal, and strongly collisional), determined by a dimensionless number D{sub *}, a quantity analogous to the Reynolds number. The marginal state is marked by a critical number D{sub *}=D{sub 0} that is preserved in time. Turbulence initialized above this value become increasingly inertial in time, evolving toward larger and larger D{sub *}; turbulence initialized below D{sub 0} become more and more collisional, decaying to progressively smaller D{sub *}.
Electromagnetic gyrokinetic turbulence in finite-beta helical plasmasa)
NASA Astrophysics Data System (ADS)
Ishizawa, A.; Watanabe, T.-H.; Sugama, H.; Maeyama, S.; Nakajima, N.
2014-05-01
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.
Gyrokinetic Particle Simulation of Alfven Eigenmodes with Zonal Fields
NASA Astrophysics Data System (ADS)
Wang, Zhixuan
2012-03-01
Effects of collective Shear Alfven wave instabilities on the energetic particle confinement in tokamak depend ultimately on the nonlinear evolution of the turbulence with spontaneously generated zonal fields (zonal flows and zonal currents). In this work, we study nonlinear interaction of Alfv'en eigenmodes with zonal fields using global gyrokinetic toroidal code GTC. We choose to start from the simplest case, linear electrostatic eigenmodes in cylindrical geometry, and then gradually extend our study into electromagnetic eigenmode in toroidal geometry. We have verified GTC for linear simulation in cylindrical geometry with the ExB flow shear. Ion temperature gradient instability is observed to be suppressed when ExB flow shear is strong enough. A good agreement has also been achieved between our simulation result of kinetic Alfv'en wave and LAPD experimental result. Now we are doing TAE (torodicity-induced Alfv'en eigenmodes) simulation using the DIII-D equilibrium data. Linear simulation result agrees well with DIII-D data. Our next step is to include nonlinear effects to study the interaction between zonal fields and TAEs. Work supported by DOE SciDAC GSEP Center and Plasma Science Center.
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.
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.
Nonlinear Gyrokinetic Simulation of Electron-Driven Turbulence in HSX
NASA Astrophysics Data System (ADS)
Faber, Benjamin; Pueschel, M. J.; Weir, Gavin; Likin, Konstatin; Talmadge, Joseph; Anderson, Simon; Anderson, David
2014-10-01
The first nonlinear gyrokinetic simulations of plasmas in the Helically Symmetric eXperiment (HSX) are presented. Due to large electron cyclotron resonance heating (ECRH) and little ion heating, microtubulence in HSX is driven by electron dynamics and thus the simulations performed require two kinetic species. Linear growth rate calculations of plasmas at experimental parameters indicate HSX is unstable at low kyρs to the Trapped Electron Mode (TEM) and the Electron Temperature Gradient (ETG) mode at high kyρs , especially in the core region where the normalized temperature gradient is significantly larger than the normalized density gradient. Nonlinear flux tube simulations show heat fluxes shift to smaller scales than for ion-driven turbulence, with the flux spectrum peaking at kyρs ~ 0 . 9 for TEM turbulence. Nonlinear simulations also show the evolution of zonal flows, which are a possible candidate for the nonlinear saturation mechanism. Calculation of the dependence of the saturated heat flux on the normalized electron temperature gradient provides a computational comparison with the stiffness measurements obtained in heat pulse propagation experiments. Work supported by U.S. DOE Contract No. DE-FG02-93ER54222.
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.
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.
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.
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.
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.
Hatch, David R.; Del-Castillo-Negrete, Diego B; Terry, P.W.
2012-01-01
Higher order singular value decomposition (HOSVD) is explored as a tool for analyzing and compressing gyrokinetic data. An efficient numerical implementation of an HOSVD algorithm is described. HOSVD is used to analyze the full six-dimensional (three spatial, two velocity space, and time dimensions) gyrocenter distribution function from gyrokinetic simulations of ion temperature gradient, electron temperature gradient, and trapped electron mode driven turbulence. The HOSVD eigenvalues for the velocity space coordinates decay very rapidly, indicating that only a few structures in velocity space can capture the most important dynamics. In almost all of the cases studied, HOSVD extracts parallel velocity space structures which are very similar to orthogonal polynomials. HOSVD is also used to compress gyrokinetic datasets, an application in which it is shown to significantly outperform the more commonly used singular value decomposition. It is shown that the effectiveness of the HOSVD compression improves as the dimensionality of the dataset increases. (C) 2012 Elsevier Inc. All rights reserved.
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.
Kinetic Equations for the Plasma Edge
NASA Astrophysics Data System (ADS)
Abel, Ian; Hammett, Greg
2015-11-01
A hybrid fluid-kinetic framework for studying large-amplitude fluctuations in the edge of tokamak plasmas is presented. We derive equations for the behavior of an anisotropic plasma in the presence of both large fluctuations and steep gradients. The system consists of kinetic equations for electrons and ions, supplemented with fluid equations for the electromagnetic fields. In this way it builds upon both kinetic MHD and from the use of vorticity equations in gyrokinetics. This framework, by including both Alfvénic (including current-driven modes) and drift wave dynamics, can handle fully nonlinear perturbations such as erupting ELM filaments and blob-based turbulence. We not only present equations for such fast behavior, but also develop higher order equations that describe pedestal equilibria and slow scrape-off-layer dynamics. The relationship between this framework and existing collisional edge models is made clear.
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.
Electromagnetic gyrokinetic turbulence in high-beta helical plasmas
NASA Astrophysics Data System (ADS)
Ishizawa, Akihiro
2013-10-01
Gyrokinetic simulation of electromagnetic turbulence in finite-beta plasmas is important for predicting the performance of fusion reactors. Whereas in low-beta tokamaks the zonal flow shear acts to regulate ion temperature gradient (ITG) driven turbulence, it has often been observed that the kinetic ballooning mode (KBM) and, at moderate-beta, the ITG mode continue to grow without reaching a physically relevant level of saturation. The corresponding problem in helical high-beta plasmas, the identification of a saturation mechanism for microturbulence in regimes where zonal flow generation is too weak, is the subject of the present work. 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 present study identifies a new saturation process of the KBM turbulence originating from the spatial structure of the KBM instabilities in a high-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 finite radial wave-number in flux tube coordinates, in contrast to KBMs in tokamaks as well as ITG 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, rather than by the zonal flow shear. The mechanism is quantitatively evaluated by analysis of the nonlinear entropy transfer.
Electromagnetic gyrokinetic simulation of turbulence in torus plasmas
NASA Astrophysics Data System (ADS)
Ishizawa, A.; Maeyama, S.; Watanabe, T.-H.; Sugama, H.; Nakajima, N.
2015-04-01
Gyrokinetic simulations of electromagnetic turbulence in magnetically confined torus plasmas including tokamak and heliotron/stellarator are reviewed. Numerical simulation of turbulence in finite beta plasmas is an important task for predicting the performance of fusion reactors and a great challenge in computational science due to multiple spatio-temporal scales related to electromagnetic ion and electron dynamics. The simulation becomes further challenging in non-axisymmetric plasmas. In finite beta plasmas, magnetic perturbation appears and influences some key mechanisms of turbulent transport, which include linear instability and zonal flow production. Linear analysis shows that the ion-temperature gradient (ITG) instability, which is essentially an electrostatic instability, is unstable at low beta and its growth rate is reduced by magnetic field line bending at finite beta. On the other hand, the kinetic ballooning mode (KBM), which is an electromagnetic instability, is destabilized at high beta. In addition, trapped electron modes (TEMs), electron temperature gradient (ETG) modes, and micro-tearing modes (MTMs) can be destabilized. These instabilities are classified into two categories: ballooning parity and tearing parity modes. These parities are mixed by nonlinear interactions, so that, for instance, the ITG mode excites tearing parity modes. In the nonlinear evolution, the zonal flow shear acts to regulate the ITG driven turbulence at low beta. On the other hand, at finite beta, interplay between the turbulence and zonal flows becomes complicated because the production of zonal flow is influenced by the finite beta effects. When the zonal flows are too weak, turbulence continues to grow beyond a physically relevant level of saturation in finite-beta tokamaks. Nonlinear mode coupling to stable modes can play a role in the saturation of finite beta ITG mode and KBM. Since there is a quadratic conserved quantity, evaluating nonlinear transfer of the
Performance of particle in cell methods on highly concurrent computational architectures
M.F. Adams; S. Ethier; N. Wichmann
2007-08-01
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.
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.
Linear dispersion relation for the mirror instability in context of the gyrokinetic theory
Porazik, Peter; Johnson, Jay R.
2013-10-15
The linear dispersion relation for the mirror instability is discussed in context of the gyrokinetic theory. The objective is to provide a coherent view of different kinetic approaches used to derive the dispersion relation. The method based on gyrocenter phase space transformations is adopted in order to display the origin and ordering of various terms.
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.
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.
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
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).
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.
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.
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.
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.
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.
NASA Astrophysics Data System (ADS)
Holod, I.; Lin, Z.
2013-03-01
The fluid-kinetic hybrid electron model is verified in global gyrokinetic particle simulation of linear electromagnetic drift-Alfvénic instabilities in tokamak. In particular, we have recovered the β-stabilization of the ion temperature gradient mode, transition to collisionless trapped electron mode, and the onset of kinetic ballooning mode as βe (ratio of electron kinetic pressure to magnetic pressure) increases.
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.
A gyrokinetic approach to modeling mirror and firehose instabilites in the solar wind
NASA Astrophysics Data System (ADS)
Johnson, J.; Porazik, P.
2015-12-01
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. Finally, we present new insights into the nature of the parallel and oblique firehose instability by considering how the topology of the dispersion surfaces change as an anisotropic population is added to an isotropic plasma. We discuss the role of resonant and nonresonant particles in the instability and show that a gyrokinetic description is in good agreement with a fully kinetic description.
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.
Fluid electron, gyrokinetic ion simulations of linear internal kink and energetic particle modes
Cole, Michael Mishchenko, Alexey; Könies, Axel; Kleiber, Ralf; Borchardt, Matthias
2014-07-15
The internal kink mode is an important plasma instability responsible for a broad class of undesirable phenomena in tokamaks, including the sawtooth cycle and fishbones. To predict and discover ways to mitigate this behaviour in current and future devices, numerical simulations are necessary. The internal kink mode can be modelled by reduced magnetohydrodynamics (MHD). Fishbone modes are an inherently kinetic and non-linear phenomenon based on the n = 1 Energetic Particle Mode (EPM), and have been studied using hybrid codes that combine a reduced MHD bulk plasma model with a kinetic treatment of fast ions. In this work, linear simulations are presented using a hybrid model which couples a fluid treatment of electrons with a gyrokinetic treatment of both bulk and fast ions. Studies of the internal kink mode in geometry relevant to large tokamak experiments are presented and the effect of gyrokinetic ions is considered. Interaction of the kink with gyrokinetic fast ions is also considered, including the destabilisation of the linear n = 1 EPM underlying the fishbone.
NASA Astrophysics Data System (ADS)
Zhang, Wenlu; Holod, Ihor; Lin, Zhihong; Xiao, Yong
2012-02-01
Linear properties of toroidal Alfvén eigenmode (TAE) is studied in global gyrokinetic particle simulations using both fast ion and antenna excitations. A synthetic antenna provides a precise measurement of the Alfvén continuum gap width and the TAE eigenmode frequency, damping rate, and mode structures. The measured gap width exhibits a linear dependence on the aspect ratio, in agreement to a local analytic theory. The TAE frequency and mode structure excited by fast ions show a significant radial symmetry breaking relative to the ideal magnetohydrodynamic theory due to the non-perturbative contributions from the fast ions. The electromagnetic capability of the global gyrokinetic toroidal code (GTC) is verified through these global gyrokinetic simulations of Alfvén eigenmode in cylindrical and toroidal geometries.
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.
A 5D gyrokinetic full- f global semi-Lagrangian code for flux-driven ion turbulence simulations
NASA Astrophysics Data System (ADS)
Grandgirard, V.; Abiteboul, J.; Bigot, J.; Cartier-Michaud, T.; Crouseilles, N.; Dif-Pradalier, G.; Ehrlacher, Ch.; Esteve, D.; Garbet, X.; Ghendrih, Ph.; Latu, G.; Mehrenberger, M.; Norscini, C.; Passeron, Ch.; Rozar, F.; Sarazin, Y.; Sonnendrücker, E.; Strugarek, A.; Zarzoso, D.
2016-10-01
This paper addresses non-linear gyrokinetic simulations of ion temperature gradient (ITG) turbulence in tokamak plasmas. The electrostatic GYSELA code is one of the few international 5D gyrokinetic codes able to perform global, full- f and flux-driven simulations. Its has also the numerical originality of being based on a semi-Lagrangian (SL) method. This reference paper for the GYSELA code presents a complete description of its multi-ion species version including: (i) numerical scheme, (ii) high level of parallelism up to 500k cores and (iii) conservation law properties.
Phase space scales of free energy dissipation in gradient-driven gyrokinetic turbulence
NASA Astrophysics Data System (ADS)
Hatch, D. R.; Jenko, F.; Bratanov, V.; Navarro, A. Bañón; Navarro
2014-08-01
A reduced four-dimensional (integrated over perpendicular velocity) gyrokinetic model of slab ion temperature gradient-driven turbulence is used to study the phase-space scales of free energy dissipation in a turbulent kinetic system over a broad range of background gradients and collision frequencies. Parallel velocity is expressed in terms of Hermite polynomials, allowing for a detailed study of the scales of free energy dynamics over the four-dimensional phase space. A fully spectral code - the DNA code - that solves this system is described. Hermite free energy spectra are significantly steeper than would be expected linearly, causing collisional dissipation to peak at large scales in velocity space even for arbitrarily small collisionality. A key cause of the steep Hermite spectra is a critical balance - an equilibration of the parallel streaming time and the nonlinear correlation time - that extends to high Hermite number n. Although dissipation always peaks at large scales in all phase space dimensions, small-scale dissipation becomes important in an integrated sense when collisionality is low enough and/or nonlinear energy transfer is strong enough. Toroidal full-gyrokinetic simulations using the Gene code are used to verify results from the reduced model. Collision frequencies typically found in present-day experiments correspond to turbulence regimes slightly favoring large-scale dissipation, while turbulence in low-collisionality systems like ITER and space and astrophysical plasmas is expected to rely increasingly on small-scale dissipation mechanisms. This work is expected to inform gyrokinetic reduced modeling efforts like Large Eddy Simulation and gyrofluid techniques.
Gyrokinetic Particle Simulation of Turbulent Transport in Burning Plasmas (GPS - TTBP) Final Report
Chame, Jacqueline
2011-05-27
The goal of this project is the development of the Gyrokinetic Toroidal Code (GTC) Framework and its applications to problems related to the physics of turbulence and turbulent transport in tokamaks,. The project involves physics studies, code development, noise effect mitigation, supporting computer science efforts, diagnostics and advanced visualizations, verification and validation. Its main scientific themes are mesoscale dynamics and non-locality effects on transport, the physics of secondary structures such as zonal flows, and strongly coherent wave-particle interaction phenomena at magnetic precession resonances. Special emphasis is placed on the implications of these themes for rho-star and current scalings and for the turbulent transport of momentum. GTC-TTBP also explores applications to electron thermal transport, particle transport; ITB formation and cross-cuts such as edge-core coupling, interaction of energetic particles with turbulence and neoclassical tearing mode trigger dynamics. Code development focuses on major initiatives in the development of full-f formulations and the capacity to simulate flux-driven transport. In addition to the full-f -formulation, the project includes the development of numerical collision models and methods for coarse graining in phase space. Verification is pursued by linear stability study comparisons with the FULL and HD7 codes and by benchmarking with the GKV, GYSELA and other gyrokinetic simulation codes. Validation of gyrokinetic models of ion and electron thermal transport is pursed by systematic stressing comparisons with fluctuation and transport data from the DIII-D and NSTX tokamaks. The physics and code development research programs are supported by complementary efforts in computer sciences, high performance computing, and data management.
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
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.
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.
NASA Astrophysics Data System (ADS)
White, A. E.; Howard, N. T.; Creely, A. J.; Chilenski, M. A.; Greenwald, M.; Hubbard, A. E.; Hughes, J. W.; Marmar, E.; Rice, J. E.; Sierchio, J. M.; Sung, C.; Walk, J. R.; Whyte, D. G.; Mikkelsen, D. R.; Edlund, E. M.; Kung, C.; Holland, C.; Candy, J.; Petty, C. C.; Reinke, M. L.; Theiler, C.
2015-05-01
For the first time, nonlinear gyrokinetic simulations of I-mode plasmas are performed and compared with experiment. I-mode is a high confinement regime, featuring energy confinement similar to H-mode, but without enhanced particle and impurity particle confinement [D. G. Whyte et al., Nucl. Fusion 50, 105005 (2010)]. As a consequence of the separation between heat and particle transport, I-mode exhibits several favorable characteristics compared to H-mode. The nonlinear gyrokinetic code GYRO [J. Candy and R. E. Waltz, J Comput. Phys. 186, 545 (2003)] is used to explore the effects of E × B shear and profile stiffness in I-mode and compare with L-mode. The nonlinear GYRO simulations show that I-mode core ion temperature and electron temperature profiles are more stiff than L-mode core plasmas. Scans of the input E × B shear in GYRO simulations show that E × B shearing of turbulence is a stronger effect in the core of I-mode than L-mode. The nonlinear simulations match the observed reductions in long wavelength density fluctuation levels across the L-I transition but underestimate the reduction of long wavelength electron temperature fluctuation levels. The comparisons between experiment and gyrokinetic simulations for I-mode suggest that increased E × B shearing of turbulence combined with increased profile stiffness are responsible for the reductions in core turbulence observed in the experiment, and that I-mode resembles H-mode plasmas more than L-mode plasmas with regards to marginal stability and temperature profile stiffness.
On the definition of a kinetic equilibrium in global gyrokinetic simulations
Angelino, P.; Bottino, A.; Hatzky, R.; Jolliet, S.; Sauter, O.; Tran, T.M.; Villard, L.
2006-05-15
Nonlinear electrostatic global gyrokinetic simulations of collisionless ion temperature gradient (ITG) turbulence and ExB zonal flows in axisymmetric toroidal plasmas are examined for different choices of the initial distribution function. Using a local Maxwellian leads to the generation of axisymmetric ExB flows that can be so strong as to prevent ITG mode growth. A method using a canonical Maxwellian is shown to avoid this spurious generation of ExB flows. In addition, a revised {delta}f scheme is introduced and compared to the standard {delta}f method.
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.
2007-08-15
The GEM gyrokinetic {delta}f simulation code [Y. Chen and S. Parker, J. Comput. Phys. 189, 463 (2003); and ibid.220, 839 (2007)] is shown to reproduce electron temperature gradient turbulence at the benchmark operating point established in previous work [W. M. Nevins, J. Candy, S. Cowley, T. Dannert, A. Dimits, W. Dorland, C. Estrada-Mila, G. W. Hammett, F. Jenko, M. J. Pueschel, and D. E. Shumaker, Phys. Plasmas 13, 122306 (2006)]. The electron thermal transport is within 10% of the expected value, while the turbulent fluctuation spectrum is shown to have the expected intensity and two-point correlation function.
Gyrokinetic simulations of off-axis minimum-q profile corrugations
Waltz, R.E.; Austin, M.E.; Burrell, K.H.; Candy, J.
2006-05-15
Quasiequilibrium radial 'profile corrugations' in the electron temperature gradient are found at lowest-order singular surfaces in global gyrokinetic code simulations of both monotonic-q and off-axis minimum-q discharges. The profile corrugations in the temperature and density gradients are time-averaged components of zonal flows. The m/n=2/1 electron temperature gradient corrugation is measurably large and appears to trigger an internal transport barrier as the off-axis minimum-q=2 surfaces enter the plasma.
Multiscale Nature of the Dissipation Range in Gyrokinetic Simulations of Alfvénic Turbulence.
Told, D; Jenko, F; TenBarge, J M; Howes, G G; Hammett, G W
2015-07-10
Nonlinear energy transfer and dissipation in Alfvén wave turbulence are analyzed in the first gyrokinetic simulation spanning all scales from the tail of the MHD range to the electron gyroradius scale. For typical solar wind parameters at 1 AU, about 30% of the nonlinear energy transfer close to the electron gyroradius scale is mediated by modes in the tail of the MHD cascade. Collisional dissipation occurs across the entire kinetic range k(⊥)ρ(I)≳1. Both mechanisms thus act on multiple coupled scales, which have to be retained for a comprehensive picture of the dissipation range in Alfvénic turbulence. PMID:26207474
Verification of a magnetic island in gyro-kinetics by comparison with analytic theory
Zarzoso, D. Casson, F. J.; Poli, E.; Hornsby, W. A.; Peeters, A. G.
2015-02-15
A rotating magnetic island is imposed in the gyrokinetic code GKW, when finite differences are used for the radial direction, in order to develop the predictions of analytic tearing mode theory and understand its limitations. The implementation is verified against analytics in sheared slab geometry with three numerical tests that are suggested as benchmark cases for every code that imposes a magnetic island. The convergence requirements to properly resolve physics around the island separatrix are investigated. In the slab geometry, at low magnetic shear, binormal flows inside the island can drive Kelvin-Helmholtz instabilities which prevent the formation of the steady state for which the analytic theory is formulated.
Simulation of 3-D Magnetic Reconnection by Gyrokinetic Electron and Fully Kinetic Ion Particle Model
NASA Astrophysics Data System (ADS)
Wang, X.; Lin, Y.; Chen, L.
2015-12-01
3-D collisionless magnetic reconnection is investigated using the gyrokinetic electron and fully-kinetic ion (GeFi) particle simulation model. The simulation is carried out for cases with various finite guide field BG in a current sheet as occurring in space and laboratory plasmas. Turbulence power spectrum of magenetic field is found in the reconnection current sheet, with a clear k-5/3 dependence. The wave properties are analyzed. The anomalous resistivity in the electron diffusion region is estimated. The Dependence of the reconnection physics on the ion-to-electron mass ratio mi/me, beta values, and the half-width of the current sheet are also investigated.
Effects of the magnetic equilibrium on gyrokinetic simulations of tokamak microinstabilities
Wan, Weigang; Chen, Yang; Parker, Scott E.; Groebner, Richard J.
2015-06-15
The general geometry of the experimental tokamak magnetic equilibrium is implemented in the global gyrokinetic simulation code GEM. Compared to the general geometry, the well used Miller parameterization of the magnetic equilibrium is a good approximation in the core region and up to the top of the pedestal. Linear simulations indicate that results with the two geometries agree for r/a ≤ 0.9. However, in the edge region, the instabilities are sensitive to the magnetic equilibrium in both the L-mode and the H-mode plasmas. A small variation of the plasma shaping parameters leads to large changes to the edge instability.
J. Squire, H. Qin and W.M. Tang
2012-09-25
We present a new variational principle for the gyrokinetic system, similar to the Maxwell-Vlasov action presented in Ref. 1. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid velocity and particle distribution function. Using a Legendre transform, we explicitly derive the field theoretic Hamiltonian structure of the system. This is carried out with the Dirac theory of constraints, which is used to construct meaningful brackets from those obtained directly from Euler-Poincare theory. Possible applications of these formulations include continuum geometric integration techniques, large-eddy simulation models and Casimir type stability methods. __________________________________________________
A gyrokinetic continuum code based on the numerical Lie transform (NLT) method
NASA Astrophysics Data System (ADS)
Ye, Lei; Xu, Yingfeng; Xiao, Xiaotao; Dai, Zongliang; Wang, Shaojie
2016-07-01
In this work, we report a novel gyrokinetic simulation method named numerical Lie transform (NLT), which depends on a new physical model derived from the I-transform theory. In this model, the perturbed motion of a particle is decoupled from the unperturbed motion. Due to this property, the unperturbed orbit can be computed in advance and saved as numerical tables for real-time computation. A 4D tensor B-spline interpolation module is developed and applied with the semi-Lagrangian scheme to avoid operator splitting. The NLT code is verified by the Rosenbluth-Hinton test and the linear ITG Cyclone test.
NASA Astrophysics Data System (ADS)
Howard, N. T.; Greenwald, M.; Mikkelsen, D. R.; Reinke, M. L.; White, A. E.; Ernst, D.; Podpaly, Y.; Candy, J.
2012-06-01
Nonlinear gyrokinetic simulations of impurity transport are compared to experimental impurity transport for the first time. The GYRO code (Candy and Waltz 2003 J. Comput. Phys. 186 545) was used to perform global, nonlinear gyrokinetic simulations of impurity transport for a standard Alcator C-Mod, L-mode discharge. The laser blow-off technique was combined with soft x-ray measurements of a single charge state of calcium to provide time-evolving profiles of this non-intrinsic, non-recycling impurity over a radial range of 0.0 ⩽ r/a ⩽ 0.6. Experimental transport coefficient profiles and their uncertainties were extracted from the measurements using the impurity transport code STRAHL and rigorous Monte Carlo error analysis. To best assess the agreement of gyrokinetic simulations with the experimental profiles, the sensitivity of the GYRO predicted impurity transport to a wide range of turbulence-relevant plasma parameters was investigated. A direct comparison of nonlinear gyrokinetic simulation and experiment is presented with an in depth discussion of error sources and a new data analysis methodology.
Howard, N. T.; Greenwald, M.; White, A. E.; Reinke, M. L.; Ernst, D.; Podpaly, Y.; Mikkelsen, D. R.; Candy, J.
2012-05-15
Measured impurity transport coefficients are found to demonstrate a strong dependence on plasma current in the core of Alcator C-Mod. These measurements are compared directly with linear and nonlinear gyrokinetic simulation in an attempt to both qualitatively and quantitatively reproduce the measured impurity transport. Discharges constituting a scan of plasma current from 0.6 to 1.2 MA were performed during the 2010 run campaign. The impurity transport from these discharges was determined using a novel set of spectroscopic diagnostics available on Alcator C-Mod. This diagnostic suite allowed for the effective constraint of impurity transport coefficient profiles inside of r/a = 0.6. A decrease in the measured impurity diffusivity and inward convection is found with increased plasma current. Global, nonlinear gyrokinetic simulations were performed using the GYRO code [J. Candy and R. E. Waltz, J Comput. Phys. 186, 545 (2003)] for all discharges in the experimental scan and are found to reproduce the experimental trends, while demonstrating good quantitative agreement with measurement. A more comprehensive quantitative comparison was performed on the 0.8 MA discharge of the current scan which demonstrates that simultaneous agreement between experiment and simulation in both the impurity particle transport and ion heat transport channels is attainable within experimental uncertainties.
The linear tearing instability in three dimensional, toroidal gyro-kinetic simulations
Hornsby, W. A. Migliano, P.; Buchholz, R.; Kroenert, L.; Weikl, A.; Peeters, A. G.; Zarzoso, D.; Poli, E.; Casson, F. J.
2015-02-15
Linear gyro-kinetic simulations of the classical tearing mode in three-dimensional toroidal geometry were performed using the global gyro-kinetic turbulence code, GKW. The results were benchmarked against a cylindrical ideal MHD and analytical theory calculations. The stability, growth rate, and frequency of the mode were investigated by varying the current profile, collisionality, and the pressure gradients. Both collisionless and semi-collisional tearing modes were found with a smooth transition between the two. A residual, finite, rotation frequency of the mode even in the absence of a pressure gradient is observed, which is attributed to toroidal finite Larmor-radius effects. When a pressure gradient is present at low collisionality, the mode rotates at the expected electron diamagnetic frequency. However, the island rotation reverses direction at high collisionality. The growth rate is found to follow a η{sup 1∕7} scaling with collisional resistivity in the semi-collisional regime, closely following the semi-collisional scaling found by Fitzpatrick. The stability of the mode closely follows the stability analysis as performed by Hastie et al. using the same current and safety factor profiles but for cylindrical geometry, however, here a modification due to toroidal coupling and pressure effects is seen.
NASA Astrophysics Data System (ADS)
Weir, G. M.; Faber, B. J.; Likin, K. M.; Talmadge, J. N.; Anderson, D. T.; Anderson, F. S. B.
2015-05-01
Stiffness measurements are presented in the quasi-helically symmetric experiment (HSX), in which the neoclassical transport is comparable to that in a tokamak and turbulent transport dominates throughout the plasma. Electron cyclotron emission is used to measure the local electron temperature response to modulated electron cyclotron resonant heating. The amplitude and phase of the heat wave through the steep electron temperature gradient (ETG) region of the plasma are used to determine a transient electron thermal diffusivity that is close to the steady-state diffusivity. The low stiffness in the region between 0.2 ≤ r/a ≤ 0.4 agrees with the scaling of the steady-state heat flux with temperature gradient in this region. These experimental results are compared to gyrokinetic calculations in a flux-tube geometry using the gyrokinetic electromagnetic numerical experiment code with two kinetic species. Linear simulations show that the ETG mode may be experimentally relevant within r/a ≤ 0.2, while the Trapped Electron Mode (TEM) is the dominant long-wavelength microturbulence instability across most of the plasma. The TEM is primarily driven by the density gradient. Non-linear calculations of the saturated heat flux driven by the TEM and ETG bracket the experimental heat flux.
Gyrokinetic simulations of momentum transport and fluctuation spectra for ICRF-heated L-Mode plasmas
NASA Astrophysics Data System (ADS)
Sierchio, J. M.; White, A. E.; Howard, N. T.; Sung, C.; Ennever, P.; Porkolab, M.; Candy, J.
2014-10-01
We examine ICRF-heated L-mode plasmas in Alcator C-Mod, with differing momentum transport (hollow vs. peaked radial profiles of intrinsic toroidal rotation) but similar heat and particle transport. Nonlinear gyrokinetic simulations of heat and particle transport with GYRO [Candy and Waltz, J. Comp. Phys. 186, 545 (2003)] have previously been compared with these experiments [White et al., Phys. Plasmas 20, 056106 (2013); Howard et al. PPCF submitted (2014)] as part of an effort to validate the gyrokinetic model for core turbulent transport in C-Mod plasmas. To further test the model for these plasmas, predicted core turbulence characteristics such as fluctuation spectra will be compared with experiment. Using synthetic diagnostics for the CECE, reflectometry, and PCI systems at C-Mod, synthetic spectra and, when applicable, fluctuation amplitudes, are generated. We compare these generated results with fluctuation measurements from the experiment. We also report the momentum transport results from simulations of these plasmas and compare them to experiment. Supported by USDoE award DE-FC02-99ER54512.
NASA Astrophysics Data System (ADS)
Howard, N. T.; Greenwald, M.; Mikkelsen, D. R.; White, A. E.; Reinke, M. L.; Ernst, D.; Podpaly, Y.; Candy, J.
2012-05-01
Measured impurity transport coefficients are found to demonstrate a strong dependence on plasma current in the core of Alcator C-Mod. These measurements are compared directly with linear and nonlinear gyrokinetic simulation in an attempt to both qualitatively and quantitatively reproduce the measured impurity transport. Discharges constituting a scan of plasma current from 0.6 to 1.2 MA were performed during the 2010 run campaign. The impurity transport from these discharges was determined using a novel set of spectroscopic diagnostics available on Alcator C-Mod. This diagnostic suite allowed for the effective constraint of impurity transport coefficient profiles inside of r/a = 0.6. A decrease in the measured impurity diffusivity and inward convection is found with increased plasma current. Global, nonlinear gyrokinetic simulations were performed using the GYRO code [J. Candy and R. E. Waltz, J Comput. Phys. 186, 545 (2003)] for all discharges in the experimental scan and are found to reproduce the experimental trends, while demonstrating good quantitative agreement with measurement. A more comprehensive quantitative comparison was performed on the 0.8 MA discharge of the current scan which demonstrates that simultaneous agreement between experiment and simulation in both the impurity particle transport and ion heat transport channels is attainable within experimental uncertainties.
Weir, G. M.; Faber, B. J.; Likin, K. M.; Talmadge, J. N.; Anderson, D. T.; Anderson, F. S. B.
2015-05-15
Stiffness measurements are presented in the quasi-helically symmetric experiment (HSX), in which the neoclassical transport is comparable to that in a tokamak and turbulent transport dominates throughout the plasma. Electron cyclotron emission is used to measure the local electron temperature response to modulated electron cyclotron resonant heating. The amplitude and phase of the heat wave through the steep electron temperature gradient (ETG) region of the plasma are used to determine a transient electron thermal diffusivity that is close to the steady-state diffusivity. The low stiffness in the region between 0.2 ≤ r/a ≤ 0.4 agrees with the scaling of the steady-state heat flux with temperature gradient in this region. These experimental results are compared to gyrokinetic calculations in a flux-tube geometry using the gyrokinetic electromagnetic numerical experiment code with two kinetic species. Linear simulations show that the ETG mode may be experimentally relevant within r/a ≤ 0.2, while the Trapped Electron Mode (TEM) is the dominant long-wavelength microturbulence instability across most of the plasma. The TEM is primarily driven by the density gradient. Non-linear calculations of the saturated heat flux driven by the TEM and ETG bracket the experimental heat flux.
NASA Astrophysics Data System (ADS)
Zhang, Shi-Chang; Liu, Yaowu
2001-02-01
By making use of the gyrokinetics of free-electron masers, the efficiency formula of a cylindrical-cavity gyrotron oscillator is presented, where the misalignment of the electron-beam axis to the cavity axis has been taken into account. Comparison with a recent experimental report [Int. J. Infrared and Millimeter Waves 19, 1303 (1998)] is made, which confirms the creditability of the gyrokinetic theory.
Search for gyrokinetic dependencies in helium transport at Alcator C-Mod
NASA Astrophysics Data System (ADS)
Liao, Kenneth; Rowan, William; Hatch, David; Bespamyatnov, Igor; Horton, Wendell
2013-10-01
Helium-3 and helium-4 impurity transport measurements and density profile measurements have been obtained on Alcator C-Mod in a variety of discharge conditions, using the core Charge Exchange Recombination Spectroscopy (CXRS) diagnostic. The helium concentrations range from trace (< 2 %) to large minority (~ 20 %). L-mode, H-mode, and I-mode results are included, with Ohmic heated, ICRF heated, and LH heated plasmas. Helium profiles are observed to vary with plasma current, and also change in time during ICRF shots. Linear and nonlinear gyrokinetic simulations are performed for some shots using the GENE code. Sensitivity scans are done for magnetic shear, impurity density, and other plasma parameters and transport scalings are compared with experimental results. Simulated transport flux is compared with experimentally derived D and v parameters. Supported by USDoE awards DE-FG03-96ER-54373 and DE-FC02-99ER54512.
Singh, Rameswar; Brunner, S.; Ganesh, R.; Jenko, F.
2014-03-15
This paper presents effects of finite ballooning angles on linear ion temperature gradient (ITG) driven mode and associated heat and momentum flux in Gyrokinetic flux tube simulation GENE. It is found that zero ballooning angle is not always the one at which the linear growth rate is maximum. The ITG mode acquires a short wavelength (SW) branch (k{sub ⊥}ρ{sub i} > 1) when growth rates maximized over all ballooning angles are considered. However, the SW branch disappears on reducing temperature gradient showing characteristics of zero ballooning angle SWITG in case of extremely high temperature gradient. Associated heat flux is even with respect to ballooning angle and maximizes at nonzero ballooning angle while the parallel momentum flux is odd with respect to the ballooning angle.
Particle pinch and collisionality in gyrokinetic simulations of tokamak plasma turbulence
Angioni, C.; Candy, J.; Waltz, R. E.; Fable, E.; Maslov, M.; Weisen, H.; Peeters, A. G.
2009-06-15
The generic problem of how, in a turbulent plasma, the experimentally relevant conditions of a particle flux very close to the null are achieved, despite the presence of strong heat fluxes, is addressed. Nonlinear gyrokinetic simulations of plasma turbulence in tokamaks reveal a complex dependence of the particle flux as a function of the turbulent spatial scale and of the velocity space as collisionality is increased. At experimental values of collisionality, the particle flux is found close to the null, in agreement with the experiment, due to the balance between inward and outward contributions at small and large scales, respectively. These simulations provide full theoretical support to the prediction of a peaked density profile in a future nuclear fusion reactor.
Collision-dependent power law scalings in two dimensional gyrokinetic turbulence
Cerri, S. S. Bañón Navarro, A.; Told, D.; Jenko, F.
2014-08-15
Nonlinear gyrokinetics provides a suitable framework to describe short-wavelength turbulence in magnetized laboratory and astrophysical plasmas. In the electrostatic limit, this system is known to exhibit a free energy cascade towards small scales in (perpendicular) real and/or velocity space. The dissipation of free energy is always due to collisions (no matter how weak the collisionality), but may be spread out across a wide range of scales. Here, we focus on freely decaying two dimensional electrostatic turbulence on sub-ion-gyroradius scales. An existing scaling theory for the turbulent cascade in the weakly collisional limit is generalized to the moderately collisional regime. In this context, non-universal power law scalings due to multiscale dissipation are predicted, and this prediction is confirmed by means of direct numerical simulations.
Faber, B. J.; Pueschel, M. J.; Terry, P. W.; Proll, J. H. E.; Hegna, C. C.; Weir, G. M.; Likin, K. M.; Talmadge, J. N.
2015-07-15
Gyrokinetic simulations of plasma microturbulence in the Helically Symmetric eXperiment are presented. Using plasma profiles relevant to experimental operation, four dominant drift wave regimes are observed in the ion wavenumber range, which are identified as different flavors of density-gradient-driven trapped electron modes. For the most part, the heat transport exhibits properties associated with turbulence driven by these types of modes. Additionally, long-wavelength, radially localized, nonlinearly excited coherent structures near the resonant central flux surface, not predicted by linear simulations, can further enhance flux levels. Integrated heat fluxes are compatible with experimental observations in the corresponding density gradient range. Despite low shearing rates, zonal flows are observed to regulate turbulence but can be overwhelmed at higher density gradients by the long-wavelength coherent structures.
Gyrokinetic simulations of solar wind turbulence from ion to electron scales.
Howes, G G; TenBarge, J M; Dorland, W; Quataert, E; Schekochihin, A A; Numata, R; Tatsuno, T
2011-07-15
A three-dimensional, nonlinear gyrokinetic simulation of plasma turbulence resolving scales from the ion to electron gyroradius with a realistic mass ratio is presented, where all damping is provided by resolved physical mechanisms. The resulting energy spectra are quantitatively consistent with a magnetic power spectrum scaling of k(-2.8) as observed in in situ spacecraft measurements of the "dissipation range" of solar wind turbulence. Despite the strongly nonlinear nature of the turbulence, the linear kinetic Alfvén wave mode quantitatively describes the polarization of the turbulent fluctuations. The collisional ion heating is measured at subion-Larmor radius scales, which provides evidence of the ion entropy cascade in an electromagnetic turbulence simulation.
Grid-based Parallel Data Streaming Implemented for the Gyrokinetic Toroidal Code
S. Klasky; S. Ethier; Z. Lin; K. Martins; D. McCune; R. Samtaney
2003-09-15
We have developed a threaded parallel data streaming approach using Globus to transfer multi-terabyte simulation data from a remote supercomputer to the scientist's home analysis/visualization cluster, as the simulation executes, with negligible overhead. Data transfer experiments show that this concurrent data transfer approach is more favorable compared with writing to local disk and then transferring this data to be post-processed. The present approach is conducive to using the grid to pipeline the simulation with post-processing and visualization. We have applied this method to the Gyrokinetic Toroidal Code (GTC), a 3-dimensional particle-in-cell code used to study microturbulence in magnetic confinement fusion from first principles plasma theory.
Simulations of 4D edge transport and dynamics using the TEMPEST gyro-kinetic code
NASA Astrophysics Data System (ADS)
Rognlien, T. D.; Cohen, B. I.; Cohen, R. H.; Dorr, M. R.; Hittinger, J. A. F.; Kerbel, G. D.; Nevins, W. M.; Xiong, Z.; Xu, X. Q.
2006-10-01
Simulation results are presented for tokamak edge plasmas with a focus on the 4D (2r,2v) option of the TEMPEST continuum gyro-kinetic code. A detailed description of a variety of kinetic simulations is reported, including neoclassical radial transport from Coulomb collisions, electric field generation, dynamic response to perturbations by geodesic acoustic modes, and parallel transport on open magnetic-field lines. Comparison is made between the characteristics of the plasma solutions on closed and open magnetic-field line regions separated by a magnetic separatrix, and simple physical models are used to qualitatively explain the differences observed in mean flow and electric-field generation. The status of extending the simulations to 5D turbulence will be summarized. The code structure used in this ongoing project is also briefly described, together with future plans.
NASA Astrophysics Data System (ADS)
Storelli, A.; Vermare, L.; Hennequin, P.; Gürcan, Ö. D.; Dif-Pradalier, G.; Sarazin, Y.; Garbet, X.; Görler, T.; Singh, Rameswar; Morel, P.; Grandgirard, V.; Ghendrih, P.
2015-06-01
In a dedicated collisionality scan in Tore Supra, the geodesic acoustic mode (GAM) is detected and identified with the Doppler backscattering technique. Observations are compared to the results of a simulation with the gyrokinetic code GYSELA. We found that the GAM frequency in experiments is lower than predicted by simulation and theory. Moreover, the disagreement is higher in the low collisionality scenario. Bursts of non harmonic GAM oscillations have been characterized with filtering techniques, such as the Hilbert-Huang transform. When comparing this dynamical behaviour between experiments and simulation, the probability density function of GAM amplitude and the burst autocorrelation time are found to be remarkably similar. In the simulation, where the radial profile of GAM frequency is continuous, we observed a phenomenon of radial phase mixing of the GAM oscillations, which could influence the burst autocorrelation time.
Storelli, A. Vermare, L.; Hennequin, P.; Gürcan, Ö. D.; Singh, Rameswar; Morel, P.; Dif-Pradalier, G.; Sarazin, Y.; Garbet, X.; Grandgirard, V.; Ghendrih, P.; Görler, T.
2015-06-15
In a dedicated collisionality scan in Tore Supra, the geodesic acoustic mode (GAM) is detected and identified with the Doppler backscattering technique. Observations are compared to the results of a simulation with the gyrokinetic code GYSELA. We found that the GAM frequency in experiments is lower than predicted by simulation and theory. Moreover, the disagreement is higher in the low collisionality scenario. Bursts of non harmonic GAM oscillations have been characterized with filtering techniques, such as the Hilbert-Huang transform. When comparing this dynamical behaviour between experiments and simulation, the probability density function of GAM amplitude and the burst autocorrelation time are found to be remarkably similar. In the simulation, where the radial profile of GAM frequency is continuous, we observed a phenomenon of radial phase mixing of the GAM oscillations, which could influence the burst autocorrelation time.
Flux- and gradient-driven global gyrokinetic simulation of tokamak turbulence
Goerler, Tobias; Jenko, Frank; Marcus, Patrick; Merz, Florian; Told, Daniel; Lapillonne, Xavier; Brunner, Stephan; Aghdam, Sohrab Khosh; McMillan, Ben F.; Sauter, Olivier; Villard, Laurent; Dannert, Tilman
2011-05-15
The Eulerian gyrokinetic turbulence code gene has recently been extended to a full torus code. Moreover, it now provides Krook-type sources for gradient-driven simulations where the profiles are maintained on average as well as localized heat sources for a flux-driven type of operation. Careful verification studies and benchmarks are performed successfully. This setup is applied to address three related transport issues concerning nonlocal effects. First, it is confirmed that in gradient-driven simulations, the local limit can be reproduced--provided that finite aspect ratio effects in the geometry are treated carefully. In this context, it also becomes clear that the profile widths (not the device width) may constitute a more appropriate measure for finite-size effects. Second, the nature and role of heat flux avalanches are discussed in the framework of both local and global, flux- and gradient-driven simulations. Third, simulations dedicated to discharges with electron internal barriers are addressed.
NASA Astrophysics Data System (ADS)
Faber, B. J.; Pueschel, M. J.; Proll, J. H. E.; Xanthopoulos, P.; Terry, P. W.; Hegna, C. C.; Weir, G. M.; Likin, K. M.; Talmadge, J. N.
2015-07-01
Gyrokinetic simulations of plasma microturbulence in the Helically Symmetric eXperiment are presented. Using plasma profiles relevant to experimental operation, four dominant drift wave regimes are observed in the ion wavenumber range, which are identified as different flavors of density-gradient-driven trapped electron modes. For the most part, the heat transport exhibits properties associated with turbulence driven by these types of modes. Additionally, long-wavelength, radially localized, nonlinearly excited coherent structures near the resonant central flux surface, not predicted by linear simulations, can further enhance flux levels. Integrated heat fluxes are compatible with experimental observations in the corresponding density gradient range. Despite low shearing rates, zonal flows are observed to regulate turbulence but can be overwhelmed at higher density gradients by the long-wavelength coherent structures.
Squire, J.; Tang, W. M.; Qin, H.; Chandre, C.
2013-02-15
We present a new variational principle for the gyrokinetic system, similar to the Maxwell-Vlasov action presented in H. Cendra et al., [J. Math. Phys. 39, 3138 (1998)]. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid velocity and particle distribution function. Using a Legendre transform, we explicitly derive the field theoretic Hamiltonian structure of the system. This is carried out with a modified Dirac theory of constraints, which is used to construct meaningful brackets from those obtained directly from Euler-Poincare theory. Possible applications of these formulations include continuum geometric integration techniques, large-eddy simulation models, and Casimir type stability methods.
Gyrokinetic Calculations of Microturbulence and Transport for NSTX and Alcator-CMOD H-modes
M.H. Redi; W. Dorland; R. Bell; P. Bonoli; C. Bourdelle; J. Candy; D. Ernst; C. Fiore; D. Gates; G. Hammett; K. Hill; S. Kaye; B. LeBlanc; J. Menard; D. Mikkelsen; G. Rewoldt; J. Rice; R. Waltz; S. Wukitch
2003-07-08
Recent H-mode experiments on NSTX [National Spherical Torus Experiment] and experiments on Alcator-CMOD, which also exhibit internal transport barriers (ITB), have been examined with gyrokinetic simulations with the GS2 and GYRO codes to identify the underlying key plasma parameters for control of plasma performance and, ultimately, the successful operation of future reactors such as ITER [International Thermonuclear Experimental Reactor]. On NSTX the H-mode is characterized by remarkably good ion confinement and electron temperature profiles highly resilient in time. On CMOD, an ITB with a very steep electron density profile develops following off-axis radio-frequency heating and establishment of H-mode. Both experiments exhibit ion thermal confinement at the neoclassical level. Electron confinement is also good in the CMOD core.
NASA Astrophysics Data System (ADS)
White, A. E.; Howard, N. T.; Mikkelsen, D. R.; Greenwald, M.; Candy, J.; Waltz, R. E.
2011-11-01
This paper describes the use of nonlinear gyrokinetic simulations to assess the feasibility of a new correlation electron cyclotron emission (CECE) diagnostic that has been proposed for the Alcator C-Mod tokamak (Marmar et al 2009 Nucl. Fusion 49 104014). This work is based on a series of simulations performed with the GYRO code (Candy and Waltz 2003 J. Comput. Phys. 186 545). The simulations are used to predict ranges of fluctuation level, peak poloidal wavenumber and radial correlation length of electron temperature fluctuations in the core of the plasma. The impact of antenna pattern and poloidal viewing location on measurable turbulence characteristics is addressed using synthetic diagnostics. An upper limit on the CECE sample volume size is determined. The modeling results show that a CECE diagnostic capable of measuring transport-relevant, long-wavelength (kθρs < 0.5) electron temperature fluctuations is feasible at Alcator C-Mod.
Multi-scale gyrokinetic simulation of Alcator C-Mod tokamak discharges
Howard, N. T. White, A. E.; Greenwald, M.; Holland, C.; Candy, J.
2014-03-15
Alcator C-Mod tokamak discharges have been studied with nonlinear gyrokinetic simulation simultaneously spanning both ion and electron spatiotemporal scales. These multi-scale simulations utilized the gyrokinetic model implemented by GYRO code [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] and the approximation of reduced electron mass (μ = (m{sub D}/m{sub e}){sup .5} = 20.0) to qualitatively study a pair of Alcator C-Mod discharges: a low-power discharge, previously demonstrated (using realistic mass, ion-scale simulation) to display an under-prediction of the electron heat flux and a high-power discharge displaying agreement with both ion and electron heat flux channels [N. T. Howard et al., Nucl. Fusion 53, 123011 (2013)]. These multi-scale simulations demonstrate the importance of electron-scale turbulence in the core of conventional tokamak discharges and suggest it is a viable candidate for explaining the observed under-prediction of electron heat flux. In this paper, we investigate the coupling of turbulence at the ion (k{sub θ}ρ{sub s}∼O(1.0)) and electron (k{sub θ}ρ{sub e}∼O(1.0)) scales for experimental plasma conditions both exhibiting strong (high-power) and marginally stable (low-power) low-k (k{sub θ}ρ{sub s} < 1.0) turbulence. It is found that reduced mass simulation of the plasma exhibiting marginally stable low-k turbulence fails to provide even qualitative insight into the turbulence present in the realistic plasma conditions. In contrast, multi-scale simulation of the plasma condition exhibiting strong turbulence provides valuable insight into the coupling of the ion and electron scales.
NASA Astrophysics Data System (ADS)
Ernst, Darin; Long, A.; Basse, N.; Lin, L.; Porkolab, M.; Dorland, W.
2006-04-01
We have developed a synthetic diagnostic^1 for the GS2 gyrokinetic code for direct comparisons with phase contrast imaging (PCI) measurements of density fluctuations in Alcator C-Mod. The gyrokinetic simulation is carried out in a local, field line following flux-tube, while PCI measures density fluctuations along 32 chords passing vertically through the plasma cross-section.^2 Transforming from Clebsch to cartesian coordinates, and integrating appropriately over portions of the flux tube viewed by the diagnostic, yields a density fluctuation spectrum versus wavenumber kR in the major radius direction. To achieve vertical localization, we examine an ITB case in which the spectrum is dominated by a strong trapped electron mode, localized near the half-radius. The wavelength spectrum from the simulations, using the synthetic diagnostic, closely reproduces the PCI spectrum. Contributions from kψ, where B=∇αx∇ψ, downshift the GS2 kα spectrum to improve upon our previous raw comparison with the PCI kR spectrum.^3 ^1A. Long, D. R. Ernst et al., Bull. Am. Phys. Soc. 50(8) p. 153, GP1.48, also p. 235, LP1.37 http://www.psfc.mit.edu/research/alcator/pubs/APS/APS2005/ernst.pdf. ^2N. P. Basse et al., Phys. Plasmas 12, 052512 (2005). ^3D. R. Ernst et al., 2004 IAEA Fusion Energy Conference, IAEA-CN116/TH/4-1 http://www-naweb.iaea.org/napc/physics/fec/fec2004/datasets/TH4-1.html, see also Phys. Plasmas 11 (2004) 2637.
Comparison of Measurements of Profile Stiffness in HSX to Nonlinear Gyrokinetic Calculations
NASA Astrophysics Data System (ADS)
Weir, Gavin
2014-10-01
Tokamaks and stellarators have observed significant differences in profile stiffness, defined as the ratio of the transient thermal diffusivity obtained from heat pulse propagation to the diffusivity obtained from steady-state power balance. Typically, stellarators have measured stiffness values below 2 and tokamaks have observed stiffness greater than 4. In this paper we present the first results on stiffness measurements in the quasihelically symmetric experiment HSX in which the neoclassical transport is comparable to that in a tokamak and turbulent transport dominates throughout the plasma. Electron Cyclotron Emission (ECE) is used to measure the local electron temperature perturbation from modulating the ECRH system on HSX. Spectral analysis of the ECE data yields a profile of the perturbed amplitude and a resulting transient electron thermal diffusivity that is close to the steady-state diffusivity. This evidence of a lack of stiffness in HSX agrees with the scaling of the steady-state heat flux with temperature gradient. The experimental data is compared to gyrokinetic calculations using the GENE code with two kinetic species. Linear calculations demonstrate that the Trapped Electron Mode (TEM) is the dominant long-wavelength microturbulence instability with growth rates that scale linearly with electron temperature gradient. Nonlinear gyrokinetic flux tube simulations indicate that the TEM contributes significantly to the saturated heat fluxes in HSX, shifting the transport-carrying wavenumbers to larger values than in typical Ion Temperature Gradient (ITG) turbulence. A set of nonlinear simulations are being executed, examining the saturated nonlinear heat flux as a function of the electron temperature gradient, to obtain a stiffness value from the simulations to compare with experimental results. This work is supported by DOE Grant DE-FG02-93ER54222.
Multi-scale gyrokinetic simulation of Alcator C-Mod tokamak discharges
NASA Astrophysics Data System (ADS)
Howard, N. T.; White, A. E.; Greenwald, M.; Holland, C.; Candy, J.
2014-03-01
Alcator C-Mod tokamak discharges have been studied with nonlinear gyrokinetic simulation simultaneously spanning both ion and electron spatiotemporal scales. These multi-scale simulations utilized the gyrokinetic model implemented by GYRO code [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] and the approximation of reduced electron mass (μ = (mD/me).5 = 20.0) to qualitatively study a pair of Alcator C-Mod discharges: a low-power discharge, previously demonstrated (using realistic mass, ion-scale simulation) to display an under-prediction of the electron heat flux and a high-power discharge displaying agreement with both ion and electron heat flux channels [N. T. Howard et al., Nucl. Fusion 53, 123011 (2013)]. These multi-scale simulations demonstrate the importance of electron-scale turbulence in the core of conventional tokamak discharges and suggest it is a viable candidate for explaining the observed under-prediction of electron heat flux. In this paper, we investigate the coupling of turbulence at the ion (kθρs˜O(1.0)) and electron (kθρe˜O(1.0)) scales for experimental plasma conditions both exhibiting strong (high-power) and marginally stable (low-power) low-k (kθρs < 1.0) turbulence. It is found that reduced mass simulation of the plasma exhibiting marginally stable low-k turbulence fails to provide even qualitative insight into the turbulence present in the realistic plasma conditions. In contrast, multi-scale simulation of the plasma condition exhibiting strong turbulence provides valuable insight into the coupling of the ion and electron scales.
Static cylindrically symmetric spacetimes
NASA Astrophysics Data System (ADS)
Fjällborg, Mikael
2007-05-01
We prove the existence of static solutions to the cylindrically symmetric Einstein Vlasov system, and we show that the matter cylinder has finite extension in two of the three spatial dimensions. The same results are also proved for a quite general class of equations of state for perfect fluids coupled to the Einstein equations, extending the class of equations of state considered by Bicak et al (2004 Class. Quantum Grav.21 1583). We also obtain this result for the Vlasov Poisson system.
Hager, Robert; Chang, C. S.
2016-04-08
As a follow-up on the drift-kinetic study of the non-local bootstrap current in the steep edge pedestal of tokamak plasma by Koh et al. [Phys. Plasmas 19, 072505 (2012)], a gyrokinetic neoclassical study is performed with gyrokinetic ions and drift-kinetic electrons. Besides the gyrokinetic improvement of ion physics from the drift-kinetic treatment, a fully non-linear Fokker-Planck collision operator—that conserves mass, momentum, and energy—is used instead of Koh et al.'s linearized collision operator in consideration of the possibility that the ion distribution function is non-Maxwellian in the steep pedestal. An inaccuracy in Koh et al.'s result is found in the steepmore » edge pedestal that originated from a small error in the collisional momentum conservation. The present study concludes that (1) the bootstrap current in the steep edge pedestal is generally smaller than what has been predicted from the small banana-width (local) approximation [e.g., Sauter et al., Phys. Plasmas 6, 2834 (1999) and Belli et al., Plasma Phys. Controlled Fusion 50, 095010 (2008)], (2) the plasma flow evaluated from the local approximation can significantly deviate from the non-local results, and (3) the bootstrap current in the edge pedestal, where the passing particle region is small, can be dominantly carried by the trapped particles in a broad trapped boundary layer. In conclusion, a new analytic formula based on numerous gyrokinetic simulations using various magnetic equilibria and plasma profiles with self-consistent Grad-Shafranov solutions is constructed.« less
Gyrokinetic particle-in-cell simulations of Alfvén eigenmodes in presence of continuum effects
Mishchenko, Alexey Könies, Axel; Hatzky, Roman
2014-05-15
First-principle gyrokinetic particle-in-cell simulations of a global Toroidal Alfvén Eigenmode (TAE) are undertaken in the presence of a strong coupling with the continuum. Effects of the bulk plasma temperature on the interplay between the TAE and Kinetic Alfvén Waves (KAWs) are investigated. A global TAE-KAW structure is identified which appears to be more unstable with respect to the fast ions than a simple (fluid-like) TAE mode.
NASA Astrophysics Data System (ADS)
Hager, Robert; Chang, C. S.
2016-04-01
As a follow-up on the drift-kinetic study of the non-local bootstrap current in the steep edge pedestal of tokamak plasma by Koh et al. [Phys. Plasmas 19, 072505 (2012)], a gyrokinetic neoclassical study is performed with gyrokinetic ions and drift-kinetic electrons. Besides the gyrokinetic improvement of ion physics from the drift-kinetic treatment, a fully non-linear Fokker-Planck collision operator—that conserves mass, momentum, and energy—is used instead of Koh et al.'s linearized collision operator in consideration of the possibility that the ion distribution function is non-Maxwellian in the steep pedestal. An inaccuracy in Koh et al.'s result is found in the steep edge pedestal that originated from a small error in the collisional momentum conservation. The present study concludes that (1) the bootstrap current in the steep edge pedestal is generally smaller than what has been predicted from the small banana-width (local) approximation [e.g., Sauter et al., Phys. Plasmas 6, 2834 (1999) and Belli et al., Plasma Phys. Controlled Fusion 50, 095010 (2008)], (2) the plasma flow evaluated from the local approximation can significantly deviate from the non-local results, and (3) the bootstrap current in the edge pedestal, where the passing particle region is small, can be dominantly carried by the trapped particles in a broad trapped boundary layer. A new analytic formula based on numerous gyrokinetic simulations using various magnetic equilibria and plasma profiles with self-consistent Grad-Shafranov solutions is constructed.
White, A. E.; Schmitz, L.; Peebles, W. A.; Carter, T. A.; Doyle, E. J.; Rhodes, T. L.; Wang, G.; McKee, G. R.; Shafer, M. W.; Holland, C.; Tynan, G. R.; Austin, M. E.; Burrell, K. H.; Candy, J.; DeBoo, J. C.; Prater, R.; Staebler, G. M.; Waltz, R. E.; Makowski, M. A.
2008-05-15
For the first time, profiles (0.3<{rho}<0.9) of electron temperature and density fluctuations in a tokamak have been measured simultaneously and the results compared to nonlinear gyrokinetic simulations. Electron temperature and density fluctuations measured in neutral beam-heated, sawtooth-free low confinement mode (L-mode) plasmas in DIII-D [J. L. Luxon, Nucl. Fusion 42, 614 (2002)] are found to be similar in frequency and normalized amplitude, with amplitude increasing with radius. The measured radial profile of two fluctuation fields allows for a new and rigorous comparison with gyrokinetic results. Nonlinear gyrokinetic flux-tube simulations predict that electron temperature and density fluctuations have similar normalized amplitudes in L-mode. At {rho}=0.5, simulation results match experimental heat diffusivities and density fluctuation amplitude, but overestimate electron temperature fluctuation amplitude and particle diffusivity. In contrast, simulations at {rho}=0.75 do not match either the experimentally derived transport properties or the measured fluctuation levels.
Kluy, N.; Angioni, C.; Camenen, Y.; Peeters, A. G.
2009-12-15
The toroidal momentum transport in the presence of trapped electron mode microinstabilities in tokamak plasmas is studied by means of quasilinear gyrokinetic calculations. In particular, the role of the Coriolis drift in producing an inward convection of toroidal momentum is investigated. The Coriolis drift term has been implemented in the gyrokinetic code GS2 [W. Dorland et al., Phys. Rev. Lett. 85, 5579 (2000)] specifically for the completion of this work. A benchmark between the GS2 implementation of the Coriolis drift and the implementations included in two other gyrokinetic codes is presented. The numerical calculations show that in the presence of trapped electron modes, despite of a weaker symmetry breaking of the eigenfunctions with respect to the case of ion temperature gradient modes, a pinch of toroidal momentum is produced in most conditions. The toroidal momentum viscosity is also computed, and found to be small as compared with the electron heat conductivity, but significantly larger than the ion heat conductivity. In addition, interesting differences are found in the dependence of the toroidal momentum pinch as a function of collisionality between trapped electron modes and ion temperature gradient modes. The results identify also parameter domains in which the pinch is predicted to be small, which are also of interest for comparisons with the experiments.
NASA Astrophysics Data System (ADS)
Waltz, R. E.; Waelbroeck, F. L.
2012-03-01
Static external resonant magnetic field perturbations (RMPs) have been added to the gyrokinetic code GYRO [J. Candy and R. E. Waltz, J. Comp. Phys. 186, 545 (2003)]. This allows nonlinear gyrokinetic simulations of the nonambipolar radial current flow jr, and the corresponding j→×B→ plasma torque (density) R[jrBp/c], induced by magnetic islands that break the toroidal symmetry of a tokamak. This extends the previous GYRO formulation for the transport of toroidal angular momentum (TAM) [R. E. Waltz, G. M. Staebler, J. Candy, and F. L. Hinton, Phys. Plasmas 14, 122507 (2007); errata 16, 079902 (2009)]. The focus is on electrostatic full torus radial slice simulations of externally induced q =m/n=6/3 islands with widths 5% of the minor radius or about 20 ion gyroradii. Up to moderately strong E ×B rotation, the island torque scales with the radial electric field at the resonant surface Er, the island width w, and the intensity I of the high-n micro-turbulence, as Erw√I . The radial current inside the island is carried (entirely in the n =3 component) and almost entirely by the ion E ×B flux, since the electron E ×B and magnetic flutter particle fluxes are cancelled. The net island torque is null at zero Er rather than at zero toroidal rotation. This means that while the expected magnetic braking of the toroidal plasma rotation occurs at strong co- and counter-current rotation, at null toroidal rotation, there is a small co-directed magnetic acceleration up to the small diamagnetic (ion pressure gradient driven) co-rotation corresponding to the zero Er and null torque. This could be called the residual stress from an externally induced island. At zero Er, the only effect is the expected partial flattening of the electron temperature gradient within the island. Finite-beta GYRO simulations demonstrate almost complete RMP field screening and n =3 mode unlocking at strong Er.
Search for the Missing L-mode Edge Transport and Possible Breakdown of Gyrokinetics
NASA Astrophysics Data System (ADS)
Waltz, R. E.
2012-10-01
While GYRO simulations of typical core (0 < r/a < 0.7) DIII-D L-modes seems to be in good agreement with experiment, simulated low-k (kθρs< 1) transport and turbulence intensity is more than 5-fold lower than experimentally inferred levels in the near edge L-mode (r/a=0.7-0.95) DIII-D shot 128913 [1]. Global edge slice GYRO simulations of this and the well-studied discharge 101391 [2] are presented here to document the shortfall. TGLF transport code simulations over a large L-mode database indicate this short fall is not atypical so that L-mode edges transit to H-like pedestal profiles contrary to experiment. High edge e-i collisionality stabilizes the TEM modes so that diffusivities (χ) decrease like T^7/2/n to the cold edge. The very high magnetic shear and density gradients stabilize the ITG despite the very high temperature gradient drive and high q. High-k ETG can make-up for the shortfall in the electron but increases ion transport very little. Near L-edge transport is highly local. Focusing on local simulations at r/a=0.9, the ion channel short fall can exceed 10-fold. An artificial 10-fold increase in collisionality is needed to reach the expected resistive g-mode scaling with χ increasing like nT-1/2. Identical GYRO drift kinetic ion simulations (suppressing the gyroaverage) are close to experiment levels suggesting a possible breakdown of low-frequency gyrokinetics. Formulation of a nonlinear theory of 6D drift-cyclotron kinetics following the fast time scale of the gyrophase to test the breakdown of 5D gyrokinetics with reduced model simulations is presented. 6pt [1] C. Holland, A.E. White, et al., Phys. Plasmas 16, 052301 (2009). [2] R.E. Waltz, J. Candy, C.C. Petty, Phys. Plasmas 13, 072304 (2006).
NASA Astrophysics Data System (ADS)
Howard, N. T.; Holland, C.; White, A. E.; Greenwald, M.; Candy, J.; Creely, A. J.
2016-05-01
To better understand the role of cross-scale coupling in experimental conditions, a series of multi-scale gyrokinetic simulations were performed on Alcator C-Mod, L-mode plasmas. These simulations, performed using all experimental inputs and realistic ion to electron mass ratio ((mi/me)1/2 = 60.0), simultaneously capture turbulence at the ion ( kθρs˜O (1.0 ) ) and electron-scales ( kθρe˜O (1.0 ) ). Direct comparison with experimental heat fluxes and electron profile stiffness indicates that Electron Temperature Gradient (ETG) streamers and strong cross-scale turbulence coupling likely exist in both of the experimental conditions studied. The coupling between ion and electron-scales exists in the form of energy cascades, modification of zonal flow dynamics, and the effective shearing of ETG turbulence by long wavelength, Ion Temperature Gradient (ITG) turbulence. The tightly coupled nature of ITG and ETG turbulence in these realistic plasma conditions is shown to have significant implications for the interpretation of experimental transport and fluctuations. Initial attempts are made to develop a "rule of thumb" based on linear physics, to help predict when cross-scale coupling plays an important role and to inform future modeling of experimental discharges. The details of the simulations, comparisons with experimental measurements, and implications for both modeling and experimental interpretation are discussed.
Particle-in-cell δf gyrokinetic simulations of the microtearing mode
NASA Astrophysics Data System (ADS)
Chowdhury, J.; Chen, Yang; Wan, Weigang; Parker, Scott E.; Guttenfelder, W.; Canik, J. M.
2016-01-01
The linear stability properties of the microtearing mode are investigated in the edge and core regimes of the National Spherical Torus Experiment (NSTX) using the particle-in-cell method based gyrokinetic code GEM. The dependence of the mode on various equilibrium quantities in both regions is compared. While the microtearing mode in the core depends upon the electron-ion collisions, in the edge region, it is found to be weakly dependent on the collisions and exists even when the collision frequency is zero. The electrostatic potential is non-negligible in each of the cases. It plays opposite roles in the core and edge of NSTX. While the microtearing mode is partially stabilized by the electrostatic potential in the core, it has substantial destabilizing effect in the edge. In addition to the spherical tokamak, we also study the microtearing mode for parameters relevant to the core of a standard tokamak. The fundamental characteristics of the mode remain the same; however, the electrostatic potential in this case is destabilizing as opposed to the core of NSTX. The velocity dependence of the collision frequency, which is crucial for the mode to grow in slab calculations, is not required to destabilize the mode in toroidal devices.
NASA Astrophysics Data System (ADS)
Ernst, D. R.; Bergerson, W.; Ennever, P.; Greenwald, M.; Hubbard, A.; Irby, J.; Phillips, P.; Porkolab, M.; Rowan, W.; Terry, J. L.; Xu, P.; Alcator C-Mod Team
2013-10-01
Three new synthetic turbulence diagnostics are implemented in GS2 and compared with measurements: phase contrast imaging, polarimetry, and electron-cyclotron (ECE) emission. Our new synthetic diagnostic framework is based on transforming to a real-space annulus in Cartesian coordinates. This allows straightforward convolution with diagnostic point-spread functions, or integration over viewing chords. Wavenumber spectra and fluctuation amplitudes, as well as transport fluxes, are compared with measurements. Both phase contrast imaging and newly observed ECE electron temperature fluctuations, closely follow the electron temperature in an internal transport barrier during on-axis heating pulses, consistent with the role of TEM turbulence. New C-Mod polarimetry measurements, showing strong broadband core magnetic fluctuations, will also be examined against gyrokinetic simulations. The new framework is readily extended to other fluctuation measurements such as two-color interferometry, beam emission spectroscopy, Doppler back-scattering, ECE imaging, and microwave imaging reflectometry. Supported by U.S. DoE awards DE-FC02-08ER54966, DE-FC02-99ER54512, DE-FG03-96ER54373.
Fable, E.; Sauter, O.; Angioni, C.
2008-11-01
Peaked density profiles are observed in the core of Tokamak plasmas in regimes where the core particle sources and neoclassical transport are negligible. Gyrokinetic theory predicts that microinstabilities can produce a net inward particle convection balancing outward diffusion and thus explaining the experimental observations. In this work we present a general methodology that allows to calculate the particle pinch coefficients, i.e. the off-diagonal elements of the transport matrix. We adopt this procedure to perform a systematic study of the parametric dependence of these coefficients for electron particle transport in different plasma conditions. Once the coefficients are computed, one can reconstruct the predicted gradient and compare with the experimental observations in regimes with parameters similar to the ones employed in these calculations. The procedure can predict the density logarithmic gradient at zero particle flux in a self-consistent way, based on first principles. The results can be helpful in understanding the possible range of variation of the predicted gradients as a function of the main plasma parameters and in clarifying the relevant dependencies for electrons. Finally, as instructive example, we discuss how this procedure can effectively help to interpret measurements of peaked density profiles in TCV electron Internal Transport Barriers and the significant thermodiffusive inward convection that is observed.
Gyrokinetic Particle Simulation of Compressible Electromagnetic Turbulence in High-β Plasmas
Lin, Zhihong
2014-03-13
Supported by this award, the PI and his research group at the University of California, Irvine (UCI) have carried out computational and theoretical studies of instability, turbulence, and transport in laboratory and space plasmas. Several massively parallel, gyrokinetic particle simulation codes have been developed to study electromagnetic turbulence in space and laboratory plasmas. In space plasma projects, the simulation codes have been successfully applied to study the spectral cascade and plasma heating in kinetic Alfven wave turbulence, the linear and nonlinear properties of compressible modes including mirror instability and drift compressional mode, and the stability of the current sheet instabilities with finite guide field in the context of collisionless magnetic reconnection. The research results have been published in 25 journal papers and presented at many national and international conferences. Reprints of publications, source codes, and other research-related information are also available to general public on the PI’s webpage (http://phoenix.ps.uci.edu/zlin/). Two PhD theses in space plasma physics are highlighted in this report.
Four-Dimensional Continuum Gyrokinetic Code: Neoclassical Simulation of Fusion Edge Plasmas
NASA Astrophysics Data System (ADS)
Xu, X. Q.
2005-10-01
We are developing a continuum gyrokinetic code, TEMPEST, to simulate edge plasmas. Our code represents velocity space via a grid in equilibrium energy and magnetic moment variables, and configuration space via poloidal magnetic flux and poloidal angle. The geometry is that of a fully diverted tokamak (single or double null) and so includes boundary conditions for both closed magnetic flux surfaces and open field lines. The 4-dimensional code includes kinetic electrons and ions, and electrostatic field-solver options, and simulates neoclassical transport. The present implementation is a Method of Lines approach where spatial finite-differences (higher order upwinding) and implicit time advancement are used. We present results of initial verification and validation studies: transition from collisional to collisionless limits of parallel end-loss in the scrape-off layer, self-consistent electric field, and the effect of the real X-point geometry and edge plasma conditions on the standard neoclassical theory, including a comparison of our 4D code with other kinetic neoclassical codes and experiments.
Gyrokinetic Calculations of Microinstabilities and Transport During RF H-Modes on Alcator C-Mod
M.H. Redi; C. Fiore; P. Bonoli; C. Bourdelle; R. Budny; W.D. Dorland; D. Ernst; G. Hammett; D. Mikkelsen; J. Rice; S. Wukitch
2002-06-18
Physics understanding for the experimental improvement of particle and energy confinement is being advanced through massively parallel calculations of microturbulence for simulated plasma conditions. The ultimate goal, an experimentally validated, global, non-local, fully nonlinear calculation of plasma microturbulence is still not within reach, but extraordinary progress has been achieved in understanding microturbulence, driving forces and the plasma response in recent years. In this paper we discuss gyrokinetic simulations of plasma turbulence being carried out to examine a reproducible, H-mode, RF heated experiment on the Alcator CMOD tokamak3, which exhibits an internal transport barrier (ITB). This off axis RF case represents the early phase of a very interesting dual frequency RF experiment, which shows density control with central RF heating later in the discharge. The ITB exhibits steep, spontaneous density peaking: a reduction in particle transport occurring without a central particle source. Since the central temperature is maintained while the central density is increasing, this also suggests a thermal transport barrier exists. TRANSP analysis shows that ceff drops inside the ITB. Sawtooth heat pulse analysis also shows a localized thermal transport barrier. For this ICRF EDA H-mode, the minority resonance is at r/a * 0.5 on the high field side. There is a normal shear profile, with q monotonic.
Particle Pinch in Gyrokinetic Simulations of Closed Field-Line Systems
Kobayashi, Sumire; Rogers, Barrett N.; Dorland, William
2010-12-03
Gyrokinetic simulations of small-scale turbulent transport in a closed magnetic field-line plasma geometry are presented. The simulations are potentially applicable to dipolar systems such as the levitated dipole experiment (LDX) [J. Kesner et al., Plasma Phys. Rep. 23, 742 (1997)] and planetary magnetospheres, as well as simpler systems such as the Z pinch. We report here for the first time the existence of a robust particle (and weaker temperature) pinch regime, in which the particles are transported up the density gradient. The particle pinch is driven by non-MHD entropy-mode turbulence at k{sub perpendicular{rho}i}{approx}1 and particle pinch appears at larger {eta}{identical_to}L{sub n}/L{sub T} > or approx. 0.7, consistent with quasilinear theory. Our results suggest that entropy-mode transport will drive the LDX plasma profiles toward a state with {eta}{approx}0.7 and pressure gradients that are near marginal ideal MHD interchange-mode stability.
Joiner, N.; Hirose, A.
2008-08-15
The kinetic ballooning mode (KBM) has been shown in previous work to be unstable within the magnetohydrodynamic (MHD) region (in s-{alpha} space) of second stability [Hirose et al., Phys. Rev. Lett. 72, 3993 (2004)]. In this work we verify this result using the gyrokinetic code GS2 [Kotschenreuther et al., Comput. Phys. Commun. 88, 128 (1996)] treating both ions and electrons as kinetic species and retaining the magnetosonic perturbation B{sub parallel}. Growth rates calculated using GS2 differ significantly from the previous differential/shooting code analysis. Calculations without B{sub parallel} find the stability region is preserved, while the addition of B{sub parallel} causes the mode to be more unstable than previously calculated within the region of MHD second stability. The inclusion of parallel ion current and B{sub parallel} into the shooting code does not account for the GS2 results. The evidence presented in this paper leads us to the conclusion that the adiabatic electron approximation employed in previous studies is found to be unsuitable for this type of instability. Based on the findings of this work, the KBM becomes an interesting instability in the context of internal transport barriers, where {alpha} is often large and magnetic shear is small (positive or negative)
NASA Astrophysics Data System (ADS)
Hornsby, W. A.; Migliano, P.; Buchholz, R.; Grosshauser, S.; Weikl, A.; Zarzoso, D.; Casson, F. J.; Poli, E.; Peeters, A. G.
2016-01-01
The non-linear evolution of a magnetic island is studied using the Vlasov gyro-kinetic code GKW. The interaction of electromagnetic turbulence with a self-consistently growing magnetic island, generated by a tearing unstable {{Δ }\\prime}>0 current profile, is considered. The turbulence is able to seed the magnetic island and bypass the linear growth phase by generating structures that are approximately an ion gyro-radius in width. The non-linear evolution of the island width and its rotation frequency, after this seeding phase, is found to be modified and is dependent on the value of the plasma beta and equilibrium pressure gradients. At low values of beta the island evolves largely independent of the turbulence, while at higher values the interaction has a dramatic effect on island growth, causing the island to grow exponentially at the growth rate of its linear phase, even though the island is larger than linear theory validity. The turbulence forces the island to rotate in the ion-diamagnetic direction as opposed to the electron diamagnetic direction in which it rotates when no turbulence is present. In addition, it is found that the mode rotation slows as the island grows in size.
Three-dimensional gyrokinetic simulation of the relaxation of a magnetized temperature filament
Sydora, R. D.; Morales, G. J.; Maggs, J. E.; Van Compernolle, B.
2015-10-15
An electromagnetic, 3D gyrokinetic particle code is used to study the relaxation of a magnetized electron temperature filament embedded in a large, uniform plasma of lower temperature. The study provides insight into the role played by unstable drift-Alfvén waves observed in a basic electron heat transport experiment [D. C. Pace et al., Phys. Plasmas 15, 122304 (2008)] in which anomalous cross-field transport has been documented. The simulation exhibits the early growth of temperature-gradient-driven, drift-Alfvén fluctuations that closely match the eigenmodes predicted by linear theory. At the onset of saturation, the unstable fluctuations display a spiral spatial pattern, similar to that observed in the laboratory, which causes the rearrangement of the temperature profile. After saturation of the linear instability, the system exhibits a markedly different behavior depending on the inclusion in the computation of modes without variation along the magnetic field, i.e., k{sub z} = 0. In their absence, the initial filament evolves into a broadened temperature profile, self-consistent with undamped, finite amplitude drift-Alfvén waves. But the inclusion of k{sub z} = 0 modes causes the destruction of the filament and damping of the drift-Alfvén modes leading to a final state consisting of undamped convective cells and multiple, smaller-scale filaments.
A study of self organized criticality in ion temperature gradient mode driven gyrokinetic turbulence
Mavridis, M.; Isliker, H.; Vlahos, L.; Görler, T.; Jenko, F.; Told, D.
2014-10-15
An investigation on the characteristics of self organized criticality (Soc) in ITG mode driven turbulence is made, with the use of various statistical tools (histograms, power spectra, Hurst exponents estimated with the rescaled range analysis, and the structure function method). For this purpose, local non-linear gyrokinetic simulations of the cyclone base case scenario are performed with the GENE software package. Although most authors concentrate on global simulations, which seem to be a better choice for such an investigation, we use local simulations in an attempt to study the locally underlying mechanisms of Soc. We also study the structural properties of radially extended structures, with several tools (fractal dimension estimate, cluster analysis, and two dimensional autocorrelation function), in order to explore whether they can be characterized as avalanches. We find that, for large enough driving temperature gradients, the local simulations exhibit most of the features of Soc, with the exception of the probability distribution of observables, which show a tail, yet they are not of power-law form. The radial structures have the same radial extent at all temperature gradients examined; radial motion (transport) though appears only at large temperature gradients, in which case the radial structures can be interpreted as avalanches.
Comparisons between global and local gyrokinetic simulations of an ASDEX Upgrade H-mode plasma
NASA Astrophysics Data System (ADS)
Navarro, Alejandro Bañón; Told, Daniel; Jenko, Frank; Görler, Tobias; Happel, Tim
2016-04-01
We investigate by means of local and global nonlinear gyrokinetic GENE simulations an ASDEX Upgrade H-mode plasma. We find that for the outer core positions (i.e., ρ tor ≈ 0.5 - 0.7 ), nonlocal effects are important. For nominal input parameters local simulations over-predict the experimental heat fluxes by a large factor, while a good agreement is found with global simulations. This was a priori not expected, since the values of 1 / ρ ⋆ were large enough that global and local simulations should have been in accordance. Nevertheless, due to the high sensitivity of the heat fluxes with respect to the input parameters, it is still possible to match the heat fluxes in local simulations with the experimental and global results by varying the ion temperature gradient within the experimental uncertainties. In addition to that, once an agreement in the transport quantities between local (flux-matched) and global simulations is achieved, an agreement for other quantities, such as density and temperature fluctuations, is also found. The case presented here clearly shows that even in the presence of global size-effects, the local simulation approach is still a valid and accurate approach.
Spong, D. A.; Bass, E. M.; Deng, W.; Heidbrink, W. W.; Lin, Z.; Tobias, B.; Van Zeeland, M. A.; Austin, M. E.; Domier, C. W.; Luhmann, N. C. Jr.
2012-08-15
A verification and validation study is carried out for a sequence of reversed shear Alfven instability time slices. The mode frequency increases in time as the minimum (q{sub min}) in the safety factor profile decreases. Profiles and equilibria are based upon reconstructions of DIII-D discharge (no. 142111) in which many such frequency up-sweeping modes were observed. Calculations of the frequency and mode structure evolution from two gyrokinetic codes, GTC and GYRO, and a gyro-Landau fluid code TAEFL are compared. The experimental mode structure of the instability was measured using time-resolved two-dimensional electron cyclotron emission imaging. The three models reproduce the frequency upsweep event within {+-}10% of each other, and the average of the code predictions is within {+-}8% of the measurements; growth rates are predicted that are consistent with the observed spectral line widths. The mode structures qualitatively agree with respect to radial location and width, dominant poloidal mode number, ballooning structure, and the up-down asymmetry, with some remaining differences in the details. Such similarities and differences between the predictions of the different models and the experimental results are a valuable part of the verification/validation process and help to guide future development of the modeling efforts.
Gyrokinetic Transport Database and Comparisons to the TGLF Theory-Based Transport Model
NASA Astrophysics Data System (ADS)
Kinsey, J. E.; Staebler, G. M.; Waltz, R. E.; Candy, J.
2006-10-01
A database with over 300 nonlinear gyrokinetic simulations has been created using the GYRO code [1,2]. Using a parameterized equilibrium model for shaped geometry, simulations show that the GYRO normalized ITG/TEM diffusivities exhibit an inverse linear dependence on elongation at fixed midplane minor radius. Kinetic electron simulations show the ExB shear quench rule is robust in shifted circle geometry. With real geometry, the quench point varies systematically with elongation and aspect ratio. Using the results, a new version of the quench rule is found that captures the variation of the quench point with these two geometric quantities. Finally, we compare the results from the TGLF driftwave model [3] with the GYRO simulations. Using the TGLF eigenmodes, we compute quasilinear fluxes using a turbulence saturation model and assess the quality of the fit to the GYRO transport database. 4pt[1] J. Candy, R.E. Waltz, Phys. Rev. Lett. 91, 45001 (2003). [2] http://fusion.gat.com/comp/parallel [3] G.M. Staebler, J.E. Kinsey, R.E. Waltz, Phys. Plasmas 12, 102508 (2005).
Fully nonlinear δf gyrokinetics for scrape-off layer parallel transport
NASA Astrophysics Data System (ADS)
Pan, Q.; Told, D.; Jenko, F.
2016-10-01
Edge plasmas present a few challenges for gyrokinetic simulations that are absent in tokamak cores. Among them are large fluctuation amplitudes and plasma-wall interactions in the open field line region. In this paper, the widely used core turbulence code GENE, which employs a δf-splitting technique, is extended to simulate open systems with large electrostatic fluctuations. With inclusion and proper discretization of the parallel nonlinear term, it becomes equivalent to a full-f code and the δf-splitting causes no fundamental difficulty in handling large fluctuations. The loss of particles to the wall is accounted for by using a logical sheath boundary, which is implemented in the context of a finite-volume method. The extended GENE code is benchmarked for the well-established one-dimensional parallel transport problem in the scrape-off layer during edge-localized modes. The parallel heat flux deposited onto the divertor target is compared with previous simulation results and shows good agreement.
Bounce-Averaged Gyrokinetic Simulation of Current-Collection Feedback in a Laboratory Magnetosphere
NASA Astrophysics Data System (ADS)
Roberts, T. M.; Garnier, D.; Kesner, J.; Mauel, M. E.
2014-10-01
A self-consistent, nonlinear simulation of interchange dynamics including the bounce-averaged gyro-kinetics of trapped electrons was previously used to understand frequency sweeping and the turbulent cascades observed in dipole-confined plasmas. Through adjustment of the particle and heat sources this code reproduces dynamics that resemble the turbulence measured experimentally, both in spectral power-law trends and in the onset of a steepened density profile. Time stepping is performed in an explicit leap-frog manner and a flux-corrected transport algorithm is implemented. In this presentation, we discuss the physics and numerical methods of the simulations as well as plans for including the effects of a biasing electrode which can collect or inject electrons. By varying this source/sink of electrons at the electrode location based on the potential fluctuations occurring elsewhere, we study the effects of current-collection feedback to compare to recent experiments observed to regulate interchange turbulence. Supported by NSF-DOE Partnership for Plasma Science and DOE Grant DE-FG02-00ER54585 and NSF Award PHY-1201896.
Angioni, C.
2015-10-15
A gyrokinetic study based on numerical and analytical calculations is presented, which computes the dependence of the turbulent diffusion of highly charged impurities on the ratio of the electron to the ion heat flux of the plasma. Nonlinear simulations show that the size of the turbulent diffusion of heavy impurities can vary by one order of magnitude with fixed total heat flux and is an extremely sensitive function of the electron to ion heat flux ratio. Numerical linear calculations are found to reproduce the nonlinear results. Thereby, a quasi-linear analytical approach is used to explain the origin of this dependence.
NASA Astrophysics Data System (ADS)
Altukhov, A. B.; Gurchenko, A. D.; Gusakov, E. Z.; Esipov, L. A.; Irzak, M. A.; Kantor, M. Yu; Kouprienko, D. V.; Lashkul, S. I.; Leerink, S.; Niskala, P.; Stepanov, A. Yu; Teplova, N. V.
2016-11-01
The poloidal dependence of the drift-wave turbulence characteristics is investigated at the FT-2 tokamak by radial correlation Doppler reflectometry (RCDR) technique and using the full distribution function global gyrokinetic modelling by ELMFIRE code. The poloidal variation of the turbulence radial correlation length from 0.2-0.55 cm is demonstrated both by measurement and computation. The turbulence correlation length rapidly decreases from the top of the poloidal cross-section to the high field side and then steadily grows in the poloidal direction. A well-pronounced excess of the turbulence radial correlation length in deuterium over its value in hydrogen discharges is demonstrated.
Krommes, J. A.
2013-12-15
Some physical interpretations are given of the well-known second-order gyrokinetic Hamiltonian in the magnetohydrodynamic limit. Its relations to the conservation of the true (Galilean-invariant) magnetic moment and fluid nonlinearities are described. Subtleties about its derivation as a cold-ion limit are explained; it is important to take that limit in the frame moving with the E×B velocity. The discussion also provides some geometric understanding of certain well-known Lie generating functions, and it makes contact with general discussions of ponderomotive potentials and the thermodynamics of dielectric media.
Multi-channel transport experiments at Alcator C-Mod and comparison with gyrokinetic simulations
White, A. E.; Howard, N. T.; Greenwald, M.; Reinke, M. L.; Sung, C.; Baek, S.; Barnes, M.; Dominguez, A.; Ernst, D.; Gao, C.; Hubbard, A. E.; Hughes, J. W.; Lin, Y.; Parra, F.; Porkolab, M.; Rice, J. E.; Walk, J.; Wukitch, S. J.; Team, Alcator C-Mod; Candy, J.; and others
2013-05-15
Multi-channel transport experiments have been conducted in auxiliary heated (Ion Cyclotron Range of Frequencies) L-mode plasmas at Alcator C-Mod [Marmar and Alcator C-Mod Group, Fusion Sci. Technol. 51(3), 3261 (2007)]. These plasmas provide good diagnostic coverage for measurements of kinetic profiles, impurity transport, and turbulence (electron temperature and density fluctuations). In the experiments, a steady sawtoothing L-mode plasma with 1.2 MW of on-axis RF heating is established and density is scanned by 20%. Measured rotation profiles change from peaked to hollow in shape as density is increased, but electron density and impurity profiles remain peaked. Ion or electron heat fluxes from the two plasmas are the same. The experimental results are compared directly to nonlinear gyrokinetic theory using synthetic diagnostics and the code GYRO [Candy and Waltz, J. Comput. Phys. 186, 545 (2003)]. We find good agreement with experimental ion heat flux, impurity particle transport, and trends in the fluctuation level ratio (T(tilde sign){sub e}/T{sub e})/(ñ{sub e}/n{sub e}), but underprediction of electron heat flux. We find that changes in momentum transport (rotation profiles changing from peaked to hollow) do not correlate with changes in particle transport, and also do not correlate with changes in linear mode dominance, e.g., Ion Temperature Gradient versus Trapped Electron Mode. The new C-Mod results suggest that the drives for momentum transport differ from drives for heat and particle transport. The experimental results are inconsistent with present quasilinear models, and the strong sensitivity of core rotation to density remains unexplained.
Baumgaertel, J. A.; Redi, M. H.; Budny, R. V.; Rewoldt, G.; Dorland, W.
2005-10-19
Insight into plasma microturbulence and transport is being sought using linear simulations of drift waves on the National Spherical Torus Experiment (NSTX), following a study of drift wave modes on the Alcator C-Mod Tokamak. Microturbulence is likely generated by instabilities of drift waves, which cause transport of heat and particles. Understanding this transport is important because the containment of heat and particles is required for the achievement of practical nuclear fusion. Microtearing modes may cause high heat transport through high electron thermal conductivity. It is hoped that microtearing will be stable along with good electron transport in the proposed low collisionality International Thermonuclear Experimental Reactor (ITER). Stability of the microtearing mode is investigated for conditions at mid-radius in a high density NSTX high performance (H-mode) plasma, which is compared to the proposed ITER plasmas. The microtearing mode is driven by the electron temperature gradient, and believed to be mediated by ion collisions and magnetic shear. Calculations are based on input files produced by TRXPL following TRANSP (a time-dependent transport analysis code) analysis. The variability of unstable mode growth rates is examined as a function of ion and electron collisionalities using the parallel gyrokinetic computational code GS2. Results show the microtearing mode stability dependence for a range of plasma collisionalities. Computation verifies analytic predictions that higher collisionalities than in the NSTX experiment increase microtearing instability growth rates, but that the modes are stabilized at the highest values. There is a transition of the dominant mode in the collisionality scan to ion temperature gradient character at both high and low collisionalities. The calculations suggest that plasma electron thermal confinement may be greatly improved in the low-collisionality ITER.
Gyrokinetic study of the role of {beta} on electron particle transport in tokamaks
Hein, T.; Angioni, C.; Fable, E.; Candy, J.
2010-10-15
Electromagnetic effects on the radial transport of electrons in the core of tokamak plasmas are studied by means of linear and nonlinear gyrokinetic simulations with the code GYRO[J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] and by an analytical derivation. The impact of a finite {beta}, that is, a finite ratio of the plasma pressure to the magnetic pressure, is considered on the fluctuations of the magnetic field through Ampere's law, as well as on the geometrical modification of the vertical drift produced by the Shafranov shift in the magnetic equilibrium, which, for realistic descriptions, has to be included in both electrostatic and electromagnetic modeling. The condition of turbulent particle flux at the null, which allows the determination of stationary logarithmic density gradients when neoclassical transport and particle sources are negligible, is investigated for increasing values of {beta}, in regimes of ion temperature gradient and trapped electron mode turbulence. The loss of adiabaticity of passing electrons produced by fluctuations in the magnetic vector potential produces an outward convection. When the magnetic equilibrium geometry is kept fixed, this induces a strong reduction of the stationary logarithmic density gradient with increasing {beta}. This effect is partly compensated by the geometrical effect on the vertical drift. This compensation effect, however, is significantly weaker in nonlinear simulations as compared to quasilinear calculations. A detailed comparison between quasilinear and nonlinear results reveals that the predicted value of the logarithmic density gradient is highly sensitive on the assumptions on the wave number spectrum applied in the quasilinear model. The qualitative consistency of the theoretical predictions with the experimental results obtained so far on the dependence of density peaking on {beta} is discussed by considering the additional impact, with increasing {beta}, of a particle source delivered
Waltz, R. E.; Staebler, G. M.; Solomon, W. M.
2011-04-15
Residual stress refers to the remaining toroidal angular momentum (TAM) flux (divided by major radius) when the shear in the equilibrium fluid toroidal velocity (and the velocity itself) vanishes. Previously [Waltz et al., Phys. Plasmas 14, 122507 (2007); errata 16, 079902 (2009)], we demonstrated with GYRO [Candy and Waltz, J. Comp. Phys. 186, 545 (2003)] gyrokinetic simulations that TAM pinching from (ion pressure gradient supported or diamagnetic level) equilibrium ExB velocity shear could provide some of the residual stress needed to support spontaneous toroidal rotation against normal diffusive loss. Here we show that diamagnetic level shear in the intrinsic drift wave velocities (or ''profile shear'' in the ion and electron density and temperature gradients) provides a comparable residual stress. The individual signed contributions of these small (rho-star level) ExB and profile velocity shear rates to the turbulence level and (rho-star squared) ion energy transport stabilization are additive if the rates are of the same sign. However because of the additive stabilization effect, the contributions to the small (rho-star cubed) residual stress is not always simply additive. If the rates differ in sign, the residual stress from one can buck out that from the other (and in some cases reduce the stabilization.) The residual stress from these diamagnetic velocity shear rates is quantified by the ratio of TAM flow to ion energy (power) flow (M/P) in a global GYRO core simulation of a ''null'' toroidal rotation DIII-D [Mahdavi and Luxon, Fusion Sci. Technol. 48, 2 (2005)] discharge by matching M/P profiles within experimental uncertainty. Comparison of global GYRO (ion and electron energy as well as particle) transport flow balance simulations of TAM transport flow in a high-rotation DIII-D L-mode quantifies and isolates the ExB shear and parallel velocity (Coriolis force) pinching components from the larger ''diffusive'' parallel velocity shear driven component and
Guttenfelder, W.; Kaye, S. M.; Ren, Y.; Solomon, W.; Bell, R. E.; Candy, J.; Gerhardt, S. P.; LeBlanc, B. P.; Yuh, H.
2016-05-11
This paper presents quasi-linear gyrokinetic predictions of the Coriolis momentum pinch for low aspect-ratio National Spherical Torus Experiment (NSTX) H-modes where previous experimental measurements were focused. Local, linear calculations predict that in the region of interest (just outside the mid-radius) of these relatively high-beta plasmas, profiles are most unstable to microtearing modes that are only effective in transporting electron energy. However, sub-dominant electromagnetic and electrostaticballooning modes are also unstable, which are effective at transporting energy, particles, and momentum. The quasi-linear prediction of transport from these weaker ballooning modes, assuming they contribute transport in addition to that from microtearing modes inmore » a nonlinear turbulent state, leads to a very small or outward convection of momentum, inconsistent with the experimentally measured inward pinch, and opposite to predictions in conventional aspect ratio tokamaks. Additional predictions of a low beta L-mode plasma, unstable to more traditional electrostatic ion temperature gradient-trapped electron mode instability, show that the Coriolis pinch is inward but remains relatively weak and insensitive to many parameter variations. The weak or outward pinch predicted in NSTX plasmas appears to be at least partially correlated to changes in the parallel mode structure that occur at a finite beta and low aspect ratio, as discussed in previous theories. The only conditions identified where a stronger inward pinch is predicted occur either in the purely electrostatic limit or if the aspect ratio is increased. Lastly, as the Coriolis pinch cannot explain the measured momentum pinch, additional theoretical momentum transport mechanisms are discussed that may be potentially important.« less
SciDAC Center for Gyrokinetic Particle Simulation of Turbulent Transport in Burning Plasmas
Lin, Zhihong
2013-12-18
During the first year of the SciDAC gyrokinetic particle simulation (GPS) project, the GPS team (Zhihong Lin, Liu Chen, Yasutaro Nishimura, and Igor Holod) at the University of California, Irvine (UCI) studied the tokamak electron transport driven by electron temperature gradient (ETG) turbulence, and by trapped electron mode (TEM) turbulence and ion temperature gradient (ITG) turbulence with kinetic electron effects, extended our studies of ITG turbulence spreading to core-edge coupling. We have developed and optimized an elliptic solver using finite element method (FEM), which enables the implementation of advanced kinetic electron models (split-weight scheme and hybrid model) in the SciDAC GPS production code GTC. The GTC code has been ported and optimized on both scalar and vector parallel computer architectures, and is being transformed into objected-oriented style to facilitate collaborative code development. During this period, the UCI team members presented 11 invited talks at major national and international conferences, published 22 papers in peer-reviewed journals and 10 papers in conference proceedings. The UCI hosted the annual SciDAC Workshop on Plasma Turbulence sponsored by the GPS Center, 2005-2007. The workshop was attended by about fifties US and foreign researchers and financially sponsored several gradual students from MIT, Princeton University, Germany, Switzerland, and Finland. A new SciDAC postdoc, Igor Holod, has arrived at UCI to initiate global particle simulation of magnetohydrodynamics turbulence driven by energetic particle modes. The PI, Z. Lin, has been promoted to the Associate Professor with tenure at UCI.
Mikkelsen, D. R. Bitter, M.; Delgado-Aparicio, L.; Hill, K. W.; Greenwald, M.; Howard, N. T.; Hughes, J. W.; Rice, J. E.; Reinke, M. L.; Podpaly, Y.; Ma, Y.; Candy, J.; Waltz, R. E.
2015-06-15
Peaked density profiles in low-collisionality AUG and JET H-mode plasmas are probably caused by a turbulently driven particle pinch, and Alcator C-Mod experiments confirmed that collisionality is a critical parameter. Density peaking in reactors could produce a number of important effects, some beneficial, such as enhanced fusion power and transport of fuel ions from the edge to the core, while others are undesirable, such as lower beta limits, reduced radiation from the plasma edge, and consequently higher divertor heat loads. Fundamental understanding of the pinch will enable planning to optimize these impacts. We show that density peaking is predicted by nonlinear gyrokinetic turbulence simulations based on measured profile data from low collisionality H-mode plasma in Alcator C-Mod. Multiple ion species are included to determine whether hydrogenic density peaking has an isotope dependence or is influenced by typical levels of low-Z impurities, and whether impurity density peaking depends on the species. We find that the deuterium density profile is slightly more peaked than that of hydrogen, and that experimentally relevant levels of boron have no appreciable effect on hydrogenic density peaking. The ratio of density at r/a = 0.44 to that at r/a = 0.74 is 1.2 for the majority D and minority H ions (and for electrons), and increases with impurity Z: 1.1 for helium, 1.15 for boron, 1.3 for neon, 1.4 for argon, and 1.5 for molybdenum. The ion temperature profile is varied to match better the predicted heat flux with the experimental transport analysis, but the resulting factor of two change in heat transport has only a weak effect on the predicted density peaking.
NASA Astrophysics Data System (ADS)
Guttenfelder, W.; Kaye, S. M.; Ren, Y.; Solomon, W.; Bell, R. E.; Candy, J.; Gerhardt, S. P.; LeBlanc, B. P.; Yuh, H.
2016-05-01
This paper presents quasi-linear gyrokinetic predictions of the Coriolis momentum pinch for low aspect-ratio National Spherical Torus Experiment (NSTX) H-modes where previous experimental measurements were focused. Local, linear calculations predict that in the region of interest (just outside the mid-radius) of these relatively high-beta plasmas, profiles are most unstable to microtearing modes that are only effective in transporting electron energy. However, sub-dominant electromagnetic and electrostatic ballooning modes are also unstable, which are effective at transporting energy, particles, and momentum. The quasi-linear prediction of transport from these weaker ballooning modes, assuming they contribute transport in addition to that from microtearing modes in a nonlinear turbulent state, leads to a very small or outward convection of momentum, inconsistent with the experimentally measured inward pinch, and opposite to predictions in conventional aspect ratio tokamaks. Additional predictions of a low beta L-mode plasma, unstable to more traditional electrostatic ion temperature gradient-trapped electron mode instability, show that the Coriolis pinch is inward but remains relatively weak and insensitive to many parameter variations. The weak or outward pinch predicted in NSTX plasmas appears to be at least partially correlated to changes in the parallel mode structure that occur at a finite beta and low aspect ratio, as discussed in previous theories. The only conditions identified where a stronger inward pinch is predicted occur either in the purely electrostatic limit or if the aspect ratio is increased. As the Coriolis pinch cannot explain the measured momentum pinch, additional theoretical momentum transport mechanisms are discussed that may be potentially important.
Multi-channel transport experiments at Alcator C-Mod and comparison with gyrokinetic simulationsa)
NASA Astrophysics Data System (ADS)
White, A. E.; Howard, N. T.; Greenwald, M.; Reinke, M. L.; Sung, C.; Baek, S.; Barnes, M.; Candy, J.; Dominguez, A.; Ernst, D.; Gao, C.; Hubbard, A. E.; Hughes, J. W.; Lin, Y.; Mikkelsen, D.; Parra, F.; Porkolab, M.; Rice, J. E.; Walk, J.; Wukitch, S. J.; Team, Alcator C-Mod
2013-05-01
Multi-channel transport experiments have been conducted in auxiliary heated (Ion Cyclotron Range of Frequencies) L-mode plasmas at Alcator C-Mod [Marmar and Alcator C-Mod Group, Fusion Sci. Technol. 51(3), 3261 (2007)]. These plasmas provide good diagnostic coverage for measurements of kinetic profiles, impurity transport, and turbulence (electron temperature and density fluctuations). In the experiments, a steady sawtoothing L-mode plasma with 1.2 MW of on-axis RF heating is established and density is scanned by 20%. Measured rotation profiles change from peaked to hollow in shape as density is increased, but electron density and impurity profiles remain peaked. Ion or electron heat fluxes from the two plasmas are the same. The experimental results are compared directly to nonlinear gyrokinetic theory using synthetic diagnostics and the code GYRO [Candy and Waltz, J. Comput. Phys. 186, 545 (2003)]. We find good agreement with experimental ion heat flux, impurity particle transport, and trends in the fluctuation level ratio (T˜e/Te)/(n ˜e/ne), but underprediction of electron heat flux. We find that changes in momentum transport (rotation profiles changing from peaked to hollow) do not correlate with changes in particle transport, and also do not correlate with changes in linear mode dominance, e.g., Ion Temperature Gradient versus Trapped Electron Mode. The new C-Mod results suggest that the drives for momentum transport differ from drives for heat and particle transport. The experimental results are inconsistent with present quasilinear models, and the strong sensitivity of core rotation to density remains unexplained.
NASA Astrophysics Data System (ADS)
Mikkelsen, D. R.; Bitter, M.; Delgado-Aparicio, L.; Hill, K. W.; Greenwald, M.; Howard, N. T.; Hughes, J. W.; Rice, J. E.; Reinke, M. L.; Podpaly, Y.; Ma, Y.; Candy, J.; Waltz, R. E.
2015-06-01
Peaked density profiles in low-collisionality AUG and JET H-mode plasmas are probably caused by a turbulently driven particle pinch, and Alcator C-Mod experiments confirmed that collisionality is a critical parameter. Density peaking in reactors could produce a number of important effects, some beneficial, such as enhanced fusion power and transport of fuel ions from the edge to the core, while others are undesirable, such as lower beta limits, reduced radiation from the plasma edge, and consequently higher divertor heat loads. Fundamental understanding of the pinch will enable planning to optimize these impacts. We show that density peaking is predicted by nonlinear gyrokinetic turbulence simulations based on measured profile data from low collisionality H-mode plasma in Alcator C-Mod. Multiple ion species are included to determine whether hydrogenic density peaking has an isotope dependence or is influenced by typical levels of low-Z impurities, and whether impurity density peaking depends on the species. We find that the deuterium density profile is slightly more peaked than that of hydrogen, and that experimentally relevant levels of boron have no appreciable effect on hydrogenic density peaking. The ratio of density at r/a = 0.44 to that at r/a = 0.74 is 1.2 for the majority D and minority H ions (and for electrons), and increases with impurity Z: 1.1 for helium, 1.15 for boron, 1.3 for neon, 1.4 for argon, and 1.5 for molybdenum. The ion temperature profile is varied to match better the predicted heat flux with the experimental transport analysis, but the resulting factor of two change in heat transport has only a weak effect on the predicted density peaking.
Tangri, Varun; Terry, P. W.; Waltz, R. E.
2011-05-15
A simple large-aspect-ratio (R{sub 0}/r) circular equilibrium model is developed for low-beta reversed field pinch (RFP) geometry. The model is suitable for treating small scale instability and turbulent transport driven by ion temperature gradient (ITG) and related electron drift modes in gyrokinetic simulations. The equilibrium model is an RFP generalization of the common tokamak s-{alpha} model to small safety factor (q), where the poloidal field dominates the toroidal field. The model accommodates the RFP toroidal field reversal (where q vanishes) by generalizing the cylindrical force-free Bessel function model (BFM) [J. B. Taylor, Phys. Rev. Lett. 33, 1139 (1974)] to toroidal geometry. The global equilibrium can be described in terms of the RFP field reversal and pinch parameters [F,{Theta}]. This new toroidal Bessel function model (TBFM) has been incorporated into the gyrokinetic code GYRO [J. Candy and R. E. Waltz, J.Comput. Phys. 186, 545 (2003)] and used here to explore local electrostatic ITG adiabatic electron instability rates for typical low-q RFP parameters.
Coury, M.; Guttenfelder, W.; Mikkelsen, D. R.; Canik, J. M.; Canal, G. P.; Diallo, A.; Kaye, S.; Kramer, G. J.; Maingi, R.
2016-06-30
Linear (local) gyrokinetic predictions of edge microinstabilities in highly shaped, lithiated and non-lithiated NSTX discharges are reported using the gyrokinetic code GS2. Microtearing modes dominate the non-lithiated pedestal top. The stabilization of these modes at the lithiated pedestal top enables the electron temperature pedestal to extend further inwards, as observed experimentally. Kinetic ballooning modes are found to be unstable mainly at the mid-pedestal of both types of discharges, with un- stable trapped electron modes nearer the separatrix region. At electron wavelengths, ETG modes are found to be unstable from mid-pedestal outwards for ηe, exp ~2.2 with higher growth rates formore » the lithiated discharge. Near the separatrix, the critical temperature gradient for driving ETG modes is reduced in the presence of lithium, re ecting the reduction of the lithiated density gradients observed experimentally. A preliminary linear study in the edge of non-lithiated discharges shows that the equilibrium shaping alters the electrostatic modes stability, found more unstable at high plasma shaping.« less
NASA Astrophysics Data System (ADS)
Tangri, Varun; Terry, P. W.; Waltz, R. E.
2011-05-01
A simple large-aspect-ratio (R0/r) circular equilibrium model is developed for low-beta reversed field pinch (RFP) geometry. The model is suitable for treating small scale instability and turbulent transport driven by ion temperature gradient (ITG) and related electron drift modes in gyrokinetic simulations. The equilibrium model is an RFP generalization of the common tokamak s-α model to small safety factor (q), where the poloidal field dominates the toroidal field. The model accommodates the RFP toroidal field reversal (where q vanishes) by generalizing the cylindrical force-free Bessel function model (BFM) [J. B. Taylor, Phys. Rev. Lett. 33, 1139 (1974)] to toroidal geometry. The global equilibrium can be described in terms of the RFP field reversal and pinch parameters [F ,Θ]. This new toroidal Bessel function model (TBFM) has been incorporated into the gyrokinetic code GYRO [J. Candy and R. E. Waltz, J.Comput. Phys. 186, 545 (2003)] and used here to explore local electrostatic ITG adiabatic electron instability rates for typical low-q RFP parameters.
Fahey, Mark R.; Candy, Jeff
2013-11-07
This project initiated the development of TGYRO ? a steady-state Gyrokinetic transport code (SSGKT) that integrates micro-scale GYRO turbulence simulations into a framework for practical multi-scale simulation of conventional tokamaks as well as future reactors. Using a lightweight master transport code, multiple independent (each massively parallel) gyrokinetic simulations are coordinated. The capability to evolve profiles using the TGLF model was also added to TGYRO and represents a more typical use-case for TGYRO. The goal of the project was to develop a steady-state Gyrokinetic transport code (SSGKT) that integrates micro-scale gyrokinetic turbulence simulations into a framework for practical multi-scale simulation of a burning plasma core ? the International Thermonuclear Experimental Reactor (ITER) in particular. This multi-scale simulation capability will be used to predict the performance (the fusion energy gain, Q) given the H-mode pedestal temperature and density. At present, projections of this type rely on transport models like GLF23, which are based on rather approximate fits to the results of linear and nonlinear simulations. Our goal is to make these performance projections with precise nonlinear gyrokinetic simulations. The method of approach is to use a lightweight master transport code to coordinate multiple independent (each massively parallel) gyrokinetic simulations using the GYRO code. This project targets the practical multi-scale simulation of a reactor core plasma in order to predict the core temperature and density profiles given the H-mode pedestal temperature and density. A master transport code will provide feedback to O(16) independent gyrokinetic simulations (each massively parallel). A successful feedback scheme offers a novel approach to predictive modeling of an important national and international problem. Success in this area of fusion simulations will allow US scientists to direct the research path of ITER over the next two
Young, C.W.
1997-10-01
In 1967, Sandia National Laboratories published empirical equations to predict penetration into natural earth materials and concrete. Since that time there have been several small changes to the basic equations, and several more additions to the overall technique for predicting penetration into soil, rock, concrete, ice, and frozen soil. The most recent update to the equations was published in 1988, and since that time there have been changes in the equations to better match the expanding data base, especially in concrete penetration. This is a standalone report documenting the latest version of the Young/Sandia penetration equations and related analytical techniques to predict penetration into natural earth materials and concrete. 11 refs., 6 tabs.
NASA Astrophysics Data System (ADS)
Schekochihin, A. A.; Highcock, E. G.; Cowley, S. C.
2012-05-01
Differential rotation is known to suppress linear instabilities in fusion plasmas. However, numerical experiments show that even in the absence of growing eigenmodes, subcritical fluctuations that grow transiently can lead to sustained turbulence, limiting the ability of the velocity shear to suppress anomalous transport. Here transient growth of electrostatic fluctuations driven by the parallel velocity gradient (PVG) and the ion temperature gradient (ITG) in the presence of a perpendicular (E × B) velocity shear is considered. The maximally simplified (but, as numerical simulations suggest, most promising for transport reduction) case of zero magnetic shear is treated in the framework of a local shearing box approximation. In this case there are no linearly growing eigenmodes, so all excitations are transient. In the PVG-dominated regime, the maximum amplification factor is found to be eN with N ∝ q/ɛ (safety factor/inverse aspect ratio), the maximally amplified wavenumbers perpendicular and parallel to the magnetic field are related by kyρi ≈ (ɛ/q)1/3k∥vthi/S, where ρi is the ion Larmor radius, vthi the ion thermal speed and S the E × B shear. In the ITG-dominated regime, N is independent of wavenumber and N ∝ vthi/(LTS), where LT is the ion-temperature scale length. Intermediate ITG-PVG regimes are also analysed and N is calculated as a function of q/ɛ, LT and S. Analytical results are corroborated and supplemented by linear gyrokinetic numerical tests. Regimes with N ≲ 1 for all wavenumbers are possible for sufficiently low values of q/ɛ (≲7 in our model); ion-scale turbulence is expected to be fully suppressed in such regimes. For cases when it is not suppressed, an elementary heuristic theory of subcritical PVG turbulence leading to a scaling of the associated ion heat flux with q, ɛ, S and LT is proposed; it is argued that the transport is much less ‘stiff’ than in the ITG regime.
NASA Astrophysics Data System (ADS)
Waltz, R. E.; Staebler, G. M.; Solomon, W. M.
2011-04-01
Residual stress refers to the remaining toroidal angular momentum (TAM) flux (divided by major radius) when the shear in the equilibrium fluid toroidal velocity (and the velocity itself) vanishes. Previously [Waltz et al., Phys. Plasmas 14, 122507 (2007); errata 16, 079902 (2009)], we demonstrated with GYRO [Candy and Waltz, J. Comp. Phys. 186, 545 (2003)] gyrokinetic simulations that TAM pinching from (ion pressure gradient supported or diamagnetic level) equilibrium E ×B velocity shear could provide some of the residual stress needed to support spontaneous toroidal rotation against normal diffusive loss. Here we show that diamagnetic level shear in the intrinsic drift wave velocities (or "profile shear" in the ion and electron density and temperature gradients) provides a comparable residual stress. The individual signed contributions of these small (rho-star level) E ×B and profile velocity shear rates to the turbulence level and (rho-star squared) ion energy transport stabilization are additive if the rates are of the same sign. However because of the additive stabilization effect, the contributions to the small (rho-star cubed) residual stress is not always simply additive. If the rates differ in sign, the residual stress from one can buck out that from the other (and in some cases reduce the stabilization.) The residual stress from these diamagnetic velocity shear rates is quantified by the ratio of TAM flow to ion energy (power) flow (M/P) in a global GYRO core simulation of a "null" toroidal rotation DIII-D [Mahdavi and Luxon, Fusion Sci. Technol. 48, 2 (2005)] discharge by matching M/P profiles within experimental uncertainty. Comparison of global GYRO (ion and electron energy as well as particle) transport flow balance simulations of TAM transport flow in a high-rotation DIII-D L-mode quantifies and isolates the E ×B shear and parallel velocity (Coriolis force) pinching components from the larger "diffusive" parallel velocity shear driven component and
NASA Astrophysics Data System (ADS)
Bravenec, Ronald; Citrin, Jonathan; Mantica, Paola; Garcia, Jeronimo; Pueschel, M. J.; Goerler, Tobias; Barnes, Michael; Candy, Jeff; Belli, Emily; Staebler, Gary; JET contributors Team
2015-11-01
Comparing results (linear frequencies, eigenfunctions, and nonlinear fluxes) from different gyrokinetic codes as a means of verification (benchmarking) is only convincing if the codes agree over a wide range of plasma conditions. Otherwise, agreement may simply be fortuitous. We present here linear and nonlinear comparisons of the Eulerian codes GENE, GS2, and GYRO for a variety of JET discharges. The discharges include a simplified, 2-species, circular geometry case based on an actual JET discharge, an L-mode discharge with a significant fast ion pressure fraction, and a carbon-wall low triangularity hybrid discharge. All discharges were studied at rho=0.33 where significant ion temperature peaking is observed. The benchmarking is carried out to verify the GENE predictions that fast-ion-enhanced electromagnetic stabilization is the main contributor to the low ion heat flux. Supported by U.S. Dept. of Energy through grant DE-FG02-08ER54978 and EUROfusion No. 633053.
Waltz, R. E.; Candy, J.; Petty, C. C.
2006-07-15
Global gyrokinetic simulations of DIII-D [M. A. Mahdavi and J. L. Luxon, in 'DIII-D Tokamak Special Issue', Fusion Sci. Technol. 48, 2 (2005)] L- and H-mode dimensionally similar discharge pairs are treated in detail. The simulations confirm the Bohm scaling of the well-matched L-mode pair. The paradoxical but experimentally apparent gyro-Bohm scaling of the H-mode pair at larger relative gyroradius (rho-star) and lower transport levels is due to poor profile similarity. Simulations of projected experimental plasma profiles with perfect similarity show both the L- and H-mode pairs to have Bohm scaling. A {rho}{sub *} stabilization rule for predicting the breakdown of gyro-Bohm scaling from simulations of a single discharge is presented.
NASA Astrophysics Data System (ADS)
Ku, S.; Chang, C.-S.; Adams, M.; Cummings, J.; Hinton, F.; Keyes, D.; Klasky, S.; Lee, W.; Lin, Z.; Parker, S.; CPES Team
2006-09-01
A gyrokinetic neoclassical solution for a diverted tokamak edge plasma has been obtained for the first time using the massively parallel Jaguar XT3 computer at Oak Ridge National Laboratory. The solutions show similar characteristics to the experimental observations: electric potential is positive in the scrape-off layer and negative in the H-mode layer, and the parallel rotation is positive in the scrape-off layer and at the inside boundary of the H-mode layer. However, the solution also makes a new physical discovery that there is a strong ExB convective flow in the scrape-off plasma. A general introduction to the edge simulation problem is also presented.
Adams, Mark; Chang, C. S.; Cummings, J.; Hinton, F.; Keyes, David E; Klasky, Scott A; Ku, S.; Lee, W. W.; Lin, Z.; Parker, Scott; CPES Team, the
2006-01-01
A gyrokinetic neoclassical solution for a diverted tokamak edge plasma has been obtained for the first time using the massively parallel Jaguar XT3 computer at Oak Ridge National Laboratory. The solutions show similar characteristics to the experimental observations: electric potential is positive in the scrape-off layer and negative in the H-mode layer, and the parallel rotation is positive in the scrape-off layer and at the inside boundary of the H-mode layer. However, the solution also makes a new physical discovery that there is a strong ExB convective flow in the scrape-off plasma. A general introduction to the edge simulation problem is also presented.
Xu, Yingfeng Dai, Zongliang; Wang, Shaojie
2014-04-15
The nonlinear gyrokinetic theory in the tokamak configuration based on the two-step transform is developed; in the first step, we transform the magnetic potential perturbation to the Hamiltonian part, and in the second step, we transform away the gyroangle-dependent part of the perturbed Hamiltonian. Then the I-transform method is used to decoupled the perturbation part of the motion from the unperturbed motion. The application of the I-transform method to the computation of the guiding-center orbit and the guiding-center distribution function in tokamaks is presented. It is demonstrated that the I-transform method of the orbit computation which involves integrating only along the unperturbed orbit agrees with the conventional method which integrates along the full orbit. A numerical code based on the I-transform method is developed and two numerical examples are given to verify the new method.
NASA Astrophysics Data System (ADS)
Kim, Kyuho; Chang, C. S.; Ku, Seunghoe; Hager, Robert
2015-11-01
The edge gyrokinetic code XGC1 has been used to study impurity transport from combined neoclassical and turbulent mechanisms in and across a steep H-mode pedestal, in realistic magnetic separatrix geometry. Both low-Z and high-Z impurity transport are studied.. The effect on the turbulence and transport is found to be different whether the impurity radial profile gradient is in the same or opposite direction to the main ion profile gradient. Co-existence of the low- and high-Z impurities also makes difference in the transport of each species. Edge impurity behavior in NSTX, JET, and DIII-D tokamak plasma will be discussed. Work funded by National Research Foundation of Korea and US DOE. Computing time was supported by NERSC.
White, A. E.; Schmitz, L.; Peebles, W. A.; Carter, T. A.; Rhodes, T. L.; Doyle, E. J.; Gourdain, P. A.; Hillesheim, J. C.; Wang, G.; Holland, C.; Tynan, G. R.; Austin, M. E.; McKee, G. R.; Shafer, M. W.; Burrell, K. H.; Candy, J.; DeBoo, J. C.; Prater, R.; Staebler, G. M.; Waltz, R. E.
2008-10-15
A correlation electron cyclotron emission (CECE) diagnostic has been used to measure local, turbulent fluctuations of the electron temperature in the core of DIII-D plasmas. This paper describes the hardware and testing of the CECE diagnostic and highlights the importance of measurements of multifield fluctuation profiles for the testing and validation of nonlinear gyrokinetic codes. The process of testing and validating such codes is critical for extrapolation to next-step fusion devices. For the first time, the radial profiles of electron temperature and density fluctuations are compared to nonlinear gyrokinetic simulations. The CECE diagnostic at DIII-D uses correlation radiometry to measure the rms amplitude and spectrum of the electron temperature fluctuations. Gaussian optics are used to produce a poloidal spot size with w{sub o}{approx}1.75 cm in the plasma. The intermediate frequency filters and the natural linewidth of the EC emission determine the radial resolution of the CECE diagnostic, which can be less than 1 cm. Wavenumbers resolved by the CECE diagnostic are k{sub {theta}}{<=}1.8 cm{sup -1} and k{sub r}{<=}4 cm{sup -1}, relevant for studies of long-wavelength turbulence associated with the trapped electron mode and the ion temperature gradient mode. In neutral beam heated L-mode plasmas, core electron temperature fluctuations in the region 0.5
White, A E; Schmitz, L; Peebles, W A; Carter, T A; Rhodes, T L; Doyle, E J; Gourdain, P A; Hillesheim, J C; Wang, G; Holland, C; Tynan, G R; Austin, M E; McKee, G R; Shafer, M W; Burrell, K H; Candy, J; DeBoo, J C; Prater, R; Staebler, G M; Waltz, R E; Makowski, M A
2008-10-01
A correlation electron cyclotron emission (CECE) diagnostic has been used to measure local, turbulent fluctuations of the electron temperature in the core of DIII-D plasmas. This paper describes the hardware and testing of the CECE diagnostic and highlights the importance of measurements of multifield fluctuation profiles for the testing and validation of nonlinear gyrokinetic codes. The process of testing and validating such codes is critical for extrapolation to next-step fusion devices. For the first time, the radial profiles of electron temperature and density fluctuations are compared to nonlinear gyrokinetic simulations. The CECE diagnostic at DIII-D uses correlation radiometry to measure the rms amplitude and spectrum of the electron temperature fluctuations. Gaussian optics are used to produce a poloidal spot size with w(o) approximately 1.75 cm in the plasma. The intermediate frequency filters and the natural linewidth of the EC emission determine the radial resolution of the CECE diagnostic, which can be less than 1 cm. Wavenumbers resolved by the CECE diagnostic are k(theta) < or = 1.8 cm(-1) and k(r) < or = 4 cm(-1), relevant for studies of long-wavelength turbulence associated with the trapped electron mode and the ion temperature gradient mode. In neutral beam heated L-mode plasmas, core electron temperature fluctuations in the region 0.5 < r/a < 0.9, increase with radius from approximately 0.5% to approximately 2%, similar to density fluctuations that are measured simultaneously with beam emission spectroscopy. After incorporating "synthetic diagnostics" to effectively filter the code output, the simulations reproduce the characteristics of the turbulence and transport at one radial location r/a = 0.5, but not at a second location, r/a = 0.75. These results illustrate that measurements of the profiles of multiple fluctuating fields can provide a significant constraint on the turbulence models employed by the code.
NASA Astrophysics Data System (ADS)
Burckel, A.; Sauter, O.; Angioni, C.; Candy, J.; Fable, E.; Lapillonne, X.
2010-11-01
In order to better identify the role of the magnetic topology on ITG and TEM instabilities, different MHD equilibria with increasing complexity are calculated using the CHEASE code [1]. We start from the geometry of the s-α cyclone benchmark case [2], consider the corresponding circular numerical equilibrium, and then successively add a non zero value of a consistent with the kinetic profiles, an elongation of 1.68, a triangularity of 0.15, and finally an up-down asymmetry corresponding to a single-null diverted geometry. This gives the opportunity to study separately the effect of each main characteristics of the equilibrium on microinstabilities in core plasmas. Linear local electrostatic gyrokinetic simulations of these different numerical equilibria and of their corresponding analytical descriptions (Miller-type representations [3]) are performed using the codes GS2 [4, 5] and GYRO[6]. It is observed that each modification of the equilibrium has an influence on the results of gyrokinetic simulations. The effect of the α parameter can compensate the stabilizing effect of an increase in the elongation. A comparison between the up-down symmetric shaped equilibrium and its corresponding diverted configuration show a non negligible effect on the growth rate of ITG and TEM turbulence. The comparison between the local Miller model and using a full equilibrium shows that it is mainly the indirect change of elongation in the plasma core which influences the results. The global aim is to provide well defined benchmark cases including real geometry and kinetic electrons physics, since this is not analyzed by the cyclone case. In addition, the goal is to define a procedure for testing of local simulations inspired by experimental constraints and results.
Bass, E. M.; Waltz, R. E.
2013-01-15
The unstable spectrum of Alfven eigenmodes (AEs) driven by neutral beam-sourced energetic particles (EPs) in a benchmark DIII-D discharge (142111) is calculated in a fully gyrokinetic model using the GYRO code's massively parallel linear eigenvalue solver. One cycle of the slow (equilibrium scale) frequency sweep of the reverse shear Alfven eigenmode (RSAE) at toroidal mode number n=3 is mapped. The RSAE second harmonic and an unstable beta-induced Alfven eigenmode (BAE) are simultaneously tracked alongside the primary RSAE. An observed twist in the eigenmode pattern, caused mostly by shear in the driving EP profile, is shown through artificially varying the E Multiplication-Sign B rotational velocity shear to depend generally on shear in the local wave phase velocity. Coupling to the BAE and to the toroidal Alfven eigenmode limit the RSAE frequency sweeps at the lower and upper end, respectively. While the present fully gyrokinetic model (including thermal ions and electrons) constitutes the best treatment of compressibility physics available, the BAE frequency is overpredicted by about 20% against experiment here and is found to be sensitive to energetic beam ion pressure. The RSAE frequency is more accurately matched except when it is limited by the BAE. Simulations suggest that the experiment is very close to marginal AE stability at points of RSAE-BAE coupling. A recipe for comparing the radial profile of quasilinear transport flux from local modes to that from global modes paves the way for the development of a stiff (critical gradient) local AE transport model based on local mode stability thresholds.
Brizard, Alain J
2009-12-31
Final Technical Report for U.S. Department of Energy Grant No. DE-FG02-09ER55005 Nonlinear FLR Effects in Reduced Fluid Models Alain J. Brizard, Saint Michael's College The above-mentioned DoE grant was used to support research activities by the PI during a sabbatical leave from Saint Michael's College in 2009. The major focus of the work was the role played by guiding-center and gyrocenter (linear and nonlinear) polarization and magnetization effects in understanding transport processes in turbulent magnetized plasmas. The theoretical tools used for this work include Lie-transform perturbation methods and Lagrangian (variational) methods developed by the PI in previous work. The present final technical report lists (I) the peer-reviewed publications that were written based on work funded by the Grant; (II) invited and contributed conference presentations during the period funded by the Grant; and (III) seminars presented during the period funded by the Grant. I. Peer-reviewed Publications A.J. Brizard and N. Tronko, 2011, Exact momentum conservation for the gyrokinetic Vlasov- Poisson equations, Physics of Plasmas 18 , 082307:1-14 [http://dx.doi.org/10.1063/1.3625554 ]. J. Decker, Y. Peysson, A.J. Brizard, and F.-X. Duthoit, 2010, Orbit-averaged guiding-center Fokker-Planck operator for numerical applications, Physics of Plasmas 17, 112513:1-12 [http://dx.doi.org/10.1063/1.3519514]. A.J. Brizard, 2010, Noether derivation of exact conservation laws for dissipationless reduced fluid models, Physics of Plasmas 17, 112503:1-8 [http://dx.doi.org/10.1063/1.3515303]. F.-X. Duthoit, A.J. Brizard, Y. Peysson, and J. Decker, 2010, Perturbation analysis of trapped particle dynamics in axisymmetric dipole geometry, Physics of Plasmas 17, 102903:1-9 [http://dx.doi.org/10.1063/1.3486554]. A.J. Brizard, 2010, Exact energy conservation laws for full and truncated nonlinear gyrokinetic equations, Physics of Plasmas 17, 042303:1-11 [http://dx.doi.org/10.1063/1.3374428]. A
Center for Gyrokinetic Particle Simulations of Turbulent Transport in Burning Plasmas
Scott, Parker
2011-05-02
This is the Final Technical Report for University of Colorado's portion of the SciDAC project 'Center for Gyrokinetic Particle Simulation of Turbulent Transport.' This is funded as a multi-institutional SciDAC Center and W.W. Lee at the Princeton Plasma Physics Laboratory is the lead Principal Investigator. Scott Parker is the local Principal Investigator for University of Colorado and Yang Chen is a Co-Principal Investigator. This is Cooperative Agreement DE-FC02-05ER54816. Research personnel include Yang Chen (Senior Research Associate), Jianying Lang (Graduate Research Associate, Ph.D. Physics Student) and Scott Parker (Associate Professor). Research includes core microturbulence studies of NSTX, simulation of trapped electron modes, development of efficient particle-continuum hybrid methods and particle convergence studies of electron temperature gradient driven turbulence simulations. Recently, the particle-continuum method has been extended to five-dimensions in GEM. We find that actually a simple method works quite well for the Cyclone base case with either fully kinetic or adiabatic electrons. Particles are deposited on a 5D phase-space grid using nearest-grid-point interpolation. Then, the value of delta-f is reset, but not the particle's trajectory. This has the effect of occasionally averaging delta-f of nearby (in the phase space) particles. We are currently trying to estimate the dissipation (or effective collision operator). We have been using GEM to study turbulence and transport in NSTX with realistic equilibrium density and temperature profiles, including impurities, magnetic geometry and ExB shear flow. Greg Rewoldt, PPPL, has developed a TRANSP interface for GEM that specifies the equilibrium profiles and parameters needed to run realistic NSTX cases. Results were reported at the American Physical Society - Division of Plasma Physics, and we are currently running convergence studies to ensure physical results. We are also studying the effect of
NASA Astrophysics Data System (ADS)
Viljamaa, Panu; Jacobs, J. Richard; Chris; JamesHyman; Halma, Matthew; EricNolan; Coxon, Paul
2014-07-01
In reply to a Physics World infographic (part of which is given above) about a study showing that Euler's equation was deemed most beautiful by a group of mathematicians who had been hooked up to a functional magnetic-resonance image (fMRI) machine while viewing mathematical expressions (14 May, http://ow.ly/xHUFi).
DOE R&D Accomplishments Database
1998-09-21
In the late 1950s to early 1960s Rudolph A. Marcus developed a theory for treating the rates of outer-sphere electron-transfer reactions. Outer-sphere reactions are reactions in which an electron is transferred from a donor to an acceptor without any chemical bonds being made or broken. (Electron-transfer reactions in which bonds are made or broken are referred to as inner-sphere reactions.) Marcus derived several very useful expressions, one of which has come to be known as the Marcus cross-relation or, more simply, as the Marcus equation. It is widely used for correlating and predicting electron-transfer rates. For his contributions to the understanding of electron-transfer reactions, Marcus received the 1992 Nobel Prize in Chemistry. This paper discusses the development and use of the Marcus equation. Topics include self-exchange reactions; net electron-transfer reactions; Marcus cross-relation; and proton, hydride, atom and group transfers.
Lauber, Ph. Guenter, S.; Koenies, A.; Pinches, S.D.
2007-09-10
In a plasma with a population of super-thermal particles generated by heating or fusion processes, kinetic effects can lead to the additional destabilisation of MHD modes or even to additional energetic particle modes. In order to describe these modes, a new linear gyrokinetic MHD code has been developed and tested, LIGKA (linear gyrokinetic shear Alfven physics) [Ph. Lauber, Linear gyrokinetic description of fast particle effects on the MHD stability in tokamaks, Ph.D. Thesis, TU Muenchen, 2003; Ph. Lauber, S. Guenter, S.D. Pinches, Phys. Plasmas 12 (2005) 122501], based on a gyrokinetic model [H. Qin, Gyrokinetic theory and computational methods for electromagnetic perturbations in tokamaks, Ph.D. Thesis, Princeton University, 1998]. A finite Larmor radius expansion together with the construction of some fluid moments and specification to the shear Alfven regime results in a self-consistent, electromagnetic, non-perturbative model, that allows not only for growing or damped eigenvalues but also for a change in mode-structure of the magnetic perturbation due to the energetic particles and background kinetic effects. Compared to previous implementations [H. Qin, mentioned above], this model is coded in a more general and comprehensive way. LIGKA uses a Fourier decomposition in the poloidal coordinate and a finite element discretisation in the radial direction. Both analytical and numerical equilibria can be treated. Integration over the unperturbed particle orbits is performed with the drift-kinetic HAGIS code [S.D. Pinches, Ph.D. Thesis, The University of Nottingham, 1996; S.D. Pinches et al., CPC 111 (1998) 131] which accurately describes the particles' trajectories. This allows finite-banana-width effects to be implemented in a rigorous way since the linear formulation of the model allows the exchange of the unperturbed orbit integration and the discretisation of the perturbed potentials in the radial direction. Successful benchmarks for toroidal Alfven
NASA Astrophysics Data System (ADS)
Lauber, Ph.; Günter, S.; Könies, A.; Pinches, S. D.
2007-09-01
In a plasma with a population of super-thermal particles generated by heating or fusion processes, kinetic effects can lead to the additional destabilisation of MHD modes or even to additional energetic particle modes. In order to describe these modes, a new linear gyrokinetic MHD code has been developed and tested, LIGKA (linear gyrokinetic shear Alfvén physics) [Ph. Lauber, Linear gyrokinetic description of fast particle effects on the MHD stability in tokamaks, Ph.D. Thesis, TU München, 2003; Ph. Lauber, S. Günter, S.D. Pinches, Phys. Plasmas 12 (2005) 122501], based on a gyrokinetic model [H. Qin, Gyrokinetic theory and computational methods for electromagnetic perturbations in tokamaks, Ph.D. Thesis, Princeton University, 1998]. A finite Larmor radius expansion together with the construction of some fluid moments and specification to the shear Alfvén regime results in a self-consistent, electromagnetic, non-perturbative model, that allows not only for growing or damped eigenvalues but also for a change in mode-structure of the magnetic perturbation due to the energetic particles and background kinetic effects. Compared to previous implementations [H. Qin, mentioned above], this model is coded in a more general and comprehensive way. LIGKA uses a Fourier decomposition in the poloidal coordinate and a finite element discretisation in the radial direction. Both analytical and numerical equilibria can be treated. Integration over the unperturbed particle orbits is performed with the drift-kinetic HAGIS code [S.D. Pinches, Ph.D. Thesis, The University of Nottingham, 1996; S.D. Pinches et al., CPC 111 (1998) 131] which accurately describes the particles' trajectories. This allows finite-banana-width effects to be implemented in a rigorous way since the linear formulation of the model allows the exchange of the unperturbed orbit integration and the discretisation of the perturbed potentials in the radial direction. Successful benchmarks for toroidal Alfv
NASA Astrophysics Data System (ADS)
Bonanomi, N.; Mantica, P.; Szepesi, G.; Hawkes, N.; Lerche, E.; Migliano, P.; Peeters, A.; Sozzi, C.; Tsalas, M.; Van Eester, D.; Contributors, JET
2015-09-01
The main purpose of this work is to study the dependence of trapped electron modes (TEM) threshold and of electron stiffness on the most relevant plasma parameters. Dedicated transport experiments based on heat flux scans and Te modulation have been performed in JET in TEM dominated plasmas with pure ICRH electron heating and a numerical study using gyrokinetic simulations has been performed with the code GKW. Using multilinear regressions on the experimental data, the stabilizing effect of magnetic shear predicted by theory for our plasma parameters is confirmed while no significant effect of safety factor was found. Good quantitative agreement is found between the TEM thresholds found in the experiments and calculated with linear GKW simulations. Non-linear simulations have given further confirmation of the threshold values and allowed comparison with the values of stiffness found experimentally. Perturbative studies using RF power modulation indicate the existence of an inward convective term for the electron heat flux. Adding NBI power, ion temperature gradient (ITG) modes become dominant and a reduction of |\
Wang, Zhenyu; Lin, Yu; Wang, Xueyi; Tummel, Kurt; Chen, Liu
2016-07-07
The eigenmode stability properties of three-dimensional lower-hybrid-drift-instabilities (LHDI) in a Harris current sheet with a small but finite guide magnetic field have been systematically studied by employing the gyrokinetic electron and fully kinetic ion (GeFi) particle-in-cell (PIC) simulation model with a realistic ion-to-electron mass ratio mi/me. In contrast to the fully kinetic PIC simulation scheme, the fast electron cyclotron motion and plasma oscillations are systematically removed in the GeFi model, and hence one can employ the realistic mi/me. The GeFi simulations are benchmarked against and show excellent agreement with both the fully kinetic PIC simulation and the analytical eigenmode theory. Our studies indicate that, for small wavenumbers, ky, along the current direction, the most unstable eigenmodes are peaked at the location wheremore » $$\\vec{k}$$• $$\\vec{B}$$ =0, consistent with previous analytical and simulation studies. Here, $$\\vec{B}$$ is the equilibrium magnetic field and $$\\vec{k}$$ is the wavevector perpendicular to the nonuniformity direction. As ky increases, however, the most unstable eigenmodes are found to be peaked at $$\\vec{k}$$ •$$\\vec{B}$$ ≠0. Additionally, the simulation results indicate that varying mi/me, the current sheet width, and the guide magnetic field can affect the stability of LHDI. Simulations with the varying mass ratio confirm the lower hybrid frequency and wave number scalings.« less
Jolliet, S.; McMillan, B. F.; Vernay, T.; Villard, L.; Hatzky, R.; Bottino, A.; Angelino, P.
2009-07-15
In this paper, the influence of the parallel nonlinearity on zonal flows and heat transport in global particle-in-cell ion-temperature-gradient simulations is studied. Although this term is in theory orders of magnitude smaller than the others, several authors [L. Villard, P. Angelino, A. Bottino et al., Plasma Phys. Contr. Fusion 46, B51 (2004); L. Villard, S. J. Allfrey, A. Bottino et al., Nucl. Fusion 44, 172 (2004); J. C. Kniep, J. N. G. Leboeuf, and V. C. Decyck, Comput. Phys. Commun. 164, 98 (2004); J. Candy, R. E. Waltz, S. E. Parker et al., Phys. Plasmas 13, 074501 (2006)] found different results on its role. The study is performed using the global gyrokinetic particle-in-cell codes TORB (theta-pinch) [R. Hatzky, T. M. Tran, A. Koenies et al., Phys. Plasmas 9, 898 (2002)] and ORB5 (tokamak geometry) [S. Jolliet, A. Bottino, P. Angelino et al., Comput. Phys. Commun. 177, 409 (2007)]. In particular, it is demonstrated that the parallel nonlinearity, while important for energy conservation, affects the zonal electric field only if the simulation is noise dominated. When a proper convergence is reached, the influence of parallel nonlinearity on the zonal electric field, if any, is shown to be small for both the cases of decaying and driven turbulence.
NASA Astrophysics Data System (ADS)
Wang, Zhenyu; Lin, Yu; Wang, Xueyi; Tummel, Kurt; Chen, Liu
2016-07-01
The eigenmode stability properties of three-dimensional lower-hybrid-drift-instabilities (LHDI) in a Harris current sheet with a small but finite guide magnetic field have been systematically studied by employing the gyrokinetic electron and fully kinetic ion (GeFi) particle-in-cell (PIC) simulation model with a realistic ion-to-electron mass ratio mi/me . In contrast to the fully kinetic PIC simulation scheme, the fast electron cyclotron motion and plasma oscillations are systematically removed in the GeFi model, and hence one can employ the realistic mi/me . The GeFi simulations are benchmarked against and show excellent agreement with both the fully kinetic PIC simulation and the analytical eigenmode theory. Our studies indicate that, for small wavenumbers, ky, along the current direction, the most unstable eigenmodes are peaked at the location where k →.B → =0 , consistent with previous analytical and simulation studies. Here, B → is the equilibrium magnetic field and k → is the wavevector perpendicular to the nonuniformity direction. As ky increases, however, the most unstable eigenmodes are found to be peaked at k →.B → ≠0 . In addition, the simulation results indicate that varying mi/me , the current sheet width, and the guide magnetic field can affect the stability of LHDI. Simulations with the varying mass ratio confirm the lower hybrid frequency and wave number scalings.
NASA Astrophysics Data System (ADS)
Nakata, Motoki; Honda, Mitsuru; Yoshida, Maiko; Urano, Hajime; Nunami, Masanori; Maeyama, Shinya; Watanabe, Tomo-Hiko; Sugama, Hideo
2016-08-01
Quantitative validation studies of flux-tube gyrokinetic Vlasov simulations on ion and electron heat transport are carried out for the JT-60U tokamak experiment. The ion temperature gradient (ITG) and/or trapped electron modes (TEM) driven turbulent transport and zonal flow generations are investigated for an L-mode plasma in the local turbulence limit with a sufficiently small normalized ion thermal gyroradius and weak mean radial electric fields. Nonlinear turbulence simulations by the GKV code successfully reproduce radial profiles of the ion and electron energy fluxes in the core region. The numerical results show that the TEM-driven zonal flow generation in the outer region is more significant than that in the core region with ITG- and ITG–TEM-dominated turbulence, leading to moderate transport shortfall of the ion energy flux. Error levels in the prediction of the ion and electron temperature gradient profiles in the core region are estimated as less than +/- 30% , based on a multiple flux matching technique, where the simulated ion and electron energy fluxes are simultaneously matched to the experimental values.
Benchmark Studies of the Gyro-Landau-Fluid code and Gyro-kinetic Codes on Kinetic Ballooning Modes
NASA Astrophysics Data System (ADS)
Tang, Tengfei; Xu, Xueqiao; Ma, Chenhao; Holland, Chris; Candy, Jeff
2015-11-01
A Gyro-Landau-Fluid (GLF) 3 +1 model has been implemented in BOUT + + framework recently, which contains full Finite-Larmor-Radius (FLR) effects, Landau damping and toroidal resonance. A linear global beta scan has been done using the cbm18 series equilibriums, showing that the unstable modes are kinetic ballooning modes (KBMs). In this work, we use the GYRO code, which is a gyrokinetic continuum code widely used for simulation of the plasma microturbulence, to benchmark with GLF 3 +1 code on KBMs. As the modes locate in peak pressure gradient region, a linear local beta scan using the same set of equilibriums has been done at this position for comparison. With the drift kinetic electron module in the GYRO code by including small electron-ion collision to damp electron modes, GYRO generated mode structures and parity suggest that they are kinetic ballooning modes, and the growth rate is comparable to the GLF results. However, a radial scan of the pedestal for a particular cbm18 equilibrium shows that the growth rate of the most unstable mode shifts outward to the bottom of pedestal and the real frequency of what was originally the KBMs steadily approaches and crosses over to the electron diamagnetic drift direction. Prepared by LLNL under Contract DE-AC52-07NA27344.
NASA Astrophysics Data System (ADS)
Oberparleiter, M.; Jenko, F.; Told, D.; Doerk, H.; Görler, T.
2016-04-01
Neoclassical and turbulent transport in tokamaks has been studied extensively over the past decades, but their possible interaction remains largely an open question. The two are only truly independent if the length scales governing each of them are sufficiently separate, i.e., if the ratio ρ* between ion gyroradius and the pressure gradient scale length is small. This is not the case in particularly interesting regions such as transport barriers. Global simulations of a collisional ion-temperature-gradient-driven microturbulence performed with the nonlinear global gyrokinetic code Gene are presented. In particular, comparisons are made between systems with and without neoclassical effects. In fixed-gradient simulations, the modified radial electric field is shown to alter the zonal flow pattern such that a significant increase in turbulent transport is observed for ρ*≳1 /300 . Furthermore, the dependency of the flux on the collisionality changes. In simulations with fixed power input, we find that the presence of neoclassical effects decreases the frequency and amplitude of intermittent turbulent transport bursts (avalanches) and thus plays an important role for the self-organisation behaviour.
Spong, Donald A; Bass, Eric; Deng, Wenjun; Heidbrink, W.; Lin, Zhihong; Tobias, Ben; Van Zeeland, Michael; Austin, M. E.; Domier, C. W.; Luhmann, N.C.
2012-01-01
A verification and validation study is carried out for a sequence of reversed shear Alfven instability time slices. The mode frequency increases in time as the minimum (q{sub min}) in the safety factor profile decreases. Profiles and equilibria are based upon reconstructions of DIII-D discharge (No.142111) in which many such frequency up-sweeping modes were observed. Calculations of the frequency and mode structure evolution from two gyrokinetic codes, GTC and GYRO, and a gyro-Landau fluid code TAEFL are compared. The experimental mode structure of the instability was measured using time-resolved two-dimensional electron cyclotron emission imaging. The three models reproduce the frequency upsweep event within {+-}10% of each other, and the average of the code predictions is within {+-}8% of the measurements; growth rates are predicted that are consistent with the observed spectral line widths. The mode structures qualitatively agree with respect to radial location and width, dominant poloidal mode number, ballooning structure, and the up-down asymmetry, with some remaining differences in the details. Such similarities and differences between the predictions of the different models and the experimental results are a valuable part of the verification/validation process and help to guide future development of the modeling efforts.
NASA Astrophysics Data System (ADS)
Nakata, Motoki; Honda, Mitsuru; Yoshida, Maiko; Urano, Hajime; Nunami, Masanori; Maeyama, Shinya; Watanabe, Tomo-Hiko; Sugama, Hideo
2016-08-01
Quantitative validation studies of flux-tube gyrokinetic Vlasov simulations on ion and electron heat transport are carried out for the JT-60U tokamak experiment. The ion temperature gradient (ITG) and/or trapped electron modes (TEM) driven turbulent transport and zonal flow generations are investigated for an L-mode plasma in the local turbulence limit with a sufficiently small normalized ion thermal gyroradius and weak mean radial electric fields. Nonlinear turbulence simulations by the GKV code successfully reproduce radial profiles of the ion and electron energy fluxes in the core region. The numerical results show that the TEM-driven zonal flow generation in the outer region is more significant than that in the core region with ITG- and ITG-TEM-dominated turbulence, leading to moderate transport shortfall of the ion energy flux. Error levels in the prediction of the ion and electron temperature gradient profiles in the core region are estimated as less than +/- 30% , based on a multiple flux matching technique, where the simulated ion and electron energy fluxes are simultaneously matched to the experimental values.
NASA Astrophysics Data System (ADS)
Bass, E. M.; Waltz, R. E.
2012-10-01
In ITER, convection of fusion-produced alpha particles by energetic particle (EP)-driven Alfv'en eigenmodes (AEs) risks wall damage and loss of alpha heating needed for ignition. We examine beam-excited AEs and induced quasilinear transport in a DIII-D AE experiment using the gyrokinetic code GYRO [1]. Global, linear eigenvalue simulations show reverse-shear AEs (RSAEs), toroidal AEs, and beta-induced AEs interacting over one (equilibrium time scale) RSAE frequency sweep. Eigenfunction modifications over MHD, including a poloidal twist and broad AE footprint observed in electron cyclotron emission imaging [2], show the value of a kinetic approach. Under a simple quasilinear saturation assumption, a sequence of comparatively inexpensive local simulations quantitatively recreates some global features, notably the quasilinear transport footprint. Accordingly, we present here a stiff EP transport model where AEs limit the EP density gradient to the local stability threshold, and a TGLF-driven quasilinear model elsewhere. The model gives some``worst case'' predictions of the AE-limited alpha profile in ITER.[4pt] [1] J. Candy and R.E. Waltz, Phys. Rev. Lett. 91, 045001 (2003). [2] B.J. Tobias, et al., Phys. Rev. Lett. 106, 075003 (2011).
Schekochihin, A. A.; Cowley, S. C.; Dorland, W.; Hammett, G. W.; Howes, G. G.; Quataert, E.; Tatsuno, T.
2009-04-23
This paper presents a theoretical framework for understanding plasma turbulence in astrophysical plasmas. It is motivated by observations of electromagnetic and density fluctuations in the solar wind, interstellar medium and galaxy clusters, as well as by models of particle heating in accretion disks. All of these plasmas and many others have turbulentmotions at weakly collisional and collisionless scales. The paper focuses on turbulence in a strong mean magnetic field. The key assumptions are that the turbulent fluctuations are small compared to the mean field, spatially anisotropic with respect to it and that their frequency is low compared to the ion cyclotron frequency. The turbulence is assumed to be forced at some system-specific outer scale. The energy injected at this scale has to be dissipated into heat, which ultimately cannot be accomplished without collisions. A kinetic cascade develops that brings the energy to collisional scales both in space and velocity. The nature of the kinetic cascade in various scale ranges depends on the physics of plasma fluctuations that exist there. There are four special scales that separate physically distinct regimes: the electron and ion gyroscales, the mean free path and the electron diffusion scale. In each of the scale ranges separated by these scales, the fully kinetic problem is systematically reduced to a more physically transparent and computationally tractable system of equations, which are derived in a rigorous way. In the "inertial range" above the ion gyroscale, the kinetic cascade separates into two parts: a cascade of Alfvenic fluctuations and a passive cascade of density and magnetic-fieldstrength fluctuations. The former are governed by the Reduced Magnetohydrodynamic (RMHD) equations at both the collisional and collisionless scales; the latter obey a linear kinetic equation along the (moving) field lines associated with the Alfvenic component (in the collisional limit, these compressive fluctuations
Byers, J.A.; Williams, T.J.; Cohen, B.I.; Dimits, A.M.
1994-04-27
One of the programs of the Magnetic fusion Energy (MFE) Theory and computations Program is studying the anomalous transport of thermal energy across the field lines in the core of a tokamak. We use the method of gyrokinetic particle-in-cell simulation in this study. For this LDRD project we employed massively parallel processing, new algorithms, and new algorithms, and new formal techniques to improve this research. Specifically, we sought to take steps toward: researching experimentally-relevant parameters in our simulations, learning parallel computing to have as a resource for our group, and achieving a 100 {times} speedup over our starting-point Cray2 simulation code`s performance.
Shore, B.W.
1981-01-30
The equations of motion are discussed which describe time dependent population flows in an N-level system, reviewing the relationship between incoherent (rate) equations, coherent (Schrodinger) equations, and more general partially coherent (Bloch) equations. Approximations are discussed which replace the elaborate Bloch equations by simpler rate equations whose coefficients incorporate long-time consequences of coherence.
Muñoz, P. A. Kilian, P.; Büchner, J.; Told, D.; Jenko, F.
2015-08-15
In this work, we compare gyrokinetic (GK) with fully kinetic Particle-in-Cell (PIC) simulations of magnetic reconnection in the limit of strong guide field. In particular, we analyze the limits of applicability of the GK plasma model compared to a fully kinetic description of force free current sheets for finite guide fields (b{sub g}). Here, we report the first part of an extended comparison, focusing on the macroscopic effects of the electron flows. For a low beta plasma (β{sub i} = 0.01), it is shown that both plasma models develop magnetic reconnection with similar features in the secondary magnetic islands if a sufficiently high guide field (b{sub g} ≳ 30) is imposed in the kinetic PIC simulations. Outside of these regions, in the separatrices close to the X points, the convergence between both plasma descriptions is less restrictive (b{sub g} ≳ 5). Kinetic PIC simulations using guide fields b{sub g} ≲ 30 reveal secondary magnetic islands with a core magnetic field and less energetic flows inside of them in comparison to the GK or kinetic PIC runs with stronger guide fields. We find that these processes are mostly due to an initial shear flow absent in the GK initialization and negligible in the kinetic PIC high guide field regime, in addition to fast outflows on the order of the ion thermal speed that violate the GK ordering. Since secondary magnetic islands appear after the reconnection peak time, a kinetic PIC/GK comparison is more accurate in the linear phase of magnetic reconnection. For a high beta plasma (β{sub i} = 1.0) where reconnection rates and fluctuations levels are reduced, similar processes happen in the secondary magnetic islands in the fully kinetic description, but requiring much lower guide fields (b{sub g} ≲ 3)
NASA Astrophysics Data System (ADS)
Merlo, G.; Brunner, S.; Sauter, O.; Camenen, Y.; Görler, T.; Jenko, F.; Marinoni, A.; Told, D.; Villard, L.
2015-05-01
The experimental observation made on the TCV tokamak of a significant confinement improvement in plasmas with negative triangularity (δ < 0) compared to those with standard positive triangularity has been interpreted in terms of different degrees of profile stiffness (Sauter et al 2014 Phys. Plasmas 21 055906) and/or different critical gradients. Employing the Eulerian gyrokinetic code GENE (Jenko et al 2000 Phys. Plasmas 7 1904), profile stiffness and critical gradients are studied under TCV relevant conditions. For the considered experimental discharges, trapped electron modes (TEMs) and electron temperature gradient (ETG) modes are the dominant microinstabilities, with the latter providing a significant contribution to the non-linear electron heat fluxes near the plasma edge. Two series of simulations with different levels of realism are performed, addressing the question of profile stiffness at various radial locations. Retaining finite collisionality, impurities and electromagnetic effects, as well as the physical electron-to-ion mass ratio are all necessary in order to approach the experimental flux measurements. However, flux-tube simulations are unable to fully reproduce the TCV results, pointing towards the need to carry out radially nonlocal (global) simulations, i.e. retaining finite machine size effects, in a future study. Some conclusions about the effect of triangularity can nevertheless be drawn based on the flux-tube results. In particular, the importance of considering the sensitivity to both temperature and density gradient is shown. The flux tube results show an increase of the critical gradients towards the edge, further enhanced when δ < 0, and they also appear to indicate a reduction of profile stiffness towards plasma edge.
Kinsey, J. E.; Waltz, R. E.; Candy, J.
2007-10-15
Nonlinear gyrokinetic simulations with kinetic electron dynamics are used to study the effects of plasma shaping on turbulent transport and ExB shear in toroidal geometry including the presence of kinetic electrons using the GYRO code. Over 120 simulations comprised of systematic scans were performed around several reference cases in the local, electrostatic, collisionless limit. Using a parameterized local equilibrium model for shaped geometry, the GYRO simulations show that elongation {kappa} (and its gradient) stabilizes the energy transport from ion temperature gradient (ITG) and trapped electron mode (TEM) instabilities at fixed midplane minor radius. For scans around a reference set of parameters, the GYRO ion energy diffusivity, in gyro-Bohm units, approximately follows a {kappa}{sup -1} scaling which is qualitatively similar to recent experimental energy confinement scalings. Most of the {kappa} scaling is due to the shear in the elongation rather than the local {kappa} itself. The {kappa} scaling for the electrons is found to vary and can be stronger or weaker than {kappa}{sup -1} depending on the wavenumber where the transport peaks. The {kappa} scaling is weaker when the energy diffusivity peaks at low wavenumbers and is stronger when the peak occurs at high wavenumbers. The simulations also demonstrate a nonlinear upshift in the critical temperature gradient as the elongation increases due to an increase in the residual zonal flow amplitude. Triangularity is found to be slightly destabilizing and its effect is strongest for highly elongated plasmas. Finally, we find less ExB shear is needed to quench the transport at high elongation and low aspect ratio. A new linear ExB shear quench rule, valid for shaped tokamak geometry, is presented.
NASA Astrophysics Data System (ADS)
Sung, C.; White, A. E.; Mikkelsen, D. R.; Greenwald, M.; Holland, C.; Howard, N. T.; Churchill, R.; Theiler, C.
2016-04-01
Long wavelength turbulent electron temperature fluctuations (kyρs < 0.3) are measured in the outer core region (r/a > 0.8) of Ohmic L-mode plasmas at Alcator C-Mod [E. S. Marmar et al., Nucl. Fusion 49, 104014 (2009)] with a correlation electron cyclotron emission diagnostic. The relative amplitude and frequency spectrum of the fluctuations are compared quantitatively with nonlinear gyrokinetic simulations using the GYRO code [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] in two different confinement regimes: linear Ohmic confinement (LOC) regime and saturated Ohmic confinement (SOC) regime. When comparing experiment with nonlinear simulations, it is found that local, electrostatic ion-scale simulations (kyρs ≲ 1.7) performed at r/a ˜ 0.85 reproduce the experimental ion heat flux levels, electron temperature fluctuation levels, and frequency spectra within experimental error bars. In contrast, the electron heat flux is robustly under-predicted and cannot be recovered by using scans of the simulation inputs within error bars or by using global simulations. If both the ion heat flux and the measured temperature fluctuations are attributed predominantly to long-wavelength turbulence, then under-prediction of electron heat flux strongly suggests that electron scale turbulence is important for transport in C-Mod Ohmic L-mode discharges. In addition, no evidence is found from linear or nonlinear simulations for a clear transition from trapped electron mode to ion temperature gradient turbulence across the LOC/SOC transition, and also there is no evidence in these Ohmic L-mode plasmas of the "Transport Shortfall" [C. Holland et al., Phys. Plasmas 16, 052301 (2009)].
Benchmark studies of the gyro-Landau-fluid code and gyro-kinetic codes on kinetic ballooning modes
NASA Astrophysics Data System (ADS)
Tang, T. F.; Xu, X. Q.; Ma, C. H.; Bass, E. M.; Holland, C.; Candy, J.
2016-03-01
A Gyro-Landau-Fluid (GLF) 3 + 1 model has been recently implemented in BOUT++ framework, which contains full Finite-Larmor-Radius effects, Landau damping, and toroidal resonance [Ma et al., Phys. Plasmas 22, 055903 (2015)]. A linear global beta scan has been conducted using the JET-like circular equilibria (cbm18 series), showing that the unstable modes are kinetic ballooning modes (KBMs). In this work, we use the GYRO code, which is a gyrokinetic continuum code widely used for simulation of the plasma microturbulence, to benchmark with GLF 3 + 1 code on KBMs. To verify our code on the KBM case, we first perform the beta scan based on "Cyclone base case parameter set." We find that the growth rate is almost the same for two codes, and the KBM mode is further destabilized as beta increases. For JET-like global circular equilibria, as the modes localize in peak pressure gradient region, a linear local beta scan using the same set of equilibria has been performed at this position for comparison. With the drift kinetic electron module in the GYRO code by including small electron-electron collision to damp electron modes, GYRO generated mode structures and parity suggest that they are kinetic ballooning modes, and the growth rate is comparable to the GLF results. However, a radial scan of the pedestal for a particular set of cbm18 equilibria, using GYRO code, shows different trends for the low-n and high-n modes. The low-n modes show that the linear growth rate peaks at peak pressure gradient position as GLF results. However, for high-n modes, the growth rate of the most unstable mode shifts outward to the bottom of pedestal and the real frequency of what was originally the KBMs in ion diamagnetic drift direction steadily approaches and crosses over to the electron diamagnetic drift direction.
Generalized parallel heat transport equations in collisional to weakly collisional plasmas
Zawaideh, E.; Kim, N.S.; Najmabadi, F.
1988-11-01
A new set of two-fluid heat transport equations that is valid from collisional to weakly collisional limits is derived. Starting from gyrokinetic equations in flux coordinates, a set of moment equations describing plasma energy transport along the field lines of a space- and time-dependent magnetic field is derived. No restrictions on the anisotropy of the ion distribution function or collisionality are imposed. In the highly collisional limit, these equations reduce to the classical heat conduction equation (e.g., Spitzer and Haerm or Braginskii), while in the weakly collisional limit, they describe a saturated heat flux (flux limited). Numerical examples comparing these equations with conventional heat transport equations show that in the limit where the ratio of the mean free path lambda to the scale length of the temperature gradient L/sub T/ approaches zero, there is no significant difference between the solutions of the new and conventional heat transport equations. As lambda/L/sub T/..-->..1, the conventional heat conduction equation contains a significantly larger error than (lambda/L/sub T/)/sup 2/. The error is found to be O(lambda/L)/sup 2/, where L is the smallest of the scale lengths of the gradient in the magnetic field, or the macroscopic plasma parameters (e.g., velocity scale length, temperature scale length, and density scale length). The accuracy of the flux-limited model depends significantly on the value of the flux limit parameter which, in general, is not known. The new set of equations shows that the flux-limited parameter is a function of the magnetic field and plasma parameter profiles.
Beyond single-stream with the Schrödinger method
NASA Astrophysics Data System (ADS)
Uhlemann, Cora; Kopp, Michael
2016-10-01
We investigate large scale structure formation of collisionless dark matter in the phase space description based on the Vlasov-Poisson equation. We present the Schrödinger method, originally proposed by \\cite{WK93} as numerical technique based on the Schrödinger Poisson equation, as an analytical tool which is superior to the common standard pressureless fluid model. Whereas the dust model fails and develops singularities at shell crossing the Schrödinger method encompasses multi-streaming and even virialization.
Modeling full radial electric field and flow shears in gyrokinetic simulations
NASA Astrophysics Data System (ADS)
Wan, Weigang; Chen, Yang; Parker, Scott; Groebner, Richard
2015-11-01
The radial electric field (Er) is important in the turbulence of tokamak plasmas. It affects the growth rate of instabilities through the E × B shear and changes the real frequency of drift waves by adding a Doppler shift. The modeling of Er in simulations, however, was usually not complete. The full profiles of the main ion toroidal and poloidal flows were not implemented. In the gyrokientic electromagnetic particle code GEM, the poloidal flow was assumed to be zero by introducing a parallel flow. However, recent experiments show that the poloidal flow could be important. In this study we add the full main ion rotation flows to GEM, following the comprehensive procedures of Sugama and Horton. The major contribution to the Er from the ion toroidal flow is used as Er 0, and the result as Er 1. The effects to the growth rate and Doppler shift of all terms in the force balance equation are demonstrated using linear simulations of edge and core tokamak plasmas.
White, A. E.; Peebles, W. A.; Rhodes, T. L.; Schmitz, L.; Carter, T. A.; Hillesheim, J. C.; Doyle, E. J.; Zeng, L.; Holland, C. H.; Wang, G.; McKee, G. R.; Staebler, G. M.; Waltz, R. E.; DeBoo, J. C.; Petty, C. C.; Burrell, K. H.
2010-05-15
This paper presents new measurements of the cross-phase angle, alpha{sub n{sub eT{sub e}}}, between long-wavelength (k{sub t}hetarho{sub s}<0.5) density, n-tilde{sub e}, and electron temperature, T-tilde{sub e}, fluctuations in the core of DIII-D [J. L. Luxon, Nucl. Fusion 42, 614 (2002)] tokamak plasmas. The coherency and cross-phase angle between n-tilde{sub e} and T-tilde{sub e} are measured using coupled reflectometer and correlation electron cyclotron emission diagnostics that view the same plasma volume. In addition to the experimental results, two sets of local, nonlinear gyrokinetic turbulence simulations that are performed with the GYRO code [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] are described. One set, called the pre-experiment simulations, was performed prior to the experiment in order to predict a change in alpha{sub n{sub eT{sub e}}} given experimentally realizable increases in the electron temperature, T{sub e}. In the experiment the cross-phase angle was measured at three radial locations (rho=0.55, 0.65, and 0.75) in both a 'Base' case and a 'High T{sub e}' case. The measured cross-phase angle is in good qualitative agreement with the pre-experiment simulations, which predicted that n-tilde{sub e} and T-tilde{sub e} would be out of phase. The pre-experiment simulations also predicted a decrease in cross-phase angle as T{sub e} is increased. Experimentally, this trend is observed at the inner two radial locations only. The second set of simulations, the postexperiment simulations, is carried out using local parameters taken from measured experimental profiles as input to GYRO. These postexperiment simulation results are in good quantitative agreement with the measured cross-phase angle, despite disagreements with transport fluxes. Directions for future modeling and experimental work are discussed.
NASA Technical Reports Server (NTRS)
Hamrock, B. J.; Dowson, D.
1981-01-01
Lubricants, usually Newtonian fluids, are assumed to experience laminar flow. The basic equations used to describe the flow are the Navier-Stokes equation of motion. The study of hydrodynamic lubrication is, from a mathematical standpoint, the application of a reduced form of these Navier-Stokes equations in association with the continuity equation. The Reynolds equation can also be derived from first principles, provided of course that the same basic assumptions are adopted in each case. Both methods are used in deriving the Reynolds equation, and the assumptions inherent in reducing the Navier-Stokes equations are specified. Because the Reynolds equation contains viscosity and density terms and these properties depend on temperature and pressure, it is often necessary to couple the Reynolds with energy equation. The lubricant properties and the energy equation are presented. Film thickness, a parameter of the Reynolds equation, is a function of the elastic behavior of the bearing surface. The governing elasticity equation is therefore presented.
Zhang, H.; Wu, S. Z.; Zhou, C. T.; He, X. T.; Zhu, S. P.
2013-09-15
The dispersion relation of one-dimensional longitudinal plasma waves in relativistic homogeneous plasmas is investigated with both linear theory and Vlasov simulation in this paper. From the Vlasov-Poisson equations, the linear dispersion relation is derived for the proper one-dimensional Jüttner distribution. Numerically obtained linear dispersion relation as well as an approximate formula for plasma wave frequency in the long wavelength limit is given. The dispersion of longitudinal wave is also simulated with a relativistic Vlasov code. The real and imaginary parts of dispersion relation are well studied by varying wave number and plasma temperature. Simulation results are in agreement with established linear theory.
2008-03-18
IMPACT-Z is a parallel particle-in-cell code whose primary purpose is to model the dynamics of charged particle beams in linear accelerators. The code includes the effects of externally applied fields from magnets and accelerating cavities as well as the effect of self-fields (space charge fields). Mathematically, the code solves the Vlasov/Poisson equations using a particle-based technique. The code, which is written in Fortran90 with MPI, runs on both single-processor and multi-processor systems.
Double layers on auroral field lines
NASA Technical Reports Server (NTRS)
Hudson, M. K.; Lotko, W.; Witt, E.
1982-01-01
Time-stationary solutions to the Vlasov-Poisson equation for ion holes and double layers were examined along with particle simulations which pertain to recent observations of small amplitude (e phi)/t sub e approx. 1 electric field structures on auroral field lines. Both the time-stationary analysis and the simulations suggest that double layers evolve from holes in ion phase space when their amplitude reaches (e phi)/t sub e approx. 1. Multiple small amplitude double layers which are seen in long simulation systems and are seen to propagate past spacecraft may account for the acceleration of plasma sheet electrons to produce the discrete aurora.
A splitting algorithm for Vlasov simulation with filamentation filtration
NASA Technical Reports Server (NTRS)
Klimas, A. J.; Farrell, W. M.
1994-01-01
A Fourier-Fourier transformed version of the splitting algorithm for simulating solutions of the Vlasov-Poisson system of equations is introduced. It is shown that with the inclusion of filamentation filtration in this transformed algorithm it is both faster and more stable than the standard splitting algorithm. It is further shown that in a scalar computer environment this new algorithm is approximately equal in speed and far less noisy than its particle-in-cell counterpart. It is conjectured that in a multiprocessor environment the filtered splitting algorithm would be faster while producing more precise results.
Hopf bifurcation and plasma instabilities
Crawford, J.D.
1983-11-01
Center manifold theory and the theory of normal forms are applied to examples of Hopf bifurcation in two models of plasma dynamics. A finite dimensional model of a 3-wave system with quadratic nonlinearities provides a simple example of both supercritical and subcritical Hopf bifurcation. In the second model, the electrostatic instabilities of a collisional plasma correspond to Hopf bifurcations. In this problem, the Vlasov-Poisson equations with a Krook collision term describe the electron dynamics in a weakly ionized gas. The one mode in instability is analyzed in detail; near criticality it always saturates in a small amplitude nonlinear oscillation.
ERIC Educational Resources Information Center
Blakley, G. R.
1982-01-01
Reviews mathematical techniques for solving systems of homogeneous linear equations and demonstrates that the algebraic method of balancing chemical equations is a matter of solving a system of homogeneous linear equations. FORTRAN programs using this matrix method to chemical equation balancing are available from the author. (JN)
Micah Beck
2008-09-14
This project focused on the use of Logistical Networking technology to address the challenges involved in rapid sharing of data from the the Center's gyrokinetic particle simulations, which can be on the order of terabytes per time step, among researchers at a number of geographically distributed locations. There is a great need to manage data on this scale in a flexible manner, with simulation code, file system, database and visualization functions requiring access. The project used distributed data management infrastructure based on Logistical Networking technology to address these issues in a way that maximized interoperability and achieved the levels of performance the required by the Center's application community. The work focused on the development and deployment of software tools and infrastructure for the storage and distribution of terascale datasets generated by simulations running at the National Center for Computational Science at Oak Ridge National Laboratory.
Kinetic energy equations for the average-passage equation system
NASA Technical Reports Server (NTRS)
Johnson, Richard W.; Adamczyk, John J.
1989-01-01
Important kinetic energy equations derived from the average-passage equation sets are documented, with a view to their interrelationships. These kinetic equations may be used for closing the average-passage equations. The turbulent kinetic energy transport equation used is formed by subtracting the mean kinetic energy equation from the averaged total instantaneous kinetic energy equation. The aperiodic kinetic energy equation, averaged steady kinetic energy equation, averaged unsteady kinetic energy equation, and periodic kinetic energy equation, are also treated.
Electromagnetic Gyrokinetic Simulations
Wan, W
2003-11-19
A new electromagnetic kinetic electron {delta} particle simulation model has been demonstrated to work well at large values of plasma {beta} times the ion-to-electron mass ratio. The simulation is three-dimensional using toroidal flux-tube geometry and includes electron-ion collisions. The model shows accurate shear Alfven wave damping and microtearing physics. Zonal flows with kinetic electrons are found to be turbulent with the spectrum peaking at zero and having a width in the frequency range of the driving turbulence. This is in contrast with adiabatic electron cases where the zonal flows are near stationary, even though the linear behavior of the zonal flow is not significantly affected by kinetic electrons. zonal fields are found to be very weak, consistent with theoretical predictions for {beta} below the kinetic ballooning limit. Detailed spectral analysis of the turbulence data is presented in the various limits.
Reflections on Chemical Equations.
ERIC Educational Resources Information Center
Gorman, Mel
1981-01-01
The issue of how much emphasis balancing chemical equations should have in an introductory chemistry course is discussed. The current heavy emphasis on finishing such equations is viewed as misplaced. (MP)
Interpretation of Bernoulli's Equation.
ERIC Educational Resources Information Center
Bauman, Robert P.; Schwaneberg, Rolf
1994-01-01
Discusses Bernoulli's equation with regards to: horizontal flow of incompressible fluids, change of height of incompressible fluids, gases, liquids and gases, and viscous fluids. Provides an interpretation, properties, terminology, and applications of Bernoulli's equation. (MVL)
NASA Astrophysics Data System (ADS)
Parker, Scott
2012-10-01
Global electromagnetic gyrokinetic simulations show the existence of near threshold conditions, for both a high-n Kinetic Ballooning Mode (KBM) and an intermediate-n kinetic version of Peeling-Ballooning Mode (PBM). The KBM and the PBM have been used to constrain the EPED model [1]. Global gyrokinetic simulations show that the H-mode pedestal, just prior to the onset of the Edge Localized Mode (ELM), is very near the KBM threshold. Two DIII-D experimental discharges are studied, one reporting KBM features in fluctuation measurements [2]. Simulations find that in addition to the high-n KBM, an intermediate-n electromagnetic mode is unstable. This kinetic version of the PBM has phase velocity in the electron diamagnetic direction, but otherwise has features similar to the MHD PBM. When the magnetic shear is reduced in a narrow region near the steep pressure gradient, the intermediate-n ``kinetic PBM'' is stabilized, while the high-n KBM becomes the most unstable mode. Global simulation results of the KBM compare favorably with flux tube simulations. The KBM transitions to an unstable electrostatic ion mode as the plasma beta is reduced. The intermediate-n ``kinetic peeling ballooning mode'' is sensitive to the q-profile and only seen in global electromagnetic simulations. Collisions increase the KBM critical beta and growth rate. These results indicate that an improved pedestal model should include, in detail, any corrections to the bootstrap current, and any other equilibrium effects that might reduce the local magnetic shear. It is known that the bootstrap current may flatten the q-profile in the steep gradient region [3]. Simulations are carried out using the global electromagnetic GEM code, including kinetic electrons, electron-ion collisions and the effects of realistic magnetic geometry. In addition to global linear analysis, nonlinear simulations will be reported showing that, while the equilibrium radial electric field has a weak effect on the linear growth
NASA Astrophysics Data System (ADS)
Kostov, Ivan; Serban, Didina; Volin, Dmytro
2008-08-01
We give a realization of the Beisert, Eden and Staudacher equation for the planar Script N = 4 supersymetric gauge theory which seems to be particularly useful to study the strong coupling limit. We are using a linearized version of the BES equation as two coupled equations involving an auxiliary density function. We write these equations in terms of the resolvents and we transform them into a system of functional, instead of integral, equations. We solve the functional equations perturbatively in the strong coupling limit and reproduce the recursive solution obtained by Basso, Korchemsky and Kotański. The coefficients of the strong coupling expansion are fixed by the analyticity properties obeyed by the resolvents.
Einstein equation at singularities
NASA Astrophysics Data System (ADS)
Stoica, Ovidiu-Cristinel
2014-02-01
Einstein's equation is rewritten in an equivalent form, which remains valid at the singularities in some major cases. These cases include the Schwarzschild singularity, the Friedmann-Lemaître-Robertson-Walker Big Bang singularity, isotropic singularities, and a class of warped product singularities. This equation is constructed in terms of the Ricci part of the Riemann curvature (as the Kulkarni-Nomizu product between Einstein's equation and the metric tensor).
Solving Ordinary Differential Equations
NASA Technical Reports Server (NTRS)
Krogh, F. T.
1987-01-01
Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.
Chowdhury, J.; Wan, Weigang; Chen, Yang; Parker, Scott E.; Groebner, Richard J.; Holland, C.; Howard, N. T.
2014-11-15
The δ f particle-in-cell code GEM is used to study the transport “shortfall” problem of gyrokinetic simulations. In local simulations, the GEM results confirm the previously reported simulation results of DIII-D [Holland et al., Phys. Plasmas 16, 052301 (2009)] and Alcator C-Mod [Howard et al., Nucl. Fusion 53, 123011 (2013)] tokamaks with the continuum code GYRO. Namely, for DIII-D the simulations closely predict the ion heat flux at the core, while substantially underpredict transport towards the edge; while for Alcator C-Mod, the simulations show agreement with the experimental values of ion heat flux, at least within the range of experimental error. Global simulations are carried out for DIII-D L-mode plasmas to study the effect of edge turbulence on the outer core ion heat transport. The edge turbulence enhances the outer core ion heat transport through turbulence spreading. However, this edge turbulence spreading effect is not enough to explain the transport underprediction.
NASA Astrophysics Data System (ADS)
Leboeuf, Jean-Noel; Decyk, Viktor; Newman, David; Sanchez, Raul
2013-10-01
The massively parallel, 2D domain-decomposed, nonlinear, 3D, toroidal, electrostatic, gyrokinetic, Particle in Cell (PIC), Cartesian geometry UCAN2 code, with particle ions and adiabatic electrons, has been ported to two emerging mainframes. These two computers, one at NERSC in the US built by Cray named Edison and the other at the Barcelona Supercomputer Center (BSC) in Spain built by IBM named MareNostrum III (MNIII) just happen to share the same Intel ``Sandy Bridge'' processors. The successful port of UCAN2 to MNIII which came online first has enabled us to be up and running efficiently in record time on Edison. Overall, the performance of UCAN2 on Edison is superior to that on MNIII, particularly at large numbers of processors (>1024) for the same Intel IFORT compiler. This appears to be due to different MPI modules (OpenMPI on MNIII and MPICH2 on Edison) and different interconnection networks (Infiniband on MNIII and Cray's Aries on Edison) on the two mainframes. Details of these ports and comparative benchmarks are presented. Work supported by OFES, USDOE, under contract no. DE-FG02-04ER54741 with the University of Alaska at Fairbanks.
Yagi, M.; Horton, W. )
1994-07-01
A set of reduced Braginskii equations is derived without assuming flute ordering and the Boussinesq approximation. These model equations conserve the physical energy. It is crucial at finite [beta] that the perpendicular component of Ohm's law be solved to ensure [del][center dot][bold j]=0 for energy conservation.
Uniqueness of Maxwell's Equations.
ERIC Educational Resources Information Center
Cohn, Jack
1978-01-01
Shows that, as a consequence of two feasible assumptions and when due attention is given to the definition of charge and the fields E and B, the lowest-order equations that these two fields must satisfy are Maxwell's equations. (Author/GA)
Octonic Massive Field Equations
NASA Astrophysics Data System (ADS)
Demir, Süleyman; Kekeç, Seray
2016-07-01
In the present paper we propose the octonic form of massive field equations based on the analogy with electromagnetism and linear gravity. Using the advantages of octon algebra the Maxwell-Dirac-Proca equations have been reformulated in compact and elegant way. The energy-momentum relations for massive field are discussed.
NASA Astrophysics Data System (ADS)
Kuksin, Sergei; Maiocchi, Alberto
In this chapter we present a general method of constructing the effective equation which describes the behavior of small-amplitude solutions for a nonlinear PDE in finite volume, provided that the linear part of the equation is a hamiltonian system with a pure imaginary discrete spectrum. The effective equation is obtained by retaining only the resonant terms of the nonlinearity (which may be hamiltonian, or may be not); the assertion that it describes the limiting behavior of small-amplitude solutions is a rigorous mathematical theorem. In particular, the method applies to the three- and four-wave systems. We demonstrate that different possible types of energy transport are covered by this method, depending on whether the set of resonances splits into finite clusters (this happens, e.g. in case of the Charney-Hasegawa-Mima equation), or is connected (this happens, e.g. in the case of the NLS equation if the space-dimension is at least two). For equations of the first type the energy transition to high frequencies does not hold, while for equations of the second type it may take place. Our method applies to various weakly nonlinear wave systems, appearing in plasma, meteorology and oceanography.
Nonlinear ordinary difference equations
NASA Technical Reports Server (NTRS)
Caughey, T. K.
1979-01-01
Future space vehicles will be relatively large and flexible, and active control will be necessary to maintain geometrical configuration. While the stresses and strains in these space vehicles are not expected to be excessively large, their cumulative effects will cause significant geometrical nonlinearities to appear in the equations of motion, in addition to the nonlinearities caused by material properties. Since the only effective tool for the analysis of such large complex structures is the digital computer, it will be necessary to gain a better understanding of the nonlinear ordinary difference equations which result from the time discretization of the semidiscrete equations of motion for such structures.
NASA Astrophysics Data System (ADS)
Pierret, Frédéric
2016-02-01
We derived the equations of Celestial Mechanics governing the variation of the orbital elements under a stochastic perturbation, thereby generalizing the classical Gauss equations. Explicit formulas are given for the semimajor axis, the eccentricity, the inclination, the longitude of the ascending node, the pericenter angle, and the mean anomaly, which are expressed in term of the angular momentum vector H per unit of mass and the energy E per unit of mass. Together, these formulas are called the stochastic Gauss equations, and they are illustrated numerically on an example from satellite dynamics.
Nonlinear differential equations
Dresner, L.
1988-01-01
This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.
Relativistic Guiding Center Equations
White, R. B.; Gobbin, M.
2014-10-01
In toroidal fusion devices it is relatively easy that electrons achieve relativistic velocities, so to simulate runaway electrons and other high energy phenomena a nonrelativistic guiding center formalism is not sufficient. Relativistic guiding center equations including flute mode time dependent field perturbations are derived. The same variables as used in a previous nonrelativistic guiding center code are adopted, so that a straightforward modifications of those equations can produce a relativistic version.
SIMULTANEOUS DIFFERENTIAL EQUATION COMPUTER
Collier, D.M.; Meeks, L.A.; Palmer, J.P.
1960-05-10
A description is given for an electronic simulator for a system of simultaneous differential equations, including nonlinear equations. As a specific example, a homogeneous nuclear reactor system including a reactor fluid, heat exchanger, and a steam boiler may be simulated, with the nonlinearity resulting from a consideration of temperature effects taken into account. The simulator includes three operational amplifiers, a multiplier, appropriate potential sources, and interconnecting R-C networks.
Set Equation Transformation System.
2002-03-22
Version 00 SETS is used for symbolic manipulation of Boolean equations, particularly the reduction of equations by the application of Boolean identities. It is a flexible and efficient tool for performing probabilistic risk analysis (PRA), vital area analysis, and common cause analysis. The equation manipulation capabilities of SETS can also be used to analyze noncoherent fault trees and determine prime implicants of Boolean functions, to verify circuit design implementation, to determine minimum cost fire protectionmore » requirements for nuclear reactor plants, to obtain solutions to combinatorial optimization problems with Boolean constraints, and to determine the susceptibility of a facility to unauthorized access through nullification of sensors in its protection system. Two auxiliary programs, SEP and FTD, are included. SEP performs the quantitative analysis of reduced Boolean equations (minimal cut sets) produced by SETS. The user can manipulate and evaluate the equations to find the probability of occurrence of any desired event and to produce an importance ranking of the terms and events in an equation. FTD is a fault tree drawing program which uses the proprietary ISSCO DISSPLA graphics software to produce an annotated drawing of a fault tree processed by SETS. The DISSPLA routines are not included.« less
Introducing Chemical Formulae and Equations.
ERIC Educational Resources Information Center
Dawson, Chris; Rowell, Jack
1979-01-01
Discusses when the writing of chemical formula and equations can be introduced in the school science curriculum. Also presents ways in which formulae and equations learning can be aided and some examples for balancing and interpreting equations. (HM)
The Bernoulli-Poiseuille Equation.
ERIC Educational Resources Information Center
Badeer, Henry S.; Synolakis, Costas E.
1989-01-01
Describes Bernoulli's equation and Poiseuille's equation for fluid dynamics. Discusses the application of the combined Bernoulli-Poiseuille equation in real flows, such as viscous flows under gravity and acceleration. (YP)
Parallel tridiagonal equation solvers
NASA Technical Reports Server (NTRS)
Stone, H. S.
1974-01-01
Three parallel algorithms were compared for the direct solution of tridiagonal linear systems of equations. The algorithms are suitable for computers such as ILLIAC 4 and CDC STAR. For array computers similar to ILLIAC 4, cyclic odd-even reduction has the least operation count for highly structured sets of equations, and recursive doubling has the least count for relatively unstructured sets of equations. Since the difference in operation counts for these two algorithms is not substantial, their relative running times may be more related to overhead operations, which are not measured in this paper. The third algorithm, based on Buneman's Poisson solver, has more arithmetic operations than the others, and appears to be the least favorable. For pipeline computers similar to CDC STAR, cyclic odd-even reduction appears to be the most preferable algorithm for all cases.
Nonlocal electrical diffusion equation
NASA Astrophysics Data System (ADS)
Gómez-Aguilar, J. F.; Escobar-Jiménez, R. F.; Olivares-Peregrino, V. H.; Benavides-Cruz, M.; Calderón-Ramón, C.
2016-07-01
In this paper, we present an analysis and modeling of the electrical diffusion equation using the fractional calculus approach. This alternative representation for the current density is expressed in terms of the Caputo derivatives, the order for the space domain is 0<β≤1 and for the time domain is 0<γ≤2. We present solutions for the full fractional equation involving space and time fractional derivatives using numerical methods based on Fourier variable separation. The case with spatial fractional derivatives leads to Levy flight type phenomena, while the time fractional equation is related to sub- or super diffusion. We show that the mathematical concept of fractional derivatives can be useful to understand the behavior of semiconductors, the design of solar panels, electrochemical phenomena and the description of anomalous complex processes.
Stochastic differential equations
Sobczyk, K. )
1990-01-01
This book provides a unified treatment of both regular (or random) and Ito stochastic differential equations. It focuses on solution methods, including some developed only recently. Applications are discussed, in particular an insight is given into both the mathematical structure, and the most efficient solution methods (analytical as well as numerical). Starting from basic notions and results of the theory of stochastic processes and stochastic calculus (including Ito's stochastic integral), many principal mathematical problems and results related to stochastic differential equations are expounded here for the first time. Applications treated include those relating to road vehicles, earthquake excitations and offshore structures.
NASA Technical Reports Server (NTRS)
Markley, F. Landis
1995-01-01
Kepler's Equation is solved over the entire range of elliptic motion by a fifth-order refinement of the solution of a cubic equation. This method is not iterative, and requires only four transcendental function evaluations: a square root, a cube root, and two trigonometric functions. The maximum relative error of the algorithm is less than one part in 10(exp 18), exceeding the capability of double-precision computer arithmetic. Roundoff errors in double-precision implementation of the algorithm are addressed, and procedures to avoid them are developed.
The Statistical Drake Equation
NASA Astrophysics Data System (ADS)
Maccone, Claudio
2010-12-01
We provide the statistical generalization of the Drake equation. From a simple product of seven positive numbers, the Drake equation is now turned into the product of seven positive random variables. We call this "the Statistical Drake Equation". The mathematical consequences of this transformation are then derived. The proof of our results is based on the Central Limit Theorem (CLT) of Statistics. In loose terms, the CLT states that the sum of any number of independent random variables, each of which may be ARBITRARILY distributed, approaches a Gaussian (i.e. normal) random variable. This is called the Lyapunov Form of the CLT, or the Lindeberg Form of the CLT, depending on the mathematical constraints assumed on the third moments of the various probability distributions. In conclusion, we show that: The new random variable N, yielding the number of communicating civilizations in the Galaxy, follows the LOGNORMAL distribution. Then, as a consequence, the mean value of this lognormal distribution is the ordinary N in the Drake equation. The standard deviation, mode, and all the moments of this lognormal N are also found. The seven factors in the ordinary Drake equation now become seven positive random variables. The probability distribution of each random variable may be ARBITRARY. The CLT in the so-called Lyapunov or Lindeberg forms (that both do not assume the factors to be identically distributed) allows for that. In other words, the CLT "translates" into our statistical Drake equation by allowing an arbitrary probability distribution for each factor. This is both physically realistic and practically very useful, of course. An application of our statistical Drake equation then follows. The (average) DISTANCE between any two neighboring and communicating civilizations in the Galaxy may be shown to be inversely proportional to the cubic root of N. Then, in our approach, this distance becomes a new random variable. We derive the relevant probability density
Comparison of Kernel Equating and Item Response Theory Equating Methods
ERIC Educational Resources Information Center
Meng, Yu
2012-01-01
The kernel method of test equating is a unified approach to test equating with some advantages over traditional equating methods. Therefore, it is important to evaluate in a comprehensive way the usefulness and appropriateness of the Kernel equating (KE) method, as well as its advantages and disadvantages compared with several popular item…
Accumulative Equating Error after a Chain of Linear Equatings
ERIC Educational Resources Information Center
Guo, Hongwen
2010-01-01
After many equatings have been conducted in a testing program, equating errors can accumulate to a degree that is not negligible compared to the standard error of measurement. In this paper, the author investigates the asymptotic accumulative standard error of equating (ASEE) for linear equating methods, including chained linear, Tucker, and…
Parallel Multigrid Equation Solver
2001-09-07
Prometheus is a fully parallel multigrid equation solver for matrices that arise in unstructured grid finite element applications. It includes a geometric and an algebraic multigrid method and has solved problems of up to 76 mullion degrees of feedom, problems in linear elasticity on the ASCI blue pacific and ASCI red machines.
Do Differential Equations Swing?
ERIC Educational Resources Information Center
Maruszewski, Richard F., Jr.
2006-01-01
One of the units of in a standard differential equations course is a discussion of the oscillatory motion of a spring and the associated material on forcing functions and resonance. During the presentation on practical resonance, the instructor may tell students that it is similar to when they take their siblings to the playground and help them on…
Modelling by Differential Equations
ERIC Educational Resources Information Center
Chaachoua, Hamid; Saglam, Ayse
2006-01-01
This paper aims to show the close relation between physics and mathematics taking into account especially the theory of differential equations. By analysing the problems posed by scientists in the seventeenth century, we note that physics is very important for the emergence of this theory. Taking into account this analysis, we show the…
ERIC Educational Resources Information Center
Fay, Temple H.
2010-01-01
Through numerical investigations, we study examples of the forced quadratic spring equation [image omitted]. By performing trial-and-error numerical experiments, we demonstrate the existence of stability boundaries in the phase plane indicating initial conditions yielding bounded solutions, investigate the resonance boundary in the [omega]…
Generalized reduced magnetohydrodynamic equations
Kruger, S.E.
1999-02-01
A new derivation of reduced magnetohydrodynamic (MHD) equations is presented. A multiple-time-scale expansion is employed. It has the advantage of clearly separating the three time scales of the problem associated with (1) MHD equilibrium, (2) fluctuations whose wave vector is aligned perpendicular to the magnetic field, and (3) those aligned parallel to the magnetic field. The derivation is carried out without relying on a large aspect ratio assumption; therefore this model can be applied to any general configuration. By accounting for the MHD equilibrium and constraints to eliminate the fast perpendicular waves, equations are derived to evolve scalar potential quantities on a time scale associated with the parallel wave vector (shear-Alfven wave time scale), which is the time scale of interest for MHD instability studies. Careful attention is given in the derivation to satisfy energy conservation and to have manifestly divergence-free magnetic fields to all orders in the expansion parameter. Additionally, neoclassical closures and equilibrium shear flow effects are easily accounted for in this model. Equations for the inner resistive layer are derived which reproduce the linear ideal and resistive stability criterion of Glasser, Greene, and Johnson. The equations have been programmed into a spectral initial value code and run with shear flow that is consistent with the equilibrium input into the code. Linear results of tearing modes with shear flow are presented which differentiate the effects of shear flow gradients in the layer with the effects of the shear flow decoupling multiple harmonics.
Structural Equation Model Trees
ERIC Educational Resources Information Center
Brandmaier, Andreas M.; von Oertzen, Timo; McArdle, John J.; Lindenberger, Ulman
2013-01-01
In the behavioral and social sciences, structural equation models (SEMs) have become widely accepted as a modeling tool for the relation between latent and observed variables. SEMs can be seen as a unification of several multivariate analysis techniques. SEM Trees combine the strengths of SEMs and the decision tree paradigm by building tree…
Brownian motion from Boltzmann's equation.
NASA Technical Reports Server (NTRS)
Montgomery, D.
1971-01-01
Two apparently disparate lines of inquiry in kinetic theory are shown to be equivalent: (1) Brownian motion as treated by the (stochastic) Langevin equation and Fokker-Planck equation; and (2) Boltzmann's equation. The method is to derive the kinetic equation for Brownian motion from the Boltzmann equation for a two-component neutral gas by a simultaneous expansion in the density and mass ratios.
Supersymmetric fifth order evolution equations
Tian, K.; Liu, Q. P.
2010-03-08
This paper considers supersymmetric fifth order evolution equations. Within the framework of symmetry approach, we give a list containing six equations, which are (potentially) integrable systems. Among these equations, the most interesting ones include a supersymmetric Sawada-Kotera equation and a novel supersymmetric fifth order KdV equation. For the latter, we supply some properties such as a Hamiltonian structures and a possible recursion operator.
Nikolaevskiy equation with dispersion.
Simbawa, Eman; Matthews, Paul C; Cox, Stephen M
2010-03-01
The Nikolaevskiy equation was originally proposed as a model for seismic waves and is also a model for a wide variety of systems incorporating a neutral "Goldstone" mode, including electroconvection and reaction-diffusion systems. It is known to exhibit chaotic dynamics at the onset of pattern formation, at least when the dispersive terms in the equation are suppressed, as is commonly the practice in previous analyses. In this paper, the effects of reinstating the dispersive terms are examined. It is shown that such terms can stabilize some of the spatially periodic traveling waves; this allows us to study the loss of stability and transition to chaos of the waves. The secondary stability diagram ("Busse balloon") for the traveling waves can be remarkably complicated. PMID:20365845
Causal electromagnetic interaction equations
Zinoviev, Yury M.
2011-02-15
For the electromagnetic interaction of two particles the relativistic causal quantum mechanics equations are proposed. These equations are solved for the case when the second particle moves freely. The initial wave functions are supposed to be smooth and rapidly decreasing at the infinity. This condition is important for the convergence of the integrals similar to the integrals of quantum electrodynamics. We also consider the singular initial wave functions in the particular case when the second particle mass is equal to zero. The discrete energy spectrum of the first particle wave function is defined by the initial wave function of the free-moving second particle. Choosing the initial wave functions of the free-moving second particle it is possible to obtain a practically arbitrary discrete energy spectrum.
Generalized reduced MHD equations
Kruger, S.E.; Hegna, C.C.; Callen, J.D.
1998-07-01
A new derivation of reduced magnetohydrodynamic (MHD) equations is presented. A multiple-time-scale expansion is employed. It has the advantage of clearly separating the three time scales of the problem associated with (1) MHD equilibrium, (2) fluctuations whose wave vector is aligned perpendicular to the magnetic field, and (3) those aligned parallel to the magnetic field. The derivation is carried out without relying on a large aspect ratio assumption; therefore this model can be applied to any general toroidal configuration. By accounting for the MHD equilibrium and constraints to eliminate the fast perpendicular waves, equations are derived to evolve scalar potential quantities on a time scale associated with the parallel wave vector (shear-alfven wave time scale), which is the time scale of interest for MHD instability studies. Careful attention is given in the derivation to satisfy energy conservation and to have manifestly divergence-free magnetic fields to all orders in the expansion parameter. Additionally, neoclassical closures and equilibrium shear flow effects are easily accounted for in this model. Equations for the inner resistive layer are derived which reproduce the linear ideal and resistive stability criterion of Glasser, Greene, and Johnson.
NASA Astrophysics Data System (ADS)
Konesky, Gregory
2009-08-01
In the almost half century since the Drake Equation was first conceived, a number of profound discoveries have been made that require each of the seven variables of this equation to be reconsidered. The discovery of hydrothermal vents on the ocean floor, for example, as well as the ever-increasing extreme conditions in which life is found on Earth, suggest a much wider range of possible extraterrestrial habitats. The growing consensus that life originated very early in Earth's history also supports this suggestion. The discovery of exoplanets with a wide range of host star types, and attendant habitable zones, suggests that life may be possible in planetary systems with stars quite unlike our Sun. Stellar evolution also plays an important part in that habitable zones are mobile. The increasing brightness of our Sun over the next few billion years, will place the Earth well outside the present habitable zone, but will then encompass Mars, giving rise to the notion that some Drake Equation variables, such as the fraction of planets on which life emerges, may have multiple values.
Double-Plate Penetration Equations
NASA Technical Reports Server (NTRS)
Hayashida, K. B.; Robinson, J. H.
2000-01-01
This report compares seven double-plate penetration predictor equations for accuracy and effectiveness of a shield design. Three of the seven are the Johnson Space Center original, modified, and new Cour-Palais equations. The other four are the Nysmith, Lundeberg-Stern-Bristow, Burch, and Wilkinson equations. These equations, except the Wilkinson equation, were derived from test results, with the velocities ranging up to 8 km/sec. Spreadsheet software calculated the projectile diameters for various velocities for the different equations. The results were plotted on projectile diameter versus velocity graphs for the expected orbital debris impact velocities ranging from 2 to 15 km/sec. The new Cour-Palais double-plate penetration equation was compared to the modified Cour-Palais single-plate penetration equation. Then the predictions from each of the seven double-plate penetration equations were compared to each other for a chosen shield design. Finally, these results from the equations were compared with test results performed at the NASA Marshall Space Flight Center. Because the different equations predict a wide range of projectile diameters at any given velocity, it is very difficult to choose the "right" prediction equation for shield configurations other than those exactly used in the equations' development. Although developed for various materials, the penetration equations alone cannot be relied upon to accurately predict the effectiveness of a shield without using hypervelocity impact tests to verify the design.
Reduction operators of Burgers equation
Pocheketa, Oleksandr A.; Popovych, Roman O.
2013-01-01
The solution of the problem on reduction operators and nonclassical reductions of the Burgers equation is systematically treated and completed. A new proof of the theorem on the special “no-go” case of regular reduction operators is presented, and the representation of the coefficients of operators in terms of solutions of the initial equation is constructed for this case. All possible nonclassical reductions of the Burgers equation to single ordinary differential equations are exhaustively described. Any Lie reduction of the Burgers equation proves to be equivalent via the Hopf–Cole transformation to a parameterized family of Lie reductions of the linear heat equation. PMID:23576819
New application to Riccati equation
NASA Astrophysics Data System (ADS)
Taogetusang; Sirendaoerji; Li, Shu-Min
2010-08-01
To seek new infinite sequence of exact solutions to nonlinear evolution equations, this paper gives the formula of nonlinear superposition of the solutions and Bäcklund transformation of Riccati equation. Based on the tanh-function expansion method and homogenous balance method, new infinite sequence of exact solutions to Zakharov-Kuznetsov equation, Karamoto-Sivashinsky equation and the set of (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov equations are obtained with the aid of symbolic computation system Mathematica. The method is of significance to construct infinite sequence exact solutions to other nonlinear evolution equations.
Evaluating Equating Results: Percent Relative Error for Chained Kernel Equating
ERIC Educational Resources Information Center
Jiang, Yanlin; von Davier, Alina A.; Chen, Haiwen
2012-01-01
This article presents a method for evaluating equating results. Within the kernel equating framework, the percent relative error (PRE) for chained equipercentile equating was computed under the nonequivalent groups with anchor test (NEAT) design. The method was applied to two data sets to obtain the PRE, which can be used to measure equating…
Differential Equations Compatible with Boundary Rational qKZ Equation
NASA Astrophysics Data System (ADS)
Takeyama, Yoshihiro
2011-10-01
We give diffierential equations compatible with the rational qKZ equation with boundary reflection. The total system contains the trigonometric degeneration of the bispectral qKZ equation of type (Cěen, Cn) which in the case of type GLn was studied by van Meer and Stokman. We construct an integral formula for solutions to our compatible system in a special case.
The compressible adjoint equations in geodynamics: equations and numerical assessment
NASA Astrophysics Data System (ADS)
Ghelichkhan, Siavash; Bunge, Hans-Peter
2016-04-01
The adjoint method is a powerful means to obtain gradient information in a mantle convection model relative to past flow structure. While the adjoint equations in geodynamics have been derived for the conservation equations of mantle flow in their incompressible form, the applicability of this approximation to Earth is limited, because density increases by almost a factor of two from the surface to the Core Mantle Boundary. Here we introduce the compressible adjoint equations for the conservation equations in the anelastic-liquid approximation. Our derivation applies an operator formulation in Hilbert spaces, to connect to recent work in seismology (Fichtner et al (2006)) and geodynamics (Horbach et al (2014)), where the approach was used to derive the adjoint equations for the wave equation and incompressible mantle flow. We present numerical tests of the newly derived equations based on twin experiments, focusing on three simulations. A first, termed Compressible, assumes the compressible forward and adjoint equations, and represents the consistent means of including compressibility effects. A second, termed Mixed, applies the compressible forward equation, but ignores compressibility effects in the adjoint equations, where the incompressible equations are used instead. A third simulation, termed Incompressible, neglects compressibility effects entirely in the forward and adjoint equations relative to the reference twin. The compressible and mixed formulations successfully restore earlier mantle flow structure, while the incompressible formulation yields noticeable artifacts. Our results suggest the use of a compressible formulation, when applying the adjoint method to seismically derived mantle heterogeneity structure.
Estimating Equating Error in Observed-Score Equating. Research Report.
ERIC Educational Resources Information Center
van der Linden, Wim J.
Traditionally, error in equating observed scores on two versions of a test is defined as the difference between the transformations that equate the quantiles of their distributions in the sample and in the population of examinees. This definition underlies, for example, the well-known approximation to the standard error of equating by Lord (1982).…
NASA Astrophysics Data System (ADS)
Taff, L. G.; Brennan, T. A.
1989-06-01
Intrigued by the recent advances in research on solving Kepler's equation, we have attacked the problem too. Our contributions emphasize the unified derivation of all known bounds and several starting values, a proof of the optimality of these bounds, a very thorough numerical exploration of a large variety of starting values and solution techniques in both mean anomaly/eccentricity space and eccentric anomaly/eccentricity space, and finally the best and simplest starting value/solution algorithm: M + e and Wegstein's secant modification of the method of successive substitutions. The very close second is Broucke's bounds coupled with Newton's second-order scheme.
The Arrhenius equation revisited.
Peleg, Micha; Normand, Mark D; Corradini, Maria G
2012-01-01
The Arrhenius equation has been widely used as a model of the temperature effect on the rate of chemical reactions and biological processes in foods. Since the model requires that the rate increase monotonically with temperature, its applicability to enzymatic reactions and microbial growth, which have optimal temperature, is obviously limited. This is also true for microbial inactivation and chemical reactions that only start at an elevated temperature, and for complex processes and reactions that do not follow fixed order kinetics, that is, where the isothermal rate constant, however defined, is a function of both temperature and time. The linearity of the Arrhenius plot, that is, Ln[k(T)] vs. 1/T where T is in °K has been traditionally considered evidence of the model's validity. Consequently, the slope of the plot has been used to calculate the reaction or processes' "energy of activation," usually without independent verification. Many experimental and simulated rate constant vs. temperature relationships that yield linear Arrhenius plots can also be described by the simpler exponential model Ln[k(T)/k(T(reference))] = c(T-T(reference)). The use of the exponential model or similar empirical alternative would eliminate the confusing temperature axis inversion, the unnecessary compression of the temperature scale, and the need for kinetic assumptions that are hard to affirm in food systems. It would also eliminate the reference to the Universal gas constant in systems where a "mole" cannot be clearly identified. Unless proven otherwise by independent experiments, one cannot dismiss the notion that the apparent linearity of the Arrhenius plot in many food systems is due to a mathematical property of the model's equation rather than to the existence of a temperature independent "energy of activation." If T+273.16°C in the Arrhenius model's equation is replaced by T+b, where the numerical value of the arbitrary constant b is substantially larger than T and T
Makkonen, Lasse
2016-04-01
Young's construction for a contact angle at a three-phase intersection forms the basis of all fields of science that involve wetting and capillary action. We find compelling evidence from recent experimental results on the deformation of a soft solid at the contact line, and displacement of an elastic wire immersed in a liquid, that Young's equation can only be interpreted by surface energies, and not as a balance of surface tensions. It follows that the a priori variable in finding equilibrium is not the position of the contact line, but the contact angle. This finding provides the explanation for the pinning of a contact line. PMID:26940644
Conservational PDF Equations of Turbulence
NASA Technical Reports Server (NTRS)
Shih, Tsan-Hsing; Liu, Nan-Suey
2010-01-01
Recently we have revisited the traditional probability density function (PDF) equations for the velocity and species in turbulent incompressible flows. They are all unclosed due to the appearance of various conditional means which are modeled empirically. However, we have observed that it is possible to establish a closed velocity PDF equation and a closed joint velocity and species PDF equation through conditions derived from the integral form of the Navier-Stokes equations. Although, in theory, the resulted PDF equations are neither general nor unique, they nevertheless lead to the exact transport equations for the first moment as well as all higher order moments. We refer these PDF equations as the conservational PDF equations. This observation is worth further exploration for its validity and CFD application
Solitons and nonlinear wave equations
Dodd, Roger K.; Eilbeck, J. Chris; Gibbon, John D.; Morris, Hedley C.
1982-01-01
A discussion of the theory and applications of classical solitons is presented with a brief treatment of quantum mechanical effects which occur in particle physics and quantum field theory. The subjects addressed include: solitary waves and solitons, scattering transforms, the Schroedinger equation and the Korteweg-de Vries equation, and the inverse method for the isospectral Schroedinger equation and the general solution of the solvable nonlinear equations. Also considered are: isolation of the Korteweg-de Vries equation in some physical examples, the Zakharov-Shabat/AKNS inverse method, kinks and the sine-Gordon equation, the nonlinear Schroedinger equation and wave resonance interactions, amplitude equations in unstable systems, and numerical studies of solitons. 45 references.
``Riemann equations'' in bidifferential calculus
NASA Astrophysics Data System (ADS)
Chvartatskyi, O.; Müller-Hoissen, F.; Stoilov, N.
2015-10-01
We consider equations that formally resemble a matrix Riemann (or Hopf) equation in the framework of bidifferential calculus. With different choices of a first-order bidifferential calculus, we obtain a variety of equations, including a semi-discrete and a fully discrete version of the matrix Riemann equation. A corresponding universal solution-generating method then either yields a (continuous or discrete) Cole-Hopf transformation, or leaves us with the problem of solving Riemann equations (hence an application of the hodograph method). If the bidifferential calculus extends to second order, solutions of a system of "Riemann equations" are also solutions of an equation that arises, on the universal level of bidifferential calculus, as an integrability condition. Depending on the choice of bidifferential calculus, the latter can represent a number of prominent integrable equations, like self-dual Yang-Mills, as well as matrix versions of the two-dimensional Toda lattice, Hirota's bilinear difference equation, (2+1)-dimensional Nonlinear Schrödinger (NLS), Kadomtsev-Petviashvili (KP) equation, and Davey-Stewartson equations. For all of them, a recent (non-isospectral) binary Darboux transformation result in bidifferential calculus applies, which can be specialized to generate solutions of the associated "Riemann equations." For the latter, we clarify the relation between these specialized binary Darboux transformations and the aforementioned solution-generating method. From (arbitrary size) matrix versions of the "Riemann equations" associated with an integrable equation, possessing a bidifferential calculus formulation, multi-soliton-type solutions of the latter can be generated. This includes "breaking" multi-soliton-type solutions of the self-dual Yang-Mills and the (2+1)-dimensional NLS equation, which are parametrized by solutions of Riemann equations.
Solving Nonlinear Coupled Differential Equations
NASA Technical Reports Server (NTRS)
Mitchell, L.; David, J.
1986-01-01
Harmonic balance method developed to obtain approximate steady-state solutions for nonlinear coupled ordinary differential equations. Method usable with transfer matrices commonly used to analyze shaft systems. Solution to nonlinear equation, with periodic forcing function represented as sum of series similar to Fourier series but with form of terms suggested by equation itself.
Successfully Transitioning to Linear Equations
ERIC Educational Resources Information Center
Colton, Connie; Smith, Wendy M.
2014-01-01
The Common Core State Standards for Mathematics (CCSSI 2010) asks students in as early as fourth grade to solve word problems using equations with variables. Equations studied at this level generate a single solution, such as the equation x + 10 = 25. For students in fifth grade, the Common Core standard for algebraic thinking expects them to…
The Forced Hard Spring Equation
ERIC Educational Resources Information Center
Fay, Temple H.
2006-01-01
Through numerical investigations, various examples of the Duffing type forced spring equation with epsilon positive, are studied. Since [epsilon] is positive, all solutions to the associated homogeneous equation are periodic and the same is true with the forcing applied. The damped equation exhibits steady state trajectories with the interesting…
Equating with Miditests Using IRT
ERIC Educational Resources Information Center
Fitzpatrick, Joseph; Skorupski, William P.
2016-01-01
The equating performance of two internal anchor test structures--miditests and minitests--is studied for four IRT equating methods using simulated data. Originally proposed by Sinharay and Holland, miditests are anchors that have the same mean difficulty as the overall test but less variance in item difficulties. Four popular IRT equating methods…
On stochastic diffusion equations and stochastic Burgers' equations
NASA Astrophysics Data System (ADS)
Truman, A.; Zhao, H. Z.
1996-01-01
In this paper we construct a strong solution for the stochastic Hamilton Jacobi equation by using stochastic classical mechanics before the caustics. We thereby obtain the viscosity solution for a certain class of inviscid stochastic Burgers' equations. This viscosity solution is not continuous beyond the caustics of the corresponding Hamilton Jacobi equation. The Hopf-Cole transformation is used to identify the stochastic heat equation and the viscous stochastic Burgers' equation. The exact solutions for the above two equations are given in terms of the stochastic Hamilton Jacobi function under a no-caustic condition. We construct the heat kernel for the stochastic heat equation for zero potentials in hyperbolic space and for harmonic oscillator potentials in Euclidean space thereby obtaining the stochastic Mehler formula.
Generalized Klein-Kramers equations
NASA Astrophysics Data System (ADS)
Fa, Kwok Sau
2012-12-01
A generalized Klein-Kramers equation for a particle interacting with an external field is proposed. The equation generalizes the fractional Klein-Kramers equation introduced by Barkai and Silbey [J. Phys. Chem. B 104, 3866 (2000), 10.1021/jp993491m]. Besides, the generalized Klein-Kramers equation can also recover the integro-differential Klein-Kramers equation for continuous-time random walk; this means that it can describe the subdiffusive and superdiffusive regimes in the long-time limit. Moreover, analytic solutions for first two moments both in velocity and displacement (for force-free case) are obtained, and their dynamic behaviors are investigated.
Multinomial Diffusion Equation
Balter, Ariel I.; Tartakovsky, Alexandre M.
2011-06-01
We have developed a novel stochastic, space/time discrete representation of particle diffusion (e.g. Brownian motion) based on discrete probability distributions. We show that in the limit of both very small time step and large concentration, our description is equivalent to the space/time continuous stochastic diffusion equation. Being discrete in both time and space, our model can be used as an extremely accurate, efficient, and stable stochastic finite-difference diffusion algorithm when concentrations are so small that computationally expensive particle-based methods are usually needed. Through numerical simulations, we show that our method can generate realizations that capture the statistical properties of particle simulations. While our method converges converges to both the correct ensemble mean and ensemble variance very quickly with decreasing time step, but for small concentration, the stochastic diffusion PDE does not, even for very small time steps.
NASA Astrophysics Data System (ADS)
Cardona, Carlos; Gomez, Humberto
2016-06-01
Recently the CHY approach has been extended to one loop level using elliptic functions and modular forms over a Jacobian variety. Due to the difficulty in manipulating these kind of functions, we propose an alternative prescription that is totally algebraic. This new proposal is based on an elliptic algebraic curve embedded in a mathbb{C}{P}^2 space. We show that for the simplest integrand, namely the n - gon, our proposal indeed reproduces the expected result. By using the recently formulated Λ-algorithm, we found a novel recurrence relation expansion in terms of tree level off-shell amplitudes. Our results connect nicely with recent results on the one-loop formulation of the scattering equations. In addition, this new proposal can be easily stretched out to hyperelliptic curves in order to compute higher genus.
On nonautonomous Dirac equation
Hovhannisyan, Gro; Liu Wen
2009-12-15
We construct the fundamental solution of time dependent linear ordinary Dirac system in terms of unknown phase functions. This construction gives approximate representation of solutions which is useful for the study of asymptotic behavior. Introducing analog of Rayleigh quotient for differential equations we generalize Hartman-Wintner asymptotic integration theorems with the error estimates for applications to the Dirac system. We also introduce the adiabatic invariants for the Dirac system, which are similar to the adiabatic invariant of Lorentz's pendulum. Using a small parameter method it is shown that the change in the adiabatic invariants approaches zero with the power speed as a small parameter approaches zero. As another application we calculate the transition probabilities for the Dirac system. We show that for the special choice of electromagnetic field, the only transition of an electron to the positron with the opposite spin orientation is possible.
Potential distribution around a test charge in a positive dust-electron plasma
NASA Astrophysics Data System (ADS)
Ali, S.
2016-06-01
The electrostatic potential caused by a test-charge particle in a positive dust-electron plasma is studied, accounting for the dust-charge fluctuations associated with ultraviolet photoelectron and thermionic emissions. For this purpose, the set of Vlasov-Poisson equations coupled with the dust charging equation is solved by using the space-time Fourier transform technique. As a consequence, a modified dielectric response function is obtained for dust-acoustic waves in a positive dust-electron plasma. By imposing certain conditions on the velocity of the test charge, the electrostatic potential is decomposed into the Debye-H¨uckel (DH), wake-field (WF), and far-field (FF) potentials that are significantly modified in the limit of a large dust-charge relaxation rate both analytically and numerically. The results can be helpful for understanding dust crystallization/coagulation in twocomponent plasmas, where positively charged dust grains are present.
A New Class of Non-Linear, Finite-Volume Methods for Vlasov Simulation
Banks, J W; Hittinger, J A
2009-11-24
Methods for the numerical discretization of the Vlasov equation should efficiently use the phase space discretization and should introduce only enough numerical dissipation to promote stability and control oscillations. A new high-order, non-linear, finite-volume algorithm for the Vlasov equation that discretely conserves particle number and controls oscillations is presented. The method is fourth-order in space and time in well-resolved regions, but smoothly reduces to a third-order upwind scheme as features become poorly resolved. The new scheme is applied to several standard problems for the Vlasov-Poisson system, and the results are compared with those from other finite-volume approaches, including an artificial viscosity scheme and the Piecewise Parabolic Method. It is shown that the new scheme is able to control oscillations while preserving a higher degree of fidelity of the solution than the other approaches.
Entwined paths, difference equations, and the Dirac equation
Ord, G.N.; Mann, R.B.
2003-02-01
Entwined space-time paths are bound pairs of trajectories which are traversed in opposite directions with respect to macroscopic time. In this paper, we show that ensembles of entwined paths on a discrete space-time lattice are simply described by coupled difference equations which are discrete versions of the Dirac equation. There is no analytic continuation, explicit or forced, involved in this description. The entwined paths are ''self-quantizing.'' We also show that simple classical stochastic processes that generate the difference equations as ensemble averages are stable numerically and converge at a rate governed by the details of the stochastic process. This result establishes the Dirac equation in one dimension as a phenomenological equation describing an underlying classical stochastic process, in the same sense that the diffusion and telegraph equations are phenomenological descriptions of stochastic processes.
Mode decomposition evolution equations
Wang, Yang; Wei, Guo-Wei; Yang, Siyang
2011-01-01
Partial differential equation (PDE) based methods have become some of the most powerful tools for exploring the fundamental problems in signal processing, image processing, computer vision, machine vision and artificial intelligence in the past two decades. The advantages of PDE based approaches are that they can be made fully automatic, robust for the analysis of images, videos and high dimensional data. A fundamental question is whether one can use PDEs to perform all the basic tasks in the image processing. If one can devise PDEs to perform full-scale mode decomposition for signals and images, the modes thus generated would be very useful for secondary processing to meet the needs in various types of signal and image processing. Despite of great progress in PDE based image analysis in the past two decades, the basic roles of PDEs in image/signal analysis are only limited to PDE based low-pass filters, and their applications to noise removal, edge detection, segmentation, etc. At present, it is not clear how to construct PDE based methods for full-scale mode decomposition. The above-mentioned limitation of most current PDE based image/signal processing methods is addressed in the proposed work, in which we introduce a family of mode decomposition evolution equations (MoDEEs) for a vast variety of applications. The MoDEEs are constructed as an extension of a PDE based high-pass filter (Europhys. Lett., 59(6): 814, 2002) by using arbitrarily high order PDE based low-pass filters introduced by Wei (IEEE Signal Process. Lett., 6(7): 165, 1999). The use of arbitrarily high order PDEs is essential to the frequency localization in the mode decomposition. Similar to the wavelet transform, the present MoDEEs have a controllable time-frequency localization and allow a perfect reconstruction of the original function. Therefore, the MoDEE operation is also called a PDE transform. However, modes generated from the present approach are in the spatial or time domain and can be
Menikoff, Ralph
2015-12-15
The JWL equation of state (EOS) is frequently used for the products (and sometimes reactants) of a high explosive (HE). Here we review and systematically derive important properties. The JWL EOS is of the Mie-Grueneisen form with a constant Grueneisen coefficient and a constants specific heat. It is thermodynamically consistent to specify the temperature at a reference state. However, increasing the reference state temperature restricts the EOS domain in the (V, e)-plane of phase space. The restrictions are due to the conditions that P ≥ 0, T ≥ 0, and the isothermal bulk modulus is positive. Typically, this limits the low temperature regime in expansion. The domain restrictions can result in the P-T equilibrium EOS of a partly burned HE failing to have a solution in some cases. For application to HE, the heat of detonation is discussed. Example JWL parameters for an HE, both products and reactions, are used to illustrate the restrictions on the domain of the EOS.
On the generalized Jacobi equation
NASA Astrophysics Data System (ADS)
Perlick, Volker
2008-05-01
The standard text-book Jacobi equation (equation of geodesic deviation) arises by linearizing the geodesic equation around some chosen geodesic, where the linearization is done with respect to the coordinates and the velocities. The generalized Jacobi equation, introduced by Hodgkinson in 1972 and further developed by Mashhoon and others, arises if the linearization is done only with respect to the coordinates, but not with respect to the velocities. The resulting equation has been studied by several authors in some detail for timelike geodesics in a Lorentzian manifold. Here we begin by briefly considering the generalized Jacobi equation on affine manifolds, without a metric; then we specify to lightlike geodesics in a Lorentzian manifold. We illustrate the latter case by considering particular lightlike geodesics (a) in Schwarzschild spacetime and (b) in a plane-wave spacetime.
Equations of the Randomizer's Dynamics
NASA Astrophysics Data System (ADS)
Strzałko, Jarosław; Grabski, Juliusz; Perlikowski, Przemysław; Stefanski, Andrzej; Kapitaniak, Tomasz
Basing on the Newton-Euler laws of mechanics we derive the equations which describe the dynamics of the coin toss, the die throw, and roulette run. The equations for full 3D models and for lower dimensional simplifications are given. The influence of the air resistance and energy dissipation at the impacts is described. The obtained equations allow for the numerical simulation of the randomizer's dynamics and define the mapping of the initial conditions into the final outcome.
Solving Differential Equations in R
NASA Astrophysics Data System (ADS)
Soetaert, Karline; Meysman, Filip; Petzoldt, Thomas
2010-09-01
The open-source software R has become one of the most widely used systems for statistical data analysis and for making graphs, but it is also well suited for other disciplines in scientific computing. One of the fields where considerable progress has been made is the solution of differential equations. Here we first give an overview of the types of differential equations that R can solve, and then demonstrate how to use R for solving a 2-Dimensional partial differential equation.
A note on "Kepler's equation".
NASA Astrophysics Data System (ADS)
Dutka, J.
1997-07-01
This note briefly points out the formal similarity between Kepler's equation and equations developed in Hindu and Islamic astronomy for describing the lunar parallax. Specifically, an iterative method for calculating the lunar parallax has been developed by the astronomer Habash al-Hasib al-Marwazi (about 850 A.D., Turkestan), which is surprisingly similar to the iterative method for solving Kepler's equation invented by Leonhard Euler (1707 - 1783).
Deformation of the Dirac equation
NASA Astrophysics Data System (ADS)
Faizal, Mir; Kruglov, Sergey I.
2016-10-01
In this paper, we will first clarify the physical meaning of having a minimum measurable time. Then we will combine the deformation of the Dirac equation due to the existence of minimum measurable length and time scales with its deformation due to the doubly special relativity. We will also analyze this deformed Dirac equation in curved spacetime, and observe that this deformation of the Dirac equation also leads to a nontrivial modification of general relativity. Finally, we will analyze the stochastic quantization of this deformed Dirac equation on curved spacetime.
Quaternion Dirac Equation and Supersymmetry
NASA Astrophysics Data System (ADS)
Rawat, Seema; Negi, O. P. S.
2009-08-01
Quaternion Dirac equation has been analyzed and its supersymmetrization has been discussed consistently. It has been shown that the quaternion Dirac equation automatically describes the spin structure with its spin up and spin down components of two component quaternion Dirac spinors associated with positive and negative energies. It has also been shown that the supersymmetrization of quaternion Dirac equation works well for different cases associated with zero mass, nonzero mass, scalar potential and generalized electromagnetic potentials. Accordingly we have discussed the splitting of supersymmetrized Dirac equation in terms of electric and magnetic fields.
Electronic representation of wave equation
NASA Astrophysics Data System (ADS)
Veigend, Petr; Kunovský, Jiří; Kocina, Filip; Nečasová, Gabriela; Šátek, Václav; Valenta, Václav
2016-06-01
The Taylor series method for solving differential equations represents a non-traditional way of a numerical solution. Even though this method is not much preferred in the literature, experimental calculations done at the Department of Intelligent Systems of the Faculty of Information Technology of TU Brno have verified that the accuracy and stability of the Taylor series method exceeds the currently used algorithms for numerically solving differential equations. This paper deals with solution of Telegraph equation using modelling of a series small pieces of the wire. Corresponding differential equations are solved by the Modern Taylor Series Method.
Graphical Solution of Polynomial Equations
ERIC Educational Resources Information Center
Grishin, Anatole
2009-01-01
Graphing utilities, such as the ubiquitous graphing calculator, are often used in finding the approximate real roots of polynomial equations. In this paper the author offers a simple graphing technique that allows one to find all solutions of a polynomial equation (1) of arbitrary degree; (2) with real or complex coefficients; and (3) possessing…
Equations for nonbonded concrete overlays
NASA Astrophysics Data System (ADS)
Chou, Y. T.
1985-09-01
The nature of the design equations for the nonbonded concrete overlays currently used by the US Army Corps of Engineers was examined and the original source of the equation was also examined. Using simple mechanics, new overlay equations were developed which are suitable for different thicknesses and elastic properties in the overlay and base concrete slabs. The difference in the computed overlay thickness between the new and existing equations is not large when the overlay thickness is equal to or greater than the base slab. The difference can become excessive when the overlay thickness is much less than that of the base slab. The new equations were compared with the finite element computer program for concrete overlays with various combinations of slab thickness, elastic property, and subgrade modulus. The comparisons were very favorable, indicating that the overlay equations developed in this report are analytically correct. It was difficult to judge whether the new equations are superior to the existing equation. This conclusion was expected because for all the seven test sections analyzed, the overlay thicknesses were either equal to or greater than those of the base slabs.
Uncertainty of empirical correlation equations
NASA Astrophysics Data System (ADS)
Feistel, R.; Lovell-Smith, J. W.; Saunders, P.; Seitz, S.
2016-08-01
The International Association for the Properties of Water and Steam (IAPWS) has published a set of empirical reference equations of state, forming the basis of the 2010 Thermodynamic Equation of Seawater (TEOS-10), from which all thermodynamic properties of seawater, ice, and humid air can be derived in a thermodynamically consistent manner. For each of the equations of state, the parameters have been found by simultaneously fitting equations for a range of different derived quantities using large sets of measurements of these quantities. In some cases, uncertainties in these fitted equations have been assigned based on the uncertainties of the measurement results. However, because uncertainties in the parameter values have not been determined, it is not possible to estimate the uncertainty in many of the useful quantities that can be calculated using the parameters. In this paper we demonstrate how the method of generalised least squares (GLS), in which the covariance of the input data is propagated into the values calculated by the fitted equation, and in particular into the covariance matrix of the fitted parameters, can be applied to one of the TEOS-10 equations of state, namely IAPWS-95 for fluid pure water. Using the calculated parameter covariance matrix, we provide some preliminary estimates of the uncertainties in derived quantities, namely the second and third virial coefficients for water. We recommend further investigation of the GLS method for use as a standard method for calculating and propagating the uncertainties of values computed from empirical equations.
Students' Understanding of Quadratic Equations
ERIC Educational Resources Information Center
López, Jonathan; Robles, Izraim; Martínez-Planell, Rafael
2016-01-01
Action-Process-Object-Schema theory (APOS) was applied to study student understanding of quadratic equations in one variable. This required proposing a detailed conjecture (called a genetic decomposition) of mental constructions students may do to understand quadratic equations. The genetic decomposition which was proposed can contribute to help…
Generalized Multilevel Structural Equation Modeling
ERIC Educational Resources Information Center
Rabe-Hesketh, Sophia; Skrondal, Anders; Pickles, Andrew
2004-01-01
A unifying framework for generalized multilevel structural equation modeling is introduced. The models in the framework, called generalized linear latent and mixed models (GLLAMM), combine features of generalized linear mixed models (GLMM) and structural equation models (SEM) and consist of a response model and a structural model for the latent…
Simplified Relativistic Force Transformation Equation.
ERIC Educational Resources Information Center
Stewart, Benjamin U.
1979-01-01
A simplified relativistic force transformation equation is derived and then used to obtain the equation for the electromagnetic forces on a charged particle, calculate the electromagnetic fields due to a point charge with constant velocity, transform electromagnetic fields in general, derive the Biot-Savart law, and relate it to Coulomb's law.…
Complete solution of Boolean equations
NASA Technical Reports Server (NTRS)
Tapia, M. A.; Tucker, J. H.
1980-01-01
A method is presented for generating a single formula involving arbitary Boolean parameters, which includes in it each and every possible solution of a system of Boolean equations. An alternate condition equivalent to a known necessary and sufficient condition for solving a system of Boolean equations is given.
Transient growth in stable linearized Vlasov-Maxwell plasmas
Podesta, J. J.
2010-12-15
Large amplitude transient growth of kinetic scale perturbations in stable collisionless magnetized plasmas has recently been demonstrated using a linearized Landau fluid model. Initial perturbations with lengthscales of the order of the ion gyroradius were shown to have transient timescales that in some cases were long compared to the ion gyroperiod, {Omega}{sub i}t>>1. Moreover, it was suggested that such perturbations are not rare but instead form a large class within the set of all possible initial conditions. For collisionless plasmas, the Vlasov-Maxwell equations provide a more complete description of kinetic physics and the existence of transient growth of solutions for the linearized Vlasov-Maxwell system is an interesting question. The existence of transient growth of solutions is demonstrated here for a special case of the Vlasov-Maxwell equations, namely, the one dimensional Vlasov-Poisson system. The analysis is different from the standard approach of nonmodal analysis since the initial value problem is described by a Volterra integral equation of the second kind, reflecting the fact that the time evolution of the system depends on the memory of the state from time zero through time t. For the case of a thermal equilibrium plasma, it is shown how initial conditions may be constructed to obtain solutions that grow linearly in time; the duration of this growth is the time required for a thermal electron to traverse the wavelength of the initial perturbation, a timescale that can last for many plasma periods 2{pi}/{omega}{sub pe}, thus demonstrating the existence of transient growth of solutions for the linearized Vlasov-Poisson system. The results suggest that the phenomenon of transient growth may be a common feature of the linearized Vlasov-Maxwell system as well as for Landau fluid models.
Transport equations for oscillating neutrinos
NASA Astrophysics Data System (ADS)
Zhang, Yunfan; Burrows, Adam
2013-11-01
We derive a suite of generalized Boltzmann equations, based on the density-matrix formalism, that incorporates the physics of neutrino oscillations for two- and three-flavor oscillations, matter refraction, and self-refraction. The resulting equations are straightforward extensions of the classical transport equations that nevertheless contain the full physics of quantum oscillation phenomena. In this way, our broadened formalism provides a bridge between the familiar neutrino transport algorithms employed by supernova modelers and the more quantum-heavy approaches frequently employed to illuminate the various neutrino oscillation effects. We also provide the corresponding angular-moment versions of this generalized equation set. Our goal is to make it easier for astrophysicists to address oscillation phenomena in a language with which they are familiar. The equations we derive are simple and practical, and are intended to facilitate progress concerning oscillation phenomena in the context of core-collapse supernova theory.
The Equations of Oceanic Motions
NASA Astrophysics Data System (ADS)
Müller, Peter
2006-10-01
Modeling and prediction of oceanographic phenomena and climate is based on the integration of dynamic equations. The Equations of Oceanic Motions derives and systematically classifies the most common dynamic equations used in physical oceanography, from large scale thermohaline circulations to those governing small scale motions and turbulence. After establishing the basic dynamical equations that describe all oceanic motions, M|ller then derives approximate equations, emphasizing the assumptions made and physical processes eliminated. He distinguishes between geometric, thermodynamic and dynamic approximations and between the acoustic, gravity, vortical and temperature-salinity modes of motion. Basic concepts and formulae of equilibrium thermodynamics, vector and tensor calculus, curvilinear coordinate systems, and the kinematics of fluid motion and wave propagation are covered in appendices. Providing the basic theoretical background for graduate students and researchers of physical oceanography and climate science, this book will serve as both a comprehensive text and an essential reference.
Equation predicts diesel cloud points
Tsang, C.Y.; Ker, V.S.F.; Miranda, R.D.; Wesch, J.C.
1988-03-28
Diesel fuel cloud points can be predicted by an empirical equation developed by NOCA/Husky Research Corp. The equation can accurately predict cloud points from feedstock and product data readily available in the refinery. The applicability of the equation to a full range of summer, winter, and arctic diesel blends was proven by studies conducted on data from four Canadian refineries that process a wide variety of conventional crude oils and synthetic crude from bitumen. Results of the studies show that the variance between equation predicted and measured cloud point values are within acceptable reproducibility of measured data. Considerable time can be saved in the refinery when the equation is used for optimizing diesel fuel blend formulations. Applicability ranges from daily blending calculations, to use in linear programs for long-term planning for distillate utilization.
Landau damping of Langmuir twisted waves with kappa distributed electrons
Arshad, Kashif Aman-ur-Rehman; Mahmood, Shahzad
2015-11-15
The kinetic theory of Landau damping of Langmuir twisted modes is investigated in the presence of orbital angular momentum of the helical (twisted) electric field in plasmas with kappa distributed electrons. The perturbed distribution function and helical electric field are considered to be decomposed by Laguerre-Gaussian mode function defined in cylindrical geometry. The Vlasov-Poisson equation is obtained and solved analytically to obtain the weak damping rates of the Langmuir twisted waves in a nonthermal plasma. The strong damping effects of the Langmuir twisted waves at wavelengths approaching Debye length are also obtained by using an exact numerical method and are illustrated graphically. The damping rates of the planar Langmuir waves are found to be larger than the twisted Langmuir waves in plasmas which shows opposite behavior as depicted in Fig. 3 by J. T. Mendoça [Phys. Plasmas 19, 112113 (2012)].
Thermodynamic Bounds on Nonlinear Electrostatic Perturbations in Intense Charged Particle Beams
Nikolas C. Logan and Ronald C. Davidson
2012-07-18
This paper places a lowest upper bound on the field energy in electrostatic perturbations in single-species charged particle beams with initial temperature anisotropy (TllT⊥ < 1). The result applies to all electrostatic perturbations driven by the natural anisotropies that develop in accelerated particle beams, including Harris-type electrostatic instabilities, known to limit the luminosity and minimum spot size attainable in experiments. The thermodynamic bound on the field perturbation energy of the instabilities is obtained from the nonlinear Vlasov-Poisson equations for an arbitrary initial distribution function, including the effects of intense self-fields, finite geometry and nonlinear processes. This paper also includes analytical estimates of the nonlinear bounds for space-charge-dominated and emittance-dominated anisotropic bi-Maxwellian distributions.
Thermodynamic bounds on nonlinear electrostatic perturbations in intense charged particle beams
Logan, Nikolas C.; Davidson, Ronald C.
2012-07-15
This paper places a lowest upper bound on the field energy in electrostatic perturbations in single-species charged particle beams with initial temperature anisotropy (T{sub Parallel-To }/T{sub Up-Tack }<1). The result applies to all electrostatic perturbations driven by the natural anisotropies that develop in accelerated particle beams, including Harris-type electrostatic instabilities, known to limit the luminosity and minimum spot size attainable in experiments. The thermodynamic bound on the field perturbation energy of the instabilities is obtained from the nonlinear Vlasov-Poisson equations for an arbitrary initial distribution function, including the effects of intense self-fields, finite geometry, and nonlinear processes. This paper also includes analytical estimates of the nonlinear bounds for space-charge-dominated and emittance-dominated anisotropic bi-Maxwellian distributions.
Lund, Steven M.; Friedman, Alex; Bazouin, Guillaume
2011-01-10
A one-dimensional Vlasov-Poisson model for sheet beams is reviewed and extended to provide a simple framework for analysis of space-charge effects. Centroid and rms envelope equations including image charge effects are derived and reasonable parameter equivalences with commonly employed 2D transverse models of unbunched beams are established. This sheet beam model is then applied to analyze several problems of fundamental interest. A sheet beam thermal equilibrium distribution in a continuous focusing channel is constructed and shown to have analogous properties to two- d three-dimensional thermal equilibrium models in terms of the equilibrium structure and Deybe screening properties. The simpler formulation for sheet beams is exploited to explicitly calculate the distribution of particle oscillation frequencies within a thermal equilibrium beam. It is shown that as space-charge intensity increases, the frequency distribution becomes broad, suggesting that beams with strong space-charge can have improved stability.
NASA Astrophysics Data System (ADS)
Coulette, David; Hirstoaga, Sever A.; Manfredi, Giovanni
2016-08-01
We develop a hybrid model to describe the parallel transport in a tokamak scrape-off layer following an edge-localized mode (ELM) event. The parallel dynamics is treated with a kinetic Vlasov-Poisson model, while the evolution of the perpendicular temperature {{T}\\bot} is governed by a fluid equation. The coupling is ensured by isotropising collisions. The model generalises an earlier approach where {{T}\\bot} was constant in space and time (Manfredi et al 2011 Plasma Phys. Control. Fusion 53 015012). Numerical results show that the main effect comes from electron-electron collisions, which limit the decrease of the parallel electron temperature and increase the potential drop in the Debye sheath in front of the surface. Ion-ion collisions have an almost negligible impact. The net effect is an increased peak power load on the target plates.
Extended Trial Equation Method for Nonlinear Partial Differential Equations
NASA Astrophysics Data System (ADS)
Gepreel, Khaled A.; Nofal, Taher A.
2015-04-01
The main objective of this paper is to use the extended trial equation method to construct a series of some new solutions for some nonlinear partial differential equations (PDEs) in mathematical physics. We will construct the solutions in many different functions such as hyperbolic function solutions, trigonometric function solutions, Jacobi elliptic function solutions, and rational functional solutions for the nonlinear PDEs when the balance number is a real number via the Zhiber-Shabat nonlinear differential equation. The balance number of this method is not constant as we shown in other methods, but it is changed by changing the trial equation derivative definition. This method allowed us to construct many new types of solutions. It is shown by using the Maple software package that all obtained solutions satisfy the original PDEs.
Higher derivative gravity: Field equation as the equation of state
NASA Astrophysics Data System (ADS)
Dey, Ramit; Liberati, Stefano; Mohd, Arif
2016-08-01
One of the striking features of general relativity is that the Einstein equation is implied by the Clausius relation imposed on a small patch of locally constructed causal horizon. The extension of this thermodynamic derivation of the field equation to more general theories of gravity has been attempted many times in the last two decades. In particular, equations of motion for minimally coupled higher-curvature theories of gravity, but without the derivatives of curvature, have previously been derived using a thermodynamic reasoning. In that derivation the horizon slices were endowed with an entropy density whose form resembles that of the Noether charge for diffeomorphisms, and was dubbed the Noetheresque entropy. In this paper, we propose a new entropy density, closely related to the Noetheresque form, such that the field equation of any diffeomorphism-invariant metric theory of gravity can be derived by imposing the Clausius relation on a small patch of local causal horizon.
Wave equations for pulse propagation
NASA Astrophysics Data System (ADS)
Shore, B. W.
1987-06-01
Theoretical discussions of the propagation of pulses of laser radiation through atomic or molecular vapor rely on a number of traditional approximations for idealizing the radiation and the molecules, and for quantifying their mutual interaction by various equations of propagation (for the radiation) and excitation (for the molecules). In treating short-pulse phenomena it is essential to consider coherent excitation phenomena of the sort that is manifest in Rabi oscillations of atomic or molecular populations. Such processes are not adequately treated by rate equations for excitation nor by rate equations for radiation. As part of a more comprehensive treatment of the coupled equations that describe propagation of short pulses, this memo presents background discussion of the equations that describe the field. This memo discusses the origin, in Maxwell's equations, of the wave equation used in the description of pulse propagation. It notes the separation into lamellar and solenoidal (or longitudinal and transverse) and positive and negative frequency parts. It mentions the possibility of separating the polarization field into linear and nonlinear parts, in order to define a susceptibility or index of refraction and, from these, a phase and group velocity.
SETS. Set Equation Transformation System
Worrell, R.B.
1992-01-13
SETS is used for symbolic manipulation of Boolean equations, particularly the reduction of equations by the application of Boolean identities. It is a flexible and efficient tool for performing probabilistic risk analysis (PRA), vital area analysis, and common cause analysis. The equation manipulation capabilities of SETS can also be used to analyze noncoherent fault trees and determine prime implicants of Boolean functions, to verify circuit design implementation, to determine minimum cost fire protection requirements for nuclear reactor plants, to obtain solutions to combinatorial optimization problems with Boolean constraints, and to determine the susceptibility of a facility to unauthorized access through nullification of sensors in its protection system.
Pavement performance equations. Final report
Mahoney, J.P.; Kay, R.K.; Jackson, N.C.
1988-03-01
The WSDOT PMS data base was used to develop regression equations for three pavement surface types: bituminous surface treatments, asphalt concrete, and portland-cement concrete. The primary regression equations developed were to predict Pavement Condition Rating (PCR) which is a measure of the pavement surface distress (ranges from 100 (no distress) to below 0 (extensive distress)). Overall, the equations fit the data rather well given the expected variation of pavement performance information. The relative effects of age (time since construction or reconstruction) were illustrated for the three surface types.
Overdetermined Systems of Linear Equations.
ERIC Educational Resources Information Center
Williams, Gareth
1990-01-01
Explored is an overdetermined system of linear equations to find an appropriate least squares solution. A geometrical interpretation of this solution is given. Included is a least squares point discussion. (KR)
Solving Differential Equations in R
Although R is still predominantly applied for statistical analysis and graphical representation, it is rapidly becoming more suitable for mathematical computing. One of the fields where considerable progress has been made recently is the solution of differential equations. Here w...
NASA Technical Reports Server (NTRS)
Shebalin, John V.
1987-01-01
The Boussinesq approximation is extended so as to explicitly account for the transfer of fluid energy through viscous action into thermal energy. Ideal and dissipative integral invariants are discussed, in addition to the general equations for thermal-fluid motion.
Friedmann equation with quantum potential
Siong, Ch'ng Han; Radiman, Shahidan; Nikouravan, Bijan
2013-11-27
Friedmann equations are used to describe the evolution of the universe. Solving Friedmann equations for the scale factor indicates that the universe starts from an initial singularity where all the physical laws break down. However, the Friedmann equations are well describing the late-time or large scale universe. Hence now, many physicists try to find an alternative theory to avoid this initial singularity. In this paper, we generate a version of first Friedmann equation which is added with an additional term. This additional term contains the quantum potential energy which is believed to play an important role at small scale. However, it will gradually become negligible when the universe evolves to large scale.
Parametric Equations, Maple, and Tubeplots.
ERIC Educational Resources Information Center
Feicht, Louis
1997-01-01
Presents an activity that establishes a graphical foundation for parametric equations by using a graphing output form called tubeplots from the computer program Maple. Provides a comprehensive review and exploration of many previously learned topics. (ASK)
IKT for quantum hydrodynamic equations
NASA Astrophysics Data System (ADS)
Tessarotto, Massimo; Ellero, Marco; Nicolini, Piero
2007-11-01
A striking feature of standard quantum mechanics (SQM) is its analogy with classical fluid dynamics. In fact, it is well-known that the Schr"odinger equation is equivalent to a closed set of partial differential equations for suitable real-valued functions of position and time (denoted as quantum fluid fields) [Madelung, 1928]. In particular, the corresponding quantum hydrodynamic equations (QHE) can be viewed as the equations of a classical compressible and non-viscous fluid, endowed with potential velocity and quantized velocity circulation. In this reference, an interesting theoretical problem, in its own right, is the construction of an inverse kinetic theory (IKT) for such a type of fluids. In this note we intend to investigate consequences of the IKT recently formulated for QHE [M.Tessarotto et al., Phys. Rev. A 75, 012105 (2007)]. In particular a basic issue is related to the definition of the quantum fluid fields.
Hidden Statistics of Schroedinger Equation
NASA Technical Reports Server (NTRS)
Zak, Michail
2011-01-01
Work was carried out in determination of the mathematical origin of randomness in quantum mechanics and creating a hidden statistics of Schr dinger equation; i.e., to expose the transitional stochastic process as a "bridge" to the quantum world. The governing equations of hidden statistics would preserve such properties of quantum physics as superposition, entanglement, and direct-product decomposability while allowing one to measure its state variables using classical methods.
Geometrical Solutions of Quadratic Equations.
ERIC Educational Resources Information Center
Grewal, A. S.; Godloza, L.
1999-01-01
Demonstrates that the equation of a circle (x-h)2 + (y-k)2 = r2 with center (h; k) and radius r reduces to a quadratic equation x2-2xh + (h2 + k2 -r2) = O at the intersection with the x-axis. Illustrates how to determine the center of a circle as well as a point on a circle. (Author/ASK)
Allidina, A.Y.; Malinowski, K.; Singh, M.G.
1982-12-01
The possibilities were explored for enhancing parallelism in the simulation of systems described by algebraic equations, ordinary differential equations and partial differential equations. These techniques, using multiprocessors, were developed to speed up simulations, e.g. for nuclear accidents. Issues involved in their design included suitable approximations to bring the problem into a numerically manageable form and a numerical procedure to perform the computations necessary to solve the problem accurately. Parallel processing techniques used as simulation procedures, and a design of a simulation scheme and simulation procedure employing parallel computer facilities, were both considered.
An Exact Mapping from Navier-Stokes Equation to Schr"odinger Equation via Riccati Equation
NASA Astrophysics Data System (ADS)
Christianto, Vic; Smarandache, Florentin
2010-03-01
In the present article we argue that it is possible to write down Schr"odinger representation of Navier-Stokes equation via Riccati equation. The proposed approach, while differs appreciably from other method such as what is proposed by R. M. Kiehn, has an advantage, i.e. it enables us extend further to quaternionic and biquaternionic version of Navier-Stokes equation, for instance via Kravchenko's and Gibbon's route. Further observation is of course recommended in order to refute or verify this proposition.
Optimization of one-way wave equations.
Lee, M.W.; Suh, S.Y.
1985-01-01
The theory of wave extrapolation is based on the square-root equation or one-way equation. The full wave equation represents waves which propagate in both directions. On the contrary, the square-root equation represents waves propagating in one direction only. A new optimization method presented here improves the dispersion relation of the one-way wave equation. -from Authors
Turbulent fluid motion 3: Basic continuum equations
NASA Technical Reports Server (NTRS)
Deissler, Robert G.
1991-01-01
A derivation of the continuum equations used for the analysis of turbulence is given. These equations include the continuity equation, the Navier-Stokes equations, and the heat transfer or energy equation. An experimental justification for using a continuum approach for the study of turbulence is given.
A hyperbolic equation for turbulent diffusion
NASA Astrophysics Data System (ADS)
Ghosal, Sandip; Keller, Joseph B.
2000-09-01
A hyperbolic equation, analogous to the telegrapher's equation in one dimension, is introduced to describe turbulent diffusion of a passive additive in a turbulent flow. The predictions of this equation, and those of the usual advection-diffusion equation, are compared with data on smoke plumes in the atmosphere and on heat flow in a wind tunnel. The predictions of the hyperbolic equation fit the data at all distances from the source, whereas those of the advection-diffusion equation fit only at large distances. The hyperbolic equation is derived from an integrodifferential equation for the mean concentration which allows it to vary rapidly. If the mean concentration varies sufficiently slowly compared with the correlation time of the turbulence, the hyperbolic equation reduces to the advection-diffusion equation. However, if the mean concentration varies very rapidly, the hyperbolic equation should be replaced by the integrodifferential equation.
How to Obtain the Covariant Form of Maxwell's Equations from the Continuity Equation
ERIC Educational Resources Information Center
Heras, Jose A.
2009-01-01
The covariant Maxwell equations are derived from the continuity equation for the electric charge. This result provides an axiomatic approach to Maxwell's equations in which charge conservation is emphasized as the fundamental axiom underlying these equations.
Differential Equations for Morphological Amoebas
NASA Astrophysics Data System (ADS)
Welk, Martin; Breuß, Michael; Vogel, Oliver
This paper is concerned with amoeba median filtering, a structure-adaptive morphological image filter. It has been introduced by Lerallut et al. in a discrete formulation. Experimental evidence shows that iterated amoeba median filtering leads to segmentation-like results that are similar to those obtained by self-snakes, an image filter based on a partial differential equation. We investigate this correspondence by analysing a space-continuous formulation of iterated median filtering. We prove that in the limit of vanishing radius of the structuring elements, iterated amoeba median filtering indeed approximates a partial differential equation related to self-snakes and the well-known (mean) curvature motion equation. We present experiments with discrete iterated amoeba median filtering that confirm qualitative and quantitative predictions of our analysis.
Integration of quantum hydrodynamical equation
NASA Astrophysics Data System (ADS)
Ulyanova, Vera G.; Sanin, Andrey L.
2007-04-01
Quantum hydrodynamics equations describing the dynamics of quantum fluid are a subject of this report (QFD).These equations can be used to decide the wide class of problem. But there are the calculated difficulties for the equations, which take place for nonlinear hyperbolic systems. In this connection, It is necessary to impose the additional restrictions which assure the existence and unique of solutions. As test sample, we use the free wave packet and study its behavior at the different initial and boundary conditions. The calculations of wave packet propagation cause in numerical algorithm the division. In numerical algorithm at the calculations of wave packet propagation, there arises the problem of division by zero. To overcome this problem we have to sew together discrete numerical and analytical continuous solutions on the boundary. We demonstrate here for the free wave packet that the numerical solution corresponds to the analytical solution.
Students' understanding of quadratic equations
NASA Astrophysics Data System (ADS)
López, Jonathan; Robles, Izraim; Martínez-Planell, Rafael
2016-05-01
Action-Process-Object-Schema theory (APOS) was applied to study student understanding of quadratic equations in one variable. This required proposing a detailed conjecture (called a genetic decomposition) of mental constructions students may do to understand quadratic equations. The genetic decomposition which was proposed can contribute to help students achieve an understanding of quadratic equations with improved interrelation of ideas and more flexible application of solution methods. Semi-structured interviews with eight beginning undergraduate students explored which of the mental constructions conjectured in the genetic decomposition students could do, and which they had difficulty doing. Two of the mental constructions that form part of the genetic decomposition are highlighted and corresponding further data were obtained from the written work of 121 undergraduate science and engineering students taking a multivariable calculus course. The results suggest the importance of explicitly considering these two highlighted mental constructions.
Fractional-calculus diffusion equation
2010-01-01
Background Sequel to the work on the quantization of nonconservative systems using fractional calculus and quantization of a system with Brownian motion, which aims to consider the dissipation effects in quantum-mechanical description of microscale systems. Results The canonical quantization of a system represented classically by one-dimensional Fick's law, and the diffusion equation is carried out according to the Dirac method. A suitable Lagrangian, and Hamiltonian, describing the diffusive system, are constructed and the Hamiltonian is transformed to Schrodinger's equation which is solved. An application regarding implementation of the developed mathematical method to the analysis of diffusion, osmosis, which is a biological application of the diffusion process, is carried out. Schrödinger's equation is solved. Conclusions The plot of the probability function represents clearly the dissipative and drift forces and hence the osmosis, which agrees totally with the macro-scale view, or the classical-version osmosis. PMID:20492677
Maxwell's mixing equation revisited: characteristic impedance equations for ellipsoidal cells.
Stubbe, Marco; Gimsa, Jan
2015-07-21
We derived a series of, to our knowledge, new analytic expressions for the characteristic features of the impedance spectra of suspensions of homogeneous and single-shell spherical, spheroidal, and ellipsoidal objects, e.g., biological cells of the general ellipsoidal shape. In the derivation, we combined the Maxwell-Wagner mixing equation with our expression for the Clausius-Mossotti factor that had been originally derived to describe AC-electrokinetic effects such as dielectrophoresis, electrorotation, and electroorientation. The influential radius model was employed because it allows for a separation of the geometric and electric problems. For shelled objects, a special axial longitudinal element approach leads to a resistor-capacitor model, which can be used to simplify the mixing equation. Characteristic equations were derived for the plateau levels, peak heights, and characteristic frequencies of the impedance as well as the complex specific conductivities and permittivities of suspensions of axially and randomly oriented homogeneous and single-shell ellipsoidal objects. For membrane-covered spherical objects, most of the limiting cases are identical to-or improved with respect to-the known solutions given by researchers in the field. The characteristic equations were found to be quite precise (largest deviations typically <5% with respect to the full model) when tested with parameters relevant to biological cells. They can be used for the differentiation of orientation and the electric properties of cell suspensions or in the analysis of single cells in microfluidic systems. PMID:26200856
Explicit integration of Friedmann's equation with nonlinear equations of state
NASA Astrophysics Data System (ADS)
Chen, Shouxin; Gibbons, Gary W.; Yang, Yisong
2015-05-01
In this paper we study the integrability of the Friedmann equations, when the equation of state for the perfect-fluid universe is nonlinear, in the light of the Chebyshev theorem. A series of important, yet not previously touched, problems will be worked out which include the generalized Chaplygin gas, two-term energy density, trinomial Friedmann, Born-Infeld, two-fluid models, and Chern-Simons modified gravity theory models. With the explicit integration, we are able to understand exactly the roles of the physical parameters in various models play in the cosmological evolution which may also offer clues to a profound understanding of the problems in general settings. For example, in the Chaplygin gas universe, a few integrable cases lead us to derive a universal formula for the asymptotic exponential growth rate of the scale factor, of an explicit form, whether the Friedmann equation is integrable or not, which reveals the coupled roles played by various physical sectors and it is seen that, as far as there is a tiny presence of nonlinear matter, conventional linear matter makes contribution to the dark matter, which becomes significant near the phantom divide line. The Friedmann equations also arise in areas of physics not directly related to cosmology. We provide some examples ranging from geometric optics and central orbits to soap films and the shape of glaciated valleys to which our results may be applied.
Universal soil loss equation and revised universal soil loss equation
Technology Transfer Automated Retrieval System (TEKTRAN)
Soil erosion has long been recognized as a serious problem. Considerable efforts have been expended to address this problem. Thousands of plot years of data were summarized by ARS researchers in producing the Universal Soil Loss Equation (USLE). This technology has been used for conservation planni...
Jourdain's variational equation and Appell's equation of motion for nonholonomic dynamical systems
NASA Astrophysics Data System (ADS)
Wang, Li-Sheng; Pao, Yih-Hsing
2003-01-01
Based on Jourdain's variational equation proposed in 1909, we deduce a minimal set of general equations of motion for nonholomic dynamical systems of particles and rigid bodies. This equation of motion for the system, which differs slightly from the Gibbs-Appell equation, appears to be the same as the equation derived by Kane in 1961. Since the same equation was established by Appell in 1903 on the basis of D'Alembert's principle, the newly derived equation is named Appell's equation.
Transport equations in tokamak plasmas
Callen, J. D.; Hegna, C. C.; Cole, A. J.
2010-05-15
Tokamak plasma transport equations are usually obtained by flux surface averaging the collisional Braginskii equations. However, tokamak plasmas are not in collisional regimes. Also, ad hoc terms are added for neoclassical effects on the parallel Ohm's law, fluctuation-induced transport, heating, current-drive and flow sources and sinks, small magnetic field nonaxisymmetries, magnetic field transients, etc. A set of self-consistent second order in gyroradius fluid-moment-based transport equations for nearly axisymmetric tokamak plasmas has been developed using a kinetic-based approach. The derivation uses neoclassical-based parallel viscous force closures, and includes all the effects noted above. Plasma processes on successive time scales and constraints they impose are considered sequentially: compressional Alfven waves (Grad-Shafranov equilibrium, ion radial force balance), sound waves (pressure constant along field lines, incompressible flows within a flux surface), and collisions (electrons, parallel Ohm's law; ions, damping of poloidal flow). Radial particle fluxes are driven by the many second order in gyroradius toroidal angular torques on a plasma species: seven ambipolar collision-based ones (classical, neoclassical, etc.) and eight nonambipolar ones (fluctuation-induced, polarization flows from toroidal rotation transients, etc.). The plasma toroidal rotation equation results from setting to zero the net radial current induced by the nonambipolar fluxes. The radial particle flux consists of the collision-based intrinsically ambipolar fluxes plus the nonambipolar fluxes evaluated at the ambipolarity-enforcing toroidal plasma rotation (radial electric field). The energy transport equations do not involve an ambipolar constraint and hence are more directly obtained. The 'mean field' effects of microturbulence on the parallel Ohm's law, poloidal ion flow, particle fluxes, and toroidal momentum and energy transport are all included self-consistently. The
Transport Equations In Tokamak Plasmas
NASA Astrophysics Data System (ADS)
Callen, J. D.
2009-11-01
Tokamak plasma transport equations are usually obtained by flux surface averaging the collisional Braginskii equations. However, tokamak plasmas are not in collisional regimes. Also, ad hoc terms are added for: neoclassical effects on the parallel Ohm's law (trapped particle effects on resistivity, bootstrap current); fluctuation-induced transport; heating, current-drive and flow sources and sinks; small B field non-axisymmetries; magnetic field transients etc. A set of self-consistent second order in gyroradius fluid-moment-based transport equations for nearly axisymmetric tokamak plasmas has been developed recently using a kinetic-based framework. The derivation uses neoclassical-based parallel viscous force closures, and includes all the effects noted above. Plasma processes on successive time scales (and constraints they impose) are considered sequentially: compressional Alfv'en waves (Grad-Shafranov equilibrium, ion radial force balance); sound waves (pressure constant along field lines, incompressible flows within a flux surface); and ion collisions (damping of poloidal flow). Radial particle fluxes are driven by the many second order in gyroradius toroidal angular torques on the plasma fluid: 7 ambipolar collision-based ones (classical, neoclassical, etc.) and 8 non-ambipolar ones (fluctuation-induced, polarization flows from toroidal rotation transients etc.). The plasma toroidal rotation equation [1] results from setting to zero the net radial current induced by the non-ambipolar fluxes. The radial particle flux consists of the collision-based intrinsically ambipolar fluxes plus the non-ambipolar fluxes evaluated at the ambipolarity-enforcing toroidal plasma rotation (radial electric field). The energy transport equations do not involve an ambipolar constraint and hence are more directly obtained. The resultant transport equations will be presented and contrasted with the usual ones. [4pt] [1] J.D. Callen, A.J. Cole, C.C. Hegna, ``Toroidal Rotation In
Transport equations in tokamak plasmasa)
NASA Astrophysics Data System (ADS)
Callen, J. D.; Hegna, C. C.; Cole, A. J.
2010-05-01
Tokamak plasma transport equations are usually obtained by flux surface averaging the collisional Braginskii equations. However, tokamak plasmas are not in collisional regimes. Also, ad hoc terms are added for neoclassical effects on the parallel Ohm's law, fluctuation-induced transport, heating, current-drive and flow sources and sinks, small magnetic field nonaxisymmetries, magnetic field transients, etc. A set of self-consistent second order in gyroradius fluid-moment-based transport equations for nearly axisymmetric tokamak plasmas has been developed using a kinetic-based approach. The derivation uses neoclassical-based parallel viscous force closures, and includes all the effects noted above. Plasma processes on successive time scales and constraints they impose are considered sequentially: compressional Alfvén waves (Grad-Shafranov equilibrium, ion radial force balance), sound waves (pressure constant along field lines, incompressible flows within a flux surface), and collisions (electrons, parallel Ohm's law; ions, damping of poloidal flow). Radial particle fluxes are driven by the many second order in gyroradius toroidal angular torques on a plasma species: seven ambipolar collision-based ones (classical, neoclassical, etc.) and eight nonambipolar ones (fluctuation-induced, polarization flows from toroidal rotation transients, etc.). The plasma toroidal rotation equation results from setting to zero the net radial current induced by the nonambipolar fluxes. The radial particle flux consists of the collision-based intrinsically ambipolar fluxes plus the nonambipolar fluxes evaluated at the ambipolarity-enforcing toroidal plasma rotation (radial electric field). The energy transport equations do not involve an ambipolar constraint and hence are more directly obtained. The "mean field" effects of microturbulence on the parallel Ohm's law, poloidal ion flow, particle fluxes, and toroidal momentum and energy transport are all included self-consistently. The
Young's Equation at the Nanoscale
NASA Astrophysics Data System (ADS)
Seveno, David; Blake, Terence D.; De Coninck, Joël
2013-08-01
In 1805, Thomas Young was the first to propose an equation to predict the value of the equilibrium contact angle of a liquid on a solid. Today, the force exerted by a liquid on a solid, such as a flat plate or fiber, is routinely used to assess this angle. Moreover, it has recently become possible to study wetting at the nanoscale using an atomic force microscope. Here, we report the use of molecular-dynamics simulations to investigate the force distribution along a 15 nm fiber dipped into a liquid meniscus. We find very good agreement between the measured force and that predicted by Young’s equation.
Investigation of the kinetic model equations
NASA Astrophysics Data System (ADS)
Liu, Sha; Zhong, Chengwen
2014-03-01
Currently the Boltzmann equation and its model equations are widely used in numerical predictions for dilute gas flows. The nonlinear integro-differential Boltzmann equation is the fundamental equation in the kinetic theory of dilute monatomic gases. By replacing the nonlinear fivefold collision integral term by a nonlinear relaxation term, its model equations such as the famous Bhatnagar-Gross-Krook (BGK) equation are mathematically simple. Since the computational cost of solving model equations is much less than that of solving the full Boltzmann equation, the model equations are widely used in predicting rarefied flows, multiphase flows, chemical flows, and turbulent flows although their predictions are only qualitatively right for highly nonequilibrium flows in transitional regime. In this paper the differences between the Boltzmann equation and its model equations are investigated aiming at giving guidelines for the further development of kinetic models. By comparing the Boltzmann equation and its model equations using test cases with different nonequilibrium types, two factors (the information held by nonequilibrium moments and the different relaxation rates of high- and low-speed molecules) are found useful for adjusting the behaviors of modeled collision terms in kinetic regime. The usefulness of these two factors are confirmed by a generalized model collision term derived from a mathematical relation between the Boltzmann equation and BGK equation that is also derived in this paper. After the analysis of the difference between the Boltzmann equation and the BGK equation, an attempt at approximating the collision term is proposed.
Sonar equations for planetary exploration.
Ainslie, Michael A; Leighton, Timothy G
2016-08-01
The set of formulations commonly known as "the sonar equations" have for many decades been used to quantify the performance of sonar systems in terms of their ability to detect and localize objects submerged in seawater. The efficacy of the sonar equations, with individual terms evaluated in decibels, is well established in Earth's oceans. The sonar equations have been used in the past for missions to other planets and moons in the solar system, for which they are shown to be less suitable. While it would be preferable to undertake high-fidelity acoustical calculations to support planning, execution, and interpretation of acoustic data from planetary probes, to avoid possible errors for planned missions to such extraterrestrial bodies in future, doing so requires awareness of the pitfalls pointed out in this paper. There is a need to reexamine the assumptions, practices, and calibrations that work well for Earth to ensure that the sonar equations can be accurately applied in combination with the decibel to extraterrestrial scenarios. Examples are given for icy oceans such as exist on Europa and Ganymede, Titan's hydrocarbon lakes, and for the gaseous atmospheres of (for example) Jupiter and Venus.
Equations of motion for superfluids
Basile, A.G.; Elser, V.
1995-06-01
To the principles of least action and minimum error, for determining the time evolution of the parameters in a variational wave function, we add a third: continuous collapse dynamics. In this formulation, exact time evolution is applied for an infinitesimal time and is followed by projection of the state back into the variational manifold (``collapse``). All three principles lead to the same equations of motion when applied to complex parameters but take two distinct forms when the parameters are real. As an application of these principles, we study the time evolution of two variational wave functions for superfluids. The first wave function, containing real parameters, was considered by Kerman and Koonin [Ann. Phys. (N.Y.) 100, 332 (1976)] and leads to the Euler equation in the hydrodynamic limit. The equation for our second wave function, a coherent state of Feynman excitations with complex parameters, has essentially the same hydrodynamic limit. The latter wave function, however, has a significant advantage in that the equation it generates is useful and meaningful on a microscopic scale as well.
Duffing's Equation and Nonlinear Resonance
ERIC Educational Resources Information Center
Fay, Temple H.
2003-01-01
The phenomenon of nonlinear resonance (sometimes called the "jump phenomenon") is examined and second-order van der Pol plane analysis is employed to indicate that this phenomenon is not a feature of the equation, but rather the result of accumulated round-off error, truncation error and algorithm error that distorts the true bounded solution onto…
Pendulum Motion and Differential Equations
ERIC Educational Resources Information Center
Reid, Thomas F.; King, Stephen C.
2009-01-01
A common example of real-world motion that can be modeled by a differential equation, and one easily understood by the student, is the simple pendulum. Simplifying assumptions are necessary for closed-form solutions to exist, and frequently there is little discussion of the impact if those assumptions are not met. This article presents a…
The solution of transcendental equations
NASA Technical Reports Server (NTRS)
Agrawal, K. M.; Outlaw, R.
1973-01-01
Some of the existing methods to globally approximate the roots of transcendental equations namely, Graeffe's method, are studied. Summation of the reciprocated roots, Whittaker-Bernoulli method, and the extension of Bernoulli's method via Koenig's theorem are presented. The Aitken's delta squared process is used to accelerate the convergence. Finally, the suitability of these methods is discussed in various cases.
Renaissance Learning Equating Study. Report
ERIC Educational Resources Information Center
Sewell, Julie; Sainsbury, Marian; Pyle, Katie; Keogh, Nikki; Styles, Ben
2007-01-01
An equating study was carried out in autumn 2006 by the National Foundation for Educational Research (NFER) on behalf of Renaissance Learning, to provide validation evidence for the use of the Renaissance Star Reading and Star Mathematics tests in English schools. The study investigated the correlation between the Star tests and established tests.…
Ordinary Differential Equation System Solver
1992-03-05
LSODE is a package of subroutines for the numerical solution of the initial value problem for systems of first order ordinary differential equations. The package is suitable for either stiff or nonstiff systems. For stiff systems the Jacobian matrix may be treated in either full or banded form. LSODE can also be used when the Jacobian can be approximated by a band matrix.
Perceptions of the Schrodinger equation
NASA Astrophysics Data System (ADS)
Efthimiades, Spyros
2014-03-01
The Schrodinger equation has been considered to be a postulate of quantum physics, but it is also perceived as the quantum equivalent of the non-relativistic classical energy relation. We argue that the Schrodinger equation cannot be a physical postulate, and we show explicitly that its second space derivative term is wrongly associated with the kinetic energy of the particle. The kinetic energy of a particle at a point is proportional to the square of the momentum, that is, to the square of the first space derivative of the wavefunction. Analyzing particle interactions, we realize that particles have multiple virtual motions and that each motion is accompanied by a wave that has constant amplitude. Accordingly, we define the wavefunction as the superposition of the virtual waves of the particle. In simple interaction settings we can tell what particle motions arise and can explain the outcomes in direct and tangible terms. Most importantly, the mathematical foundation of quantum mechanics becomes clear and justified, and we derive the Schrodinger, Dirac, etc. equations as the conditions the wavefunction must satisfy at each space-time point in order to fulfill the respective total energy equation.
The Symbolism Of Chemical Equations
ERIC Educational Resources Information Center
Jensen, William B.
2005-01-01
A question about the historical origin of equal sign and double arrow symbolism in balanced chemical equation is raised. The study shows that Marshall proposed the symbolism in 1902, which includes the use of currently favored double barb for equilibrium reactions.
The Forced Soft Spring Equation
ERIC Educational Resources Information Center
Fay, T. H.
2006-01-01
Through numerical investigations, this paper studies examples of the forced Duffing type spring equation with [epsilon] negative. By performing trial-and-error numerical experiments, the existence is demonstrated of stability boundaries in the phase plane indicating initial conditions yielding bounded solutions. Subharmonic boundaries are…
Scale Shrinkage in Vertical Equating.
ERIC Educational Resources Information Center
Camilli, Gregory; And Others
1993-01-01
Three potential causes of scale shrinkage (measurement error, restriction of range, and multidimensionality) in item response theory vertical equating are discussed, and a more comprehensive model-based approach to establishing vertical scales is described. Test data from the National Assessment of Educational Progress are used to illustrate the…
Mathematics and Reading Test Equating.
ERIC Educational Resources Information Center
Lee, Ong Kim; Wright, Benjamin D.
As part of a larger project to assess changes in student learning resulting from school reform, this study equates levels 6 through 14 of the mathematics and reading comprehension components of Form 7 of the Iowa Tests of Basic Skills (ITBS) with levels 7 through 14 of the mathematics and reading comprehension components of the CPS90 (another…
Empirical equation estimates geothermal gradients
Kutasov, I.M. )
1995-01-02
An empirical equation can estimate geothermal (natural) temperature profiles in new exploration areas. These gradients are useful for cement slurry and mud design and for improving electrical and temperature log interpretation. Downhole circulating temperature logs and surface outlet temperatures are used for predicting the geothermal gradients.
Sonar equations for planetary exploration.
Ainslie, Michael A; Leighton, Timothy G
2016-08-01
The set of formulations commonly known as "the sonar equations" have for many decades been used to quantify the performance of sonar systems in terms of their ability to detect and localize objects submerged in seawater. The efficacy of the sonar equations, with individual terms evaluated in decibels, is well established in Earth's oceans. The sonar equations have been used in the past for missions to other planets and moons in the solar system, for which they are shown to be less suitable. While it would be preferable to undertake high-fidelity acoustical calculations to support planning, execution, and interpretation of acoustic data from planetary probes, to avoid possible errors for planned missions to such extraterrestrial bodies in future, doing so requires awareness of the pitfalls pointed out in this paper. There is a need to reexamine the assumptions, practices, and calibrations that work well for Earth to ensure that the sonar equations can be accurately applied in combination with the decibel to extraterrestrial scenarios. Examples are given for icy oceans such as exist on Europa and Ganymede, Titan's hydrocarbon lakes, and for the gaseous atmospheres of (for example) Jupiter and Venus. PMID:27586766
Optimized solution of Kepler's equation
NASA Technical Reports Server (NTRS)
Kohout, J. M.; Layton, L.
1972-01-01
A detailed description is presented of KEPLER, an IBM 360 computer program used for the solution of Kepler's equation for eccentric anomaly. The program KEPLER employs a second-order Newton-Raphson differential correction process, and it is faster than previously developed programs by an order of magnitude.
Lattice Boltzmann equation method for the Cahn-Hilliard equation.
Zheng, Lin; Zheng, Song; Zhai, Qinglan
2015-01-01
In this paper a lattice Boltzmann equation (LBE) method is designed that is different from the previous LBE for the Cahn-Hilliard equation (CHE). The starting point of the present CHE LBE model is from the kinetic theory and the work of Lee and Liu [T. Lee and L. Liu, J. Comput. Phys. 229, 8045 (2010)]; however, because the CHE does not conserve the mass locally, a modified equilibrium density distribution function is introduced to treat the diffusion term in the CHE. Numerical simulations including layered Poiseuille flow, static droplet, and Rayleigh-Taylor instability have been conducted to validate the model. The results show that the predictions of the present LBE agree well with the analytical solution and other numerical results. PMID:25679741
A Versatile Technique for Solving Quintic Equations
ERIC Educational Resources Information Center
Kulkarni, Raghavendra G.
2006-01-01
In this paper we present a versatile technique to solve several types of solvable quintic equations. In the technique described here, the given quintic is first converted to a sextic equation by adding a root, and the resulting sextic equation is decomposed into two cubic polynomials as factors in a novel fashion. The resultant cubic equations are…
A Bayesian Nonparametric Approach to Test Equating
ERIC Educational Resources Information Center
Karabatsos, George; Walker, Stephen G.
2009-01-01
A Bayesian nonparametric model is introduced for score equating. It is applicable to all major equating designs, and has advantages over previous equating models. Unlike the previous models, the Bayesian model accounts for positive dependence between distributions of scores from two tests. The Bayesian model and the previous equating models are…
Local Linear Observed-Score Equating
ERIC Educational Resources Information Center
Wiberg, Marie; van der Linden, Wim J.
2011-01-01
Two methods of local linear observed-score equating for use with anchor-test and single-group designs are introduced. In an empirical study, the two methods were compared with the current traditional linear methods for observed-score equating. As a criterion, the bias in the equated scores relative to true equating based on Lord's (1980)…
On abstract degenerate neutral differential equations
NASA Astrophysics Data System (ADS)
Hernández, Eduardo; O'Regan, Donal
2016-10-01
We introduce a new abstract model of functional differential equations, which we call abstract degenerate neutral differential equations, and we study the existence of strict solutions. The class of problems and the technical approach introduced in this paper allow us to generalize and extend recent results on abstract neutral differential equations. Some examples on nonlinear partial neutral differential equations are presented.