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Title: Foundations of nonlinear gyrokinetic theory

Abstract

Nonlinear gyrokinetic equations play a fundamental role in our understanding of the long-time behavior of strongly magnetized plasmas. The foundations of modern nonlinear gyrokinetic theory are based on three pillars: (i) a gyrokinetic Vlasov equation written in terms of a gyrocenter Hamiltonian with quadratic low-frequency ponderomotivelike terms, (ii) a set of gyrokinetic Maxwell (Poisson-Ampere) equations written in terms of the gyrocenter Vlasov distribution that contain low-frequency polarization (Poisson) and magnetization (Ampere) terms, and (iii) an exact energy conservation law for the gyrokinetic Vlasov-Maxwell equations that includes all the relevant linear and nonlinear coupling terms. The foundations of nonlinear gyrokinetic theory are reviewed with an emphasis on rigorous application of Lagrangian and Hamiltonian Lie-transform perturbation methods in the variational derivation of nonlinear gyrokinetic Vlasov-Maxwell equations. The physical motivations and applications of the nonlinear gyrokinetic equations that describe the turbulent evolution of low-frequency electromagnetic fluctuations in a nonuniform magnetized plasmas with arbitrary magnetic geometry are discussed.

Authors:
;  [1];  [2]
  1. Department of Chemistry and Physics, Saint Michael's College, Colchester, Vermont 05439 (United States)
  2. (United States)
Publication Date:
OSTI Identifier:
21013706
Resource Type:
Journal Article
Resource Relation:
Journal Name: Reviews of Modern Physics; Journal Volume: 79; Journal Issue: 2; Other Information: DOI: 10.1103/RevModPhys.79.421; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BOLTZMANN-VLASOV EQUATION; ENERGY CONSERVATION; EVOLUTION; HAMILTONIANS; LAGRANGIAN FUNCTION; LIE GROUPS; MAGNETIZATION; MAGNETOHYDRODYNAMICS; NONLINEAR PROBLEMS; PERTURBATION THEORY; PLASMA; POLARIZATION; TURBULENCE; VARIATIONAL METHODS

Citation Formats

Brizard, A. J., Hahm, T. S., and Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543. Foundations of nonlinear gyrokinetic theory. United States: N. p., 2007. Web. doi:10.1103/REVMODPHYS.79.421.
Brizard, A. J., Hahm, T. S., & Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543. Foundations of nonlinear gyrokinetic theory. United States. doi:10.1103/REVMODPHYS.79.421.
Brizard, A. J., Hahm, T. S., and Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543. Sun . "Foundations of nonlinear gyrokinetic theory". United States. doi:10.1103/REVMODPHYS.79.421.
@article{osti_21013706,
title = {Foundations of nonlinear gyrokinetic theory},
author = {Brizard, A. J. and Hahm, T. S. and Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543},
abstractNote = {Nonlinear gyrokinetic equations play a fundamental role in our understanding of the long-time behavior of strongly magnetized plasmas. The foundations of modern nonlinear gyrokinetic theory are based on three pillars: (i) a gyrokinetic Vlasov equation written in terms of a gyrocenter Hamiltonian with quadratic low-frequency ponderomotivelike terms, (ii) a set of gyrokinetic Maxwell (Poisson-Ampere) equations written in terms of the gyrocenter Vlasov distribution that contain low-frequency polarization (Poisson) and magnetization (Ampere) terms, and (iii) an exact energy conservation law for the gyrokinetic Vlasov-Maxwell equations that includes all the relevant linear and nonlinear coupling terms. The foundations of nonlinear gyrokinetic theory are reviewed with an emphasis on rigorous application of Lagrangian and Hamiltonian Lie-transform perturbation methods in the variational derivation of nonlinear gyrokinetic Vlasov-Maxwell equations. The physical motivations and applications of the nonlinear gyrokinetic equations that describe the turbulent evolution of low-frequency electromagnetic fluctuations in a nonuniform magnetized plasmas with arbitrary magnetic geometry are discussed.},
doi = {10.1103/REVMODPHYS.79.421},
journal = {Reviews of Modern Physics},
number = 2,
volume = 79,
place = {United States},
year = {Sun Apr 15 00:00:00 EDT 2007},
month = {Sun Apr 15 00:00:00 EDT 2007}
}
  • Invariance properties under scale transformations of the nonlinear gyrokinetic Vlasov and heat transport equations are used to examine anomalous thermal transport due to drift wave turbulence. To fully exploit the gyrokinetic expansion, separation of the fluctuations from the background by means of averaging is required. This leads to a statistical description of the system and the use of the direct interaction approximation. The invariance properties of the gyrokinetic and statistical descriptions produce the same results for chi, the thermal diffusivity. For a collisionless, low-..beta.., quasineutral plasma in a general magnetic field, a unique result is derived for chi.
  • A self-consistent and energy-conserving set of nonlinear gyrokinetic equations, consisting of the averaged Vlasov and Maxwell's equations for finite-beta plasmas, is derived. The method utilized in the present investigation is based on the Hamiltonian formalism and Lie transformation. The resulting formulation is valid for arbitrary values of k/sub perpendicular/rho/sub i/ and, therefore, is most suitable for studying linear and nonlinear evolution of microinstabilities in tokamak plasmas as well as other areas of plasma physics where the finite Larmor radius effects are important. Because the underlying Hamiltonian structure is preserved in the present formalism, these equations are directly applicable to numericalmore » studies based on the existing gyrokinetic particle simulation techniques.« less
  • The turbulent convective flux of the toroidal angular momentum density is derived using the nonlinear toroidal gyrokinetic equation which conserves phase space density and energy [T. S. Hahm, Phys. Fluids, 31, 2670 (1988)]. A novel pinch mechanism is identified which originates from the symmetry breaking due to the magnetic field curvature. A net parallel momentum transfer from the waves to the ion guiding centers is possible when the fluctuation intensity varies on the flux surface, resulting in imperfect cancellation of the curvature drift contribution to the parallel acceleration. This mechanism is inherently a toroidal effect, and complements the k{sub parallel}more » symmetry breaking mechanism due to the mean ExB shear [O. Gurcan et al., Phys. Plasmas 14, 042306 (2007)] which exists in a simpler geometry. In the absence of ion thermal effects, this pinch velocity of the angular momentum density can also be understood as a manifestation of a tendency to homogenize the profile of ''magnetically weighted angular momentum density,'' nm{sub i}R{sup 2}{omega}{sub parallel}/B{sup 2}. This part of the pinch flux is mode-independent (whether it is trapped electron mode or ion temperature gradient mode driven), and radially inward for fluctuations peaked at the low-B-field side, with a pinch velocity typically, V{sub Ang}{sup TEP}{approx}-2{chi}{sub {phi}}/R{sub 0}. Ion thermal effects introduce an additional radial pinch flux from the coupling with the curvature and grad-B drifts. This curvature driven thermal pinch can be inward or outward, depending on the mode-propagation direction. Explicit formulas in general toroidal geometry are presented.« less
  • 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 perturbation method. The polarization drift is introduced in the gyrocenter equations of motion, and the corresponding polarization density is derived. Contrary to the widespread 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)].
  • In this comment, we show that by using the discrete particle distribution function the changes of the phase-space volume of gyrocenter coordinates due to the fluctuating ExB velocity do not explicitly appear in the Poisson equation and the [Sosenko et al., Phys. Scr. 64, 264 (2001)] result is recovered. It is demonstrated that there is no contradiction between the work presented by Sosenko et al. and the work presented by [Wang et al., Phys. Plasmas 17, 082304 (2010)].