Foundations of nonlinear gyrokinetic theory
Abstract
Nonlinear gyrokinetic equations play a fundamental role in our understanding of the longtime 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 lowfrequency ponderomotivelike terms, (ii) a set of gyrokinetic Maxwell (PoissonAmpere) equations written in terms of the gyrocenter Vlasov distribution that contain lowfrequency polarization (Poisson) and magnetization (Ampere) terms, and (iii) an exact energy conservation law for the gyrokinetic VlasovMaxwell 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 Lietransform perturbation methods in the variational derivation of nonlinear gyrokinetic VlasovMaxwell equations. The physical motivations and applications of the nonlinear gyrokinetic equations that describe the turbulent evolution of lowfrequency electromagnetic fluctuations in a nonuniform magnetized plasmas with arbitrary magnetic geometry are discussed.
 Authors:
 Department of Chemistry and Physics, Saint Michael's College, Colchester, Vermont 05439 (United States)
 (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; BOLTZMANNVLASOV 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 longtime 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 lowfrequency ponderomotivelike terms, (ii) a set of gyrokinetic Maxwell (PoissonAmpere) equations written in terms of the gyrocenter Vlasov distribution that contain lowfrequency polarization (Poisson) and magnetization (Ampere) terms, and (iii) an exact energy conservation law for the gyrokinetic VlasovMaxwell 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 Lietransform perturbation methods in the variational derivation of nonlinear gyrokinetic VlasovMaxwell equations. The physical motivations and applications of the nonlinear gyrokinetic equations that describe the turbulent evolution of lowfrequency 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}
}

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A set of the electrostatic toroidal gyrokinetic Vlasov equation and the Poisson equation, which explicitly includes the polarization drift, is derived systematically by using Lietransform 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 phasespace volume caused by the electrostatic potential [T. S. Hahm, Phys. Plasmas 3, 4658 (1996)]. 
Comment on 'Nonlinear gyrokinetic theory with polarization drift' [Phys. Plasmas 17, 082304 (2010)]
In this comment, we show that by using the discrete particle distribution function the changes of the phasespace 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)].