Foundations of Nonlinear Gyrokinetic Theory
Authors: A.J. Brizard and T.S. Hahm
Date of PPPL Report: February 2006
Submitted to: Review of Modern Physics
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 important pillars: (1) a gyrokinetic Vlasov equation written in terms of a gyrocenter Hamiltonian with quadratic low-frequency ponderomotive-like terms; (2) a set of gyrokinetic Maxwell equations written in terms of the gyrocenter Vlasov distribution that contain low-frequency polarization and magnetization terms (derived from the quadratic nonlinearities in the Hamiltonian); and (3) 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 the rigorous applications of Lagrangian and Hamiltonian methods used in the variational derivation of nonlinear gyrokinetic Vlasov-Maxwell equations. The physical motivations and applications of the nonlinear gyrokinetic equations, which describe the turbulent evolution of low-frequency electromagnetic fluctuations in non-uniform magnetized plasmas with arbitrary magnetic geometry, are also discussed.