Foundations of nonlinear gyrokinetic theory

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-Amp\`ere) equations written in terms of the gyrocenter Vlasov distribution that contain low-frequency polarization (Poisson) and magnetization (Amp\`ere) 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.