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…