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: ͑i͒ a gyrokinetic Vlasov equation written in terms of a gyrocenter Hamiltonian with quadratic low-frequency ponderomotivelike terms, ͑ii͒ a set of gyrokinetic Maxwell ͑Poisson-Ampère͒ equations written in terms of the gyrocenter Vlasov distribution that contain low-frequency polarization ͑Poisson͒ and magnetization ͑Ampère͒ terms, and ͑iii͒ 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.

Guiding-center theory provides the reduced dynamical equations for the motion of charged particles in slowly varying electromagnetic fields, when the fields have weak variations over a gyration radius ͑or gyroradius͒ in space and a gyration period ͑or gyroperiod͒ in time. Canonical and noncanonical Hamiltonian formulations of guiding-center motion offer improvements over non-Hamiltonian formulations: Hamiltonian formulations possess Noether's theorem ͑hence invariants follow from symmetries͒, and they preserve the Poincaré invariants ͑so that spurious attractors are prevented from appearing in simulations of guiding-center dynamics͒. Hamiltonian guiding-center theory is guaranteed to have an energy conservation law for time-independent fields-something that is not true of non-Hamiltonian guiding-center theories. The use of the phase-space Lagrangian approach facilitates this development, as there is no need to transform a priori to canonical coordinates, such as flux coordinates, which have less physical meaning. The theory of Hamiltonian dynamics is reviewed, and is used to derive the noncanonical Hamiltonian theory of guiding-center motion. This theory is further explored within the context of magnetic flux coordinates, including the generic form along with those applicable to systems in which the magnetic fields lie on nested tori. It is shown how to return to canonical coordinates to arbitrary accuracy by the Hazeltine-Meiss method and by a perturbation theory applied to the phase-space Lagrangian. This noncanonical Hamiltonian theory is used to derive the higher-order corrections to the magnetic moment adiabatic invariant and to compute the longitudinal adiabatic invariant. Noncanonical guiding-center theory is also developed for relativistic dynamics, where covariant and noncovariant results are presented. The latter is important for computations in which it is convenient to use the ordinary time as the independent variable rather than the proper time. The final section uses noncanonical guiding-center theory to discuss the dynamics of particles in systems in which the magnetic-field lines lie on nested toroidal flux surfaces. A hierarchy in the extent to which particles move off of flux surfaces is established. This hierarchy extends from no motion off flux surfaces for any particle to no average motion off flux surfaces for particular types of particles. Future work in magnetically confined plasmas may make use of this hierarchy in designing systems that minimize transport losses.

Because the underlying Hamiltonian structure is preserved in the present formalism, these equations are directly applicable to numerical studies based on the existing gyrokinetic particle simulation techniques.

The nonlinear gyrokinetic Vlasov equation is derived for an arbitrary magnetized plasma in a local reference frame moving with the nonuniform equilibrium fluid velocity u(r). The derivation of the guiding-center and gyrocenter Hamilton equations, which appear as the characteristics of the gyrokinetic Vlasov equation, is based on the use of Lie-transform perturbation techniques. Although a general form for u is initially used, attention is later focused on an incompressible toroidal equilibrium flow when considering axisymmetric tokamak geometry.

Nonlinear gyrofluid equations are obtained from the gyrocenter-fluid moments of the nonlinear gyrokinetic Vlasov equation, which describes an equilibrium magnetized nonuniform plasma perturbed by electromagnetic field fluctuations (δφ,δA∥,δB∥), whose space-time scales satisfy the gyrokinetic ordering: ω≪Ωi, ‖k∥‖/k⊥≪1, and ε⊥≡(k⊥ρi)2≂𝒪(1). These low-frequency (reduced) fluid equations contain terms of arbitrary order in ε⊥ and take into account the nonuniformity in the equilibrium density and temperature of the ion and electron species, as well as the nonuniformity in the equilibrium magnetic field. From the gyrofluid equations, one can systematically derive nonlinear reduced fluid equations with finite-Larmor-radius (FLR) corrections, which contain linear and nonlinear terms of 𝒪(ε⊥), by expressing the gyrocenter-fluid moments appearing in the gyrofluid equations in terms of the particle-fluid moments, and then keeping terms up to 𝒪(ε⊥) in the ε⊥ expansion of the gyrofluid equations. By using gyrocenter-fluid moments, this new gyrofluid approach effectively bypasses the issue of the gyroviscous cancellations, while retaining all the important diamagnetic effects and the gyroviscous corrections. From the present FLR-corrected reduced fluid equations, the reduced Braginskii equations are recoverd for the ion and electron species (without collisional dissipation) and the ideal reduced magnetohydrodynamic (MHD) equations (in the absence of FLR effects).