Foundations of nonlinear gyrokinetic theory

Abstract

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.

The Hamiltonian Structure and Euler-Poincare Formulation of the Valsov-Maxwell and Gyrokinetic System

“…For most of this article H gy will be taken to be a general function of (X, u, µ). Different forms exist in the literature, depending on 2,44 and also has the advantage of having the same Poisson structure as the guiding center equations.…”

Electromagnetic gyrokinetic simulation of turbulence in torus plasmas

“…In addition, the kinetic electron effect elongates the mode structure along the field line, and thus a large simulation domain along the field line is required. Large computational resources which meet the requirements are the reason why numerical simulations of turbulence based on the electromagnetic gyrokinetic model (Antonsen and Lane 1980;Frieman and Chen 1982;Hahm et al 1988;Hazeltine and Meiss 2003;Brizard and Hahm 2007) are mainly carried out in the flux-tube geometry (Kotschenreuther et al 1995;Jenko 2000;Jenko et al 2000;Jenko and Dorland 2001;Candy and Waltz 2003a,b;Candy 2005;Pueschel et al 2008;Peeters et al 2009;Pueschel and Jenko 2010;Waltz 2010;Hatch et al 2012;Pueschel et al 2013a,b) which is along a magnetic field line and is localized in the radial direction to reduce computational cost (Beer et al 1995). Although, there are some gyrokinetic simulations on electromagnetic turbulence in global domain covering a whole torus plasma and also some electromagnetic gyrokinetic studies on space plasmas (Schekochihin et al 2009), we focus on gyrokinetic simulations in local flux-tube domain in this review.…”

Modern gyrokinetic formulation of collisional and turbulent transport in toroidally rotating plasmas

“…Historically, using the assumptions mentioned above and classical perturbation methods such as recursive techniques, drift kinetic and gyrokinetic equations for perturbed distribution functions (df ) in the case of the low-flow ordering were individually derived as governing equations for neoclassical and turbulent transport processes, respectively (Hazeltine and Meiss 1992;Rutherford and Frieman 1968;Taylor and Hastie 1968;Antonsen and Lane 1980;Catto et al 1981;Frieman and Chen 1982). In the same way, the df drift kinetic and gyrokinetic equations in the high-flow ordering are derived for toroidally rotating plasmas (Hinton and Wong 1985;Catto 1987;Sugama and Horton 1997b;Artun and Tang 1994;Sugama and Horton 1998) and the derivations of these equations based on the classical methods are comprehensively reviewed by Abel et al (2013) On the other hand, the modern gyrokinetic equations derived from the Lie-transform techniques (Brizard and Hahm 2007;Hahm 1988;Brizard 1989Brizard , 1995Hahm 1996) govern behaviors of the full distribution function (full-F), and if the collision term is included, they should, in principle, simultaneously describe collisional and turbulent processes. Actually, in Sects.…”

“…Linear and nonlinear gyrokinetic equations for particle distribution functions were originally derived by recursive techniques combined with the WKB representation (Hazeltine and Meiss 1992;Rutherford and Frieman 1968;Taylor and Hastie 1968;Antonsen and Lane 1980;Catto et al 1981;Frieman and Chen 1982). Another modern derivation of the gyrokinetic equations based on the Lagrangian and/or Hamiltonian formulations (Cary and Littlejohn 1983;Brizard and Hahm 2007;Dubin et al 1983;Hahm 1988;Brizard 1989) was presented to ensure conservation laws for the phase space volume and the magnetic moment from Liouville's theorem and Noether's theorem (Goldstein et al 2002), respectively. Later, conservation of the total energy and momentum was obtained in the gyrokinetic field theory (Sugama 2000) where all governing equations for the distribution functions and the electromagnetic fields are derived from the Lagrangian which describes the whole system consisting of particles and fields.…”