Nonlinear gyrokinetics: a powerful tool for the description of microturbulence in magnetized plasmas

Abstract

Abstract. Gyrokinetics is the description of low-frequency dynamics in magnetized plasmas. In magnetic-confinement fusion, it provides the most fundamental basis for numerical simulations of microturbulence; there are astrophysical applications as well. In this tutorial, a sketch of the derivation of the novel dynamical system comprising the nonlinear gyrokinetic (GK) equation (GKE) and the coupled electrostatic GK Poisson equation will be given by using modern Lagrangian and Lie perturbation methods. No background in plasma physics is required in order to appreciate the logical development. The GKE describes the evolution of an ensemble of gyrocenters moving in a weakly inhomogeneous background magnetic field and in the presence of electromagnetic perturbations with wavelength of the order of the ion gyroradius. Gyrocenters move with effective drifts, which may be obtained by an averaging procedure that systematically, order by order, removes gyrophase dependence. To that end, the use of the Lagrangian differential one-form as well as the content and advantages of Lie perturbation theory will be explained.The electromagnetic fields follow via Maxwell's equations from the charge and current density of the particles. Particle and gyrocenter densities differ by an important polarization effect. That is calculated formally by a "pull-back" (a concept from differential geometry) of the gyrocenter distribution to the laboratory coordinate system. A natural truncation then leads to the closed GK dynamical system. Important properties such as GK energy conservation and fluctuation noise will be mentioned briefly, as will the possibility (and difficulties) of deriving nonlinear gyrofluid equations suitable for rapid numerical solution -although it is probably best to directly simulate the GKE. By the end of the tutorial, students should appreciate the GKE as an extremely powerful tool and will be prepared for later lectures describing its applications to physical problems.

Cross-verification of the global gyrokinetic codes GENE and XGC

“…Turbulent transport in the core of tokamak plasmas is nowadays addressed on a daily basis with the help of gyrokinetic simulations [1,2], using, e.g., the well-established core gyrokinetic code GENE [3,4]. This includes numerous successful quantitative comparisons with experimental measurements [5][6][7][8].…”

Gyrokinetic Statistical Absolute Equilibrium and Turbulence

“…We won't review the complete history of the linear and nonlinear gyrokinetic theories 3 but will just present the basic ideas and results, borrowing from some of the treatment and notation of Plunk et al 35,36 There are several published derivations of gyrokinetic equations with varied assumptions and techniques, including recent papers with a tutorial emphasis. 5,6 The starting point is the Boltzmann equation for the particle distribution function f s (r, v, t) for plasma species s located at r moving with velocity v at time t:…”

“…In between, for frequencies well below the ion cyclotron frequency but that may still involve scales comparable to the gyroradius, a detail of the particle helical motion around the field line, the cyclotron angle, may be averaged out, resulting in a reduced system called gyrokinetics. [1][2][3][4][5][6] With one dimension (the cyclotron angle) and the fast time scales associated with that dimension excluded, gyrokinetics helps the tractability of turbulent kinetic cascades of plasma turbulence numerically and analytically.…”

Advances and Current Challenges in the Theory of Zonal-Flow Generation

“…Since we will be concerned only with low-frequency fluctuations with ω ω ci , the nonlinear gyrokinetic formalism is a good place to start. The technical details have been thoroughly reviewed by Brizard and Hahm [4] and an introductory tutorial is by Krommes [5]; here I will need only the basics. In a constant magnetic field, the gyrokinetic equation is…”

“…Most fundamental is what can be called the "gyrokinetic closure." As described, for example, in the tutorial article by Krommes [5], the derivation of gyrokinetics begins merely with a series of variable transformations. No physics content is lost in that process; information is just packaged in a possibly unusual way.…”