Two-dimensional gyrokinetic turbulence

TL;DR

This paper develops a comprehensive theoretical framework for two-dimensional gyrokinetic turbulence, analyzing dual cascades across scales, deriving related equations, and establishing invariants and spectral laws in plasma turbulence.

Contribution

It introduces a unified theoretical approach to gyrokinetic turbulence, connecting fluid and kinetic regimes, and derives exact relations and spectral laws for the dual cascade phenomena.

Findings

Derivation of the Charney--Hasegawa--Mima equation from gyrokinetics for large scales

Identification of a dual cascade in phase space at sub-Larmor scales

Establishment of power-law spectra and exact third-order structure function relations

Abstract

Two-dimensional gyrokinetics is a simple paradigm for the study of kinetic magnetised plasma turbulence. In this paper, we present a comprehensive theoretical framework for this turbulence. We study both the inverse and direct cascades (the `dual cascade'), driven by a homogeneous and isotropic random forcing. The key characteristic length of gyrokinetics, the Larmor radius, divides scales into two physically distinct ranges. For scales larger than the Larmor radius, we derive the familiar Charney--Hasegawa--Mima (CHM) equation from the gyrokinetic system, and explain its relationship to gyrokinetics. At scales smaller than the Larmor radius, a dual cascade occurs in phase space (two dimensions in position space plus one dimension in velocity space) via a nonlinear phase-mixing process. We show that at these sub-Larmor scales, the turbulence is self-similar and exhibits power law spectra in position and velocity space. We propose a Hankel-transform formalism to characterise velocity-space spectra. We derive the exact relations for third-order structure functions, analogous to Kolmogorov's four-fifths and Yaglom's four-thirds laws and valid at both long and short wavelengths. We show how the general gyrokinetic invariants are related to the particular invariants that control the dual cascade in the long- and short-wavelength limits. We describe the full range of cascades from the fluid to the fully kinetic range.