Enstrophy non-conservation and the forward cascade of energy in two-dimensional electrostatic magnetized plasma turbulence

G G Plunk

doi:10.1017/s0022377820000872

Abstract

A fluid system is derived to describe electrostatic magnetized plasma turbulence at scales somewhat larger than the Larmor radius of a given species. It is related to the Hasegawa–Mima equation, but does not conserve enstrophy, and, as a result, exhibits a forward cascade of energy, to small scales. The inertial-range energy spectrum is argued to be shallower than a $-11/3$ power law, as compared to the $-5$ law of the Hasegawa–Mima enstrophy cascade. This property, confirmed here by direct numerical simulations of the fluid system, may help explain the fluctuation spectrum observed in gyrokinetic simulations of streamer-dominated electron-temperature-gradient driven turbulence (Plunk et al., Phys. Rev. Lett., vol. 122, 2019, 035002), and also possibly some cases of ion-temperature-gradient driven turbulence where zonal flows are suppressed (Plunk et al., Phys. Rev. Lett., vol. 118, 2017, 105002).

Highlights

The turbulent cascade, a mechanism for the nonlinear transfer of energy across scales, is a key idea for understanding kinetic magnetized plasma turbulence
The new system, we argue, should exhibit distinct nonlinear behaviour, including a shallower energy spectrum when the effect of the nonlinear FLR terms is sufficiently strong
Note that the spatial coordinate is R in the gyrokinetic equation and, formally, the spatial derivatives are to be interpreted in this variable, but for simplicity we avoid making the distinction explicit.) We mostly ignore the collision operator but note that some mechanism of coarse graining will be necessary to get sensible solutions out of the equation

Summary

Introduction

The turbulent cascade, a mechanism for the nonlinear transfer of energy across scales, is a key idea for understanding kinetic magnetized plasma turbulence. In a uniform magnetic geometry, one can obtain a theoretical prediction for the spectrum of fluctuations, valid across an ‘inertial range’ of scales, free from energy sources and sinks Such a theory is not able to fully describe the behaviour of realistic turbulence, which hosts instabilities, damped modes, complicated magnetic geometries, etc., it constitutes a quantitative prediction of nonlinear behaviour of the underlying gyrokinetic equation, an equation which generally governs actual systems of practical interest. Similar systems as the one presented here have been proposed and studied in the past, most notably the Hasegawa–Mima (HM) equation (Hasegawa & Mima 1978) the present derivation takes special care in considering the consequences of the appearance of nonlinear finite-Larmor-radius (FLR) terms that appear in the dynamical equation for the electrostatic potential – i.e. the ‘vorticity’ equation Such terms introduce a closure problem in the fluid moment hierarchy, where lower moments are coupled to ever higher ones, generally without end. We discuss possible examples, including cases explored in previous gyrokinetic turbulence simulations in tokamak and stellarator geometries

Equations and definitions

Fluid limit

Naive expansion

Decaying turbulence

Direct numerical simulations

Discussion