Electron MHD: dynamics and turbulence

TL;DR

This paper investigates the dynamics and turbulence of whistler modes in inertialess electron MHD, revealing unique properties of wave interactions, cascade behavior, and differences from MHD turbulence, with implications for understanding plasma turbulence.

Contribution

It provides a Hamiltonian formulation of EMHD, analyzes three-wave interactions, and numerically solves the kinetic equation to characterize turbulence spectra and cascade properties.

Findings

Whistler modes are exact non-linear solutions.

Co-linear whistlers do not interact.

The turbulence cascade often remains weak and develops over a broad range of angles.

Abstract

(Abridged) We consider dynamics and turbulent interaction of whistler modes within the framework of inertialess electron MHD (EMHD). We argue there is no energy principle in EMHD: any stationary closed configuration is neutrally stable. We consider the turbulent cascade of whistler modes. We show that (i) harmonic whistlers are exact non-linear solutions; (ii) co-linear whistlers do not interact (including counter-propagating); (iii) waves with the same value of the wave vector, $k_1=k_2$, do not interact; (iv) whistler modes have a dispersion that allows a three-wave decay, including into a zero frequency mode; (v) the three-wave interaction effectively couples modes with highly different wave numbers and propagation angles. In addition, linear interaction of a whistler with a single zero-mode can lead to spatially divergent structures via parametric instability. All these properties are drastically different from MHD, so that the qualitative properties of the Alfven turbulence cannot be transferred to the EMHD turbulence. We derive the Hamiltonian formulation of EMHD, and using Bogolyubov transformation reduce it to the canonical form; we calculate the matrix elements for the three-wave interaction of whistlers. We solve numerically the kinetic equation and show that, generally, the EMHD cascade develops within a broad range of angles, while transiently it may show anisotropic, nearly two dimensional structures. Development of a cascade depends on the forcing (non-universal) and often fails to reach a steady state. Analytical estimates predict the spectrum of magnetic fluctuations for the quasi-isotropic cascade $\propto k^{-2}$. The cascade remains weak (not critically-balanced). The cascade is UV-local, while the infrared locality is weakly (logarithmically) violated.