Wave Turbulence in Inertial Electron Magnetohydrodynamics

TL;DR

This paper develops a wave turbulence theory for inertial electron magnetohydrodynamics (IEMHD) under a strong magnetic field, revealing anisotropic energy transfer and specific spectral laws relevant for space plasma scales smaller than the electron inertial length.

Contribution

It introduces the first kinetic equations for three-wave interactions in IEMHD and derives exact stationary solutions with explicit spectral laws and Kolmogorov constants.

Findings

Energy cascade is anisotropic and mainly perpendicular to the magnetic field.

Derived spectra show a steeper magnetic energy spectrum than previous EMHD predictions.

Exact Kolmogorov-Zakharov spectra with proven locality and calculated constants.

Abstract

A wave turbulence theory is developed for inertial electron magnetohydrodynamics (IEMHD) in the presence of a relatively strong and uniform external magnetic field $\boldsymbol{B_0} = B_0 \hat{\boldsymbol{e}}_\|$. This regime is relevant for scales smaller than the electron inertial length $d_e$. We derive the kinetic equations that describe the three-wave interactions between inertial whistler or kinetic Alfv\'en waves. We show that for both invariants, energy and momentum, the transfer is anisotropic (axisymmetric) with a direct cascade mainly in the direction perpendicular ($\perp$) to $\boldsymbol{B_0}$. The exact stationary solutions (Kolmogorov-Zakharov spectra) are obtained for which we prove the locality. We also found the Kolmogorov constant $C_K \simeq 8.474$. In the simplest case, the study reveals an energy spectrum in $k_\perp^{-5/2} k_\|^{-1/2}$ and a momentum spectrum enslaved to the energy dynamics in $k_\perp^{-3/2} k_\|^{-1/2}$. These solutions correspond to a magnetic energy spectrum $\sim k_\perp^{-9/2}$, which is steeper than the EMHD prediction made for scales larger than $d_e$. We conclude with a discussion on the application of the theory to space plasmas.