Wave Turbulence in Inertial Electron Magnetohydrodynamics

Abstract

A wave turbulence theory is developed for inertial electron magnetohydrodynamics (IEMHD) in the presence of a relatively strong and uniform external magnetic field ${\mathbf{B}}_0=B_0{\hat{\mathbf{e}}}_{\Vert }$. 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 Alfvén 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 ${\mathbf{B}}_0$. 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 $k_{\perp }^{-5/2}{k}_{\Vert }^{-1/2}$ and a momentum spectrum enslaved to the energy dynamics in $k_{\perp }^{-3/2}{k}_{\Vert }^{-1/2}$. 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.

31 pages, 7 figures

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