Wave turbulence in inertial electron magnetohydrodynamics

A wave turbulence theory is developed for inertial electron magnetohydrodynamics (IEMHD) in the presence of a relatively strong and uniform external magnetic field B₀ = B₀e_∥. 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 (⊥) to B₀. The exact stationary solutions (Kolmogorov–Zakharov spectra) are obtained for which we prove the locality. We also found the Kolmogorov constant C_K ≃ 8.474. In the simplest case, the study reveals an energy spectrum in k_⊥^−5/2k_∥^−1/2 (with k the wavenumber) and a momentum spectrum enslaved to the energy dynamics in k_⊥^−3/2k_∥^−1/2. These solutions correspond to a magnetic energy spectrum ∼k_⊥^−9/2, which is steeper than the EMHD prediction made for scales larger than d_e.