Dynamics and turbulence in electron MHD

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) whistler modes have a dispersion that allows a three-wave decay, including into a zero frequency mode; (iv) 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. We derive the Hamiltonian formulation of EMHD, and using Bogolyubov transformation reduce it to a 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 depends on the forcing and often fails to reach a steady state. Analytical estimates predict the spectrum of magnetic fluctuations for the quasi-isotropic cascade $\sim$k$^{-2}$. The cascade remains weak (not critically-balanced). The cascade is UV-local, while the infrared locality is weakly (logarithmically) violated.