Universal energy cascade and relaxation in three-dimensional inertial electron magnetohydrodynamic turbulence

Electron magnetohydrodynamics (EMHD) provides a realistic model for electron-scale heating and acceleration in weakly collisional space plasmas. A divergence-free Banerjee-Galtier type (Banerjee and Galtier, JoPA, 2017) exact relation is derived for three-dimensional homogeneous and not necessarily isotropic EMHD turbulence. By explicit calculation, it has been shown that the energy cascade is not affected by the presence of a uniform background magnetic field Bo. Using direct numerical simulations, a Kolmogorov-like energy cascade with a constant flux rate is observed across the electron inertial scale $d_e$. However, as expected, for length scales greater than $d_e$, a magnetic power spectra of $k^{-7/3}$ is obtained whereas for scales smaller than $d_e$, a $k^{-5/3}$ spectra is obtained. Similar universal cascade rate is also calculated from the scale-by-scale budget in Fourier space and is found to be equal to the one calculated using the exact law in real space. Finally, quenching the turbulence drive, the relaxation of a fully-developed EMHD turbulence is studied using the recently proposed principle of vanishing nonlinear transfers (Banerjee, Halder and Pan, PRE(L), 2023) which convincingly shows the existence of a pressure-balanced relaxed state.