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

Electron magnetohydrodynamics (EMHD) provides a useful single-fluid model of plasmas across the electron-scales. The heating and acceleration of weakly collisional space plasmas may be described in terms of a scale-by-scale energy cascade of EMHD turbulence. In this paper, for three-dimensional homogeneous and not necessarily isotropic EMHD turbulence, we derive a divergence-free Banerjee–Galtier type [Banerjee and Galtier, J. Physics A: Mathematical and Theoretical, 50, 015501 (2016)] exact, law where the energy cascade rate is expressed as a function of scales. By explicit calculation, it has been shown that the energy cascade rate is not affected by the presence of a uniform background magnetic field B0. Using direct numerical simulations, a Kolmogorov-like energy cascade with a constant flux rate is observed across the electron inertial scale de. However, as expected, for length scales greater than de, the total energy power spectra of k−7/3 is obtained, whereas for scales smaller than de, a k−5/3 spectra is obtained. A 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 et al., Phys. Rev. E 107, L043201 (2023)], which convincingly shows the existence of a pressure-balanced relaxed state.