Spectral transfer and K\'arm\'an-Howarth-Monin equations for compressible Hall magnetohydrodynamics

TL;DR

This paper introduces new forms of the Kármán-Howarth-Monin equations for compressible Hall MHD turbulence, validated through simulations, and compares them with spectral transfer equations to analyze energy transfer and cascade processes.

Contribution

The paper derives and tests two new forms of the Kármán-Howarth-Monin equations for compressible Hall MHD turbulence, providing tools to analyze energy transfer in such systems.

Findings

Hall cascade appears near the ion inertial range.

KHM and spectral transfer equations are consistent and complementary.

Pressure-dilation energy exchange oscillates with no net effect.

Abstract

We derive two new forms of the K\'arm\'an-Howarth-Monin equation for decaying compressible Hall magnetohydrodynamic (MHD) turbulence. We test them on results of a weakly-compressible, two-dimensional, moderate-Reynolds-number Hall MHD simulation and compare them with an isotropic spectral transfer (ST) equation. The KHM and ST equations are automatically satisfied during the whole simulation owing to the periodic boundary conditions and have complementary cumulative behavior. They are used here to analyze the onset of turbulence and its properties when it is fully developed. These approaches give equivalent results characterizing: the decay of the kinetic + magnetic energy at large scales, the MHD and Hall cross-scale energy transfer/cascade, the pressure dilatation, and the dissipation. The Hall cascade appears when the MHD one brings the energy close to the ion inertial range and is related to the formation of reconnecting current sheets. At later times, the pressure-dilation energy-exchange rate oscillates around zero with no net effect on the cross-scale energy transfer when averaged over a period of its oscillations. A reduced one-dimensional analysis suggests that all three methods may be useful to estimate the energy cascade rate from in situ observations.