An in-depth numerical study of exact laws for compressible Hall magnetohydrodynamic turbulence

TL;DR

This study uses high-resolution numerical simulations to analyze and validate exact laws governing compressible Hall MHD turbulence, revealing the relationship between the Hall effect and energy cascade rates at sub-ion scales.

Contribution

It provides a detailed numerical validation of recent exact laws for compressible Hall MHD turbulence using large-scale DNS data, highlighting the effects of initial Mach number and magnetic field.

Findings

The two exact laws are equivalent in the inertial range.

The Hall effect strength correlates with the cascade rate at sub-ion scales.

The reduced form of the laws remains valid even without strict stationarity.

Abstract

Various exact laws governing compressible magnetohydrodynamic (MHD) and Hall-MHD (CHMHD) turbulence have been derived in recent years. Other than their fundamental theoretical interest, these laws are generally used to estimate the energy dissipation rate from spacecraft observations in order to address diverse problems related, e.g., to heating of the solar wind (SW) and magnetospheric plasmas. Here we use various $1024^3$ direct numerical simulation (DNS) data of free-decay isothermal CHMHD turbulence obtained with the GHOST code (Geophysical High-Order Suite for Turbulence) to analyze two of the recently derived exact laws. The simulations reflect different intensities of the initial Mach number and the background magnetic field. The analysis demonstrates the equivalence of the two laws in the inertial range and relates the strength of the Hall effect to the amplitude of the cascade rate at sub-ion scales. When taken in their general form (i.e., not limited to the inertial range) some subtleties regarding the validity of the stationarity assumption or the absence of the forcing in the simulations are discussed. We show that the free-decay nature of the turbulence induces a shift from a large scale forcing towards the presence of a scale-dependent reservoir of energy fueling the cascade or dissipation. The reduced form of the exact laws (valid in the inertial range) ultimately holds even if the stationarity assumption is not fully verified.