Hall effect in a strong magnetic field: Direct comparisons of compressible magnetohydrodynamics and the reduced Hall magnetohydrodynamic equations

TL;DR

This study compares the reduced Hall MHD model with full Hall MHD turbulence simulations under strong magnetic fields, demonstrating high agreement and validating the model's effectiveness in capturing key physical effects.

Contribution

The paper provides the first direct numerical comparison showing the validity of the weak compressible RHMHD model against full Hall MHD turbulence simulations.

Findings

High agreement between RHMHD and Hall MHD when assumptions are satisfied

Asymptotic agreement achieved in isotropic initial conditions with strong magnetic field

Compressibility effects are captured by the RHMHD model

Abstract

In this work we numerically test a model of Hall magnetohydrodynamics in the presence of a strong mean magnetic field, the reduced Hall MHD model (RHMHD) derived by Gomez et al., with the addition of weak compressible effects. The main advantage of this model lies in the reduction of computational cost. Nevertheless, up until now the degree of agreement with the original Hall MHD system and the range of validity in a regime of turbulence was not established. In this work direct numerical simulations of three-dimensional Hall MHD turbulence in the presence of a strong mean magnetic field are compared with simulations of the weak compressible RHMHD model. The results show that the degree of agreement is very high (when the different assumptions of RHMHD, like spectral anisotropy, are satisfied). Nevertheless, when the initial conditions are isotropic but the mean magnetic field is maintained strong, the results differ at the beginning but asymptotically reach a good agreement at relatively short times. We also found evidence that the compressibility still plays a role in the dynamics of these systems, and the weak compressible RHMHD model is able to capture these effects. In conclusion the weak compressible RHMHD model is a valid approximation of the Hall MHD turbulence in the relevant physical context.