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In this work we numerically test a model of Hall magnetohydrodynamics in thepresence of a strong mean magnetic field, the reduced Hall MHD model (RHMHD)derived by Gomez et al., with the addition of weak compressible effects. Themain advantage of this model lies in the reduction of computational cost.Nevertheless, up until now the degree of agreement with the original Hall MHDsystem and the range of validity in a regime of turbulence was not established.In this work direct numerical simulations of three-dimensional Hall MHDturbulence in the presence of a strong mean magnetic field are compared withsimulations of the weak compressible RHMHD model. The results show that thedegree of agreement is very high (when the different assumptions of RHMHD, likespectral anisotropy, are satisfied). Nevertheless, when the initial conditionsare isotropic but the mean magnetic field is maintained strong, the resultsdiffer at the beginning but asymptotically reach a good agreement at relativelyshort times. We also found evidence that the compressibility still plays a rolein the dynamics of these systems, and the weak compressible RHMHD model is ableto capture these effects. In conclusion the weak compressible RHMHD model is avalid approximation of the Hall MHD turbulence in the relevant physicalcontext.

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