  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.
