Magnetohydrodynamic turbulence at moderate magnetic Reynolds number

We consider the case of homogeneous turbulence in a conducting fluid that is exposed to a uniform external magnetic field at low to moderate magnetic Reynolds numbers (by moderate we mean here values as high as 20). When the magnetic Reynolds number is vanishingly small ($R_m \ll 1$), it is customary to simplify the governing magnetohydrodynamic (MHD) equations using what is known as the quasi-static (QS) approximation. As the magnetic Reynolds number is increased, a progressive transition between the physics described by the QS approximation and the MHD equations occurs. We show here that this intermediate regime can be described by another approximation which we call the quasi-linear (QL) approximation. For the numerical simulations performed, the predictions of the QL approximation are in good agreement with those of MHD for magnetic Reynolds number up to $R_m \sim 20$.