Magnetohydrodynamic turbulence at low magnetic Reynolds number

This article reviews the main established ideas on the influence of a magnetic field on turbulence in electrically conducting fluids. We limit our discussion to the asymptotic range of very small values of the magnetic Reynolds number, characterized by the fact that the induced magnetic field remains very small in comparison with the applied magnetic field. We consider three kinds of flows here. The simplest one is freely decaying homogeneous turbulence, which serves as a test bed to analyze the development of anisotropy resulting from the linear damping by the Lorentz force. We then discuss flows between walls perpendicular to the magnetic field and emphasize the influence of the Hartmann layers that develop in their vicinity. We then review the main features of the possible quasi-two-dimensional regime that can arise in that context. Finally, we consider magnetohydrodynamic turbulent shear flows. These are frequent in industrial applications involving molten metals, such as in metal processing or in the blanket of future nuclear fusion reactors. We pay particular attention to recent attempts to develop specific RAINS (Reynolds-averaged Navier-Stokes) models for these flows.