Exact two-dimensionalization of low-magnetic-Reynolds-number flows subject to a strong magnetic field

Abstract

We investigate the behaviour of flows, including turbulent flows, driven by a horizontal body force and subject to a vertical magnetic field, with the following question in mind: for a very strong applied magnetic field, is the flow mostly two-dimensional, with remaining weak three-dimensional fluctuations, or does it become exactly 2-D, with no dependence along the vertical direction? We first focus on the quasi-static approximation, i.e. the asymptotic limit of vanishing magnetic Reynolds number, $\mathit{Rm}\ll 1$: we prove that the flow becomes exactly 2-D asymptotically in time, regardless of the initial condition and provided that the interaction parameter $N$ is larger than a threshold value. We call this property absolute two-dimensionalization: the attractor of the system is necessarily a (possibly turbulent) 2-D flow. We then consider the full magnetohydrodynamic (MHD) equations and prove that, for low enough $\mathit{Rm}$ and large enough $N$, the flow becomes exactly 2-D in the long-time limit provided the initial vertically dependent perturbations are infinitesimal. We call this phenomenon linear two-dimensionalization: the (possibly turbulent) 2-D flow is an attractor of the dynamics, but it is not necessarily the only attractor of the system. Some 3-D attractors may also exist and be attained for strong enough initial 3-D perturbations. These results shed some light on the existence of a dissipation anomaly for MHD flows subject to a strong external magnetic field.