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

TL;DR

This paper proves that under strong magnetic fields and low magnetic Reynolds number, flows become exactly two-dimensional over time, with the flow's long-term behavior being dominated by 2D attractors, including turbulent flows.

Contribution

It introduces the concepts of absolute and linear two-dimensionalization, providing rigorous proofs for the conditions under which flows become exactly 2D in magnetohydrodynamics.

Findings

Flow becomes exactly 2D asymptotically in time for high interaction parameter N.

For low Rm and high N, flows become 2D if initial vertical perturbations are infinitesimal.

Results suggest a dissipation anomaly in strongly magnetized MHD flows.

Abstract

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