Breaking Kelvin: Circulation conservation and vortex breakup in MHD at low Magnetic Prandtl Number

TL;DR

This paper investigates how weak magnetic fields cause vortex breakup and violate Kelvin's circulation theorem in low magnetic Prandtl number MHD flows, revealing the dependence on vortex shape and magnetic effects.

Contribution

It introduces scaling laws for circulation loss due to magnetic fields in 2D MHD, highlighting the impact of vortex geometry and magnetic diffusion.

Findings

Magnetic fields cause initial circulation loss proportional to field strength.

Vortex patches experience greater circulation loss than Gaussian vortices.

Elliptical vortices lose circulation more efficiently due to larger swept area.

Abstract

In this paper we examine the role of weak magnetic fields in breaking Kelvin's circulation theorem and in vortex breakup in two-dimensional magnetohydrodynamics for the physically important case of a low magnetic Prandtl number (low $Pm$) fluid. We consider three canonical inviscid solutions for the purely hydrodynamical problem, namely a Gaussian vortex, a circular vortex patch and an elliptical vortex patch. We examine how magnetic fields lead to an initial loss of circulation $\Gamma$ and attempt to derive scaling laws for the loss of circulation as a function of field strength and diffusion as measured by two non-dimensional parameters. We show that for all cases the loss of circulation depends on the integrated effects of the Lorentz force, with the patch cases leading to significantly greater circulation loss. For the case of the elliptical vortex the loss of circulation depends on the total area swept out by the rotating vortex and so this leads to more efficient circulation loss than for a circular vortex.