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In this paper we examine the role of weak magnetic fields in breakingKelvin's circulation theorem and in vortex breakup in two-dimensionalmagnetohydrodynamics for the physically important case of a low magneticPrandtl number (low $Pm$) fluid. We consider three canonical inviscid solutionsfor the purely hydrodynamical problem, namely a Gaussian vortex, a circularvortex patch and an elliptical vortex patch. We examine how magnetic fieldslead to an initial loss of circulation $\Gamma$ and attempt to derive scalinglaws for the loss of circulation as a function of field strength and diffusionas measured by two non-dimensional parameters. We show that for all cases theloss of circulation depends on the integrated effects of the Lorentz force,with the patch cases leading to significantly greater circulation loss. For thecase of the elliptical vortex the loss of circulation depends on the total areaswept out by the rotating vortex and so this leads to more efficientcirculation loss than for a circular vortex.

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