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Circulation conservation and vortex breakup in magnetohydrodynamics at low magnetic Prandtl number

Published online by Cambridge University Press:  15 October 2018

D. G. Dritschel Open the ORCID record for D. G. Dritschel [Opens in a new window] ,
P. H. Diamond  and
S. M. Tobias
D. G. Dritschel
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK
P. H. Diamond
Affiliation:
Center for Astrophysics and Space Sciences, University of California, San Diego, La Jolla, CA 92093, USA Department of Physics, University of California, San Diego, La Jolla, CA 92093, USA Center for Fusion Sciences, Southwestern Institute of Physics, Chengdu, Sichuan 610041, PR China
S. M. Tobias*
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
*
†Email address for correspondence: smt@maths.leeds.ac.uk
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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 fluid with low magnetic Prandtl number (low $Pm$). 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 $\unicode[STIX]{x1D6E4}$ 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.

JFM classification

Vortex Flows: Contour dynamics MHD and Electrohydrodynamics: MHD and Electrohydrodynamics Vortex Flows: Vortex dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 857 , 25 December 2018 , pp. 38 - 60
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Copyright
© 2018 Cambridge University Press 

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