The motion of point vortices on closed surfaces
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We develop a mathematical framework for the dynamics of a set of point vortices on a class of differentiable surfaces conformal to the unit sphere. When the sum of the vortex circulations is non-zero, a compensating uniform vorticity field is required to satisfy the Gauss condition (that the integral of the Laplace-Beltrami operator must vanish). On variable Gaussian curvature surfaces, this results in self-induced vortex motion, a feature entirely absent on the plane, the sphere or the hyperboloid.We derive explicit equations of motion for vortices on surfaces of revolution and compute their solutions for a variety of surfaces. We also apply these equations to study the linear stability of a ring of vortices on any surface of revolution. On an ellipsoid of revolution, as few as 2 vortices can be unstable on oblate surfaces or sufficiently prolate ones. This extends known results for the plane, where 7 vortices are marginally unstable [1,2], and the sphere, where 4 vortices may be unstable if sufficiently close to the equator .
Dritschel , D G & Boatto , S 2015 , ' The motion of point vortices on closed surfaces ' Proceedings of the Royal Society A - Mathematical, Physical & Engineering Sciences , vol 471 , 20140890 . DOI: 10.1098/rspa.2014.0890
Proceedings of the Royal Society A - Mathematical, Physical & Engineering Sciences
Copyright. 2015 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author andvsource are credited.
Date of Acceptance: 29/01/2015
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