The motion of point vortices on closed surfaces
Abstract
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 two vortices can be unstable on oblate surfaces or sufficiently prolate ones. This extends known results for the plane, where seven vortices are marginally unstable (Thomson 1883 A treatise on the motion of vortex rings, pp. 94–108; Dritschel 1985 J. Fluid Mech. 157 , 95–134 (doi:10.1017/S0022112088003088)), and the sphere, where four vortices may be unstable if sufficiently close to the equator (Polvani & Dritschel 1993 J. Fluid Mech. 255 , 35–64 (doi:10.1017/S0022112093002381)).
Citation
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 , no. 2176 , 20140890 , pp. 1-25 . https://doi.org/10.1098/rspa.2014.0890
Publication
Proceedings of the Royal Society A - Mathematical, Physical & Engineering Sciences
Status
Peer reviewed
ISSN
1364-5021Type
Journal article
Rights
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.
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