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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 nonzero, 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 selfinduced 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)).
Original language  English 

Article number  20140890 
Pages (fromto)  125 
Number of pages  25 
Journal  Proceedings of the Royal Society A  Mathematical, Physical & Engineering Sciences 
Volume  471 
Issue number  2176 
Early online date  8 Apr 2015 
DOIs  
Publication status  Published  Apr 2015 
Keywords
 Vortex dynamics
 Point vortices
 Closed surfaces
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Dive into the research topics of 'The motion of point vortices on closed surfaces'. Together they form a unique fingerprint.Projects
 1 Finished

Geophysical Vortices: The Structure, stability and interaction of geophysical vortices
Reinaud, J. N., Dritschel, D. G. & Scott, R. K.
5/01/10 → 1/11/13
Project: Standard