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)).
Highlights
The original development of the theory of vortex motion goes back to Helmholtz [1] and Kelvin [2], who formulated various theorems concerning vorticity, in particular the conservation of circulation in a ‘perfect’ fluid [3,4]
We have sought to reveal the effects of surface geometry on vortex motion, in particular the self-induced motion associated with variable surface curvature
This self-induced motion stems from the mathematical requirement that the integral of the vorticity must vanish on a compact surface
Summary
The original development of the theory of vortex motion goes back to Helmholtz [1] and Kelvin [2], who formulated various theorems concerning vorticity, in particular the conservation of circulation in a ‘perfect’ fluid (e.g. inviscid, incompressible and subject to forces derived from a single-valued potential) [3,4]. We develop a general theory for point vortex dynamics on a general compact, differentiable surface This makes direct use of the Hamiltonian formalism, for which we state the general form of the vortex interaction energy for a system of n vortices. 2. Energy of a point vortex system on a compact surface (a) Incompressible flow induced by vorticity. See appendix B. (c) Lin [25] derives a directly analogous expression for H, called the ‘Kirchhoff–Routh’ function (see (4.4) in that paper), for describing vortex motion in planar domains
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