Abstract
This paper discusses the problem of finding the equilibrium positions of four point vortices, of generally unequal circulations, on the surface of a sphere. A random search method is developed which uses a modification of the linearized equations to converge on distinct equilibria. Many equilibria (47 and possibly more) may exist for prescribed circulations and angular impulse. A linear stability analysis indicates that they are generally unstable, though stable equilibria do exist. Overall, there is a surprising diversity of equilibria, including those which rotate about an axis opposite to the angular impulse vector.
Highlights
The general motion of point vortices on the surface of a sphere is a fascinating and difficult mathematical problem
This is done by randomly throwing the vortices on the sphere, adjusting them so that they have the correct angular impulse, relaxing them to a nearby equilibrium using a modification of the linearized dynamics
We describe a general procedure to determine the stability of any uniformly rotating equilibrium configuration of point vortices on the unit sphere
Summary
The general motion of point vortices on the surface of a sphere is a fascinating and difficult mathematical problem. An alternative approach, described here, instead fixes the vortex circulations and the angular impulse vector (without loss of generality along the z-axis), searches for the vortex positions which are in relative equilibrium. This is done by randomly throwing the vortices on the sphere (though one can always be placed at 0 longitude), adjusting them so that they have the correct angular impulse, relaxing them to a nearby equilibrium using a modification of the linearized dynamics. Zero-circulation points are placed at one or more co-rotating points of the configuration, their circulations are slowly increased while solving a Newton relaxation problem to determine the new vortex positions In this way, they discovered entirely asymmetric configurations of 8 or more vortices.
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