Numerical tests on vortex polygons

Point vortex, first introduced by Helmholtz in 1958, is a classic model in two-dimensional incompressible fluid dynamics. The solutions to the equations of various configurations of point vortices have provided many insightful findings in various physical fields involved with vortex dynamics, such as atomic structure, large scale weather patterns and superconductors, etc (Aref 1988). One kind of such configurations of great interest is vortex polygon, in which N identical vortices sit on the corners of a regular polygon (Aref 1983, 1995). Because of its symmetry, vortex polygon rotates as a whole without change of shape, and it is often referred as the best known equilibrium for identical vortices. The problem of vortex polygon was first studied by J.J. Thomson in his essay for the Adams Prize of 1882 (Thomson 1883). He proved that this configuration is linearly stable for N<=6 and unstable for N>=7, to infinitesimal perturbations. Since then a lot of work has been done on this problem. Morikawa et al. (1971) and Aref (1995) found that the heptagon (N=7) is actually neutrally stable in linear theory. As a natural extension of regular vortex polygon, body-centered (N+1) vortex polygon is also well studied. Mertz (1978) showed that a strong enough central vortex can stabilize a regular vortex polygon. Cabral et al. (1999) added a upper limit to the strength of the central vortex in question. Except the body-centered vortex polygons, Campell et al. (1978, 1979) found some other stable equilibrium configurations for identical vortices by placing them on concentric circles. Aref (1995) established a connection between vortex polygon and vortices in an infinite row by showing that the stability problem for both configurations can be solved by the same eigenvalue problem for a certain symmetric matrix.