K. O'Neil

Equilibrium configurations of point vortices with circulations of two discrete values are associated with the zeros of a sequence of polynomials having many continuous parameters: the Adler–Moser polynomials in the case of circulation ratio −1, and the Loutsenko polynomials in the case of ratio −2. In this paper a new set of polynomial sequences generalizing the vortex system to three values of circulations is constructed. These polynomials are extensions of the previously known polynomials in the sense that they are special cases of the latter when the third circulation is zero. The polynomials are naturally connected with rational functions that satisfy a second-order differential equation.

Point vortex equilibria in which the vortices are arranged in clusters are examined. The vortex velocities in these configurations are all equal. Necessary conditions for their existence are established that relate the circulations within the clusters to the cluster radius. A method for generating these configurations by singular continuation is proved to be valid for the generic case. Finally, a partial analysis of exceptional cases is given and their connection to the existence of parametrized families of equilibria is described.

Configurations of point vortices on the sphere are considered in which all vortex velocities are zero. A sharp upper bound for the number of equilibria lying on a great circle is found, valid for generic circulations, and some unusual equilibrium configurations with a free real parameter are described. Equilibria of rings (vortices evenly spaced along circles of latitude) are also discussed. All equilibrium configurations of four vortices are determined.

Relative equilibrium configurations of point vortices in the plane can be related to a system of polynomial equations in the vortex positions and circulations. For systems of four vortices the solution set to this system is proved to be finite, so long as a number of polynomial expressions in the vortex circulations are nonzero, and the number of relative equilibrium configurations is thereby shown to have an upper bound of 56. A sharper upper bound is found for the special case of vanishing total circulation. The polynomial system is simple enough to allow the complete set of relative equilibrium configurations to be found numerically when the circulations are chosen appropriately. Collapse configurations of four vortices are also considered; while finiteness is not proved, the approach provides an effective computational method that yields all configurations with a given ratio of velocity to position.