A transformation between stationary point vortex equilibria

Abstract

A new transformation between stationary point vortex equilibria in the unbounded plane is presented. Given a point vortex equilibrium involving only vortices with negative circulation normalized to −1 and vortices with positive circulations that are either integers or half-integers, the transformation produces a new equilibrium with a free complex parameter that appears as an integration constant. When iterated the transformation can produce infinite hierarchies of equilibria, or finite sequences that terminate after a finite number of iterations, each iteration generating equilibria with increasing numbers of point vortices and free parameters. In particular, starting from an isolated point vortex as a seed equilibrium, we recover two known infinite hierarchies of equilibria corresponding to the Adler–Moser polynomials and a class of polynomials found, using very different methods, by Loutsenko (Loutsenko 2004 J. Phys. A: Math. Gen. 37 , 1309–1321 (doi:10.1088/0305-4470/37/4/017)). For the latter polynomials, the existence of such a transformation appears to be new. The new transformation, therefore, unifies a wide range of disparate results in the literature on point vortex equilibria.