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[J. Phys. A: Math. Gen. 37, (2004)]. 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.