A transformation between stationary point vortex equilibria. (26th August 2020)

• Record Type: Journal Article
• Authors: Krishnamurthy, Vikas S.; Wheeler, Miles H.; Crowdy, Darren G.; Constantin, Adrian
• Is Part Of: Proceedings. Volume 476:Number 2240(2020)
• Publication Date: 2020-08-26
• Subjects: point vortex equilibria -- Adler–Moser polynomials -- Burchnall–Chaundy; Physical sciences -- Periodicals; Engineering -- Periodicals; Mathematics -- Periodicals; 500
• DOI: 10.1098/rspa.2020.0310
• Languages: English
• ISSNs: 1364-5021

Abstract

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.