A transformation between stationary point vortex equilibria
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Accepted version
Author(s)
Krishnamurthy, V
Wheeler, M
Crowdy, D
Constantin, A
Type
Journal Article
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.
Date Issued
2020-08
Date Acceptance
2020-07-16
ISSN
1364-5021
Publisher
The Royal Society
Start Page
1
End Page
21
Journal / Book Title
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume
476
Issue
2240
Copyright Statement
© 2020 The Author(s)
Published by the Royal Society. All rights reserved.
Published by the Royal Society. All rights reserved.
Identifier
https://royalsocietypublishing.org/doi/10.1098/rspa.2020.0310
Subjects
01 Mathematical Sciences
02 Physical Sciences
09 Engineering
Publication Status
Published
Date Publish Online
2020-08-26