Liouville chains: new hybrid vortex equilibria of the 2D Euler equation
File(s)
Accepted version
Author(s)
Crowdy, Darren
Krishnamurthy, Vikas
Constantin, Adrian
Wheeler, Miles
Type
Journal Article
Abstract
A large class of new exact solutions to the steady, incompressible Euler equation on the plane is presented. These hybrid solutions consist of a set of stationary point vortices embedded in a background sea of Liouville-type vorticity that is exponentially related to the stream function. The input to the construction is a “pure” point vortex equilibrium in a background irrotational flow. Pure point vortex equilibria also appear as a parameter A in the hybrid solutions approaches the limits A → 0, ∞. While A → 0 reproduces the input equilibrium, A → ∞ produces a new pure point vortex equilibrium. We refer to the family of hybrid equilibria continuously parametrised by A as a “Liouville link”. In some cases, the emergent point vortex equilibrium as A → ∞ can itself be the input for a second family of hybrid equilibria linking, in a limit, to yet another pure point vortex equilibrium. In this way, Liouville links together form a “Liouville chain”. We discuss several examples of Liouville chains and demonstrate that they can have a finite or an infinite number of links. We show here that the class of hybrid solutions found by Crowdy (2003) and by Krishnamurthy et al. (2019) form the first two links in one such infinite chain. We also show that the stationary point vortex equilibria recently studied by Krishnamurthy et al. (2020) can be interpreted as the limits of a Liouville link. Our results point to a rich theoretical structure underlying this class of equilibria of the 2D Euler equation.
Date Issued
2021-08-25
Date Acceptance
2021-03-08
Citation
Journal of Fluid Mechanics, 2021, 921
ISSN
0022-1120
Publisher
Cambridge University Press
Journal / Book Title
Journal of Fluid Mechanics
Volume
921
Copyright Statement
© The Author(s), 2021. Published by Cambridge University Press. This paper has been accepted for publication and will appear in a revised form, subsequent to peer-review and/or editorial input by Cambridge University Press.
Subjects
Fluids & Plasmas
01 Mathematical Sciences
09 Engineering
Publication Status
Published
Article Number
ARTN A1
Date Publish Online
2021-06-22