Liouville chains: new hybrid vortex equilibria of the two-dimensional Euler equation

Vikas S. Krishnamurthy, Miles Wheeler, Darren G. Crowdy, Adrian Constantin

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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 in the hybrid solutions approaches the limits. While reproduces the input equilibrium, produces a new pure point vortex equilibrium. We refer to the family of hybrid equilibria continuously parametrised by as a 'Liouville link'. In some cases, the emergent point vortex equilibrium as 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 (Phys. Fluids, vol. 15, 2003, pp. 3710-3717) and by Krishnamurthy et al. (J. Fluid Mech., vol. 874, 2019, R1) 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. (Proc. R. Soc. A, vol. 476, 2020, 20200310) 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 two-dimensional Euler equation.

Original languageEnglish
Article numberA1
Pages (from-to)A1
JournalJournal of Fluid Mechanics
Early online date22 Jun 2021
Publication statusPublished - 25 Aug 2021

Bibliographical note

Funding Information:
V.S.K. was supported by the WWTF research grant number MA16-009. D.G.C. acknowledges support from the European Partners Funds at Imperial College London. V.S.K. and D.G.C. received support from EPSRC grant number EP/R014604/1 during the program ‘Complex analysis: techniques, applications and computations’ held at the Isaac Newton Institute in Cambridge during September–December 2019.

Publisher Copyright:
© The Author(s), 2021. Published by Cambridge University Press.


  • General fluid mechanics
  • Vortex dynamics
  • Vortex interactions

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering


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