Liouville chains: new hybrid vortex equilibria of the 2D Euler equation

TL;DR

This paper introduces Liouville chains, a new class of exact steady solutions to the 2D Euler equation, linking point vortex equilibria through a family of hybrid solutions with Liouville-type vorticity.

Contribution

It constructs the concept of Liouville chains, connecting known vortex equilibria via hybrid solutions parameterized by a continuous variable, revealing a rich underlying structure.

Findings

Liouville chains can have finite or infinite links.

Existing solutions form the first two links of an infinite chain.

Stationary vortex equilibria are limits of Liouville links.

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 $A\to 0,\infty$. While $A\to 0$ reproduces the input equilibrium, $A\to\infty$ 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 $A\to\infty$ 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.