Liouville links and chains on the plane and associated stationary point vortex equilibria

Abstract

<p style='text-indent:20px;'>Liouville links and chains are exact steady solutions of the Euler equation for two-dimensional, incompressible, homogeneous and planar fluid flow, uncovered recently in [<xref ref-type="bibr" rid="b11">11</xref>,<xref ref-type="bibr" rid="b12">12</xref>,<xref ref-type="bibr" rid="b13">13</xref>]. These solutions consist of a set of stationary point vortices embedded in a smooth non-zero and non-uniform background vorticity described by a Liouville-type partial differential equation. The solutions contain several arbitrary parameters and possess a rich structure. The background vorticity can be varied with one of the parameters, resulting in two limiting cases where it concentrates into some point vortex equilibrium configuration in one limit and another distinct point vortex equilibrium in the other limit. By a simple scaling of the point vortex strengths at a limit, a new steady solution can be constructed, and the procedure iterated indefinitely in some cases. The resulting sequence of solutions has been called a Liouville chain [<xref ref-type="bibr" rid="b13">13</xref>]. A transformation exists that can produce the limiting point vortex equilibria from a given seed equilibrium. In this paper, we collect together all these results in a review and present selected new examples corresponding to special sequences of 'collapse configurations.' The final section discusses possible applications to different geophysical flow scenarios.</p>