Multipole vortex patch equilibria for active scalar equations

TL;DR

This paper develops a method to transform steady configurations of point vortices into vortex patches for active scalar equations, enabling the construction of new asymmetric and symmetric vortex structures with specific motions.

Contribution

It introduces a novel approach to desingularize point vortex equilibria into vortex patches using the implicit function theorem, extending previous techniques to more complex configurations.

Findings

Constructed new asymmetric translating and rotating vortex pairs.

Established existence of stationary tripoles.

Adapted techniques for symmetric polygonal vortex configurations.

Abstract

We study how a general steady configuration of finitely-many point vortices, with Newtonian interaction or generalized surface quasi-geostrophic interactions, can be desingularized into a steady configuration of vortex patches. The configurations can be uniformly rotating, uniformly translating, or completely stationary. Using a technique first introduced by Hmidi and Mateu \cite{Hmidi-Mateu} for vortex pairs, we reformulate the problem for the patch boundaries so that it no longer appears singular in the point-vortex limit. Provided the point vortex equilibrium is non-degenerate in a natural sense, solutions can then be constructed directly using the implicit function theorem, yielding asymptotics for the shape of the patch boundaries. As an application, we construct new families of asymmetric translating and rotating pairs, as well as stationary tripoles. We also show how the techniques can be adapted for highly symmetric configurations such as regular polygons, body-centered polygons and nested regular polygons by integrating the appropriate symmetries into the formulation of the problem.