Multipole Vortex Patch Equilibria for Active Scalar Equations

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 [36] 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.

Steady asymmetric vortex pairs for Euler equations

“…It is important to note that these solutions exhibit extreme singularity and do not adhere to the governing Euler equations. As an illustrative example, this approach was applied to pairs of rotating vortex points in [37,46], and subsequently extended to cover Thomson and nested Thomson polygons, Kármán vortex streets as described in [28,29,41]. An alternative approach based on variational arguments was developed before in [57].…”