Co-rotating vortices with N fold symmetry for the inviscid surface quasi-geostrophic equation

TL;DR

This paper constructs special steady solutions for the generalized surface quasi-geostrophic equations using a variational approach, focusing on N vortex patches with symmetry and their asymptotic behavior as patches shrink.

Contribution

It introduces a variational method to build symmetric vortex patch solutions and analyzes their limit as patches become point vortices.

Findings

Solutions form N vortex patches with N-fold symmetry

As patches shrink, solutions approximate N Dirac masses

Patches rotate at a constant angular velocity

Abstract

We provide a variational construction of special solutions to the generalized surface quasi-geostrophic equations. These solutions take the form of N vortex patches with N-fold symmetry , which are steady in a uniformly rotating frame. Moreover, we investigate their asymptotic properties when the size of the corresponding patches vanishes. In this limit, we prove these solutions to be a desingularization of N Dirac masses with the same intensity, located on the N vertices of a regular polygon rotating at a constant angular velocity.