Desingularization of a steady vortex pair in the lake equation

TL;DR

This paper constructs steady vortex pair solutions in the lake equation, showing how they depend on physical parameters and providing a method to desingularize singular vortex pairs through energy maximization.

Contribution

It introduces a novel approach to desingularize vortex pairs in the lake equation by energy maximization, accounting for Coriolis effects and depth variations.

Findings

Desingularized vortex pairs depend on depth and Coriolis parameter.

Constructed solutions converge to singular vortex pairs.

Method applies to rotation-invariant lake domains.

Abstract

We construct a family of steady solutions of the lake model perturbed by some small Coriolis force, that converge to a singular vortex pair. The desingularized solutions are obtained by maximization of the kinetic energy over a class of rearrangements of sign changing functions. The precise localization of the asymptotic singular vortex pair is proved to depend on the depth function and the Coriolis parameter, and it is independent on the geometry of the lake domain. We apply our result to construct a singular rotating vortex pair in a rotation invariant lake.