Asymptotic of steady vortex pairs in the lake equation

TL;DR

This paper investigates the detailed asymptotic behavior of shrinking vortex pairs in a lake model, revealing their localization and shape properties through advanced variational analysis.

Contribution

It advances understanding of vortex pair asymptotics by analyzing second order properties and shape, extending previous first order results in lake equations.

Findings

Vortices localize according to an adapted Kirchhoff-Routh function.

Optimal vortex shapes are characterized asymptotically.

A relaxed maximization yields vortex patches with specific distribution properties.

Abstract

We bring new results in the study the asymptotic behavior of shrinking vortex pairs obtained by maximization of the kinetic energy in a 2-dimensional lake over a class of rearrangements. After improving recent results obtained for the first order asymptotic behavior of such pairs, we focus on second order asymptotic properties. We show that among all points of maximal depth, the vortex locates according to an adaptation of the Kirchoff-Routh function, and we study the asymptotic shape of optimal vortices. We also explore a relaxed maximization problem with uniform constraints, for which we prove that the distribution consists of two vortex patches.