Lake equations with an evanescent or emergent island

TL;DR

This paper investigates the asymptotic behavior of lake equations with shrinking or emerging islands, deriving simplified models and addressing mathematical challenges related to singularities when water depth vanishes.

Contribution

It introduces new asymptotic lake-type equations for both shrinking and emerging islands, handling singularities with novel uniform estimates in weighted spaces.

Findings

Derived asymptotic lake equations for both island scenarios

Identified Dirac mass contribution in vorticity for shrinking islands

Established uniform estimates to handle singular water depth conditions

Abstract

We study the asymptotic dynamics of the lake equations in the following two cases, an island shrinking to a point and an emerging island. For both cases, we derive an asymptotic lake-type equation. In the former case, the asymptotic dynamics includes an additional Dirac mass in the vorticity. The main mathematical difficulty is that the equations are singular when the water depth vanishes. We provide new uniform estimates in weighted spaces for the related stream functions which will imply the compactness result.