Filamentation near monotone zonal vortex caps

TL;DR

This paper investigates the stability of vortex caps on a rotating sphere, revealing how certain initial disturbances can lead to interface perimeter growth due to instability mechanisms in Euler flows.

Contribution

It demonstrates the instability of nearly zonal vortex caps with localized perturbations, expanding understanding of vortex dynamics on a rotating sphere.

Findings

Growth of interface perimeter in perturbed vortex caps

Identification of instability mechanism via flow comparison

Validation of $L^1$-stability for monotone profiles

Abstract

We study the Euler equations on a rotating unit sphere, focusing on the dynamics of vortex caps. Leveraging the $L^1$-stability of monotone, longitude-independent profiles, we demonstrate that certain ill-prepared initial data within the vortex cap class exhibit an instability characterized by the growth of the interface perimeter. These configurations are nearly equivalent in area to a zonal vortex cap but are perturbed by a localized latitudinal bump. By comparing the longitudinal flows at points along the zonal interface and within the bump region, we track the induced stretching and capture the underlying instability mechanism.