Confinement results near point vortices on the rotating sphere

TL;DR

This paper investigates vorticity confinement near point vortices on a rotating sphere, extending planar Euler equation results to spherical geometry with new mathematical insights.

Contribution

It provides new results on collision improbability, confinement duration, and configuration examples for vorticity on the rotating sphere, with a unified proof approach.

Findings

Collision improbability logarithmic in time

Logarithmic vorticity confinement for general configurations

Existence of configurations with power-law long confinement

Abstract

We study the Euler equation on the rotating sphere in the case where the absolute vorticity is initially sharply concentrated around several points. We follow the literature already concerning vorticity confinement for the planar Euler equations, and obtain similar results on the rotating sphere, with new challenges due to the geometry. More precisely, we show the improbability of collisions for point-vortices, logarithmic in time absolute vorticity confinement for general configurations, the optimality of this last result in general, and the existence of configurations with power-law long confinement. We take this opportunity to write a unified, self-contained, and improved version of all the proofs, previously scattered across multiple papers on the planar case, with detailed exposition for pedagogical clarity.