Time quasi-periodic vortex patches for quasi-geostrophic shallow-water equations

TL;DR

This paper applies KAM theory to quasi-geostrophic shallow-water equations to construct time quasi-periodic vortex solutions with boundaries confined to thin annuli, revealing complex fluid dynamics behaviors.

Contribution

It introduces a novel application of KAM theory to construct invariant tori and vortex patches in quasi-geostrophic models with Rossby deformation length in a Cantor set.

Findings

Existence of time quasi-periodic vortex patches

Boundaries localized in thin annuli for all times

Construction of invariant tori in fluid dynamics models

Abstract

In this paper, we shall implement KAM theory in order to construct a large class of time quasi-periodic solutions for an active scalar model arising in fluid dynamics. More precisely, the construction of invariant tori is performed for quasi-geostrophic shallow-water equations when the {\it Rossby deformation length} belongs to a massive Cantor set. As a consequence, we construct pulsating vortex patches whose boundary is localized in a thin annulus for any time.