Stability of the Rankine Vortex and Perimeter Growth in Vortex Patches

TL;DR

This paper establishes bounds on the deviation of vortex patches from the Rankine vortex and confirms through analysis that perimeter growth is linear over time for certain perturbations.

Contribution

It provides new bounds relating vorticity deviations to energy and angular momentum, and proves linear perimeter growth for symmetric vortex perturbations.

Findings

Bound on $L^1$ deviation from Rankine vortex in terms of pseudo-energy and angular momentum

Linear perimeter growth demonstrated for symmetric vortex perturbations

Dependence on angular momentum can be eliminated under symmetry conditions

Abstract

We prove that for $\omega: \mathbb{R}^2 \to [0,1]$ sharing the same total vorticity and center of vorticity as the Rankine vortex, the $L^1$ deviation from the Rankine patch can be bounded by a function of the pseudo-energy deviation and the angular momentum of $\omega$. In the case of $m-$fold symmetry, the dependence on the angular momentum can be dropped. Using this, we affirm the results of prior simulations by demonstrating linear in time perimeter growth for a simply connected perturbation of the Rankine vortex.