Stability of the Rankine Vortex and Perimeter Growth in Vortex Patches

John Brownfield

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arXiv · 2511.09772

Stability of the Rankine Vortex and Perimeter Growth in Vortex Patches

John Brownfield

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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.

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John Brownfield. 2025-11-12. Stability of the Rankine Vortex and Perimeter Growth in Vortex Patches. https://arxiv.org/abs/2511.09772

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