Stability of the Rankine Vortex and Perimeter Growth in Vortex Patches
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915613381754880 |
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| author | Brownfield, John |
| author_facet | Brownfield, John |
| contents | We prove that for $ω: \mathbb{R}^2 \to [0,1]$ 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 $ω$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_09772 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability of the Rankine Vortex and Perimeter Growth in Vortex Patches Brownfield, John Analysis of PDEs We prove that for $ω: \mathbb{R}^2 \to [0,1]$ 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 $ω$. 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. |
| title | Stability of the Rankine Vortex and Perimeter Growth in Vortex Patches |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.09772 |