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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2310.20674 |
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| _version_ | 1866912752579117056 |
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| author | Albritton, Dallas Ożański, Wojciech |
| author_facet | Albritton, Dallas Ożański, Wojciech |
| contents | We consider vortex column solutions $v = V(r) e_θ+ W(r) e_z$ to the $3$D Euler equations. We give a mathematically rigorous construction of the countable family of unstable modes discovered by Liebovich and Stewartson (J. Fluid Mech. 126, 1983) via formal asymptotic analysis. The unstable modes exhibit $O(1)$ growth rates and concentrate on a ring $r= r_0$ asymptotically as the azimuthal and axial wavenumbers $n, α\to \infty$ with a fixed ratio. We construct these so-called ring modes with an inner-outer gluing procedure. Finally, we prove that each linear instability implies nonlinear instability for vortex columns. In particular, our analysis yields nonlinear instability for the Batchelor trailing line vortex $V(r) :=\frac{q}{r} (1-\mathrm{e}^{-r^2})$ and $W(r) :=\mathrm{e}^{-r^2}$ when $0<q <\log 2 / \sqrt{1-\log 2} \approx 1.251$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_20674 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Linear and nonlinear instability of vortex columns Albritton, Dallas Ożański, Wojciech Analysis of PDEs We consider vortex column solutions $v = V(r) e_θ+ W(r) e_z$ to the $3$D Euler equations. We give a mathematically rigorous construction of the countable family of unstable modes discovered by Liebovich and Stewartson (J. Fluid Mech. 126, 1983) via formal asymptotic analysis. The unstable modes exhibit $O(1)$ growth rates and concentrate on a ring $r= r_0$ asymptotically as the azimuthal and axial wavenumbers $n, α\to \infty$ with a fixed ratio. We construct these so-called ring modes with an inner-outer gluing procedure. Finally, we prove that each linear instability implies nonlinear instability for vortex columns. In particular, our analysis yields nonlinear instability for the Batchelor trailing line vortex $V(r) :=\frac{q}{r} (1-\mathrm{e}^{-r^2})$ and $W(r) :=\mathrm{e}^{-r^2}$ when $0<q <\log 2 / \sqrt{1-\log 2} \approx 1.251$. |
| title | Linear and nonlinear instability of vortex columns |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2310.20674 |