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| Natura: | Preprint |
| Pubblicazione: |
2018
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| Accesso online: | https://arxiv.org/abs/1810.09523 |
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| _version_ | 1866908353203011584 |
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| author | Shimizu, Yuuki |
| author_facet | Shimizu, Yuuki |
| contents | We derive an analytic formula for the hydrodynamic Green function and the Robin function on every orientable surface admitting a hydrodynamic Killing vector field. Closed-form expressions are provided for all fourteen canonical Riemann surfaces, covering both compact and non-compact cases; the formulae satisfy the slip boundary condition and generate complete Hamiltonian vector fields. As an application, we clarify the mechanism whereby the curvature affects a point vortex in both qualitative and quantitative viewpoints. Qualitatively, we show a single point vortex is governed by a Hamiltonian flow whose vorticity is given by the curvature up to area constant. Quantitatively, on a rectangular torus with periodic curvature we use the analytic formula to describe two regimes: linear response that mirrors the curvature wave when the mean component is small, and a nonlinear response with amplitude resonance. The results supply a unified tool for detailed studies of point vortex dynamics and Euler-Arnold flows on surfaces with continuous symmetry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1810_09523 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Point vortex on surfaces with continuous symmetry Shimizu, Yuuki Differential Geometry 76B47, 35Q31, 58J05, 53C21 We derive an analytic formula for the hydrodynamic Green function and the Robin function on every orientable surface admitting a hydrodynamic Killing vector field. Closed-form expressions are provided for all fourteen canonical Riemann surfaces, covering both compact and non-compact cases; the formulae satisfy the slip boundary condition and generate complete Hamiltonian vector fields. As an application, we clarify the mechanism whereby the curvature affects a point vortex in both qualitative and quantitative viewpoints. Qualitatively, we show a single point vortex is governed by a Hamiltonian flow whose vorticity is given by the curvature up to area constant. Quantitatively, on a rectangular torus with periodic curvature we use the analytic formula to describe two regimes: linear response that mirrors the curvature wave when the mean component is small, and a nonlinear response with amplitude resonance. The results supply a unified tool for detailed studies of point vortex dynamics and Euler-Arnold flows on surfaces with continuous symmetry. |
| title | Point vortex on surfaces with continuous symmetry |
| topic | Differential Geometry 76B47, 35Q31, 58J05, 53C21 |
| url | https://arxiv.org/abs/1810.09523 |