Vortex-carrying solitary gravity waves of large amplitude
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| Format: | Preprint |
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2024
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| _version_ | 1866914750279974912 |
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| author | Chen, Robin Ming Varholm, Kristoffer Walsh, Samuel Wheeler, Miles H. |
| author_facet | Chen, Robin Ming Varholm, Kristoffer Walsh, Samuel Wheeler, Miles H. |
| contents | In this paper, we study two-dimensional traveling waves in finite-depth water that are acted upon solely by gravity. We prove that, for any supercritical Froude number (non-dimensionalized wave speed), there exists a continuous one-parameter family $\mathcal{C}$ of solitary waves in equilibrium with a submerged point vortex. This family bifurcates from an irrotational uniform flow, and, at least for large Froude numbers, extends up to the development of a surface singularity. These are the first rigorously constructed gravity wave-borne point vortices without surface tension, and notably our formulation allows the free surface to be overhanging. We also provide a numerical bifurcation study of traveling periodic gravity waves with submerged point vortices, which strongly suggests that some of these waves indeed overturn. Finally, we prove that at generic solutions on $\mathcal{C}$ $\unicode{x2013}$ including those that are large amplitude or even overhanging $\unicode{x2013}$ the point vortex can be desingularized to obtain solitary waves with a submerged hollow vortex. Physically, these can be thought of as traveling waves carrying spinning bubbles of air. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08074 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Vortex-carrying solitary gravity waves of large amplitude Chen, Robin Ming Varholm, Kristoffer Walsh, Samuel Wheeler, Miles H. Analysis of PDEs In this paper, we study two-dimensional traveling waves in finite-depth water that are acted upon solely by gravity. We prove that, for any supercritical Froude number (non-dimensionalized wave speed), there exists a continuous one-parameter family $\mathcal{C}$ of solitary waves in equilibrium with a submerged point vortex. This family bifurcates from an irrotational uniform flow, and, at least for large Froude numbers, extends up to the development of a surface singularity. These are the first rigorously constructed gravity wave-borne point vortices without surface tension, and notably our formulation allows the free surface to be overhanging. We also provide a numerical bifurcation study of traveling periodic gravity waves with submerged point vortices, which strongly suggests that some of these waves indeed overturn. Finally, we prove that at generic solutions on $\mathcal{C}$ $\unicode{x2013}$ including those that are large amplitude or even overhanging $\unicode{x2013}$ the point vortex can be desingularized to obtain solitary waves with a submerged hollow vortex. Physically, these can be thought of as traveling waves carrying spinning bubbles of air. |
| title | Vortex-carrying solitary gravity waves of large amplitude |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.08074 |