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arXiv · 2404.04523

On the vorticity threshold for steady water waves

Evgeniy Lokharu·Miles H. Wheeler
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Abstract

One of the most striking features of two-dimensional steady water waves with adverse vorticity is the emergence of stagnation points in the flow, even along branches of solutions that are initially unidirectional. In this paper we show that, for constant adverse vorticity above a certain explicit threshold, the first stagnation point must appear on the bed directly below the crest. This is in contrast to waves with favourable or even small adverse vorticity, where it is known that bottom stagnation cannot occur, leading to extreme waves exhibiting surface singularities. We also establish several related inequalities for unidirectional waves with strong adverse vorticity, including a new upper bound on the amplitude.

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Evgeniy Lokharu, Miles H. Wheeler. 2024-04-06. On the vorticity threshold for steady water waves. https://arxiv.org/abs/2404.04523

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