Symmetric doubly periodic gravity-capillary waves with small vorticity

Seth, Douglas Svensson; Varholm, Kristoffer; Wahlén, Erik

doi:10.48550/arxiv.2204.13093

Cited by 1 publication

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“…Despite these advances in the two-dimensional case, the understanding of three-dimensional rotational waves remains comparatively rudimentary. Currently, there are only two regimes in which existence is known: Lokharu, Seth, and Wahlén [20] have constructed small-amplitude threedimensional waves with Beltrami-type flow, and Seth, Varholm, and Wahlén [28] obtained symmetric diamond waves with small vorticity. The first result is proved using a careful multi-parameter Lyapunov-Schmidt reduction, while the second involves a delicate fixed-point argument inspired by related problems in plasma physics.…”

Section: Introductionmentioning

confidence: 99%

Rigidity of three-dimensional internal waves with constant vorticity

Chen¹,

Fan²,

Walsh³

et al. 2022

Preprint

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This paper studies the structural implications of constant vorticity for steady threedimensional internal water waves. It is known that in many physical regimes, water waves beneath vacuum that have constant vorticity are necessarily two dimensional. The situation is more subtle for internal waves that traveling along the interface between two immiscible fluids. When the layers have the same density, there is a large class of explicit steady waves with constant vorticity that are three-dimensional in that the velocity field and pressure depend on one horizontal variable while the interface is an arbitrary function of the other. We prove the following rigidity result: every three-dimensional traveling internal wave with bounded velocity for which the vorticities in the upper and lower layers are nonzero, constant, and parallel must belong to this family. If the densities in each layer are distinct, then in fact the flow is fully two dimensional.

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Section: Introductionmentioning

confidence: 99%