Rigidity of three-dimensional internal waves with constant vorticity

TL;DR

This paper investigates the structure of steady three-dimensional internal water waves with constant vorticity, revealing conditions under which such waves are necessarily two-dimensional or belong to a specific three-dimensional family.

Contribution

It proves a rigidity theorem showing that three-dimensional internal waves with constant, parallel vorticities are either part of a known family or are two-dimensional if densities differ.

Findings

Three-dimensional waves with constant vorticity are constrained to a specific family.

Distinct densities in layers imply the flow is two-dimensional.

The result extends understanding of wave structure under constant vorticity conditions.

Abstract

This paper studies the structural implications of constant vorticity for steady three-dimensional 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.