Gravity water waves over constant vorticity flows: From laminar flows to touching waves

TL;DR

This paper extends the existence theory of gravity water waves with constant vorticity, demonstrating a continuous transition from laminar flows to touching and breaking waves, including analysis of critical layers.

Contribution

It generalizes prior results to finite depth and arbitrary vorticity, establishing a continuous wave family connecting laminar and touching wave profiles.

Findings

Existence of a continuous wave curve from laminar to touching waves

Presence of breaking waves with vertical tangents

Behavior of critical layers near the surface

Abstract

In a recent paper, Hur & Wheeler [J. Differential Equations, 338:572-590, 2022] proved the existence of periodic steady water waves over an infinitely deep, two-dimensional and constant vorticity flow under the influence of gravity. These solutions include overhanging wave profiles, some of which exhibit surfaces that touch at a point and thereby enclose a bubble of air. We extend these results by formulating a problem that encompasses both infinitely deep and finitely deep flows, and by proving the existence of a continuous curve of water waves that connects a laminar flow to a touching wave for fixed, nonzero gravity. This implies the existence of a wave profile featuring a vertical tangent at a point, which is not overhanging, and is referred to as a breaking wave. We also study the behaviour of critical layers, which are points where the horizontal velocity vanishes, near the surface. In particular, this result holds for arbitrary vorticity.