Large-amplitude steady gravity water waves with general vorticity and critical layers

TL;DR

This paper develops a comprehensive mathematical framework for analyzing large-amplitude steady gravity water waves with arbitrary vorticity, including overhanging surfaces and critical layers, using bifurcation theory and conformal mappings.

Contribution

It introduces a novel reformulation of Bernoulli's equation and constructs a global solution set for complex water wave configurations with general vorticity.

Findings

Established existence of a connected set of solutions bifurcating from laminar flows.

Analyzed the structure of solutions including overhanging waves and critical layers.

Provided insights into downstream wave behaviors.

Abstract

We consider two-dimensional steady periodic gravity waves on water of finite depth with a prescribed but arbitrary vorticity distribution. The water surface is allowed to be overhanging and no assumptions regarding the absence of stagnation points and critical layers are made. Using conformal mappings and a new reformulation of Bernoulli's equation, we uncover an equivalent formulation as "identity plus compact," which is amenable to Rabinowitz's global bifurcation theorem. This allows us to construct a global connected set of solutions, bifurcating from laminar flows with a flat surface. Moreover, a nodal analysis is carried out for these solutions under a certain spectral assumption involving the vorticity function. Lastly, downstream waves are investigated in more detail.