Stratified Large-Amplitude Steady Periodic Water Waves with Critical Layers

Susanna V Haziot

doi:10.1007/s00220-020-03892-0

Abstract

By means of a conformal mapping and bifurcation theory, we prove the existence of large-amplitude steady stratified periodic water waves, with a density function depending linearly on the streamfunction, which may have critical layers and overhanging profiles. We also provide certain conditions for which these waves cannot overturn.

Highlights

By means of a conformal mapping and bifurcation theory, we prove the existence of large-amplitude steady stratified periodic water waves, with a density function depending linearly on the streamfunction, which may have critical layers and overhanging profiles
We will study two-dimensional steady water waves which propagate on the surface of an inviscid fluid and over a flat, impermeable bed
The term stratification here refers to the presence of a non-constant density function ρ(X, Y ), and throughout this paper, we will choose ρ to depend linearly on the streamfunction

Summary

Introduction

We will study two-dimensional steady water waves which propagate on the surface of an inviscid fluid and over a flat, impermeable bed. In [7,12] the authors managed to construct small- and large-amplitude water waves with constant vorticity, by considering the fluid domain as the image of a strip via a conformal map This map makes no assumption on the geometry of the physical domain, or on the streamfunction, and so the solutions obtained in this way may contain critical layers, stagnation points and overturning profiles. We boil down the whole problem to studying Bernoulli’s boundary condition at the surface of the fluid domain by reformulating it in terms of the periodic Hilbert transform for the strip, applied to the imaginary conformal coordinate This will give us a nonlinear pseudo-differential equation containing several parameters. First proving that there exist solutions with overturning profiles and use a maximum principle argument involving the pressure of the fluid to show that, for our assumptions, such solutions cannot exist in the physical plane

Formulation of the Problem

Existence Theory

Nodal Analysis

Overhanging Waves