On stratified water waves with critical layers and Coriolis forces

Miles H. Wheeler

doi:10.3934/dcds.2019193

Article Contents

2019, Volume 39, Issue 8: 4747-4770. Doi: 10.3934/dcds.2019193

This issue Previous Article Infinity-harmonic potentials and their streamlines Next Article Study of a nonlinear boundary-value problem of geophysical relevance

On stratified water waves with critical layers and Coriolis forces

Miles H. Wheeler

University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

The paper is for the special theme: Mathematical Aspects of Physical Oceanography, organized by Adrian Constantin

Received: October 2018

Revised: January 2019

Published: May 2019

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Abstract

We consider nonlinear traveling waves in a two-dimensional fluid subject to the effects of vorticity, stratification, and in-plane Coriolis forces. We first observe that the terms representing the Coriolis forces can be completely eliminated by a change of variables. This does not appear to be well-known, and helps to organize some of the existing literature.

Second we give a rigorous existence result for periodic waves in a two-layer system with a free surface and constant densities and vorticities in each layer, allowing for the presence of critical layers. We augment the problem with four physically-motivated constraints, and phrase our hypotheses directly in terms of the explicit dispersion relation for the problem. This approach smooths the way for further generalizations, some of which we briefly outline at the end of the paper.

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Mathematics Subject Classification: Primary: 76B15, 35Q35; Secondary: 35R35.

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Figure 1. Fluid configurations with multiple layers using the notation (1.1) and (1.2). (a) A configuration with N = 4 layers and a rigid lid. (b) A configuration with N = 2 layers and a free surface. This is the type of configuration which will be considered in Section 3

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Figure 2. Shear flows $\overline U(z)$ corresponding to the stream functions $\overline\Psi_1, \overline\Psi_2$ in (3.9). Both flows have $\omega_2 < 0 < \omega_1$ and $c > 0$. (a) A flow with a critical layer at the marked point in $D_1$ where $\overline U_1 = c$. (b) A flow without a critical layer

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