Existence and qualitative aspects of geophysical water waves

Abstract

The object of geophysical fluid dynamics is the study of naturally occurring flows on the Earth. Most of the problems that arise are at the large-scale end, where either the rotation of the Earth or density differences (warm and cold air masses, fresh and saline waters) or both are of importance. In this regard, geophysical fluid dynamics comprises rotating-stratified fluid flows. This thesis consists of eleven research papers which are grouped into two parts. The first part is dedicated to the study of nonlinear capillary-gravity water waves traveling at the surface of rotational flows with rough vorticities or of flows with a vertical layering (continuous resp. discontinuous) of density, the emphasis being on developing a rigorous existence theory. The papers collected in the second part of the thesis investigate the existence and the qualitative properties of equatorial geophysical water flows with Coriolis effects in the so-called f-plane approximation. In this part we also present several explicit solutions describing equatorial water flows in the f-plane approximation in different geophysical scenarios −deep-water waves, edge waves, stratified flows− and we analyze their properties.