Mathematical Analysis of Vorticity Fronts

: (Alternative Format Thesis)

Abstract

This thesis analyses a particular family of solutions to the steady incompressible Euler equations known as vorticity fronts. Steady solutions are of interest both practically, and from a mathematical point of view. Practically, in nature it is common to see a wave profile moving at constant speed without change of form, and through a change of reference frame, these can be considered time-independent. More abstractly, steady solutions are of interest as potential limiting states for the time dependent case, as well as for their own sake.

We consider a channel bounded above and below by rigid walls, where the fluid within has two layers, each of constant vorticity, with an unknown interface separating them. This is the only property distinguishing the two layers; for example, there is no surface tension or change in density between the layers. The fact that the interface is unknown poses one of the key challenges for this problem, and we must find coordinates to map our problem onto a known domain. This process leaves us with a highly non-linear problem.

In Chapter 2 we find exact small amplitude solitary solutions using a centre manifold theorem, and thoroughly describe the qualitative properties of the streamlines. The loss of compactness incurred by considering solitary rather than periodic waves is a significant challenge. In Chapter 3 we construct exact large amplitude periodic solutions, using global bifurcation. Our coordinates are given by a conformal map in each layer, one of which is then composed with a “horizontal distortion” such that the coordinates agree at the interface. To our knowledge, this is the first example of a multi-layer problem with a local formulation permitting both stagnation points and overhanging wave profiles. We will also find periodic numerical solutions, and will see that for long wavelengths, these agree well the solitary solutions from Chapter 2.

Date of Award	12 Nov 2025
Original language	English
Awarding Institution	University of Bath
Supervisor	Miles Wheeler (Supervisor), Alex Doak (Supervisor) & Karsten Matthies (Supervisor)