Exponential asymptotics of parasitic capillary ripples and the complication of divergent eigenvalues

: (Alternative Format Thesis)

Abstract

Many problems across fluid dynamics include effects that are exponentially small as certain parameters tend to zero. These may be visible features in the solution profiles, or solvability conditions which are obtained only when the exponentially small components of the mathematical formulation are considered. The techniques required for the resolution of these features are known as exponential asymptotics.

In this thesis, special attention is placed on the limits of small surface tension and weak shear. Many physically-occurring water waves exist in the regime of small surface tension. We focus on the ideal formulation of an inviscid, irrotational, and incompressible fluid. In this formulation, the limit of small surface tension is a singular perturbative problem as the order of the governing equations differs from that found in the absence of surface tension. This is a sign that crucial exponentially small effects may appear under this limit. Numerical investigations are performed for both steadily travelling waves and time-dependent standing waves. In fixing the energy of these waves to be large, such that their solution profiles are highly nonlinear, exponentially small parasitic ripples are observed in the solution profile. In both of these cases, we characterise the bifurcation space that emerges. These parasitic capillary ripples are derived asymptotically for the steadily travelling solutions; in addition to describing these using exponential asymptotic techniques, a solvability condition is also derived.

The second limit of physical importance considered in this thesis is that of weak shear, which we consider for the equatorial Kelvin wave. We demonstrate analytically that the exponentially-small component of the eigenvalue of this problem is imaginary. This is an exponentially small instability, as the imaginary component of the eigenvalue destabilises the travelling wave.

The results contained within this thesis mark a significant milestone in our understanding of exponentially-small effects in water waves, both for surface waves in low-surface tension regimes for which we have uncovered delicate structures of solutions, and geophysical waves that are destabilised by the inclusion of weak shear.

Date of Award	22 Feb 2023
Original language	English
Awarding Institution	University of Bath
Supervisor	Phil Trinh (Supervisor) & Paul Milewski (Supervisor)