Internal tides and resonance

This thesis is a study of resonant internal tide motion. One common linear inviscid model represents the motion in terms of a two dimensional stream function satisfying a standard second order hyperbolic equation. This model is examined in detail and it is shown that existing solution techniques can be extended and unified with the classical theory of a single complex variable. This results in a number of new theoretical devices for the solution of the model, including c-conformal mapping - a mapping procedure that shares many of the traits of classical conformal mapping, and c-analytic continuation - a procedure for extending the domain of a solution analogous to standard analytic continuation. These techniques are applied to a study of the model on closed basins. A crucial distinction relating to the geometry of the corners of the basin is noted, and the importance of energy sources and sinks is recognized. The analysis is found both to support the conjectures of other researchers that the existence of resonant motion may be predicted through an analysis of the characteristic coordinates of the inviscid model, and to demonstrate that the inviscid model is an inadequate vehicle for the study of internal tides on basin topographies. A viscous form of the linear model is then discussed and a number of numerical solution techniques suitable for basin topographies are devised. The first of these is a straightforward application of standard collocation methods, but is found to be computationally expensive. A simpler, more efficient family of techniques based on the theory of c-analytic continuation is then derived. These methods are seen to be a natural extension of a Fourier series technique successfully employed by other authors. The predictions of the linear models are then compared with the results of laboratory experiments. Resonance effects are seen to play a major role in determining the motion of the fluid. However, the strongly resonant motions predicted by the linear models are not observed, and, in general, the agreement between model and data is poor. This is attributed to both the sensitivity of the problem and the difficulty associated with modelling the external forcing. The large amplitude strongly resonant motions predicted by the linear models prompt a study of nonlinear internal tide dynamics. A nonlinear model based on the Navier Stokes equations is considered, and numerical solutions are obtained via finite differences. These solutions show that for strongly resonant motion the \Rigid Lid\" approximation commonly imposed at the free surface is inadequate and must be replaced with a more accurate nonlinear boundary condition."