The asymptotic analysis of some problems arising in fluid dynamics

The techniques of asymptotic analysis provide a valuable framework for the investigation of physical systems. These methods are applied to three distinct problems in fluid dynamics. In the first problem a dual perturbation expansion is employed to provide a sixth order solution for the height of the groundwater table in a phreatic aquifer subject to tidal forcing. A strictly analytical result is obtained with the use of a computer algebra package; this result shows the role of higher harmonics and quantifies both the asymmetry and super-elevation of the water table for beaches of both steep and shallow incline. In the second problem asymptotic analysis is again employed, here to examine strongly nonlinear vortex instabilities in a periodically heated horizontal fluid. The end result is obtained through the numerical solution of a free boundary problem. Of specific interest is the effect of both an upper bounding layer to the fluid and of noise in the periodic heating, on the resulting structures. These effects are shown to be significant, at times enhancing, and at other times limiting, vortex growth. In the final problem a second order solution to the neutral stability curve for the onset of convection in a rapidly rotating spherical shell is obtained using asymptotic methods. Comparison of these results to previous low order solutions, and to a numerical solution of the system, demonstrates the need to apply perturbation expansions with care, as in some regimes the second order solution is shown to provide an improved solution, demonstrating that previous low order solutions consistently over-predict the stability of the system, while in other regimes the second order terms are shown to be detrimental. Collectively, these three problems demonstrate both the suitability and usefulness of asymptotic analysis as a tool for understanding complex systems.