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Title: Atmospheric-wave generation : an exponential-asymptotic analysis
Author: Ólafsdóttir, Elínborg Ingunn
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 2006
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We use the tools of exponential-asymptotic analysis to describe the solutions of the three-dimensional Boussinesq equations and of Lorenz's five-component model that arise in geophysical fluid dynamics. In particular we use both of these models to study the generation of high-frequency motions known as inertia-gravity waves from slow balanced motion via the Stokes phenomenon. To start with we give an example demonstrating the Stokes phenomenon and to give an idea of the general approach of these techniques. We derive a full asymptotic expansion of the solutions of the Scorer equation which is an inhomogeneous Airy equation. The Stokes phenomenon causes the particular integral to switch on solutions of the Airy equation as a Stokes line is crossed. We identify the Stokes lines and the anti-Stokes lines, and most importantly calculate the Stokes multipliers. We do this by studying the Borel-Laplace transform of the solutions of the homogeneous and inhomogeneous part together with examining the behaviour of late-terms in the asymptotic expansion of the particular integral. We study homoclinic solutions of Lorenz's five-component model. We linearise the solution and hence we can use methods analogous to the ones mentioned above. Here the slow motion is represented by the first function in the linear expansion of the solution. It switches on the consequent term in the expansion representing the fast motion as a Stokes line is crossed. By considering solutions to the Boussnesq equations that consist of a sheared flow and a perturbation we narrow our problem to a differential equation that governs the amplitude of the inertia-gravity waves. Analytically the particular integral represents the slow motion that switches on solutions of the homogeneous equations which represent the fast oscillation. It follows from this that we can derive a power series expansion of the Stokes multipliers where we derive an expression for the first terms. The other terms we calculate numerically. We use the Fourier transform to reduce the Boussinesq equations to an ordinary differential equation. When we then carry out the inverse transform we use our previous result to replace the the integrand with its dominant behaviour. As the saddle point of the exponent governing the integrand collides with the endpoints for certain values we use Bleistein's method together with numerical integration to estimate the inverse.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available