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Title: Complex and contact geometry in geophysical fluid dynamics
Author: Delahaies, Sylvain
ISNI:       0000 0001 3421 5518
Awarding Body: University of Surrey
Current Institution: University of Surrey
Date of Award: 2008
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Due to its conceptual simplicity and its remarkable mathematical properties, semi-geostrophic theory has been much used for the analysis of large-scale atmospheric dynamics since its introduction by Hoskins [41] in the mid-seventies. Despite its limited accuracy, its ability to tolerate contact discontinuities within the fluid makes it a useful and elegant model for the study of subsynoptic phenomenon such as fronts and jets. In their attempt to find a suitable candidate for a model whose accuracy improves over semi-geostrophic theory while retaining its essential features, McIntyre & Roulstone [59] discovered the existence of a hyper-Kahler structure for a class of Hamiltonian balanced models. In this thesis, in the context of shallow-water dynamics, we recall the formulation of f-plane semi-geostrophic theory and the derivation of McIntyre & Roulstone balanced models firstly using a Hamiltonian framework and secondly using a multisymplectic framework. Introducing the notion of contact manifold, we propose a classification of contact transformations and a characterisation of contact transformations in terms of generating functions. We then introduce the theory of Monge-Ampere operators introduced by Lychagin [54] to study the geometric properties of the Monge-Ampere equation relating the potential vorticity to the geopotential for balanced models. Using this formalism we give a systematic derivation of hyper-Kahler and hyper-para-Kahler structures associated with symplectic nondegenerate Monge-Ampere equations and we use these structures to extend some of the properties of semi-geostrophic theory to McIntyre & Roulstone's balanced models. We discuss the application of the theory of Monge-Ampere operators to the divergence equation for shallow-water model. Finally we present semi-geostrophic theory in three dimensions, and we show how the theory of Monge-Ampere operators in R3 associates a real generalised Calabi-Yau structure to this model.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available