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Complex and contact geometry in geophysical fluid dynamics.

Delahaies, Sylvain. (2008) Complex and contact geometry in geophysical fluid dynamics. Doctoral thesis, University of Surrey (United Kingdom)..

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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.

Item Type: Thesis (Doctoral)
Divisions : Theses
Authors :
Delahaies, Sylvain.
Date : 2008
Contributors :
Depositing User : EPrints Services
Date Deposited : 09 Nov 2017 12:11
Last Modified : 16 Mar 2018 19:04

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