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The Euler-Poincaré Equations in Geophysical Fluid Dynamics

Holm, Darryl D. and Marsden, Jerrold E. and Ratiu, Tudor S. (2002) The Euler-Poincaré Equations in Geophysical Fluid Dynamics. In: Large-scale Atmosphere-ocean Dynamics II: Geometric methods and models. Vol.2. Cambridge University Press , Cambridge [England], pp. 251-300. ISBN 0521807573 http://resolver.caltech.edu/CaltechAUTHORS:20100909-094611543

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Abstract

Recent theoretical work has developed the Hamilton's-principle analog of Lie-Poisson Hamiltonian systems defined on semidirect products. The main theoretical results are twofold: 1. Euler-Poincaré equations (the Lagrangian analog of Lie-Poisson Hamiltonian equations) are derived for a parameter dependent Lagrangian from a general variational principle of Lagrange d'Alembert type in which variations are constrained; 2. an abstract Kelvin-Noether theorem is derived for such systems. By imposing suitable constraints on the variations and by using invariance properties of the Lagrangian, as one does for the Euler equations for the rigid body and ideal fluids, we cast several standard Eulerian models of geophysical fluid dynamics (GFD) at various levels of approximation into Euler-Poincaré form and discuss their corresponding Kelvin-Noether theorems and potential vorticity conservation laws. The various levels of GFD approximation are related by substituting a sequence of velocity decompositions and asymptotic expansions into Hamilton's principle for the Euler equations of a rotating stratified ideal incompressible fluid. We emphasize that the shared properties of this sequence of approximate ideal GFD models follow directly from their Euler-Poincaré formulations. New modifications of the Euler-Boussinesq equations and primitive equations are also proposed in which nonlinear dispersion adaptively filters high wavenumbers and thereby enhances stability and regularity without compromising either low wavenumber behavior or geophysical balances.


Item Type:Book Section
Additional Information:We would like to extend our gratitude to John Allen, Ciprian Foias, Rodney Kinney, HansfPeter Kruse, Len Margolin, Jim McWilliams, Balu Nadiga, Ian Roulstone, Steve Shkoller, Edriss Titi and Vladimir Zeitlin for their time, encouragement and invaluable input. Work by D. Holm was conducted under the auspices of the US Department of Energy, supported (in part) by funds provided by the University of California for the conduct of discretionary research by Los Alamos National Laboratory. Work of J. Marsden was supported by the California Institute of Technology and NSF grant DMS 96-33161. Work by T. Ratiu was partially supported by NSF Grant DMS-9503273 and DOE contract DE-FG03-95ER25245-A000. Some of this research was performed while two of the authors (DDH and TSR) were visiting the Isaac Newton Institute for Mathematical Sciences at Cambridge University. We gratefully Acknowledge financial support by the program “Mathematics in Atmosphere and Ocean Dynamics" as well as the stimulating atmosphere at the Isaac Newton Institute during our stay there.
Funders:
Funding AgencyGrant Number
California Institute of TechnologyUNSPECIFIED
University of CaliforniaUNSPECIFIED
Los Alamos National LaboratoryUNSPECIFIED
NSFDMS 96-33161
NSFDMS-9503273
Department of EnergyDE-FG03-95ER25245-A000
Cambridge University Isaac Newton Institute for Mathematical SciencesUNSPECIFIED
Record Number:CaltechAUTHORS:20100909-094611543
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20100909-094611543
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ID Code:19842
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:16 Sep 2010 21:13
Last Modified:26 Dec 2012 12:24

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