Holm, Darryl D. and Marsden, Jerrold E. and Ratiu, Tudor S. (2002) The EulerPoincaré Equations in Geophysical Fluid Dynamics. In: Largescale Atmosphereocean Dynamics II: Geometric methods and models. Vol.2. Cambridge University Press , Cambridge [England], pp. 251300. ISBN 0521807573 http://resolver.caltech.edu/CaltechAUTHORS:20100909094611543

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Abstract
Recent theoretical work has developed the Hamilton'sprinciple analog of LiePoisson Hamiltonian systems defined on semidirect products. The main theoretical results are twofold: 1. EulerPoincaré equations (the Lagrangian analog of LiePoisson 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 KelvinNoether 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 EulerPoincaré form and discuss their corresponding KelvinNoether 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 EulerPoincaré formulations. New modifications of the EulerBoussinesq 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.
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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 9633161. Work by T. Ratiu was partially supported by NSF Grant DMS9503273 and DOE contract DEFG0395ER25245A000. 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.  
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Record Number:  CaltechAUTHORS:20100909094611543  
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:20100909094611543  
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Deposited By:  Ruth Sustaita  
Deposited On:  16 Sep 2010 21:13  
Last Modified:  26 Dec 2012 12:24 
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