The Euler-Poincaré Equations in Geophysical Fluid Dynamics.

Holm, D. D., J. E. Marsden and T. S. Ratiu

Large-Scale Atmosphere-Ocean Dynamics II: Geometric Methods and Models, 251-300.
J. Norbury and I. Roulstone, eds., Cambridge Univ. Press, 2002.

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.