The Euler-Poincare Equations in Geophysical Fluid Dynamics

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.

42 pages, no figures, Isaac Newton Institute Proceedings (to appear)

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Cited by in corpus (7)

• Variational Principles for Stochastic Fluid Dynamics
• Averaged Lagrangians and the mean dynamical effects of fluctuations in continuum mechanics
• Implementation of the LANS-alpha turbulence model in a primitive equation ocean model
• Efficient form of the LANS-alpha turbulence model in a primitive-equation ocean model
• Nonlinear Schrodinger-Helmholtz Equation as Numerical Regularization of the Nonlinear Schrodinger Equation
• Elliptic instability in the Lagrangian-averaged Euler-Boussinesq-alpha equations
• Euler-Poincaré equations for anelastic fluid flows