Dirac-bracket Approach to Hamiltonian Balanced Models
Thursday, February 14, 2002 - 11:00am - 12:00pm
Balanced models can be viewed as constrained systems, obtained from the primitive equations by projection on a (slow) manifold devoid of inertia-gravity waves. Salmon showed how balanced models naturally inherit the Hamiltonian structure of the primitive equations if the constraints are implemented in the variational principle associated with the primitive equations. This, however, requires the introduction of an extended state space, using Lagrangian variables either as dependent or independent variables. Here, we demonstrate how this can be avoided and we derive Hamiltonian balanced models using exclusively the standard Eulerian formulation of the primitive equations. This is achieved by applying Dirac's theory of constrained systems to the Poisson structure of the fluid equations in Eulerian form. We consider multilayer primitive equations and implement general constraints which prescribe the velocity field as a pseudo-differential function of the mass field. This leads to the Poisson structure of a general class of balanced models which include (multilayer versions of) Salmon's L1 model, the semi-geostrophic model, and higher-order balanced models. The well-posedness of models in this class depends on certain invertibility issues which will be discussed.