Extending the derivation of shallow water Hamiltonian balanced models to three-dimensional, stratified and thus more physically relevant Hamiltonian balanced models turns out to be non-trivial. This non-triviality is best illustrated in terms of Dirac's theory of constraints. The principal ideas of Dirac's theory serve as guidance to resolve the conceptual and technical problems inherent in the derivation of three-dimensional Hamiltonian balanced models.
Furthermore, Hamilton's principle, conventionally given in terms of Lagrangian variables, proves in the present case to be much easier to use when expressed in terms of Clebsch variables. While retaining Lagrangian information, this facilitates treatment of quantities that are simpler in the Eulerian description, such as pressure gradients.
By combining Dirac's theory with Hamilton's principle expressed in Clebsch variables, one may derive a general expression for a three-dimensional Hamiltonian balanced model. As an example, the extension of Salmon's L_1-dynamics to three dimensions is derived. The Eady problem is chosen in order to illustrate the properties of the new three-dimesional L_1-dynamics.
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12th Conference on Atmospheric and Oceanic Fluid Dynamics