Dirac-bracket approach to nearly geostrophic Hamiltonian balanced models

Dirac’s theory of constrained Hamiltonian systems is applied to derive the Poisson structure of a class of balanced models describing the slow dynamics of geophysical flows. Working with the Poisson structure, instead of the canonical Hamiltonian structure previously considered in this context, allows the standard Eulerian description of fluids to be used, with no need for Lagrangian variables, and leads to completely explicit balanced equations of motion. The balanced models are derived for a class of multilayer, isentropic or isopycnic, hydrostatic models by constraining the velocity field to be an arbitrary pseudo-differential function of the mass field. Particularization to the geostrophic constraint and a slight modification thereof provides the Poisson formulation of (a multilayer version of) Salmon’s L1 model and of the semi-geostrophic model, respectively. A higher-order balanced model is also derived using a constraint that is more accurate than geostrophy.