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The static energy encodes all possible information about the thermodynamics and potential energy (and all related forces) of stratified geophysical fluids. In this paper, we develop a systematic methodology, called static energy asymptotics, that exploits this property for constructing energetically and thermodynamically consistent sound-proof approximations of the equations of motion. By approximating the static energy to various orders of accuracy, two main families of approximations are (re-)derived and discussed: the pseudo-incompressible (PI) approximation and the anelastic (AN) approximation. For all approximations, the background and available potential energies (in Lorenz sense) can be constructed to match their exact counterparts as closely as feasible and to be expressible in terms of the exact (as opposed to ad-hoc) thermodynamic potentials. For hydrostatic motions, the AN approximation (of which the Boussinesq approximation is a special case) has the same structure as that of legacy Seawater Boussinesq primitive equations. The energetics of such models could therefore be made transparently traceable to that of the full Navier–Stokes equations at little to no additional cost, thus allowing them to take full advantage of the Gibbs Sea Water (GSW) library developed as part of the new thermodynamic standard for seawater TEOS-10.
Tailleux et al. (Fri,) studied this question.