Derivation of the planetary geostrophic equations
Résumé
In this paper, we justify mathematically the derivation of the planetary geostrophic equations (PGE) from the hydrostatic Boussinesq equations with Coriolis force, usually named the primitive equations (PE). The planetary geostrophic equations, which are a classical model of thermohaline circulation, are obtained from the primitive equations as the Froude number $Fr$, the Rossby number $\varepsilon$, and the Burger number $Bu$ go to 0. These numbers are supposed to satisfy $Fr = O(\varepsilon^{1/2})$ and $Bu=O(\varepsilon)$ which is relevant to the thermohaline planetary dynamics. The analysis performed here does not follow the same lines as previous asymptotic studies on rotating fluids. It involves a singular operator which is not skew symmetric, and prevents classical energy estimates. To handle such operator requires to put the primitive equations under normal form, together with an appropriate use of the viscous terms.