The quasi-geostrophic theory of the thermal shallow water equations

Abstract

AbstractThe thermal shallow water equations provide a depth-averaged description of motions in a fluid layer that permits horizontal variations in material properties. They typically arise through an equivalent barotropic approximation of a two-layer system, with a spatially varying density contrast due to an evolving temperature field in the active layer. We formalize a previous derivation of the quasi-geostrophic (QG) theory of these equations, by performing a direct asymptotic expansion for small Rossby number. We then present a second derivation as the small Rossby number limit of a balanced model that projects out high-frequency dynamics due to inertia-gravity waves. This latter derivation has wider validity, not being restricted to mid-latitude $\beta $-planes. We also derive their local energy conservation equation from the QG limit of a thermal shallow water pseudo-energy conservation equation. This derivation involves the ageostrophic correction to the leading-order geostrophic velocity that is eliminated in the usual derivation of a closed evolution equation for the QG potential vorticity. Finally, we derive the non-canonical Hamiltonian structure of the thermal QG equations from a decomposition in Rossby number of a pseudo-energy and Poisson bracket for the thermal shallow water equations.