Balanced models in isentropic coordinates and the shallow water equations

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Abstract
It is shown that there is an appropriate family of three balanced models in isentropic coordinates and the shallow water equations. “Appropriate” means the equations include exact hydrostatic, continuity and heat equations and form a complete system; some family members are second-order accurate in a midlatitude Rossby number expansion; the models conserve at least one integral invariant analogous to the primitive equation energy and enstrophy invariants; and the terms retained in the vorticity and divergence equations are the minimum required for the accuracy of the model in an asymptotic scale analysis. We compare vorticity and divergence in physical and isentropic coordinates to illustrate an aspect of the difference in physical content between balanced models defined in the different coordinates. We show that the early balanced models of Bolin and Charney are similar to members of our family in isentropic coordinates, but fail some of our criteria for appropriateness. Finally, we discuss why we think the above criteria are the right ones for balanced model families, and compare and contrast similar criteria for other intermediate models. DOI: 10.1111/j.1600-0870.1984.tb00236.x