Modified quasi‐geostrophic equations using geometric height as vertical coordinate

Abstract

AbstractA quasi‐geostrophic β‐plane set using geometric height as vertical coordinate is derived without the conventional assumption that fractional changes of basic state potential temperature over a scale height of the atmosphere are small. Although the hydrostatic, continuity and vorticity equations contain terms not present in the usual set, and the rigid horizontal boundary condition is modified, the form of the implied potential vorticity equation in terms of the stream function is the same as before. Some of these conclusions are extensions to the nonlinear case of results obtained in 1968 by Lindzen in a linearized analysis. the modified equations have reasonable integral conservation properties. As regards accuracy they are closely comparable with the quasi‐geostrophic pressure coordinate equations, but since rigid horizontal boundary conditions are easily applied exactly in the modified set the most nearly equivalent pressure‐based system is evidently a (quasi‐geostrophic) σ‐coordinate set rather than the p‐coordinate set. the modified rigid horizontal boundary condition has the property of allowing for the change in the apparent vertical when a coordinate transformation to a frame moving with a uniform zonal velocity relative to the β‐plane is carried out. Application of the modified equations to an f‐plane baroclinic instability problem reveals a dependence of stability properties on the height‐averaged flow Uo as well as the shear (as found also in 1947 by Charney and by Geisler and Dickinson in 1975 in β‐plane analyses), the dependence on Uo being most marked for the longer waves.