On the non-hydrostatic equations in pressure and sigma coordinates

The non-hydrostatic equations governing the inviscid, adiabatic motion of a perfect gas are formulated using pressure as vertical coordinate; and M. J. Miller's 1974 approximate quasi-non-hydrostatic pressure coordinate equations are derived by applying a systematic scale analysis and power series expansion. The derivation makes clear that these equations are the pressure coordinate counterparts of the anelastic height coordinate equations obtained by Y. Ogura and N. A. Phillips in 1962. The two sets cannot be interconverted by coordinate transformation and so they are not physically equivalent; but the differences are small at the order of validity of both sets. Consideration of a quasi-hydrostatic approximation emphasizes the non-hydrostatic character of Miller's equations. Sigma coordinate quasi-non-hydrostatic equations are obtained by direct transformation of the pressure coordinate forms, and consistent energy equations are derived for both sets. Convenient diagnostic partial differential equations for the geopotential field are obtained for both pressure and sigma coordinate forms. As shown by Miller, the quasi-non-hydrostatic formulation does not permit vertically propagating acoustic waves. Horizontally propagating acoustic waves (Lamb waves) are in general allowed, but can be removed from the pressure coordinate system by applying suitable boundary conditions. Some aspects of the treatment of the Lamb wave problem are corrected in this study. The quasi-non-hydrostatic sigma coordinate system permits Lamb waves, but it may still be considered suitable for convective and (especially) mesoscale modelling with or without orography. The possible use of the quasi-non-hydrostatic system in large-scale theory and modelling is also discussed.