Free and rigid boundary quasigeostrophic models in pressure coordinates

Starting from the hydrostatic primitive equations in pressure coordinates, a quasigeostrophic (QG) model is derived with temperature and ground surface pressure (GSP) as the governing prognostic fields. In this model two different tendency equations for the GSP exist: one expressing mass conservation, and the other the condition of zero vertical velocity at the ground in common physical space. Comparison of these equations leads to a diagnostic relationship that provides a boundary condition for the omega equation. Equivalence of the system of equations to the QG model that employs the potential vorticity equation is established. Introduction of the GSP as one of the main prognostic fields highlights the problem of the lower boundary condition in pressure coordinates. Two models are introduced and studied, and their main features are compared - the free surface (FS) and the rigid boundary (RB) models, the latter being most common in pressure coordinate QG studies. The choice has an effect on the boundary conditions for the omega equation and affects the physical qualities of the model. The model with the FS has additional energy, similar to the energy of an elastic membrane, which the RB model lacks. Both models give similar results for synoptic-scale processes but differ essentially for scales larger than the external Rossby deformation radius ([approximately]3000 km). As the scale analysis shows, the FS model is accurate in this case, while the RB model, which filters mass fluctuations of vertical unit columns of the atmosphere, seriously distorts the phase speeds.