A Quasi-Geostrophic System of Equations in Terrain-Following Coordinates

The classic quasi-geostrophic (QG) system of equations has been very useful for studying and understanding the large-scale and synoptic-scale dynamic processes. Since the secondary circulation forced by the primary geostrophically balanced flow in the QG system plays a key role in the development of some subsynoptic-scale flow phenomena such as fronts especially at the surface, the QG system is also useful for studying the interactions between the synoptic-scale and mesoscale processes. However, the classic QG system was traditionally formulated and has been widely used in the pressure coordinate or modified pressure coordinate with a constant pressure (such as the sea level pressure) specified at the lower boundary. When the secondary circulation was diagnosed from this system, the effects of surface pressure variations and their interactions with the terrain were inevitably neglected. To solve this and many other related problems, a new QG system is derived in a generally terrain-following coordinate (such as that used in the WRF model or an operational NWP system). This QG system preserves the potential vorticity (PV) conservation, and its PV is not only invertible but also an improved approximation to the original PV (in the WRF model or an operational NWP system) due to the inclusion of the effect of surface pressure variation on the model terrain. A complete set of diagnostic equations is derived for the secondary circulation in the new QG system, and this set of diagnostic equations also contains the effect of surface pressure variation on the model terrain. The differences of the new QG system from the classic QG system will be highlighted at the conference, and its improved utilities with an operational NWP system for severe weather analyses will discussed in connection with the fundamental issues on the existence and fuzziness of the slow-manifold (constrained by a balanced dynamic system) and related predictability problems for atmospheric motions from synoptic-scale to mesoscale.