Note on simplified forms of the quasi-geostrophic potential vorticity equation

Different approximations of the general quasi-geostrophic potential vorticity equation under high Richardson number condition are obtained for different types of large-scale disturbances in the atmosphere. It is found that the complete equation is needed for the long waves in middle latitudes, but that the barotropic and non-divergent approximation is adequate for ordinary long waves in low latitude, whereas for disturbances with ultra-long horizontal length-scale in both zonal and meridional directions, the potential vorticity can be approximated by the thermal portion alone. On the other hand, for the EW-elongated ultra-long waves in the tropics, the uy term in the potential vorticity must be included.