From AMS Glossary
(Also called geostrophic approximation, pseudogeostrophic approximation.) A form of the primitive equations in which an approximation to the actual winds is selectively used in the momentum and thermodynamic equations.
Specifically, horizontal winds are replaced by their geostrophic values in the horizontal acceleration terms of the momentum equations, and horizontal advection in the thermodynamic equation is approximated by geostrophic advection. In addition, the quasigeostrophic approximation neglects vertical advection of momentum and replaces the four-dimensional static stability parameter with a basic-state static stability, which is a function of the vertical coordinate only. The quasigeostrophic approximation is used in the analysis of extratropical synoptic-scale systems, in which winds can be closely approximated by their geostrophic values. This approximation is not accurate in situations in which the ageostrophic wind plays an important advective role, for example, around fronts.