From Glossary of Meteorology
A set of mathematical approximations introduced into a system of hydrodynamical partial differential equations to filter out or exclude solutions corresponding to those physical disturbances that are believed to contribute only negligibly to the problem at hand.
In the theory of cyclonic-scale atmospheric flow, for example, the introduction of the quasi- hydrostatic approximation effectively excludes solutions of the equations of motion corresponding to atmospheric sound waves, while the selective use of the quasigeostrophic approximation, as in the vorticity equation with the divergence eliminated, effectively filters out solutions corresponding to high-speed atmospheric gravity waves. Through the use of such filtering approximations, the equations of motion can be focused upon the desired components of the flow.
Charney, J. G. 1948. On the Scale of Atmospheric Motion. Geofys. Publ.. 17. 3–17.