From AMS Glossary
A diagnostic equation by which the vertical velocity in pressure coordinates (ω = Dp/Dt) may be calculated according to quasigeostrophic theory:
for which f is the Coriolis parameter, σ is the static stability, vg is the geostrophic velocity vector, ζg is the geostrophic relative vorticity, φ is the geopotential, ∇H2 is the horizontal Laplacian operator, and ∇H is the horizontal del operator.
The right-hand side of the omega equation can also be expressed in terms of the divergence of the Q vector.
Holton, J. R. 1992. An Introduction to Dynamic Meteorology. 3d edition, Academic Press, . 166–175.