omega equation
A diagnostic equation by which the vertical velocity in pressure coordinates (ω = Dp/Dt) may be calculated according to quasigeostrophic theory:
σ
∇
H
2 ω +
f
2
∂
2 ω
∂
p
2
f
∂
∂ p
[
v
g ⋅
∇
H
(
f +
ζ
s ) ] −
∇
H
2
(
v
g ⋅
∇
H
∂ ϕ
∂ p ) {\displaystyle \sigma \nabla _{H}^{2}\omega +f^{2}{\frac {\partial ^{2}\omega }{\partial p^{2}}}=f{\frac {\partial }{\partial p}}\left[\mathbf {v} _{g}\cdot \nabla _{H}\left(f+\zeta _{s}\right)\right]-\nabla _{H}^{2}\left(\mathbf {v} _{g}\cdot \nabla _{H}{\frac {\partial \phi }{\partial p}}\right)}
![{\displaystyle \sigma \nabla _{H}^{2}\omega +f^{2}{\frac {\partial ^{2}\omega }{\partial p^{2}}}=f{\frac {\partial }{\partial p}}\left[\mathbf {v} _{g}\cdot \nabla _{H}\left(f+\zeta _{s}\right)\right]-\nabla _{H}^{2}\left(\mathbf {v} _{g}\cdot \nabla _{H}{\frac {\partial \phi }{\partial p}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/294403f070b774421d80d1e9fc5f0e5327e1b95a)
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.