The omega equation is a culminating result in synoptic-scale meteorology. It is an elliptic partial differential equation, named because its left-hand side produces an estimate of vertical velocity, customarily expressed by symbol ω {\displaystyle \omega } , in a pressure coordinate measuring height of the atmosphere. Mathematically, ω = d p d t {\displaystyle \omega ={\frac {dp}{dt}}} , where d d t {\displaystyle {d \over dt}} represents a material derivative. The underlying concept is more general, however, and can also be applied to the Boussinesq fluid equation system where vertical velocity is w = d z d t {\displaystyle w={\frac {dz}{dt}}} in altitude coordinate z.
Concept and summary Vertical wind is crucial to weather and storms of all types. Even slow, broad updrafts can create convective instability or bring air to its lifted condensation level creating stratiform cloud decks. Unfortunately, predicting vertical motion directly is difficult. For synoptic scales in Earth's broad and shallow troposphere, the vertical component of Newton's law of motion is sacrificed in meteorology's primitive equations, by accepting the hydrostatic approximation. Instead, vertical velocity must be solved through its link to horizontal laws of motion, via the mass continuity equation. But this presents further difficulties, because horizontal winds are mostly geostrophic, to a good approximation. Geostrophic winds merely circulate horizontally, and do not significantly converge or diverge in the horizontal to provide the needed link to mass continuity and thus vertical motion. The key insight embodied by the quasi-geostrophic omega equation is that thermal wind balance (the combination of hydrostatic and geostrophic force balances above) holds throughout time, even though the horizontal transport of momentum and heat by geostrophic winds will often tend to destroy that balance. Logically, then, a small non-geostrophic component of the wind (one which is divergent, and thus connected to vertical motion) must be acting as a secondary circulation to maintain balance of the geostrophic primary circulation. The quasi-geostrophic omega ω Q G {\displaystyle \omega _{QG}} is the hypothetical vertical motion whose adiabatic cooling or warming effect (based on the atmosphere's static stability) would prevent thermal wind imbalance from growing with time, by countering the balance-destroying (or imbalance-creating) effects of advection. Strictly speaking, QG theory approximates both the advected momentum and the advecting velocity as given by the geostrophic wind. In summary, one may consider the vertical velocity that results from solving the omega equation as that which would be needed to maintain geostrophy and hydrostasy in the face of advection by the geostrophic wind. The equation reads:
where f {\displaystyle f} is the Coriolis parameter, σ {\displaystyle \sigma } is related to the static stability, V g {\displaystyle \mathbf {V} _{\text{g}}} is the geostrophic velocity vector, ζ g {\displaystyle \zeta _{\text{g}}} is the geostrophic relative vorticity, ϕ {\displaystyle \phi } is the geopotential, ∇ H 2 {\displaystyle \nabla _{\text{H}}^{2}} is the horizontal Laplacian operator and ∇ H {\displaystyle \nabla _{\text{H}}} is the horizontal del operator. Its sign and sense in typical weather applications is: upward motion is produced by positive vorticity advection above the level in question (the first term), plus warm advection (the second term).
Derivation The derivation of the ω {\displaystyle \omega } equation is based on the vertical component of the vorticity equation, and the thermodynamic equation. The vertical vorticity equation for a frictionless atmosphere may be written using pressure as the vertical coordinate:
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