# Omega equation

The omega equation is of great importance in meteorology and atmospheric physics. It is a partial differential equation for the vertical velocity, $\omega$ , which is defined as a Lagrangian rate of change of pressure with time, that is, $\omega ={\frac {dp}{dt}}$ .

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## Derivation

The derivation of the $\omega$ equation is based on the vorticity equation and the thermodynamic equation. The vorticity equation for a frictionless atmosphere may be written as:

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The thermodynamic equation may be written as:

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The $\omega$ equation (Template:EquationNote) is then obtained from equation (Template:EquationNote) and (Template:EquationNote) by substituting values:

$\xi ={\frac {g}{f}}\nabla ^{2}Z$ and

${\hat {k}}\cdot \nabla \omega \times {\frac {\partial V}{\partial p}}={\frac {\partial \omega }{\partial y}}{\frac {\partial u}{\partial p}}-{\frac {\partial \omega }{\partial x}}{\frac {\partial v}{\partial p}}$ into (Template:EquationNote), which gives:

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and

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where $\omega _{1}$ is the vertical velocity due to the mean baroclinicity in the atmosphere and $\omega _{2}$ is the vertical velocity due to the non-adiabatic heating, which includes the latent heat of condensation, sensible heat radiation, etc. (Singh & Rathor, 1974).

## Interpretation

Physically, the omega equation combines the effects of vertical differential of geostrophic absolute vorticity advection (first term on the right-hand side) and three-dimensional Laplacian of thickness thermal advection (second term on the right-hand side) and determines the resulting vertical motion (as expressed by the dependent variable $\omega$ .)

The above equation is used by meteorologists and operational weather forecasters to assess development from synoptic charts. In rather simple terms, positive vorticity advection (or PVA for short) and no thermal advection results in a negative $\omega$ , that is, ascending motion. Similarly, warm advection (or WA for short) also results in a negative $\omega$ corresponding to ascending motion. Negative vorticity advection (NVA) or cold advection (CA) both result in a positive $\omega$ corresponding to descending motion.