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 the Lagrangian rate of change of pressure with time. Mathematically, ${\displaystyle \omega ={\frac {dp}{dt}}}$, where ${\displaystyle {d \over dt}}$ represents a material derivative. It is valid for large scale flows under the conditions of quasi-geostrophy and hydrostatic balance. In fact, one may consider the vertical velocity that results from solving the omega equation as that which is needed to maintain quasi-geostrophy and hydrostasy.^[1]
The equation reads:

${\displaystyle \sigma \nabla _{H}^{2}\omega +f^{2}{\frac {\partial ^{2}\omega }{\partial p^{2}}}=f{\frac {\partial }{\partial p}}\left[\mathbf {V} _{g}\cdot \nabla _{H}(\zeta _{g}+f)\right]-\nabla _{H}^{2}\left(\mathbf {V} _{g}\cdot \nabla _{H}{\frac {\partial \phi }{\partial p}}\right)}$

(1)

where $f$ is the Coriolis parameter, $\sigma$ is related to the static stability, ${\displaystyle \mathbf {V} _{g}}$ is the geostrophic velocity vector, ${\displaystyle \zeta _{g}}$ is the geostrophic relative vorticity, $\phi$ is the geopotential, ${\displaystyle \nabla _{H}^{2}}$ is the horizontal Laplacian operator and ${\displaystyle \nabla _{H}}$ is the horizontal del operator.^[2]

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 using pressure as the vertical coordinate:

${\displaystyle {\frac {\partial \xi }{\partial t}}+V\cdot \nabla \eta -f{\frac {\partial \omega }{\partial p}}=\left(\xi {\frac {\partial \omega }{\partial p}}-\omega {\frac {\partial \xi }{\partial p}}\right)+k\cdot \nabla \omega \times {\frac {\partial V}{\partial p}}}$

(2)

Here $\xi$ is the relative vorticity, $V$ the horizontal wind velocity vector, whose components in the $x$ and $y$ directions are $u$ and $v$ respectively, $\eta$ the absolute vorticity, $f$ the Coriolis parameter, ${\displaystyle \omega ={\frac {dp}{dt}}}$ the material derivative of pressure $p$. $k$ is the unit vertical vector, $\nabla$ is the isobaric Del (grad) operator, ${\displaystyle \left(\xi {\frac {\partial \omega }{\partial p}}-\omega {\frac {\partial \xi }{\partial p}}\right)}$ is the vertical advection of vorticity and ${\displaystyle k\cdot \nabla \omega \times {\frac {\partial V}{\partial p}}}$ represents the transformation of horizontal vorticity into vertical vorticity.^[3]

The thermodynamic equation may be written as:

${\displaystyle {\frac {\partial }{\partial t}}\left(-{\frac {\partial Z}{\partial p}}\right)+V\cdot \nabla \left(-{\frac {\partial Z}{\partial p}}\right)-k\omega ={\frac {R}{C_{p}\cdot g}}\cdot {\frac {q}{p}}}$

(3)

where ${\displaystyle k\equiv \left({\frac {\partial Z}{\partial p}}\right){\frac {\partial }{\partial p}}\ln \theta }$, in which $q$ is the supply of heat per unit-time and mass, $C_{p}$the specific heat of dry air, $R$ the gas constant for dry air, $\theta$ is the potential temperature and $\phi$ is geopotential ${\displaystyle (gZ)}$.

The $\omega$ equation (1) is then obtained from equation (2) and (3) by substituting values:

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

and

${\displaystyle {\hat {k}}\cdot \nabla \omega \times {\frac {\partial V}{\partial p}}={\frac {\partial \omega }{\partial y}}{\frac {\partial u}{\partial p}}-{\frac {\partial \omega }{\partial x}}{\frac {\partial v}{\partial p}}}$

into (2), which gives:

${\displaystyle {\frac {\partial }{\partial t}}\left({\frac {g}{f}}\nabla ^{2}Z\right)+V\cdot \nabla \eta -f{\frac {\partial \omega }{\partial p}}=\left(\xi {\frac {\partial \omega }{\partial p}}-\omega {\frac {\partial \xi }{\partial p}}\right)+\left({\frac {\partial \omega }{\partial x}}{\frac {\partial v}{\partial p}}\right)}$

(4)

Differentiating (4) with respect to $p$ gives:

${\displaystyle {\frac {g}{f}}{\frac {\partial }{\partial t}}\nabla ^{2}\left({\frac {\partial Z}{\partial p}}\right)+{\frac {\partial }{\partial p}}(V\cdot \nabla \eta )-f{\frac {\partial ^{2}\omega }{\partial p^{2}}}-{\frac {\partial f}{\partial p}}{\frac {\partial \omega }{\partial p}}={\frac {\partial }{\partial p}}\left(\xi {\frac {\partial \omega }{\partial p}}-\omega {\frac {\partial \xi }{\partial p}}\right)+{\frac {\partial }{\partial p}}\left({\frac {\partial \omega }{\partial y}}\cdot {\frac {\partial u}{\partial p}}-{\frac {\partial \omega }{\partial x}}\cdot {\frac {\partial v}{\partial p}}\right)}$

(5)

Taking the Laplacian ($\nabla ^{2}$) of (3) gives:

${\displaystyle \nabla ^{2}\left(-{\frac {\partial Z}{\partial p}}\right)+\nabla ^{2}V\cdot \nabla \left(-{\frac {\partial Z}{\partial p}}\right)-\nabla ^{2}k\omega ={\frac {R}{C_{p}\cdot g}}\cdot {\frac {\nabla ^{2}q}{p}}}$

(6)

Adding (5) and (6), simplifying and substituting ${\displaystyle gk=\sigma }$, gives:

${\displaystyle \nabla ^{2}\omega +{\frac {f^{2}}{\sigma }}{\frac {\partial ^{2}\omega }{\partial p^{2}}}={\frac {1}{\sigma }}\left[{\frac {\partial }{\partial p}}J(\phi ,\eta )+{\frac {1}{f}}\nabla ^{2}J\left(\phi ,-{\frac {\partial \phi }{\partial p}}\right)\right]-{\frac {f}{\sigma }}{\frac {\partial }{\partial p}}\left({\frac {\partial \omega }{\partial y}}\cdot {\frac {\partial u}{\partial p}}-{\frac {\partial \omega }{\partial x}}\cdot {\frac {\partial v}{\partial p}}\right)-{\frac {f}{\sigma }}{\frac {\partial }{\partial p}}\left(\xi {\frac {\partial \omega }{\partial p}}-\omega {\frac {\partial \xi }{\partial p}}\right){\frac {R\cdot \nabla ^{2}q}{C_{p}\cdot S\cdot p}}}$

(7)

Equation (7) is now a linear differential equation in $\omega$, such that it can be split into two part, namely $\omega _{1}$ and $\omega _{2}$, such that:

${\displaystyle \nabla ^{2}\omega _{1}+{\frac {f^{2}}{\sigma }}{\frac {\partial ^{2}\omega _{1}}{\partial p^{2}}}={\frac {1}{\sigma }}\left[{\frac {\partial }{\partial p}}J(\phi ,\eta )+{\frac {1}{f}}\nabla ^{2}J\left(\phi ,-{\frac {\partial \phi }{\partial p}}\right)\right]-{\frac {f}{\sigma }}{\frac {\partial }{\partial p}}\left({\frac {\partial \omega }{\partial y}}\cdot {\frac {\partial u}{\partial p}}-{\frac {\partial \omega }{\partial x}}\cdot {\frac {\partial v}{\partial p}}\right)-{\frac {f}{\sigma }}{\frac {\partial }{\partial p}}\left(\xi {\frac {\partial \omega }{\partial p}}-\omega {\frac {\partial \xi }{\partial p}}\right)}$

(8)

and

${\displaystyle \nabla ^{2}\omega _{2}+{\frac {f^{2}}{\sigma }}{\frac {\partial ^{2}\omega _{2}}{\partial p^{2}}}={\frac {R\cdot \nabla ^{2}q}{C_{p}\cdot \sigma \cdot p}}}$

(9)

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.

References

1. Holton, James (2004). An Introduction to Dynamic Meteorology. Elsevier Academic Press. ISBN 0123540151.
2. Holton, J.R., 1992, An Introduction to Dynamic Meteorology Academic Press, 166-175
3. Singh & Rathor, 1974, Reduction of the Complete Omega Equation to the Simplest Form, Pure and Applied Geophysics, 112, 219-223

External links

• American Meteorological Society definition