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CLIMATE/EARTH 401�Geophysical Fluid Dynamics�Dynamical Meteorology

Richard B. Rood

rbrood@umich.edu

https://clasp.engin.umich.edu/people/richard-b-rood/

https://openclimate.org/

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Omega Equation

The omega equation is a diagnostic equation for vertical velocity for large-scale, mid-latitude systems. It spans multiple processes that influence vertical velocity.

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Quasi-geostrophic equations in terms of geopotential and omega.

THERMODYNAMIC EQUATION

VORTICITY EQUATION

ELIMINATE THE GEOPOTENTIAL AND GET AN EQUATION FOR OMEGA

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QG-omega equation: Eliminate geopotential

1.) Apply the horizontal Laplacian operator ( ²) to the QG thermodynamic equation

2.) Differentiate the geopotential height tendency� equation with respect to p

3.) Combine 1) and 2)

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QG-omega equation: divergence / stretching

Link between vertical derivative of vorticity advection divergence/stretching and vertical motion

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QG-omega equation: temperature advection

Link between temperature advection and vertical motion

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QG-omega equation: diabatic heating

Link between diabatic heating and vertical motion

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QG-omega equation: simplification

This is still a complicated equation to analyze directly—simplify it
Assume small diabatic heating (scale out last term)
Use the chain rule on the first and second terms on the right-hand side and combine the remaining terms

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QG-omega equation (simplified)

Advection of absolute vorticity�by the thermal wind

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Analysis of omega equation

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“Advection” by thermal wind?

How to analyze this on a map?

Thermal�Wind

Perpendicular �to

Thickness

Look at contours of constant thickness

Gradient of

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What about that Laplacian?

QG omega equation relates vorticity advection by the thermal wind with the Laplacian of omega
Assume omega has a wave-like form

This leads to

which means

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Vertical Motion on Weather Maps

Laplacian of omega is proportional to -ω
Omega can be analyzed as:

Remember, from definition of �omega and scale analysis

Positive vorticity advection by the thermal wind indicates rising motion

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Vertical velocity

In all the estimates for vertical velocity there is no time derivative
The vertical velocity, in this large-scale, mid-latitude description of dynamics, is exactly what is needed to maintain hydrostatic balance.
If vertical acceleration is important, we need to revisit scaling and the important forces.

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