1. Dynamics
Vorticity
In the previous lecture, we used “scaling” to simplify the equations of motion and found that, to first order, horizontal winds are geostrophic while, in the vertical, the system is “static” (i.e. hydrostatic)
This does not imply that there are no vertical motions, just that these vertical motions are not driven by vertical pressure gradients
One of the limitations of the geostrophic approximation, however, is that it does not allow us to predict the evolution of the system because there is no term that is related to the time rate of change
In order to include the time rate of change of the winds, we need to look at scaling associated with the next order of magnitude

2. Dynamics
Vorticity
If we now keep all terms of order 10-4 and 10-5, the horizontal equations become:
(Eqn.)
We take the “curl” of these two equations by taking the x-derivative of the meridional wind equation and the y-derivative of the zonal wind equation, and then subtract the two
We can now define the “relative vorticity” as:
By this definition, cyclonic circulation is defined to have positive relative vorticity, while anti-cyclonic circulation is defined to have negative relative vorticity

3. Dynamics
Vorticity
Putting the definition for vorticity into the pervious equation gives:
(Eqn.)
From before, we write the the hydrostatic equation as:
We can define the “height” of a given pressure level as (x,y,t):
If we then integrate the hydrostatic equation with depth and use the conservation of mass we find that:

4. Dynamics
Vorticity
We can put this equation into the previous equation for vorticity:
(Eqn.)
Finally, assume H>> :
This equation is called the “Conservation of Potential Vorticity”

5. Dynamics
Vorticity
New term in the vorticity equation is “Planetary vorticity”
From before, we defined the Coriolis parameter as:
(Eqn.)
We can approximate this dependence as:
Using this “Beta approximation”, the Coriolis parameter can be written as:

6. Dynamics
Vorticity
Putting this approximation into the potential vorticity equation gives:
(Eqn.)
Only holds following the path of given air parcel
To simplify a bit more, we add two approximations:
H>> 
fo~10-4; by~10-5
Using these approximations, it can be shown that:

7. Dynamics Vorticity Ex.1 Lee-side low pressures Ex.2 Planetary waves
Typically we find that low pressures tend to form on the lee-side of mountains
This can be seen in daily weather maps as well as in the general circulation of the atmosphere
Ex.2 Planetary waves
In the large-scale circulation, it is possible to find wave-like patterns that are fairly stationary or move only very slowly
We can understand the dynamics of these waves by looking at the behavior of an air parcel that is initially moving to the north
As with the lee-side low, these features can be seen both in the daily fields as well as the climate fields

8. Dynamics
Vorticity
Waves of this kind are called “planetary” or “Rossby” waves.
Short Rossby waves
Long Rossby waves
a balance between planetary vorticity and relative vorticity, as described above
a balance between planetary vorticity and height
This produces global-scale wave-like features seen earlier

9. Dynamics
Vorticity

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Vorticity

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Vorticity

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Vorticity

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Vorticity

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Vorticity

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Vorticity

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Vorticity

17. Dynamics
Vorticity Z E N L H E

18. Dynamics
Vorticity

19. Dynamics
Vorticity