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
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 xderivative of the meridional wind equation and the yderivative of the zonal wind equation, and then subtract the
two
(Eqn.)
We can now define the “relative vorticity” as:
(Eqn.)
By this definition, cyclonic circulation is defined to have
positive relative vorticity, while anti-cyclonic circulation is
defined to have negative relative vorticity
Dynamics
Vorticity
Putting the definition for vorticity into the pervious equation
gives:
(Eqn.)
From before, we write the the hydrostatic equation as:
(Eqn.)
We can define the “height” of a given pressure level as
(x,y,t):
(Eqn.)
If we then integrate the hydrostatic equation with depth and
use the conservation of mass we find that:
(Eqn.)
Dynamics
Vorticity
We can put this equation into the previous
equation for vorticity:
(Eqn.)
Finally, assume H>> :
(Eqn.)
This equation is called the “Conservation of
Potential Vorticity”
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:
(Eqn.)
Using this “Beta approximation”, the Coriolis
parameter can be written as:
(Eqn.)
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:
(Eqn.)
Dynamics
Vorticity
Ex.1 Lee-side low pressures
• 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 wavelike 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
Dynamics
Vorticity
Waves of this kind are called “planetary” or
“Rossby” waves.
• Short Rossby waves
• Long Rossby waves
Short Rossby waves
• a balance between planetary vorticity and relative
vorticity, as described above
Long Rossby waves
• a balance between planetary vorticity and height
• This produces global-scale wave-like features seen
earlier
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