1. Atmospheric Fluid Dynamics
(the sequel)
References:
1 “Atmospheric Physics” Andrews, Chapter 5 (5.1;5.4)
2 “Introduction to Dynamic Meteorology” Chapter 4 (4.2;4.3)

2. Variables Describing the Atmosphere
Thermodynamic variables
Pressure, p
Density, r
Temperature, T
Dynamic variables
Velocity, (three parameters!)
Vorticity, (three parameters!)

3. Vorticity Definition:
What does it mean? It is a measure of the local (microscopic) rotation in a fluid.
Explicitly this is:

4. 2D flow
Consider a motion in the atmosphere that is only in a horizontal direction (xy-plane).
In terms of the atmosphere this means: ignore vertical winds and vertical variations of the horizontal winds.
w is perpendicular to the plane of the flow!
x is the vertical component of the vorticity (you can see different letters for it, like z)
Large scale meteorology models work mostly with x.

5. Examples: 2D flow vorticity
Constant flow
Constant gradient
Linear shear x y x y x y

6. Visualization-thought experiment
Place a pinwheel in the flow and watch for rotation. (x>0 counterclockwise)
x>0 x y x y
x<0
x=0
x=0 x y x y

7. Jet streams xr<0 xr>0 xr<0 xr>0 Jupiter
adapted from Limaye, Icarus, 1984

8. Examples: 2D circular flowz f r
V(r)
We used the curl in cylindrical coordinates.

9. 2D Rotation
Solid body rotation:
Point vortex rotation
V(r) r
V(r) r

10. Vorticity in a rotating frame of reference
When we consider the vorticity in the atmosphere of a planet we have to remember that the planet itself rotates. It is a rotation of a solid body and it has some vorticity associated with it.
In a rotating reference frame:
Take the rotation:

11. We live on a sphere! f z W
The vertical component of the vorticity is

12. Vorticity in a rotating frame of reference II
The vertical component of the vorticity at a given latitude f is:
Absolute vorticity xa, relative vorticity xr, planetary vorticity f.
Note that for the Earth f is positive in NH and negative in SH.
The planetary vorticity is typically much bigger than the relative vorticity

13. Relative vorticity
The relative vorticity is positive for counterclockwise rotation:
In the NH this is the circulation around low pressure systems has positive relative vorticity (hurricanes, typical tornadoes, mid-latitude cyclones).
In the SH the relative vorticity is negative for the low pressure centers as the motion is clockwise.

14. Potential vorticity Definition Alternative expression
P is conserved in absence of friction and diabatic heating when following the fluid motion. It can be used as a tracer of motion.
If you move on surfaces of constant potential temperaure (adiabatic motion) and there is no friction the potential vorticity is conserved!

15. Column stretching
As the column stretches the absolute vorticity increases so that the potential vorticity remains constant. If f is constant then this results in increased relative vorticity (the air parcel spins up).

16. Some examples
You follow the spinning motion of a leaf on the surface of a shallow river. Can you guess what is the depth of the water?
The leaf will spin up in deeper water.

17. What about hurricanes?
A Hurricane is a system with large positive relative vorticity (counterclockwise rotation at the surface in NH).
Can we follow a similar analogy with the hurricanes? Is the potential vorticity conserved as the hurricanes move northward?
It is not an adiabatic system! Hurricanes are fueled by latent heat due to water evaporation/condensation.

18. Vorticity equations General approach:
Use the momentum equations and take the corresponding partial derivatives to form the curl of the velocity field on the LHS and whatever is the result goes on the RHS.
And … you have a vorticity equation.
Exactly how this is done, we will see next time.