1. PV Thinking
CH 06

2. What is PV thinking?
Two main approaches to solving fluid flow problems:
We can integrate the momentum, continuity and thermodynamic equations (the primitive equations) directly.
In certain cases we can use the vorticity – streamfunction formulation
Often (2) is more insightful than (1).
The vorticity – streamfunction approach can provide a neat conceptual framework in which to understand the dynamics.
In certain circumstances the approach can be generalized.

3. How can PV thinking help the forecaster?
Davies and Emanuel (1991)
“ … a proper integration of the equations of motion is not synonymous with a conceptual grasp of the phenomena being predicted”
Answer
“PV thinking” can provide the forecaster with a sound conceptual framework in which to interpret numerical analyses and prognoses.

4. Vorticity – Streamfunction method
Assumptions: homogeneous, two-dimensional, inviscid, non-rotating flow. 1
conservation
material derivative
vorticity
continuity
invertibility 2

5. Rotating flow on a b-plane
Assumptions: homogeneous, two-dimensional, inviscid, rotating flow on a b-plane. 1
conservation
absolute vorticity
continuity
invertibility 2

6. Tropical cyclone thought experiment
cyclonic asymmetry . .
f  z
f 
z   
anticyclonic asymmetry
10oN

7. Tropical cyclone thought experiment
Asymmetries induce a poleward flow across the vortex centre . .
The symmetric vortex rotates the asymmetry
10oN

8. Tropical cyclone thought experiment
The symmetric vortex rotates the asymmetry . .
The asymmetric flow across the symmetric vortex advects the the vortex northwestwards
10oN

9. Relative vorticity
Streamfunction

10. Relative vorticity
Streamfunction

11. Relative vorticity
Streamfunction

12. Relative vorticity
Absolute vorticity

13. Divergent flow, variable depth
Assumptions: homogeneous, divergent, inviscid, variable depth h(x,y,t).
h(x,y,t) 1
Conservation
absolute vorticity
continuity
No invertibility!
Except for QG-flow

14. Quasi-geostrophic motion
Assumptions: stratified (Boussinesq), rotating, three-dimensional, adiabatic, inviscid, flow. 1
Conservation
potential vorticity
material derivative
continuity 2
Invertibility

15. General adiabatic motion of a rotating stratified fluid
Assumptions: compressible, stratified, rotating, three-dimensional, adiabatic, inviscid, flow. 1
Conservation
Ertel potential vorticity
material derivative
continuity
No invertibility!
unless …flow balanced

16. Isentropic coordinates
Assumptions: compressible, stratified, rotating, three-dimensional, adiabatic, inviscid, flow.
Conservation
Ertel potential vorticity
material derivative

17. Formulation of an invertibility principle
To formulate an invertibility principle one must:
Specify some kind of balance condition, the simplest, but least accurate option being ordinary quasi-geostrophic balance,
Specify some sort of reference state, expressing the mass distribution of q, and
Solve the inversion principle globally, with proper attention to boundary conditions.

18. Diabatic and frictional effects
PV is no longer materially conserved:
za = absolute vorticity .
q = absolute vorticity
K =   F
Suppose each string of particles along lines of latitude (y = constant) is given a sinusoidal meridional (north-south) displacement (x), as shown. The corresponding perturbation in potential vorticity for each string, q(x), is as shown also. From this, and using the string analogy (keeping in mind that a positive force is analogous to a negative q) we deduce the streamfunction distribution and finally, using the second component of (6.3), v = x, we infer the meridional velocity component to have the variation shown. Comparison of the meridional displacement with the meridional velocity shows:
(I) that the disturbance propagates westwards, and
(ii) since the longitudes of displacement maxima are places of zero velocity, the wave amplitude does not vary with time; in particular, the wave is stable. V
Frictionless, adiabatic

19. Physical interpretation
Diabatic term za za   
heat 
cool
Friction term . .

20. Example
Consider the case of an axisymmetric vortex with tangential velocity distribution v(r).
Assume that a linear frictional force F = - mv(r) acts at the ground z = 0.
Then K = -mzk, where z is the vertical component of relative vorticity.
Then PV is destroyed at the rate

21. The End