1. PV Budgeting on tropopause polar vortices Steven Cavallo University of Washington Group Meeting September 27, 2005

2. Topics Why PV? How am I calculating PV changes? Does the calculation work? Can we make any sense out of it?

3. Why PV? Ertel’s theorem says that in height coordinates: where is the potential vorticity, and where the absolute vorticity is Ertel’s PV equation gives us the mechanisms for which PV is created and destroyed…which can be through either diabatic or frictional processes.

4. PV vertical profile inside core Recall what we saw when we took a look deep into the vortex core…

5. PV vertical profile inside core Well, not really, but we did see that when there is condensation in the low levels:  PV is destroyed above (but since it is very high to begin with, it does not fall below 2 PVU)  PV is created below  since it is marginal to begin with, the increases from this can push it above 2 PVU  BUT the largest effect was from the clouds that formed from the condensation  A significant amount of cooling above the cloud  vortex strengthening.

6. How am I calculating PV changes? In pressure coordinates, if we make the hydrostatic approximation, and neglect friction, then the EPV equation is where if we recall from last time, WRF calculates the thermodynamic equation as a conservation equation:

7. How am I calculating PV changes (cont’d)? We can then rewrite the EPV equation as which upon expanding the total derivative gives Thus we have everything we need to calculate and check the changes in PV due to any component! This gives us a PV Budget for which we can see the mechanisms contributing to changes in the cyclones intensity.

8. Checking the Calculations: Method I My grandmother once said that we must not blindly follow calculations from WRF and MATLAB, and that we must first check them. She said one way we can do this by discretizing the pv tendency using a centered difference: where  x =  y = 30 km, and  t = 6 hours is the WRF model output time. To compare more visually on the  t = 6 hour model output time step:

9. Checking the Calculations I (cont’d) Using these ideas to check the calculations, the following will be on a plot overlay: (1) The left hand side: (2) The diabatic (non-conserved) PV contribution (3) The diabatic + advective PV contributions. Should roughly equal (1). * ? *The estimate will be crude given that it is calculated from a model output time of 6 hours and assumes a constant diabatic tendency.

10. Checking the Calculations I (cont’d) Looking at the results, we see at best: So fine, it looks like a noisy approximation which is what we’d expect…but are the diabatic contributions really that small compared to the advection? Diabatic PV component 12-hour change in EPV Diabatic + Advection components

11. Checking the Calculations: Method II Ok, so let’s take a deeper look at the diabatic contributions. Let’s redo the calculation following the vortex. This way we need not calculate the advection term. That is,

12. Checking the Calculations: Method II (cont’d) As even Grandma Cavallo can see, this did not turn out very well:  Sort of fits general profile…at times.  Especially off in upper levels.  Vertical displacements. What happened?  Is it because we used a vertical tendency profile at one grid point?  Does averaging the tendency over the 3 times help?  Maybe follow within a closed contour instead of at a single gridpoint?

13. What next?

14. Preliminary results…sneak peak It appears that at the present time, the scales of the PV contributions look to be small…however the profiles are often looking ok.  Is there a constant that I am missing in the PV equation in pressure coordinates?  Could this be done in isentropic coordinates instead? So we can take a sneak peak at some qualitative results, if we agree that the patterns are more or less on the right track but ignoring the numbers for the time being. Some interesting plots would be:  Vertical profiles averaged within a tropopause theta closed contour  PV changes in pressure layers  Cross sections through the vortex core!!