AOSS 401, Fall 2006 Lecture 17 October 22 , 2007

1. AOSS 401, Fall 2006Lecture 17October 22, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu 734-936-0502

2. Class News • Final exam will be last day of class

3. Material from Chapter 4 • Vorticity, Vorticity, Vorticity • Definition of barotropic and baroclinic • Review • Scaling

4. Weather • National Weather Service • http://www.nws.noaa.gov/ • Model forecasts: http://www.hpc.ncep.noaa.gov/basicwx/day0-7loop.html • Weather Underground • http://www.wunderground.com/cgi-bin/findweather/getForecast?query=ann+arbor • Model forecasts: http://www.wunderground.com/modelmaps/maps.asp?model=NAM&domain=US

5. Vorticity, Vorticity, Vorticity

6. Two important definitions • barotropic – density depends only on pressure. And by the ideal gas equation, surfaces of constant pressure, are surfaces of constant density, are surfaces of constant temperature. • baroclinic – density depends on pressure and temperature.

7. Attributes of important dynamical features • There is rotation of wind • Which is due to the rotation of the Earth • Vertical wind requires divergence of the horizontal wind • Which requires an ageostrophic part of the wind.

8. Want to formalize the representation of the role of rotation and divergence

9. Suppose we have some flow

10. Imagine at the point flow decomposed into two “components” A “component” that flows into or away from the point.

11. Imagine at the point flow decomposed into two “components” A “component” that flows around the point.

12. Lets consider a spinning skater Motion is in the (x,y) plane Axis of rotation is in the vertical plane

13. Vorticity

14. Vertical Component of Vorticity In what plane is the motion? In what direction is the vorticity?

15. Relative Vorticity

16. Planetary Vorticity

17. Absolute (or total) Vorticity

18. Vorticity and Divergence • Related to shear of the velocity field. ∂v/∂x-∂u/∂y • Related to stretching of the velocity field. ∂u/∂x+∂v/∂y

19. Full equations of motionRemember these? How would you calculate of the time rate change of the vertical component of vorticity?

20. The scaled horizontal momentum equation in z coordinates no viscosity

21. Relative Vorticity

22. Take derivatives

23. Take derivatives

24. Take derivatives

25. Take derivatives Pay attention to details of calculus here

26. Subtract these equationsConservation of vorticity What are physical interpretations of these terms? Where do the definitions barotropic and baroclinic make a difference?

27. Scale factors for “large-scale” mid-latitude

28. Scale of relative vorticity

29. Compare relative vorticity to planetary vorticity In general planetary vorticity is larger than relative vorticity.

30. Time rate of change of relative vorticity

31. Rest of the terms

32. Consider divergence term This term scales larger than all of the other terms. This suggests that the divergence of the horizontal wind must, in actuality, be small; hence, quasi-nondivergent. (∂u/∂x+∂v/∂y)~<10-6s-1

33. Consider divergence term We saw that the relative vorticity is less than the planetary vorticity. So

34. So for quasi-nondivergent flow

35. Compare relative vorticity to planetary vorticity andto divergence Again we see the importance of the rotation of the Earth.

36. Assume balance among terms of 10-10s-2

37. A nuance on vorticity and the scaled equation: potential vorticity

38. A simple version of potential vorticity returned relative vorticity to equation

39. A simple version of potential vorticity what’s this?

40. A simple version of potential vorticity Assume constant density and temperature.

41. A simple version of potential vorticity Integrate with height,z1 z2 over a layer of depth H.

42. A simple version of potential vorticity Integrate with height,z1 z2 over a layer of depth H. Why can we do this?

43. A simple version of potential vorticity This is the potential vorticity under the set of assumptions that we used to derive the equation. Constant density, constant temperature  so only in a shallow layer might this be relevant to the atmosphere. Potential vorticity is a measure of absolute vorticity relative to the depth of the vortex.

44. Relative vorticity with change of depth

45. What happens when the vortex meets the mountain? Surface with a hill.

46. A less simple version of potential vorticity This is the isentropic potential vorticity which is conserved for isentropic motion. Potential vorticity is a measure of absolute vorticity relative to the depth of the vortex.

47. Vorticity and depth • We can see that there is a relationship between depth and vorticity. • As the depth of the vortex changes, the relative vorticity has to change in order to conserve the potential vorticity. • This is the play between relative and planetary vorticity.

48. How are divergence and vorticity related? • We have gotten to a situation where we have linked the rotational and irrotational components of the wind. divergence and curl vorticity and divergence

49. Scaled vorticity equation

50. Pure constant vorticity flow.