AOSS 401, Fall 2007 Lecture 22 November 02 , 2007

1. AOSS 401, Fall 2007Lecture 22November 02, 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 November 02, 2007 • Homework 5 (Due Monday) • Posted to web • Computing assignment posted to ctools under the Homework section of Resources • Next Test: November 16

3. Seminars Today • Professor Cecilia Bitz is giving the Dept. of Geological Sciences’ Smith Lecture on Friday (tomorrow, Nov. 2) from 4-5pm. The lecture is held in room 1528 in C.C. Little and is followed by a reception. Cecilia is an expert in high-latitude climate, climate change and variability. The title of her lecture will be: • “Future thermohaline collapse and its impact are unlike the past”

4. Seminars Today • Dr. Guy Brasseur - Professor and Associate Director, National Center for Atmospheric Research and Director of the Earth and Sun Systems Laboratory will visit U of M on Thursday/Friday. • Impact of solar variability and anthropogenic forcing on the whole atmosphere: Simulations with the HAMMONIA Model • Friday, November 2, 3:30 pm  -- refreshments at 3 pm North Campus AOSS Auditorium, Room #2246

5. 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

6. Material from Chapter 6 • Quasi-geostrophic theory • Quasi-geostrophic vorticity • Relation between vorticity and geopotential

7. Going way back

8. Mathematics • Remember the idea that mathematics is a language to use to help us explore a complex system. • Verb: equal, • Qualification: not equal, greater than, less than, approximately • We own the equations and can do to them what we want, as long as we remember equal and not equal.

9. Tangential coordinate system Place a coordinate system on the surface. x = east – west (longitude) y = north – south (latitude) z = local vertical or p = local vertical Ω R=acos(f) R a Φ Earth

10. Tangential coordinate system Relation between latitude, longitude and x and y dx = acos(f)dl l is longitude dy = adf f is latitude dz = dr r is distance from center of a “spherical earth” f=2Ωsin(f) Ω b=2Ωcos(f)/a R a Φ Earth

11. Equations of motion in pressure coordinates(using Holton’s notation)

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

13. Scaled equations of motion in pressure coordinates Definition of geostrophic wind Momentum equation Continuity equation ThermodynamicEnergy equation

14. Approximate horizontal momentum equation • This equation states that the time rate of change of the geostrophic wind is related to • the coriolis force due to the ageostrophic wind and • the part of the coriolis force due to the variability of the coriolis force with latitude and the geostrophic wind. • Both of these terms are smaller than the geostrophic wind itself.

15. Derived a vorticity equation • Provides a “suitable” prognostic equation because need to include div(ageostrophic wind) in the prognostics. • Remember the importance of divergence in vorticity equations.

16. Scaled horizontal momentum in pressure coordinates

17. Use definition of vorticity vorticity equation

18. One interesting way to rewrite this equation Expand material derivative

19. One interesting way to rewrite this equation Equation of continuity Understand how this is equivalent

20. One interesting way to rewrite this equation Advection of vorticity

21. One interesting way to rewrite this equation Advection of vorticity Advection of relative vorticity Advection of planetary vorticity

22. Let’s take this to the atmosphere

23. Geopotential Map (Northern Hemisphere) Where is geostrophic approximation valid? What other force balance is important? What is the sign of the geostrophic wind? ٠ ΔΦ > 0 B Φ0 - ΔΦ L L H Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east

24. Geostrophic wind ٠ ΔΦ > 0 vg < 0 vg > 0 B Φ0 - ΔΦ L L H Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east

25. Geopotential Map (Northern Hemisphere) What is the sign of planetary vorticity? What is the sign of the relative vorticity? ٠ ΔΦ > 0 B Φ0 - ΔΦ L L H Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east

26. Vorticity ζ < 0; anticyclonic ٠ ΔΦ > 0 β > 0 β > 0 B Φ0 - ΔΦ L L H Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic

27. Advection of planetary vorticity ζ < 0; anticyclonic ٠ ΔΦ > 0 vg < 0 ; β > 0 vg > 0 ; β > 0 B Φ0 - ΔΦ L L H Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic

28. Advection of planetary vorticity ζ < 0; anticyclonic ٠ ΔΦ > 0 -vgβ > 0 -vgβ < 0 B Φ0 - ΔΦ L L H Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic

29. Advection of relative vorticity ζ < 0; anticyclonic ٠ ζ from >0 to <0 vg > 0; ug > 0 ΔΦ > 0 ζ from <0 to >0 vg < 0; ug > 0 B Φ0 - ΔΦ L L H Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic

30. Advection of relative vorticity ζ < 0; anticyclonic ٠ Advection of ζ > 0 ΔΦ > 0 Advection of ζ < 0 B Φ0 - ΔΦ L L H Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic

31. Advection of vorticity ζ < 0; anticyclonic Advection of ζ > 0 Advection of f < 0 Advection of ζ < 0 Advection of f > 0 ٠ ΔΦ > 0 B Φ0 - ΔΦ L L H Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic

32. Summary: Vorticity Advection in Wave • Planetary and relative vorticity advection in a wave oppose each other. • This is consistent with the balance that we intuitively derived from the conservation of absolute vorticity over the mountain.

33. Summary: Vorticity Advection in Wave • What does this do to the wave.

34. Advection of vorticity ζ < 0; anticyclonic  Advection of ζ tries to propagate the wave this way  ٠ ΔΦ > 0 B Φ0 - ΔΦ L L H Φ0  Advection of f tries to propagate the wave this way  ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic

35. Remember the relation to geopotential

36. An equation for geopotential tendency

37. Barotropic fluid

38. Perturbation equation

39. Wave like solutions Dispersion relation. Relates frequency and wave number to flow. Must be true for waves.

40. Stationary wave Wind must be positive, from the west, for a wave.

41. Geopotential Nuanced

42. Assume that the geopotential is a wave

43. Remember the relation to geopotential

44. Remember the relation to geopotential

45. Advection of relative vorticity

46. Advection of planetary vorticity

47. Compare advection of planetary and relative vorticity

48. Advection of vorticity ζ < 0; anticyclonic  Advection of ζ tries to propagate the wave this way  ٠ ΔΦ > 0 B Φ0 - ΔΦ L L H Φ0  Advection of f tries to propagate the wave this way  ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic

49. Compare advection of planetary and relative vorticity  Short waves, advection of relative vorticity is larger   Long waves, advection of planetary vorticity is larger 

50. Advection of vorticity ζ < 0; anticyclonic  Short waves  ٠ ΔΦ > 0 B Φ0 - ΔΦ L L H Φ0 • Long waves  ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic