1. AOSS 401, Fall 2007 Lecture 23 November 05, 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 05, 2007 Homework 6 (Posted this evening) –Due Next Monday Important Dates: –November 16: Next Exam (Review on 14 th ) –November 21: No Class –December 10: Final Exam

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

4. Couple of Links you should know about http://www.lib.umich.edu/ejournals/ –Library electronic journals http://portal.isiknowledge.com/portal.cgi?In it=Yes&SID=4Ajed7dbJbeGB3KcpBhhttp://portal.isiknowledge.com/portal.cgi?In it=Yes&SID=4Ajed7dbJbeGB3KcpBh –Web o’ Science

5. Material from Chapter 6 Quasi-geostrophic theory Quasi-geostrophic vorticity –Relation between vorticity and geopotential Geopotential prognostic equation Relationship to mid-latitude cyclones

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

7. Let’s take this to the atmosphere

8. Advection of planetary vorticity Φ 0 - Δ Φ Φ 0 + Δ Φ Φ0Φ0 Δ Φ > 0 A B C ٠ ٠ ٠ x, east y, north LL H ζ < 0; anticyclonic ζ > 0; cyclonic v g > 0 ; β > 0 v g 0

9. Advection of planetary vorticity Φ 0 - Δ Φ Φ 0 + Δ Φ Φ0Φ0 Δ Φ > 0 A B C ٠ ٠ ٠ x, east y, north LL H ζ < 0; anticyclonic ζ > 0; cyclonic -v g β < 0 -v g β > 0

10. Advection of relative vorticity Φ 0 - Δ Φ Φ 0 + Δ Φ Φ0Φ0 Δ Φ > 0 A B C ٠ ٠ ٠ x, east y, north LL H ζ < 0; anticyclonic ζ > 0; cyclonic Advection of ζ > 0 Advection of ζ < 0

11. Advection of vorticity Φ 0 - Δ Φ Φ 0 + Δ Φ Φ0Φ0 Δ Φ > 0 A B C ٠ ٠ ٠ x, east y, north LL H ζ < 0; anticyclonic ζ > 0; cyclonic Advection of ζ > 0 Advection of f < 0 Advection of ζ < 0 Advection of f > 0

12. 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.

13. Advection of vorticity Φ 0 - Δ Φ Φ 0 + Δ Φ Φ0Φ0 Δ Φ > 0 A B C ٠ ٠ ٠ x, east y, north LL H ζ < 0; anticyclonic ζ > 0; cyclonic  Advection of ζ tries to propagate the wave this way   Advection of f tries to propagate the wave this way 

14. Geopotential Nuanced

15. Assume that the geopotential is a wave

16. Remember the relation to geopotential

18. Advection of relative vorticity

19. Advection of planetary vorticity

20. Compare advection of planetary and relative vorticity

21. Advection of vorticity Φ 0 - Δ Φ Φ 0 + Δ Φ Φ0Φ0 Δ Φ > 0 A B C ٠ ٠ ٠ x, east y, north LL H ζ < 0; anticyclonic ζ > 0; cyclonic  Advection of ζ tries to propagate the wave this way   Advection of f tries to propagate the wave this way 

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

23. Advection of vorticity Φ 0 - Δ Φ Φ 0 + Δ Φ Φ0Φ0 Δ Φ > 0 A B C ٠ ٠ ٠ x, east y, north LL H ζ < 0; anticyclonic ζ > 0; cyclonic  Short waves   Long waves 

24. Go to the real atmosphere

25. An estimate of the January mean zonal wind north winter south summer -- u

26. Advection of relative vorticity for our idealized wave

27. An estimate of the January mean zonal wind north winter south summer What is the difference in the advection of vorticity at the two levels?

28. An estimate of the January mean zonal wind

29. Vertical Structure The waves propagate at different speeds at different altitudes. The waves do not align perfectly in the vertical. (This example shows that there is vertical structure, but it is only a (small) part of the story.)

30. A more general equation for geopotential

31. An equation for geopotential tendency

32. Another interesting way to rewrite vorticity equation (Flirting with) An equation for geopotential tendency An equation in geopotential and omega. (2 unknowns, 1 equation)

33. Quasi-geostrophic Geostrophic ageostrophic

34. Previous analysis In our discussion of the advection of vorticity, we completely ignored the term that had the vertical velocity. Go back to our original vorticity equation –Tilting –Divergence –Thermodynamic... (solenoidal, baroclinic) Which still exist after our scaling and assumptions?

35. We used these equations to get previous equation for geopotential tendency

36. Now let’s use this equation

37. Rewrite the thermodynamic equation to get geopotential tendency

38. Rewrite this equation to relate to our first equation for geopotential tendency.

39. Scaled equations of motion in pressure coordinates Note this is, through continuity, related to the divergence of the ageostrophic wind Note that it is the divergence of the horizontal wind, which is related to the vertical wind, that links the momentum (vorticity equation) to the thermodynamic equation

40. Scaled equations of motion in pressure coordinates Note that this looks something like the time rate of change of static stability

41. Explore this a bit. So this is a measure of how far the atmosphere moves away from its background equilibrium state

42. Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency (assume J=0)

43. Vorticity Advection

44. Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency (assume J=0) Thickness Advection

45. How do you interpret this figure in terms of geopotential? Φ 0 - Δ Φ Φ 0 + Δ Φ Φ0Φ0 Δ Φ > 0 A B C ٠ ٠ ٠ x, east y, north LL H ζ < 0; anticyclonic ζ > 0; cyclonic  Short waves   Long waves 

46. Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency (assume J=0) This is, in fact, an equation that given a geopotential distribution at a given time, then it is a linear partial differential equation for geopotential tendency. Right hand side is like a forcing. You now have a real equation for forecasting the height (the pressure field), and we know that the pressure gradient force is really the key, the initiator, of motion.

47. Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency (assume J=0) An equation like this was very important for weather forecasting before we had comprehensive numerical models. It is still important for field forecasting, and knowing how to adapt a forecast to a particular region given, for instance, local information.

48. Think about thickness advection Thickness Advection

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

50. Cold and warm advection cold warm

51. Question What happens when warm air is advected towards cool air? COOLWARM

52. Question What happens when warm air is advected towards cool air? COOL WARM

53. Question What happens the warm air? –Tell me at least two things. COOL WARM

54. Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency (assume J=0) Thickness Advection

55. Lifting and sinking

56. Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency (assume J=0) Thickness Advection

57. A nice schematic http://atschool.eduweb.co.uk/kingworc/dep artments/geography/nottingham/atmosphe re/pages/depressionsalevel.htmlhttp://atschool.eduweb.co.uk/kingworc/dep artments/geography/nottingham/atmosphe re/pages/depressionsalevel.html

58. More in the atmosphere (northern hemisphere) South North Warm Cool Temperature What can you say about the wind?

59. Idealized vertical cross section

60. Increasing the pressure gradient force

61. Relationship between upper troposphere and surface divergence over low enhances surface low // increases vorticity

62. Relationship between upper troposphere and surface vertical stretching // increases vorticity

63. Relationship between upper troposphere and surface vorticity advection thickness advection

64. Relationship between upper troposphere and surface note tilt with height

65. Mid-latitude cyclones: Norwegian Cyclone Model

66. Fronts and Precipitation CloudSat Radar Norwegian Cyclone Model

67. What’s at work here?

68. Mid-latitude cyclone development

69. Mid-latitude cyclones: Norwegian Cyclone Model http://www.srh.weather.gov/jetstream/syno ptic/cyclone.htmhttp://www.srh.weather.gov/jetstream/syno ptic/cyclone.htm

70. Below Basic Background Material

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

72. Tangential coordinate system Ω R Earth Relation between latitude, longitude and x and y dx = acos(  ) d  is longitude dy = ad   is latitude dz = dr r is distance from center of a “spherical earth” Φ a f=2Ωsin(  )  =2Ωcos(  )/a

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

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

75. Scaled equations of motion in pressure coordinates Definition of geostrophic wind Momentum equation Continuity equation Thermodynamic Energy equation