1. AOSS 401, Fall 2007 Lecture 12 October 3, 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 Homework –Homework and some review questions were posted last night. Homework due Monday We will go over the review questions on Friday –Think about them Exam next Wednesday –Today’s lecture is the last fundamentally new material that will be on the exam Friday we will talk about vertical velocity some more Friday and Monday we will look at the material in different ways and more thoroughly Also have your questions Mid-term evaluation –“students will be notified soon thereafter that they can fill out the midterm evaluations between October 8 and October 14”

3. Material from Chapter 3 Balanced flow Examples of flows –Stratospheric Vortex Ozone hole –Surface Flow Friction Thermal wind

4. Picture of Earth f=2Ωsin(Φ) 1.4X10 -4 s -1 1.0X10 -4 s -1 0.0 s -1

5. Picture of Earth Ω k k k Ω Ω Maximum rotation of vertical column. No rotation of vertical column.

6. Rotation When a fluid is in rotation, the rotation comes to define the flow field; it provides structure. That structure aligns with the vector that defines the angular velocity. –So if the flow is quasi-horizontal, then how the flow aligns in the vertical is strongly influenced by the rotation and its projection in the vertical. –On a horizontal surface the curvature of the flow is important

7. And on the Earth. Tropics are more weakly influenced, defined by rotation than middle latitudes. –This also influences the vertical structure of the dynamical features.

8. Length scales Planetary waves: 10 7 meters, 10,000 km –Have we seen one of these in our lectures? Synoptic waves: Our large-scale, middle- latitude, 10 6 meters, 1000 km –What’s a synoptic wave? What does synoptic mean? Hurricanes: 10 5 meters, 100 km Fronts: 10 4 meters, 10 km Cumulonimbus clouds: 10 3 meters, 1 km Tornadoes: 10 2 meters, 0.1 km Dust devils: 1 - 10 meters

9. Returning to our mid-latitude, large- scale flow. We saw last lecture that we could define natural coordinates that were (potentially) useful for determining the motion from maps of thermodynamic fields. That is, the pressure gradient or its analogue, geopotential height. We saw that, while a powerful constraint, geostrophy is formally true only when the lines of geopotential are straight. –It’s also a balance, steady state. Hence, while seductive, this is not adequate.

10. How do these natural coordinates relate to the tangential coordinates? They are still tangential, but the unit vectors do not point west to east and south to north. The coordinate system turns with the wind. And if it turns with the wind, what do we expect to happen to the forces? Ω Earth Φ = latitude a

11. Looking down from above

16. Balanced flows in natural coordinates (balanced, here, means steady)

17. Low Cyclostrophic Flow How do we get this kind of flow? Low Pressure gradient force Centrifugal force Do we have this balance around a high?

18. Low Gradient Flow What forces are being balanced? High Definition of normal, n, direction n n

19. Low Gradient Flow High Definition of normal, n, direction n n R>0 R<0

20. Gradient Flow Solution must be real Low ∂Φ/∂n<0 R>0 Always satisfied High ∂Φ/∂n<0 R<0 Trouble! pressure gradient MUST go to zero faster than R

21. What does this mean physically For a high, the pressure gradient weakens towards the center of the high. If pressure weakens, then wind speed weakens. Hence, highs associated with relatively weak winds. For a low, there is no similar constraint. Hence lows can spin up into strong storms.

22. Low Gradient Flow (Solutions for Lows, remember that square root.) Low Pressure gradient force Centrifugal force Coriolis Force NORMAL ANOMALOUS V V

23. High Gradient Flow (Solutions for Highs, remember that square root.) High Pressure gradient force Centrifugal force Coriolis Force V V NORMAL ANOMALOUS

24. Why do we call these flows anomalous? Where might these flows happen?

25. Normal and Anomalous Flows Normal flows are observed all the time. –Highs tend to have slower magnitude winds than lows. –Lows are storms; highs are fair weather Anomalous flows are not often observed. –Anomalous highs have been reported in the tropics –Anomalous lows are strange –Holton “clearly not a useful approximation.”

26. Balanced flow: an application of all that we know

27. Geopotential, 50 hPa surface Pressure units: hPa mbar inches of Hg Length scale? >1,000 km ~10,000 km

28. What about the wind? Pressure gradient Coriolis force What’s the latitude? Centrifugal force Wind

30. What would happen if I put dye in the low?

31. So we observe that what happens in this low stays in this low.

32. Ozone, October 23, 2006

33. Summary from ozone hole Ozone hole movie Cyclonic polar low isolates air from rest of Earth. Extreme cold temperature cause nitric acid and water clouds which changes basic chemical environment of atmosphere. Return of sun destroys ozone in isolated air with changed chemical environment.

34. Let’s move down to the surface. At 1000 mb How are things different? How would we have to modify the equations?

35. Geostrophic and observed wind 1000 mb (land)

36. Geostrophic and observed wind 1000 mb (ocean)

37. Think about this in terms of natural coordinates.

38. Our geostrophic flow. east west Φ0+ΔΦΦ0+ΔΦ Φ 0 +3 Δ Φ Φ0Φ0 Φ 0 +2 Δ Φ south north ΔnΔn

39. We have said that what’s going on near the surface is related to viscosity.

40. So what does it say if our wind crosses the height contours? east west Φ0+ΔΦΦ0+ΔΦ Φ 0 +3 Δ Φ Φ0Φ0 Φ 0 +2 Δ Φ south north Δn ?

41. So what does it say if our wind crosses the height contours? (Staying in natural coordinates.) east west Φ0+ΔΦΦ0+ΔΦ Φ 0 +3 Δ Φ Φ0Φ0 Φ 0 +2 Δ Φ south north ΔΦ t n

42. So what does it say if our wind crosses the height contours? (Staying in natural coordinates.) east west Φ0+ΔΦΦ0+ΔΦ Φ 0 +3 Δ Φ Φ0Φ0 Φ 0 +2 Δ Φ south north ΔΦ t n u v angle, α

43. Friction force

45. Balance of forces (northern hemisphere) (Staying in natural coordinates.) east west Φ0+ΔΦΦ0+ΔΦ Φ 0 +3 Δ Φ Φ0Φ0 Φ 0 +2 Δ Φ south north ΔΦ t n angle, α

46. Balance of forces (northern hemisphere) (Staying in natural coordinates.) east west Φ0+ΔΦΦ0+ΔΦ Φ 0 +3 Δ Φ Φ0Φ0 Φ 0 +2 Δ Φ south north ΔΦ t n angle, α angle, α, as well?

47. Angle in terms of forces

48. Can also be derived from Looks like a great homework problem!

49. Some basics of the atmosphere Troposphere: depth ~ 1.0 x 10 4 m Troposphere ------------------ ~ 2 Mountain Troposphere ------------------ ~ 1.6 x 10 -3 Earth radius This scale analysis tells us that the troposphere is thin relative to the size of the Earth and that mountains extend half way through the troposphere.

50. Structure of the atmospheric boundary layer (Vertical length scales) Viscous sublayer Transition layer Inertial sublayer Atmospheric Surface Layer (ASL) Planetary (Convective) Boundary Layer (PBL) Roughness sublayer ~ 10 1~2 m ~ 10 -1~1 m ~ 10 -3 m ~ 10 2-3 m Free Atmosphere Wind profile Blending height PBL height Interfacial sublayer from Bob Su ( www.itc.nl )www.itc.nl k {

51. Let’s think about balance on a different scale Going back to our equations of motion in the tangential coordinate system.

52. Equations of motion in pressure coordinates (plus hydrostatic and equation of state)

53. Linking thermal field with wind field. The Thermal Wind

54. Geostrophic wind

55. Hydrostatic Balance

56. Geostrophic wind Take derivative wrt p. Links horizontal temperature gradient with vertical wind gradient.

57. Thermal wind p is an independent variable, a coordinate. Hence, x and y derivatives are taken with p constant.

58. A excursion to the atmosphere. Zonal mean temperature - Jan north (winter) south (summer) approximate tropopause

59. A excursion to the atmosphere. Zonal mean temperature - Jan north (winter) south (summer) ∂T/∂y ?

60. A excursion to the atmosphere. Zonal mean temperature - Jan north (winter) south (summer) ∂T/∂y ? <0 >0 <0

61. A excursion to the atmosphere. Zonal mean temperature - Jan north (winter) south (summer) ∂T/∂y ? <0 >0 <0 > 0 <0 >0 ∂u g /∂p ?

62. A excursion to the atmosphere. Zonal mean wind - Jan north (winter) south (summer)

63. Relation between zonal mean temperature and wind is strong This is a good diagnostic – an excellent check of consistency of temperature and winds observations. We see the presence of jet streams in the east-west direction, which are persistent on seasonal time scales. Is this true in the tropics?

64. Thermal wind

67. ?

68. From Previous Lecture Thickness Z 2 -Z 1 = Z T ≡ Thickness - is proportional to temperature is often used in weather forecasting to determine, for instance, the rain-snow transition. (We will return to this.) Note link of thermodynamic variables, and similarity to scale heights calculated in idealized atmospheres above.

69. Similarity of the equations There is clearly a relationship between thermal wind and thickness.

70. Schematic of thermal wind. from Brad Muller Thickness of layers related to temperature. Causing a tilt of the pressure surfaces.

71. Another excursion into the atmosphere. 850 hPa surface 300 hPa surface X XX from Brad Muller

72. Another excursion into the atmosphere. 850 hPa surface 300 hPa surface X XX from Brad Muller

73. Another excursion into the atmosphere. 850 hPa surface 300 hPa surface from Brad Muller

74. Another excursion into the atmosphere. 850 hPa surface 300 hPa surface from Brad Muller

75. A summary of ideas. In general, these large-scale, middle latitude dynamical features tilt westward with height. The way the wind changes direction with altitude is related to the advection of temperature, warming or cooling in the atmosphere below a level. –This is related to the growth and decay of these disturbances. –Lifting and sinking of geopotential surfaces.

76. Balance and rotation We keep making a big deal of rotation and the balance of the coriolis force and the pressure gradient force, e.g. the geostrophic balance. We have all of these equations and scale analysis, and they keep leading use to these notions of geostrophic and hydrostatic balance. Let’s examine some of these ideas in a more visual way.

77. Rotation When a fluid is in rotation, the rotation comes to define the flow field; it provides structure. That structure aligns with the vector that defines the angular velocity. –So if the flow is quasi-horizontal, then how the flow aligns in the vertical is strongly influenced by the rotation and its projection in the vertical.

78. And on the Earth. Tropics are more weakly influenced, defined by rotation than middle latitudes. –This also influences the vertical structure of the dynamical features.

79. Some things that we learned (1) Organizing structure provided by rotation. Rotation is less important in the tropics, which is clearly observable in the atmosphere. There is a theoretical limit on pressure gradients associated with high pressure systems. –Highs tend to be smeared out; they tend to have moderate wind speeds. There is not such a limit for low pressure systems. –Lows can be very intense; The highest wind speeds are associated with lows.

80. Some things that we learned (2) There is the possibility of “anomalous” circulations. –Possibility of cyclonic highs –Possibility of anti-cyclonic lows We can estimate frictional dissipation based on the angle between lines of constant pressure, or height, and the observed wind.

81. Some things that learned (3) Dynamical features can isolate air and allow the evolution of extraordinary chemical processes.

82. Where do we need to go next? We need to understand the role of vertical motion in large-scale dynamics. We need to understand the role of thermodynamic variables in the dynamical balances.

83. Analysis of Hurricane

84. Let’s take a stab at a hurricane. (Northern hemisphere) What balance might we use?

85. Let’s take a stab at a hurricane. (Northern hemisphere) L

86. L r = radial coordinate

87. Gradient balance for hurricane

88. Rewrite gradient wind for hurricane

89. Define angular momentum (You’ve seen this before.)

90. Some analysis

93. Near ground we have friction. And hurricanes are observed to maintain themselves! Latent heat from warm ocean water.

94. Hurricane Heat Engine Hurricanes are maintained by latent heat release from water that is evaporated from the ocean. –~ 27 o C is threshold. Bring in thermodynamic equation. Hurricanes are an efficient heat engines.