1. AOSS 401, Fall 2006 Lecture 7 September 21, 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 Contract with class. –First exam October 10. Homework 3 is posted. –Due next Friday –Seven problems First four should take about ½ hour. Last three a problem you have to “solve” –Exercises a whole suite of problem solving skills

3. Outline Scale analysis –In class problem –Exploring the atmosphere Vertical Structure –Isentropic and isothermal Stability and Instability –Wave motion Balances

4. Times scales Distance = velocity * time How long does it take you to.... ?

5. So for our equations D ( )/Dt can be characterized by 1/(L/U)

6. Let us define.

7. Geostrophic wind 300 mb

8. For “large-scale” mid-latitude

9. In class problem: Scaling Above are the horizontal components of the momentum equation. In class (and text) these equations were scaled with characteristic values. We saw that away from the surface that the viscosity term was very small. Near the Earth’s surface, however, viscosity is important. What is an appropriate vertical length scale near the surface? What is the approximate value of vertical shear of the horizontal wind? (assume  is constant = 1.46x10 -5 m 2 s -1 )

10. Geostrophic and observed wind 1000 mb (land)

12. Let’s spend some time with the atmosphere. Start with a typical upper tropospheric chart. What is a good estimate of the pressure at the surface? What is a good estimate of the pressure in the upper troposphere? How could you figure out the geometric height?

13. Geostrophic wind 300 mb How does this example relate to global scales?

14. 300 mb

15. Some mid-latitude scales Mid-latitude cyclones about 1000 km and 1 day Jet stream. More on a planetary scale. 1000’s of kilometers

16. 700 mb

17. 500 mb

18. 300 mb

19. 50 mb

20. DJF 500 mb Average

21. JJA 500 mb Average

22. Anomaly 100 mb

23. What are the scales of the terms? (horizontal momentum equations, mid-latitudes, away from surface) U*U/L U*U/a U*W/a Uf Wf 10 -4 10 -5 10 -8 10 -3 10 -6 10 -12 Geostrophic terms Acceleration

24. Some carry away points (Mid-latitudes) Motions are determined to a good approximation by the balance of the pressure gradient and the coriolis force –approximately in horizontal plane Multiple scales, scales embedded within larger scales Change of scale with altitude Change of scale from winter to summer Balance of forces with altitude Balance of forces with scale

25. What about the vertical?

26. Full equations of motion We saw that the first two equations were dominated by the geostrophic balance. What do we do for the vertical motion?

27. Thermodynamic equation (Use the equation of state)

28. Thermodynamic equation (conservative, no heating, adiabatic) conservative, no heating, adiabatic all mean J=0

29. Thermodynamic equation conservative, no heating, adiabatic (solve, perfect differential) Know how to do this mathematical manipulation.

30. Definition of potential temperature This is the temperature a parcel would have if it was moved from some pressure and temperature to the surface. This is Poisson’s equation.

31. This is a very important point. Even in adiabatic motion, with no external source of heating, if a parcel moves up or down its temperature changes. What if a parcel moves about a surface of constant pressure?

32. Adiabatic lapse rate. For an adiabatic, hydrostatic atmosphere the temperature decreases with height.

33. Another important point If the atmosphere is in adiabatic balance, the temperature still changes with height. Adiabatic does not mean isothermal. It means that there is no external heating or cooling.

34. Consider the vertical structure some more. Hydrostatic Eq. of State

35. Consider the vertical structure some more. T as constant - (Isothermal) Under special consideration of T as constant. (Isothermal)

36. Consider the vertical structure some more. Under special consideration of T as constant. (Isothermal)

37. Consider the vertical structure some more. Under special consideration of T as constant. (Isothermal) Units [R] = J/(kg*K)=kg*m*m/(s*s*kg*K), [T] = K [RT]=m*m/(s*s) [RT/g]=m (unit of length) GTQ: Given the magnitude of the scale of H what is the average temperature of the atmosphere?

38. Pressure altitude Exponential decrease with height

39. Consider a different vertical structure. Under special consideration of T changing with a constant linear slope (or lapse rate).

40. Consider a different vertical structure. Under special consideration of T changing with a constant lapse rate. Or a linear slope.

41. Consider a different vertical structure. Under special consideration of T changing with a constant lapse rate. Or a linear slope.  positive, T decreases with height, pressure decreases with height.  negative, T increases with height, pressure decreases with height.

42. Pressure altitude Under virtually all conditions pressure (and density) decreases with height. ∂p/∂z < 0. That’s why it is a good vertical coordinate. If ∂p/∂z = 0, then utility as a vertical coordinate falls apart.

43. Let’s return to our linear lapse rate. Under special consideration of T changing with a constant linear slope (or lapse rate).

44. Temperature as function of height z Warmer Cooler z T ∂T/∂z is defined as lapse rate

45. Temperature as function of height z Warmer Cooler z T ∂T/∂z is defined as lapse rate

46. Temperature as function of height z Warmer Cooler z T ∂T/∂z is defined as lapse rate

47. Temperature as function of height z Warmer Cooler z T ∂T/∂z is defined as lapse rate

48. The parcel method We are going displace this parcel – move it up and down. –We are going to assume that the pressure adjusts instantaneously; that is, the parcel assumes the pressure of altitude to which it is displaced. –As the parcel is moved its temperature will change according to the adiabatic lapse rate. That is, the motion is without the addition or subtraction of energy. J is zero in the thermodynamic equation.

49. Parcel cooler than environment z Warmer Cooler If the parcel moves up and finds itself cooler than the environment then it will sink. (What is its density? larger or smaller?)

50. Parcel cooler than environment z Warmer Cooler If the parcel moves up and finds itself cooler than the environment, then it will sink. (What is its density? larger or smaller?)

51. Parcel warmer than environment z Warmer Cooler If the parcel moves up and finds itself warmer than the environment then it will go up some more. (What is its density? larger or smaller?)

52. Parcel cooler than environment z Warmer Cooler If the parcel moves up and finds itself cooler than the environment, then it will sink. (What is its density? larger or smaller?) This is our first example of “instability” – a perturbation that grows.

53. Let’s quantify this. Under consideration of T changing with a constant linear slope (or lapse rate).

54. Let’s quantify this. Under consideration of T of parcel changing with the dry adiabatic lapse rate

55. Stable: temperature of parcel cooler than environment.

56. Unstable: temperature of parcel greater than environment.

57. Stability criteria from physical argument

58. Let’s return to the vertical momentum equation

59. What are the scales of the terms? W*U/L U*U/a Uf g 10 -7 10 -5 10 10 -3 10 10 -15

60. What are the scales of the terms? W*U/L U*U/a Uf g 10 -7 10 -5 10 10 -3 10 10 -15

61. Vertical momentum equation  Hydrostatic balance

62. Hydrostatic balance

63. But our parcel experiences an acceleration Assumption of adjustment of pressure.

64. Solve for pressure gradient

65. But our parcel experiences an acceleration

66. Again, our pressure of parcel and environment are the same so

67. So go back to our definitions of temperature and temperature change above

68. Use binomial expansion

69. So go back to our definitions of temperature and temperature change above

70. Ignore terms in z 2

71. For stable situation Seek solution of the form

72. For stable situation Seek solution of the form

73. Parcel cooler than environment z Warmer Cooler If the parcel moves up and finds itself cooler than the environment then it will sink. (What is its density? larger or smaller?)

74. Example of such an oscillation

75. For unstable situation Seek solution of the form

76. Parcel cooler than environment z Warmer Cooler If the parcel moves up and finds itself cooler than the environment, then it will sink. (What is its density? larger or smaller?) This is our first example of “instability” – a perturbation that grows.

77. What are the scales of the terms? W*U/L U*U/a Uf g 10 -7 10 -5 10 10 -3 10 10 -15

78. What are the scales of the terms? W*U/L U*U/a Uf g 10 -7 10 -5 10 10 -3 10 10 -15

79. Definition of potential temperature This is essentially the thermodynamic energy equation under the assumption of adiabatic motion. If the atmosphere were stirred adiabatically, then this is how the temperature would evolve.. This is Poisson’s equation.

80. Adiabatic lapse rate. For an adiabatic, hydrostatic atmosphere the temperature decreases with height.

81. 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.

82. What are the scales of the terms? U*U/L U*U/a U*W/a Uf Wf 10 -4 10 -5 10 -8 10 -3 10 -6 10 -12

83. What are the scales of the terms? U*U/L U*U/a U*W/a Uf Wf 10 -4 10 -5 10 -8 10 -3 10 -6 10 -12 Largest Terms

84. Geostrophic balance High Pressure Low Pressure

85. Looking at the atmosphere What does this map tell you?

86. Forced Ascent/Descent Warming Cooling

87. An Eulerian Map