AOSS 401, Fall 2006 Lecture 7 September 21 , 2007

1. AOSS 401, Fall 2006Lecture 7September 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 themomentum 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 n is constant = 1.46x10-5 m2 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*W/a U*U/L 10-8 10-4 Uf Wf U*U/a 10-3 10-3 10-6 10-12 10-5 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. 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) Under special consideration of T as constant. (Isothermal) 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. g positive, T decreases with height, pressure decreases with height. g negative, T increases with height, pressure decreases with height. Under special consideration of T changing with a constant lapse rate. Or a linear slope.

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

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

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

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

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

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