AOSS 401, Fall 2007 Lecture 15 October 17 , 2007

1. AOSS 401, Fall 2007Lecture 15October 17, 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. October 17, 2007 • Exam results • Roadmap for the next month • Introduction to vorticity

3. Exam Results • Class average: 21.7 • Class median: 21.0

4. Grades? • We will not be assigning letter grades until the end of the semester • Here is some guidance on how the scores might map to a letter grade on this exam • (Remember that solid scores on homeworks will bump up your overall course grade and offset low exam scores…)

5. Grades? ~C- • There were about 10-11 points on the test that we expected every one to get. Most everyone got these, and this is good. • If you have 15 or less, we would like to make an appointment to talk with you. • There were about 18-20 points on the test if you got the problems started. Average was 21.5. So most people got the problems started. This is better than good. • At 22 points and higher, people have a very good grasp on the concepts and their application. • At > 25 points, excellent grasp of material. ~A+

6. Exam Results • Class averages for each question: • 4.9 / 5.0 • 5.1 / 6.0 • 3.6 / 6.0 • 3.1 / 6.0 • 2.8 / 4.0 • 2.2 / 3.0

7. Tendency, acceleration Viscosity (surface force) Coriolis force (apparent force) Curvature or metric terms (accept apparent force) Pressure gradient (surface force) Exam Question (5)Class Average: 4.9 1. In the equation below what is the physical meaning of each of the terms? If the term is a force, then state whether it is a surface force, body force, or apparent force.

8. z k y j x i Remember how we derived PGF r≡ density = mass per unit volume (DV) DV = DxDyDz m = rDxDyDz ------------------------------------- p ≡ pressure = force per unit area acting on the particle of atmosphere Dz Dy Dx

9. Force per unit area = surface force (x0, y0, z0) FBx = (p0 - (∂p/∂x)Dx/2) (DyDz) Dz . FAx = - (p0 + (∂p/∂x)Dx/2) (DyDz) Fx = FBx + FAx A B Dy Fx/m = - 1/r (∂p/∂x) Dx x axis

10. Questions?

11. 2. Exam Question (6)Class Average: 5.1 Above the equations of motion are provided using both z, height, as a vertical coordinate and p, pressure, as a vertical coordinate. 2) Write out the material derivative in both coordinate systems. Show explicitly whether or not z or p is held constant when partial derivatives are taken (4 points). What are the units of the vertical velocity in the two coordinate systems (2 points)?

12. Partial Derivatives… • Important to remember that partial derivation implies we are holding everything else constant • For a coordinate system that includes (x,y,z,t):

13. Partial Derivatives… • Important to remember that partial derivation implies we are holding everything else constant • For a coordinate system that includes (x,y,p,t):

14. 2. Exam Question (6)

15. 2. Exam Question (6)

16. Questions?

17. 3. Exam Question (6)Class Average: 3.6 3) Write the mass conservation equation in pressure coordinates (1 point). Let the horizontal wind in the x and y direction, (u, v) = (ug+ua, vg+va), where subscript g represents a geostrophic wind and subscript a represents the ageostrophic wind. Using the definition of ω and of the geostrophic wind, with the assumption of f = f0= constant, show that (2 points). Then, with the assumption that the divergence can be represented by a constant average value, use the definition of ω and rewrite the equation in terms of the time rate of change of surface pressure (3 points).

18. Answer to Question 3a 3) Write the mass conservation equation in pressure coordinates (1 point).

19. Answer to Question 3b 3) Let the horizontal wind in the x and y direction, (u, v) = (ug+ua, vg+va), where subscript g represents a geostrophic wind and subscript a represents the ageostrophic wind. Using the definition of ω and of the geostrophic wind, with the assumption of f = f0= constant, show that (2 points).

20. Answer to Question 3b

21. Answer to Question 3b

22. Answer to Question 3b

23. Answer to Question 3c Then, with the assumption that the divergence can be represented by a constant average value, use the definition of ω and rewrite the equation in terms of the time rate of change of surface pressure (3 points).

24. Answer to Question 3c

25. Answer to Question 3c

26. Answer to Question 3c

27. Answer to Question 3c

28. Answer to Question 3c

29. Questions?

30. 4. Exam Question (6)Class Average: 3.1 4) Refer to the figure. This is geopotential height at a constant pressure level in the troposphere in the northern hemisphere far above the Earth’s surface. At points A, B, and C, draw the direction of the geostrophic wind and indicate whether the speed (magnitude) of the geostrophic wind is the same or different at these three points (1 point)? In class and text we derived the ratio of the geostrophic wind speed to the gradient wind speed: At points A, B, and C, draw the direction of the gradient wind and indicate whether the speed of the gradient wind is the same or different at these three points (3 points)? Using the definition of horizontal divergence, show regions where the gradient wind is divergent or convergent (= - divergence) (2 points).

31. The geostrophic wind, red, is the same at A, B and C. It is parallel to the isolines. Answer for Problem 4a Geostrophic wind speed only depends on gradient of pressure/height ٠ ΔΦ > 0 B Φ0 - ΔΦ Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east

32. R > 0 R > 0 V < Vg V < Vg n n n R < 0 V > Vg t t t Answer for Problem 4b Gradient wind flows // to gradient of pressure/height ٠ ΔΦ > 0 B Φ0 - ΔΦ Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east

33. The geostrophic wind, red, is the same at A, B and C. It is parallel to the isolines. The gradient wind, blue, is less than the geostrophic wind, red, at A, C and greater than the geostrophic wind at B. It is parallel to the isolines. Answer for Problem 4b ٠ ΔΦ > 0 B Φ0 - ΔΦ Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east

34. The divergence is ∂u/∂x + ∂v/∂y. Consider A and B. Δu is > 0, Δx > 0, Δv=0; hence, gradient is positive and there is divergence between the two points. The divergence is ∂u/∂x + ∂v/∂y. Consider B and C. Δu is < 0, Δx > 0, Δv=0; hence, gradient is negative and there is convergence between the two points. Answer for Problem 4c ٠ ΔΦ > 0 B Φ0 - ΔΦ Φ0 ٠ ٠ y, north Φ0 + ΔΦ A C x, east

35. Questions?

36. 5. Exam Question (4)Class Average: 2.8 5) Refer to the figure. This figure shows a jet stream in the northern hemisphere, upper troposphere. The direction is easterly, from the east. We saw that in a hydrostatic atmosphere the vertical gradient of the geostrophic wind, was related to the horizontal gradient of temperature. That is the thermal wind relationship. What is the sign of the vertical gradient of the wind below the jet stream? (1 point) With this information, is point A warmer or colder than point B (2 points)? Where are the temperature gradients strongest (1 points)? Be sure to justify your decisions.

37. Between lower and upper point Δu is < 0, Δp < 0, hence vertical gradient is positive. Answer for Problem 5a 5) What is the sign of the vertical gradient of the wind below the jet stream? (1 point) -10 m/s -5 m/s -20 m/s -30 m/s B A - p, vertical y, north

38. Answer for Problem 5b(Pressure Coordinates) 5) With this information, is point A warmer or colder than point B (2 points)?

39. Answer for Problem 5b(Height Coordinates) 5) With this information, is point A warmer or colder than point B (2 points)?

40. Answer for Problem 5b 5) With this information, is point A warmer or colder than point B (2 points)? -10 m/s -5 m/s -20 m/s -30 m/s A, cooler B, warmer - p, vertical y, north

41. Strong shear. Answer for Problem 5c 5) Where are the temperature gradients strongest (1 points)? -10 m/s -5 m/s -20 m/s -30 m/s A, cooler B, warmer - p, vertical y, north

42. Answer for Problem 5c 5) Where are the temperature gradients strongest (1 points)? Strong temperature gradient. -10 m/s -5 m/s -20 m/s -30 m/s A, cooler B, warmer - p, vertical y, north

43. Questions?

44. 6. Exam Problem (3)Class Average: 2.2 • In several lectures we talked about the transport of trace “gases” such as ozone, smoke, or “dye.” • What is the conservation principle that governs the behavior of such tracers? (1) • Write down the conservation equation for water vapor. (1) • As water vapor changes phases between liquid, gas, and ice, energy is absorbed and released from the atmosphere. • Specifically, what term in which of the equations of motion represents this energy exchange? (1)

45. 6a. Answer • What is the conservation principle that governs the behavior of such tracers? • Conservation of Mass (1)

46. 6b. Answer • Write down the conservation equation for water vapor. (1) • PH2O: Production of water vapor (source term) • LH2O: Loss of water vapor (sink term)

47. 6c. Answer • As water vapor changes phases between liquid, gas, and ice, energy is absorbed and released from the atmosphere. • Specifically, what term in which of the equations of motion represents this energy exchange? (1) • The diabatic heating term, J, in the Thermodynamic equation

48. Questions?

49. Roadmap to the Second Exam • Exam 2 is scheduled for 16 November (Friday) • This exam will cover mostly chapter 4 in Holton, specifically: • Holton Section 4.2: Vorticity • Holton Section 4.4: Vorticity equation • tangential Cartesian coordinates • pressure coordinates • scale analysis in middle latitudes • Holton Section 4.5: Vorticity in barotropic fluids • Holton Section 4.3: Potential vorticity

50. Roadmap to the Second Exam • Exam 2 is scheduled for 16 November (Friday) • If we have time, we may delve into chapter 6, section 2: • Quasi-geostrophic approximation • Quasi-geostrophic vorticity equation