1. AOSS 401, Fall 2007 Lecture 11 October 1, 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 October 1, 2007 Ricky will be lecturing again starting Wednesday—I will lecture next on the 17 th of October There is an exam next Wednesday, but you’re all probably well aware of that…

3. Material from Chapter 3(2) Balanced flow Examples of flows in the atmosphere

4. Refresher from Friday…

5. Geostrophic & observed wind 300 mb In order to understand the flow on maps that looked like this, we introduced “natural” coordinates.

6. The horizontal momentum equation Assume no viscosity

7. Return to Geopotential (Φ) in upper troposphere east westsouth north H IGH t t t n n n L ow Do you see some notion of a radius of curvature? Sort of like a circle, but NOT a circle.

8. The horizontal momentum equation (in natural coordinates)

9. Curved flow (Centrifugal Force) CoriolisPressure Gradient One Diagnostic Equation

10. Natural Coordinates: Key Points Velocity is defined to be positive The n direction always points to the left of the velocity (remember the right hand rule: k x t = n) If n points toward the center of curvature, the radius is positive If n points away from the center of curvature, the radius is negative The pattern of isobars/height lines is assumed to be fixed in space; no movement of weather systems

11. Uses of Natural Coordinates Geostrophic balance –Definition: coriolis and pressure gradient in exact balance. –Parallel to contours  straight line  R is infinite 0

12. Geostrophic balance

13. Which actually tells us the geostrophic wind can only be equal to the real wind if the height contours are straight. east west Φ0+ΔΦΦ0+ΔΦ Φ 0 +3 Δ Φ Φ0Φ0 Φ 0 +2 Δ Φ south north ΔnΔn

14. How does curvature affect the wind? (cyclonic flow/low pressure) R t n ΔnΔn Φ0Φ0 Φ 0 +ΔΦ Φ 0 -ΔΦ H IGH L ow

15. From Holton If V g /V < 1, geostrophic wind is an overestimate of the actual wind speed Since V is always positive, in the northern hemisphere (f > 0) this only happens for R positive For typical northern hemisphere large scale flow, R is positive for cyclonic flow; flow around low pressure systems

16. Geostrophic & observed wind 300 hPa

17. Observed: 95 knots Geostrophic: 140 knots

18. How does curvature affect the wind? (anticyclonic flow/high pressure) R t n ΔnΔn Φ0Φ0 Φ 0 +ΔΦ Φ 0 -ΔΦ H IGH L ow

19. From Holton If V g /V < 1, geostrophic wind is an underestimate of the actual wind speed Since V is always positive, in the northern hemisphere (f > 0) this only happens for R negative For typical northern hemisphere large scale flow, R is negative for anticyclonic flow; flow around high pressure systems

20. Geostrophic & observed wind 300 hPa

21. Observed: 30 knots Geostrophic: 25 knots

22. Uses of Natural Coordinates: Balanced Flows Tornados Hurricanes General high and low pressure systems

23. Cyclostrophic Flow

24. A balance in the normal, as opposed to tangential, component of the momentum equation. A balance of centrifugal force and the pressure gradient force. The following are needed –steady (time derivative = 0) –coriolis force is small relative to pressure gradient and centrifugal force

25. Cyclostrophic Flow Get cyclostrophic flow with either large V small R

26. Cyclostrophic Flow Radical must be positive: two solutions

27. Cyclostrophic Flow Tornadoes: 10 2 meters, 0.1 km Dust devils: 1 - 10 meters –Small length scales –Strong winds

28. Low Cyclostrophic Flow Low Pressure gradient force Centrifugal force

29. Low Cyclostrophic Flow Low Counterclockwise Rotation Clockwise Rotation

30. http://www.youtube.com/watch?v=vgbzKF_pSXo http://www.youtube.com/watch?v=k1dZpW5aFFk http://www.youtube.com/watch?v=3jQoGm8JEPY Anticyclonic Tornado (looking up) Sunnyvale, CA 4 May 1998

31. In-Class Exercise: Compute Tornado Wind Speed Remember: P=850 mb P=750 mb R = 100 m (Assume ρ = 1 kg/m 3 )

32. In-Class Exercise: Compute Tornado Wind Speed P=850 mb P=750 mb R = 100 m

33. High Cyclostrophic Flow Around a High Pressure System? High n n

34. Gradient Flow (Momentum equation in natural coordinates) Balance in the normal, as opposed to tangential, component of the momentum equation Balance between pressure gradient, coriolis, and centrifugal force

35. Gradient Flow (Momentum equation in natural coordinates)

36. Look for real and non-negative solutions

37. Gradient Flow Solution must be real

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

39. 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

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

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

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

43. 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.” But it is possible in tornadoes…

44. Compute Wind Speed Around a Hurricane R = 100 km dP = -25 mb f = 4 x 10 -5 V = 48 m/s = 107 mph = 93 kt Category 2 hurricane…

45. We have covered a lot of material in a short time! Study and think about balances in the natural coordinate system from the point of view of 1.first, pressure gradient, 2.then coriolis force, 3.then the force due to curvature of the lines of geopotential (or pressure) Don’t confuse “curvature” in the natural coordinate system with the curvature terms derived from use of a tangential coordinate system!

46. Next time Think about adding viscosity to the balance. And return to thermal wind balance…

47. Weather NCAR Research Applications Program –http://www.rap.ucar.edu/weather/http://www.rap.ucar.edu/weather/ http://www.aos.wisc.edu/weatherdata/eta_ tempest/12UTC/eta_c850_h06.gifhttp://www.aos.wisc.edu/weatherdata/eta_ tempest/12UTC/eta_c850_h06.gif National Weather Service –http://www.nws.noaa.gov/dtx/http://www.nws.noaa.gov/dtx/