AOSS 401, Fall 2007 Lecture 11 October 1 , 2007

1. AOSS 401, Fall 2007Lecture 11October 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 NewsOctober 1, 2007 • Ricky will be lecturing again starting Wednesday—I will lecture next on the 17th 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. Do you see some notion of a radius of curvature? Sort of like a circle, but NOT a circle. n n n Return to Geopotential (Φ) in upper troposphere north Low t t HIGH t east south west

8. The horizontal momentum equation(in natural coordinates)

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

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

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. north Φ0+ΔΦ Φ0 Φ0+2ΔΦ Δn Φ0+3ΔΦ east south west

14. Low Φ0-ΔΦ R n Φ0 Φ0+ΔΦ Δn t HIGH How does curvature affect the wind?(cyclonic flow/low pressure)

15. From Holton • If Vg/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. Geostrophic & observed wind 300 hPa Observed:95 knots Geostrophic:140 knots

18. Low n Δn Φ0-ΔΦ t Φ0 R Φ0+ΔΦ HIGH How does curvature affect the wind?(anticyclonic flow/high pressure)

19. From Holton • If Vg/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. Geostrophic & observed wind 300 hPa 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. Cyclostrophic Flow • 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: 102 meters, 0.1 km • Dust devils: 1 - 10 meters • Small length scales • Strong winds

28. Cyclostrophic Flow Pressure gradient force Low Low Centrifugal force

29. Cyclostrophic Flow Low Low Counterclockwise Rotation Clockwise Rotation

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

31. In-Class Exercise: Compute Tornado Wind Speed • Remember: P=750 mb R = 100 m P=850 mb (Assume ρ = 1 kg/m3)

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

33. n n Cyclostrophic FlowAround a High Pressure System? High High

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. Gradient Flow(Momentum equation in natural coordinates) Look for real and non-negative solutions

37. Gradient FlowSolution must be real

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

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

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

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

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. R = 100 km dP = -25 mb f = 4 x 10-5 V = 48 m/s = 107 mph = 93 kt Category 2 hurricane… Compute Wind Speed Around a 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 • first, pressure gradient, • then coriolis force, • 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.aos.wisc.edu/weatherdata/eta_tempest/12UTC/eta_c850_h06.gif • National Weather Service • http://www.nws.noaa.gov/dtx/