Richard B. Rood (Room 2525, SRB) rbrood@umich 734-647-3530 Cell: 301-526-8572

1. AOSS 401Geophysical Fluid Dynamics:Atmospheric DynamicsPrepared: 20131008Balanced Flows: Gradient, Cyclostrophic Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Cell: 301-526-8572

2. Class News • Ctools site (AOSS 401 001 F13) • First Examination on October 22, 2013 • Second Examination on December 10, 2013 • Homework posted: • Ctools Assignments tab • Due Thursday October 10, 2013 • Derivations (using notes)

3. Weather • National Weather Service • Model forecasts: • Weather Underground • Model forecasts: • NCAR Research Applications Program

4. Outline • Balanced flows • Geostrophic • Gradient • Cyclostrophic

5. Atmosphere in Balance • Hydrostatic balance (no vertical acceleration) • Geostrophic balance (no horizontal acceleration or divergence) • Adiabatic lapse rate (no clouds or precipitation) • Wind Balances • Geostrophic • Gradient • Cyclostrophic

6. “Natural” Coordinate System • Follow the flow • From hydrodynamics—assumes no local changes (“steady state”) • No local change in geopotential height • No local change in wind speed or direction • Assume • Horizontal flow only (no vertical component) • No friction • This is a little like a Lagrangian parcel approach

7. Natural Coordinates Define one component of the horizontal wind as tangent to the direction of the wind. t north Φ0 t t t Φ0+3ΔΦ east south west ΔΦ > 0

8. Natural Coordinates Define the other component of the horizontal wind as normal to the direction of the wind. n north Φ0 n n t n t t Φ0+3ΔΦ east south west ΔΦ > 0

9. Cyclonic and Anticyclonic • What do these mean? • Cyclonic is flow around a low pressure system. • Anticyclonic is flow around a high pressure system. • Descriptions of rotational flows

10. The horizontal momentum equation(in natural coordinates) Along-flow direction (t) Across-flow direction (n)

11. MomentumNatural Pressure Geostrophic terms • We are only looking at steady flow parallel to geopotential height contours

12. MomentumNatural Pressure Ageostrophic terms • We are only looking at flow parallel to geopotential height contours

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

14. Gradient Flow(Momentum equation in natural coordinates) • Gradient flow is the flow that satisfies the normal component of the momentum equation in natural coordinates. • Balance in the normal, as opposed to tangential, component of the momentum equation • Balance between • pressure gradient • Coriolis • centrifugal force

15. Gradient Flow(Momentum equation in natural coordinates)

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

17. V < 0 V < 0 Anomalous Low V < 0 V < 0 Regular Low Anomalous High Regular High Gradient FlowSolution (V) must be real and non-negative8 Possible Solutions

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

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

20. Regular (Normal) and Anomalous Flows • Regular (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…

21. Some analysis of normal lows and highs

22. Gradient Flow: ImplicationsSolution (V) must be real

23. R < 0 R > 0 Gradient FlowRegular High and Low Definition of normal, n, direction Low High n n

24. Gradient Flow: ImplicationsSolution 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 goes to zero

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

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

27. Use Vg for pressure gradient then divide • 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)

28. 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 goes to zero

29. Different scale of motion: Small

30. What are the scales of the terms? For a tornado (In-class exercise)

31. What are the scales of the terms? For a tornado Largest Terms

32. Cyclostrophic Balance

33. 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

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

35. Cyclostrophic Flow • Tornadoes: 102 meters, 0.1 km • Dust devils: 1 - 10 meters • Small length scales • Strong winds

36. Cyclostrophic Flow

37. Cyclostrophic Flow • Radical must be positive: two solutions

38. Cyclostrophic Flow Pressure gradient force Low Low Centrifugal force

39. Cyclostrophic Flow Low Low Counterclockwise Rotation Clockwise Rotation

40. 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=T0lvlVxfQx8&feature=related

41. In-Class Exercise: Compute Tornado Wind Speed • Remember: P=850 mb R = 100 m P=950 mb (Assume ρ = 1 kg/m3)

42. P=850 mb R = 100 m P=950 mb In-Class Exercise: Compute Tornado Wind Speed

43. n n Solution non-real and non-physical Solution non-real and non-physical Cyclostrophic Flow Around a High Pressure System? High High

44. Balanced flows in natural coordinates(balanced, here, means steady) Good for diagnostics and interpretation. Unit vector changes in time, with flow field. Requires “balance” to be useful. These specific ones are “away from the surface.”

45. Balanced flow: an application of all that we know

46. Geopotential, 50 hPa surface Pressure units: hPa mbar inches of Hg Length scale? >1,000 km ~10,000 km

47. What about the wind? Pressure gradientforce Coriolis force What’s the latitude? Centrifugalforce Wind

48. Wind

49. What would happen if I put dye in the low?

50. 23 October 2006, Geopotential Height