AOSS 401, Fall 2006 Lecture 10 September 28 , 2007

1. AOSS 401, Fall 2006Lecture 10September 28, 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 • Homework 2 returned today • Homework 3 due today (questions?) • Homework 4 posted Monday • Exam 1 October 10—covers chapters 1-3 in Holton

3. Weather • NCAR Research Applications Program • http://www.rap.ucar.edu/weather/ • National Weather Service • http://www.nws.noaa.gov/dtx/

4. Correction… • I made a mistake in my last set of lectures (September 19th) • Geostrophic wind is only non-divergent if pressure is the vertical coordinate… • Corrected lecture 6 posted to ctools by Monday.

5. Today:Material from Chapter 3 • Natural coordinates • Balanced flow

6. Another Coordinate System? • We want to simplify the equations of motion • For horizontal motions on many scales, the atmosphere is in balance • Mass (p, Φ) fields in balance with wind (u) • It is easy to observe the pressure or geopotential height, much more difficult to observe the wind • Balance provides a way to infer the wind from the observed (p, Φ) • Wind is useful for prediction (remember the advection homework and in-class problems?)

7. The horizontal momentum equation Assume no viscosity and no vertical wind

8. Geostrophic balance Low Pressure High Pressure Flow initiated by pressure gradient Flow turned by Coriolis force

9. Geostrophic & observed wind 300 mb

10. Describe previous figure. What do we see? • At upper levels (where friction is negligible) the observed wind is parallel to geopotential height contours. • (On a constant pressure surface) • Wind is faster when height contours are close together. • Wind is slower when height contours are farther apart.

11. Geopotential (Φ) in upper troposphere ΔΦ > 0 north Φ0+ΔΦ Φ0 Φ0+2ΔΦ Φ0+3ΔΦ east south west

12. Geopotential (Φ) in upper troposphere ΔΦ > 0 north Φ0+ΔΦ Φ0 Δy Φ0+2ΔΦ Φ0+3ΔΦ east south west

13. Geopotential (Φ) in upper troposphere ΔΦ > 0 north Φ0+ΔΦ δΦ = Φ0 – (Φ0+2ΔΦ) Φ0 Δy Φ0+2ΔΦ Φ0+3ΔΦ east south west

14. Geopotential (Φ) in upper troposphere ΔΦ > 0 north Φ0+ΔΦ Φ0 Δy Φ0+2ΔΦ Φ0+3ΔΦ east south west

15. The horizontal momentum equation Assume no viscosity

16. Geostrophic approximation

17. Geopotential (Φ) in upper troposphere ΔΦ > 0 north Φ0+ΔΦ Φ0 Δy Φ0+2ΔΦ Φ0+3ΔΦ east south west

18. Geopotential (Φ) in upper troposphere ΔΦ > 0 north Φ0+ΔΦ Φ0 Δy Φ0+2ΔΦ Φ0+3ΔΦ east south west

19. Geopotential (Φ) in upper troposphere • Think about the observed wind • Flow is parallel to geopotential height lines • There is curvature in the flow

20. Geostrophic & observed wind 300 hPa

21. Geopotential (Φ) in upper troposphere • Think about the observed (upper level) wind • Flow is parallel to geopotential height lines • There is curvature in the flow • Geostrophic balance describes flow parallel to geopotential height lines • Geostrophic balance does not account for curvature • How to best describe balanced flow with curvature?

22. Another Coordinate System? • We want to simplify the equations of motion • For horizontal motions on many scales, the atmosphere is in balance • Mass (p, Φ) fields in balance with wind (u) • It is easy to observe the pressure or geopotential height, much more difficult to observe the wind • Balance provides a way to infer the wind from the observed (p, Φ) • Need to describe balance between pressure gradient, coriolis, and curvature

23. “Natural” Coordinate System • Follow the flow • From hydrodynamics—assumes no local changes • No local change in geopotential height • No local change in wind speed or direction • Assume • Horizontal flow only (no vertical component) • No friction

24. Return to Geopotential (Φ) in upper troposphere 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

25. Return to Geopotential (Φ) in upper troposphere 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

26. “Natural” Coordinate System • Regardless of position (i,j) • t always points in the direction of flow • n always points perpendicular to the direction of the flow toward the left • Remember the “right hand rule” for vectors? Take k x t to get n • Assume • Pressure as a vertical coordinate • Flow parallel to contours of geopotential height

27. “Natural” Coordinate System • Advantage: We can look at a height (on a pressure surface) and pressure (on a height surface) and estimate the wind. • It is difficult to directly measure winds • We estimate winds from pressure (or hydrostatically equivalent height), a thermodynamic variable. • Natural coordinates are useful for diagnostics and interpretation.

28. “Natural” Coordinate System • For diagnostics and interpretation of flows, we need an equation…

29. Return to Geopotential (Φ) in upper troposphere north Low n n t n t t • Geostrophic assumption. • Do you notice that those n vectors point towards something out in the distance? HIGH east south west ΔΦ > 0

30. Return to Geopotential (Φ) in upper troposphere Do you see some notion of a radius of curvature? Sort of like a circle, but NOT a circle. north Low n n t n t HIGH t east south west

31. Time to look at themathematics One direction: no (u,v) First simplification: the velocity • Always positive • Always points in the positive t direction Define velocity as: Definition of magnitude:

32. Goal: Quantify Acceleration acceleration is: (Chain Rule) Change in speed Change in Direction

33. How to get as a function of V, R ?

34. Remember our circle geometry… Δs=RΔφ this is not rotation of the Earth! It is an element of curvature in the flow. Δφ Δt t+Δt t R= radius of curvature Δs t

35. Remember our circle geometry… Δs=RΔφ this is not rotation of the Earth! It is an element of curvature in the flow. Δφ Δt t+Δt n t R= radius of curvature n Δs t

36. Remember our circle geometry… Δs=RΔφ If Δs is very small, Δt is parallel to n. So, Δt points in the direction of n Δφ Δt t+Δt n t R= radius of curvature n Δs t

37. Remember, we want an expression for From circle geometry we have: Rearrange and take the limit Use the chain rule Remember the definition of velocity

38. Goal: Quantify Acceleration acceleration defined as: (Chain Rule) We just derived: So the total acceleration is

39. Acceleration in Natural Coordinates Along-flow speed change ?

40. Acceleration in Natural Coordinates The total acceleration is Definition of wind speed angle of rotation Circle geometry angular velocity Plug in for Δs Centrifugal force

41. Acceleration in Natural Coordinates Along-flow speed change Centrifugal Acceleration

42. We have seen that Coriolis force is normal to the velocity.

43. Pressure gradient (by definition)

44. The horizontal momentum equation

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

46. Simplification? Which coordinate system is easier to interpret? 0 0 • We are only looking at flow parallel to geopotential height contours

47. Simplification? • Which coordinate system is easier to interpret? • We are only looking at flow parallel to geopotential height contours

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

49. 0 Uses of Natural Coordinates • Geostrophic balance • Definition: coriolis and pressure gradient in exact balance. • Parallel to contours  straight line  R is infinite

50. Geostrophic balance in natural coordinates