AOSS 401, Fall 2007 Lecture 21 October 31 , 2007

1. AOSS 401, Fall 2007Lecture 21October 31, 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 31, 2007 • Homework 5 (Due Friday) • Posted to web • Computing assignment posted to ctools under the Homework section of Resources • Next Test November 16 • Thanks for the comments on the Midterms evaluations

3. A Diversion • Santa Ana Winds

4. Weather • National Weather Service • http://www.nws.noaa.gov/ • Model forecasts: http://www.hpc.ncep.noaa.gov/basicwx/day0-7loop.html • Weather Underground • http://www.wunderground.com/cgi-bin/findweather/getForecast?query=ann+arbor • Model forecasts: http://www.wunderground.com/modelmaps/maps.asp?model=NAM&domain=US

5. Material from Chapter 6 • Quasi-geostrophic theory • Quasi-geostrophic vorticity • Relation between vorticity and geopotential

6. Quasi-Geostrophic • Derive a system of equations that is close to geostrophic and hydrostatic balance, but includes the effects of ageostrophic wind • Comes from scale analysis of equations of motion in pressure coordinates • Scale analysis = make assumptions(where do these assumptions break down?)

7. Equations of motion in pressure coordinates(using Holton’s notation)

8. Scale factors for “large-scale” mid-latitude

9. From the scale analysis we introduced the non-dimensional Rossby number A measure of planetary vorticity compared to relative vorticity. A measure of the importance of rotation.

10. Scale analysis of equations in pressure coordinates • Start: • horizontal flow is approximately geostrophic • vertical velocity much smaller than horizontal velocity

11. We will scale the material derivative Ignore // small This is for use in the advection of temperature and momentum. ω comes from div(ageostrophic wind)

12. Variation of Coriolis parameter • L, length scale, is small compared to the radius of the Earth • In the calculation of geostrophic wind, assume f is constant; f = f0 • We cannot assume f is constant in the Coriolis terms…

13. Variation of Coriolis parameter

14. Variation of Coriolis parameter Scale of first two terms.

15. Continuity equation becomes

16. Thermodynamic equation static stability, Sp, is large; ω cannot be ignored geostrophic wind can be used here.

17. Thermodynamic equation(use the fact that atmosphere is near hydrostatic balance) split temperatureinto basic state plus deviation

18. Thermodynamic equation(and with the hydrostatic equation) note the inverse relation of heating with pressure

19. The Momentum Equation

20. horizontal flow is approximately geostrophic • L, length scale, is small compared to the radius of the Earth • In the calculation of geostrophic wind, assume f is constant; f = f0

21. horizontal flow is approximately geostrophic

22. def’n of the full wind approx of coriolis parameter Use definition of geostrophic wind in the pressure gradient force Forcing terms in momentum equation

23. Forcing terms in momentum equation

24. Approximate horizontal momentum equation • This equation states that the time rate of change of the geostrophic wind is related to • the coriolis force due to the ageostrophic wind and • the part of the coriolis force due to the variability of the coriolis force with latitude and the geostrophic wind. • Both of these terms are smaller than the geostrophic wind itself.

25. A Point • All of the terms in the equation for the CHANGE in the geostrophic wind, which is really a measure of the difference from geostrophic balance, are order Ro (Rossby number). • Again, reflects the importance of rotation to the dynamics of the atmosphere and ocean

26. Scaled equations of motion in pressure coordinates Definition of geostrophic wind Momentum equation Continuity equation ThermodynamicEnergy equation

27. What is the point? • Set of equations that describes synoptic-scale motions and includes the effects of ageostrophic wind (vertical motion)

28. Scale Analysis = Make Assumptions Quasi-geostrophic system is good for: • Synoptic scales • Middle latitudes • Situations in which Va is important • Flows in approximate geostrophic and hydrostatic balance • Mid-latitude cyclones

29. Scale Analysis = Make Assumptions Quasi-geostrophic system is not good for: • Very small or very large scales • Flows with large vertical velocities • Situations in which Va≈ Vg • Flows not in approximate geostrophic and hydrostatic balance • Thunderstorms/convection, boundary layer, tropics, etc…

30. What will we do next? • Derive a vorticity equation for these scaled equations. • Actually provides a “suitable” prognostic equation because need to include div(ageostrophic wind) in the prognostics. • Remember the importance of divergence in vorticity equations.

31. Derive a vorticity equation • Going to spend some time with this.

32. Vorticity relative vorticity velocity in (x,y) plane shear of velocity suggests rotation

33. Vorticity view this as the definition of relative vorticity

34. If we want an equation for the conservation of vorticity, then • We want an equation that represents the time rate of change of vorticity in terms of sources and sinks of vorticity.

35. Conservation (continuity) principle • dM/dt = Production – Loss

36. Newton’s Law of Motion Which is the vector form of the momentum equation. (Conservation of momentum) Where F is the sum of forces acting on a parcel, m mass, v velocity

37. What are the forces? • Total Force is the sum of all of these forces • Pressure gradient force • Gravitational force • Viscous force • Apparent forces • Derived these forces from first principles

38. If we want an equation for the conservation of vorticity, then • We could approach it the same way as momentum, define the sources and sinks of vorticity from first principles. • But that is hard to do. What are the first principle sources of vorticity? • Or we could use the conservation of momentum, and the definition of vorticity to derive the equation.

39. Newton’s Law of Motion(components)

40. Combine definition and conservation principle

41. Operate on momentum equation

42. Subtract so a time rate of change of vorticity will come from here.

43. Subtract so details will depend on d( )/dt. For an Eulerian fluid d( )/dt = D( )Dt, material derivative. For a Lagrangian description could write immediately.

44. Expand derivative

45. Expand derivative

46. Expand derivative Dζ/Dt comes from here.

47. Expand derivative Other things comes from here.

48. Collect terms

49. Let’s return to our quasi-geostrophic formulation

50. Scaled horizontal momentum in pressure coordinates