AOSS 401, Fall 2007 Lecture 24 November 07 , 2007

1. AOSS 401, Fall 2007Lecture 24November 07, 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 November 07, 2007 • Homework 6 (Posted this evening) • Due Next Monday • Important Dates: • November 16: Next Exam (Review on 14th) • November 21: No Class • December 10: Final Exam

3. 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

4. Rest of Course • Wrap up quasi-geostrophic theory (Chapter 6) • Potential vorticity • Vertical velocity • Will NOT do Q vectors • We will have a lecture on the Eckman layer (Chapter 5) • Boundary layer, mix friction with rotation • We will have a lecture on Kelvin waves (Chapter 11) • A long wave in the tropics • There will be a joint lecture with 451 on hurricanes (Chapter 11) • Computer homework (perhaps lecture) on modeling • Special topics?

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

6. Scaled equations in pressure coordinates(The quasi-geostrophic (QG) equations) momentum equation geostrophic wind continuity equation thermodynamicequation

7. Approximations in the quasi-geostrophic (QG) theory

8. Quasi-geostrophic equations cast in terms of geopotential and omega. THERMODYNAMIC EQUATION VORTICITY EQUATION

9. Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency(assume J=0) GEOPOTENTIAL TENDENCY EQUATION

10. Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency(assume J=0) f0 * Vorticity Advection

11. Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency(assume J=0) Thickness Advection

12. Advection of vorticity ζ < 0; anticyclonic  Advection of ζ tries to propagate the wave this way  ٠ ΔΦ > 0 B Φ0 - ΔΦ L L H Φ0  Advection of f tries to propagate the wave this way  ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic

13. Relationship between upper troposphere and surface vorticity advection thickness advection

14. To think about this • Read and re-read pages 174-176 in the text.

15. Idealized vertical cross section

16. Great web page with current maps: Real baroclinic disturbances http://www.meteoblue.ch/More-Maps.79+M5fcef4ad590.0.html Personalize your maps (create a login): http://my.meteoblue.com

17. Real baroclinic disturbances:850 hPa temperature and geopot. thickness cold airadvection,enhances trough warm airadvectioneast of thesurface low,enhances the ridge

18. Real baroclinic disturbances:500 hPa rel. vorticity and mean SLP sea level pressure Positivevorticity, pos. vorticityadvection,increase incyclonicvorticity

19. Real baroclinic disturbances:500 hPa geopot. height and mean SLP Upper level systemslags behind (to the west):system stilldevelops

20. With the benefit of hindsight and foresight let’s look back.

21. QG vorticity equation THINKING ABOUT THESE TERMS Advection of relative vorticity Stretchingterm Advection of planetary vorticity Competing

22. QG vorticity equation Advection of relative vorticity WHAT ABOUT THIS TERM? Stretchingterm Advection of planetary vorticity Competing

23. Consider our simple form of potential vorticity From scaled equation, with assumption of constant density and temperature. There was the assumption that the layer of fluid was shallow.

24. Fluid of changing depth What if we have something like this, but the fluid is an ideal gas?

25. Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency(assume J=0) Still looks a lot like time rate of change of vorticity

26. Quasi-Geostrophicpotential vorticity (PV) equation Simplify the last term of the geopotential tendencyequation by applying the chain rule: = 0 Why?

27. Quasi-Geostrophicpotential vorticity (PV) equation Simplify the last term of the geopotential tendencyequation by applying the chain rule: = 0 Why? THERMAL WIND RELATION

28. Quasi-Geostrophicpotential vorticity (PV) equation Simplify the last term of the geopotential tendencyequation by applying the chain rule: = 0 Why? Leads to the conservation law: Quasi-geostrophic potential vorticity: Conserved following the geostrophic motion

29. Imagine at the point flow decomposed into two “components” A “component” that flows around the point.

30. Vorticity • Related to shear of the velocity field. ∂v/∂x-∂u/∂y

31. Imagine at the point flow decomposed into two “components” A “component” that flows into or away from the point.

32. Divergence • Related to stretching of the velocity field. ∂u/∂x+∂v/∂y

33. Potential vorticity (PV): Comparison Units: Barotropic PV: m-1s-1 Quasi-geostrophic PV: s-1 K kg-1 m2 s-1 Ertel’s PV: THESE ARE LIKE STRETCHING IN THE VERTICAL

34. QG vorticity equation Advection of relative vorticity WHAT ABOUT THIS TERM? Stretchingterm Advection of planetary vorticity Competing

35. Fluid of changing depth What if we have something like this, but the fluid is an ideal gas? Conversion of thermodynamic energy to vorticity, kinetic energy. Again the link between the thermal field and the motion field.

36. Two important definitions • barotropic – density depends only on pressure. And by the ideal gas equation, surfaces of constant pressure, are surfaces of constant density, are surfaces of constant temperature (idealized assumption).= (p) • baroclinic – density depends on pressure and temperature (as in the real world).= (p,T)

37. Barotropic/baroclinic atmosphere Barotropic: p T p + p T+T p + 2p T+2T T T+T T+2T Baroclinic: p p + p p + 2p ENERGY IN HERE THAT IS CONVERTED TO MOTION

38. Barotropic/baroclinic atmosphere Barotropic: p T p + p T+T p + 2p T+2T T T+T T+2T Baroclinic: p p + p p + 2p DIABATIC HEATING KEEPS BUILDING THIS UP

39. NOW WOULD BE A GOOD TIME FOR A SILLY STORY

40. VERTICAL VELOCITY

41. Vertical motions: The relationship between w and  = 0 hydrostatic equation ≈ 1m/s 1Pa/km≈ 1 hPa/d ≈ 100 hPa/d ≈ 10 hPa/d

42. Link between  and the ageostrophic wind = 0 Links the horizontal and vertical motions. Since geostrophy is such a good balance, the vertical motion is linked to the divergence of the ageostrophic wind (small).

43. Vertical pressure velocity  For synoptic-scale (large-scale) motions in midlatitudesthehorizontal velocity is nearly in geostrophic balance.Recall: the geostrophic wind is nondivergent (for constant Coriolis parameter), that isHorizontal divergence is mainly due to small departures from geostrophic balance (ageostrophic wind). Therefore: small errors in evaluating the winds <u> and <v> lead to large errors in . The kinematic method is inaccurate.

44. Think about this ... • If I have errors in data, noise. • What happens if you average that data? • What happens if you take an integral over the data? • What happens if you take derivatives of the data?

45. Estimating the vertical velocity: Adiabatic Method Start from thermodynamic equation in p-coordinates: Assume that the diabatic heating term J is small (J=0), re-arrange the equation - (Horizontal temperature advection term) Sp:Stability parameter

46. Estimating the vertical velocity: Adiabatic Method Horizontal temperature advection term Stability parameter • If T/t = 0 (steady state), J=0 (adiabatic) and Sp > 0 (stable): • then warm air advection:  < 0, w ≈ -/g > 0 (ascending air) • then cold air advection:  > 0, w ≈ -/g < 0 (descending air)

47. Adiabatic Method • Based on temperature advection, which is dominated by the geostrophic wind, which is large. Hence this is a reasonable way to estimate local vertical velocity when advection is strong.

48. Estimating the vertical velocity: Diabatic Method Start from thermodynamic equation in p-coordinates: If you take an average over space and time, then the advection and time derivatives tend to cancel out. Diabatic term

49. mean meridional circulation

50. Conceptual/Heuristic Model • Observed characteristic behavior • Theoretical constructs • “Conservation” • Spatial Average or Scaling • Temporal Average or Scaling • Yields • Relationship between parameters if observations and theory are correct Plumb, R. A. J. Meteor. Soc. Japan, 80, 2002