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

1. AOSS 401Geophysical Fluid Dynamics:Atmospheric DynamicsPrepared: 20131119Quasi-geostrophic / Waves / Vertical / 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) • Second Examination on December 10, 2013 • Homework • Posted Today

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

4. Outline • Waves • In Class Problem • Quasi-geostrophic Review • Vertical Velocity

5. Let’s think about waves some more • We assume that dependent variables like u and v can be represented by an average and deviation from the average.

6. Assume you have a bunch of points and at those points observations.

7. Imagine some axis

8. Add up all of the points, divide by number of points to get an average. THE AVERAGE VALUE

9. Add up all of the points, divide by number of points to get an average. CAN DEFINE, say, T = Taverage + Tdeviation

10. Add up all of the points, divide by number of points to get an average. By Definition: AVE(Taverage )= Taverage

11. Add up all of the points, divide by number of points to get an average. By Definition: AVE(Tdeviation )= 0

12. The averaging • We can define spatial averages or we can define time averages.

13. The averaging • We can define spatial averages or we can define time averages. • When we define a spatial average, say over the east-west direction, x, all the way around the globe, the average is NOT a function of variable over which we averaged. ∂ave(T)/∂x=0

14. The averaging • We can define spatial averages or we can define time averages. • When we define a time average, especially when we are developing theory, we can say that that the average changes slowly with time, while the perturbation changes rapidly in time. ∂ave(T)/∂t not required to be 0 But the perturbations still average to zero, because we assume that the average is over a long span of time compared with the perturbation time scale. THINK ABOUT IT...Define, use the definitions.

15. Linear perturbation theory • Assume: variable is equal to a mean state plus a perturbation • With these assumptions non-linear terms (like the one below) becomelinear: These terms are zero if the mean is independent of x. Terms with products of the perturbations are very small and will be ignored

16. Waves • The equations of motion contain many forms of wave-like solutions, true for the atmosphere and ocean • Some are of interest depending on the problem: Rossby waves, internal gravity (buoyancy) waves, inertial waves, inertial-gravity waves, topographic waves, shallow water gravity waves • Some are not of interest to meteorologists, e.g. sound waves • Waves transport energy, mix the air (especially when breaking)

17. Waves • Large-scale mid-latitude waves, are critical for weather forecasting and transport. • Large-scale waves in the tropics (Kelvin waves, mixed Rossby-gravity waves) are also important, but of very different character. Will introduce these after the next test; this is focus of AOSS 451. • This is true for both ocean and atmosphere. • Waves can be unstable. That is they start to grow, rather than just bounce back and forth.

18. Outline • Waves • In Class Problem • Quasi-geostrophic Review • Vertical Velocity

19. In class problems • Group (Thermal) • Scott • Anna • James • Kevin • Trent • Group (Perturb) • John • Ross • Rachel • Jordan • Justin • Alex

20. In Class Problem (go!) For a shallow fluid the linearized perturbation momentum equation and the vertically integrated continuity equation can be written as: where density and the density increment, δρ, are not a function of x. Now, assume a wave-like solution of the form and derive the dispersion equation for c (the wave frequency) Hint: you will need to find a way to eliminate u’ from the set of equations above to form a single equation for h’

21. Outline • Waves • In Class Problem • Quasi-geostrophic Review • Vertical Velocity

22. Forming the QG Equations • Assume the horizontal wind is approximately geostrophic • Scale the material derivative • Assume the north-south variation of the coriolis parameter is constant • Modify the continuity equation • Modify the thermodynamic equation

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

24. Remember the relationship between vorticity and geopotential

25. Develop an equation forgeopotential tendency Start with the QG vorticity equation Use the definition of geostrophic wind Plug this in, and we immediately have an equation for the time rate of change of the geopotential height

26. Rewrite the QG vorticity equation Expand material derivative

27. Rewrite the QG vorticity equation Use the continuity equation Remember why these are equivalent?

28. Rewrite the QG vorticity equation Advection of vorticity

29. Consider the vorticity advection Advection of vorticity Advection of relative vorticity Advection of planetary vorticity

30. Summary: Vorticity Advection in Wave • Planetary and relative vorticity advection in a wave oppose each other. • This is consistent with our observation of trade-off between relative and planetary vorticity in westerly flow over a mountain range

31. 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

32. Idealized solution: Barotropic Vorticity

33. Compare advection of planetary and relative vorticity  Short waves, advection of relative vorticity is larger   Long waves, advection of planetary vorticity is larger 

34. Short waves Strong curvature: large values of relative vorticity Relatively small amplitude: relatively small changes in coriolis parameter from trough to ridge Advection of relative vorticity > advection of planetary vorticity Wave propagates to the east Long waves Weak curvature: small values of relative vorticity Relatively large amplitude: relatively large changes in coriolis parameter from trough to ridge Advection of relative vorticity < advection of planetary vorticity Wave propagates to the west (depending on the mean wind speed) Qualitative Description

35. Advection of vorticity ζ < 0; anticyclonic  Short waves  ٠ ΔΦ > 0 B Φ0 - ΔΦ L L H Φ0 • Long waves  ٠ ٠ y, north Φ0 + ΔΦ A C x, east ζ > 0; cyclonic ζ > 0; cyclonic

36. A more general equation for geopotential tendency

37. Use the definition of geostrophic vorticity to rewrite the vorticity equation (Moving toward) An equation for geopotential tendency An equation in geopotential and omega. (2 unknowns, 1 equation)

38. ageostrophic Geostrophic “Quasi-geostrophic”

39. We used these equations to get our previous equation for geopotential tendency This worked fine for the barotropic system, but we want to describe baroclinic systems (horizontal temperature/density gradients)

40. Now let’s use this equation

41. Thermodynamic Vorticity/momentum Compare the two equations • Through continuity, both equations are related to the divergence of the ageostrophic wind • The divergence of the horizontal wind, which is related to the vertical wind, links the momentum (vorticity equation) to the thermodynamic equation

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

43. Geopotential tendency equation • Linear partial differential equation for geopotential tendency. • Given a geopotential distribution at an initial time, can compute geopotential distribution at a later time. • Right hand side is like a forcing. • We now have a real equation for forecasting the height (pressure field), and we know that the pressure gradient force is really the key to understanding the motion.

44. Thickness Advection What about the thickness advection?

45. warm Cold and warm advection(Related to thickness advection term?) cold

46. What about the thickness advection?

47. Outline • Waves • In Class Problem • Quasi-geostrophic Review • Vertical Velocity

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

49. 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).

50. 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.