1. Richard B. Rood (Room 2525, SRB)
AOSS 401 Geophysical Fluid Dynamics: Atmospheric Dynamics Prepared: Quasi-geostrophic / Waves / Vertical /
Richard B. Rood (Room 2525, SRB)
Cell:

2. Class News Ctools site (AOSS 401 001 F13)
Second Examination on December 10, 2013
Homework
Posted Today

3. Weather National Weather Service Weather Underground
Model forecasts:
Weather Underground
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) become linear:
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) Group (Perturb) 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
Thermodynamic Energy equation

24. Remember the relationship between vorticity and geopotential

25. Develop an equation for geopotential 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 Φ0 H
 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. Qualitative Description
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)

35. Advection of vorticity
ζ < 0; anticyclonic
 Short waves  ٠
ΔΦ > 0 B
Φ0 - ΔΦ L L Φ0 H
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. “Quasi-geostrophic”
Geostrophic
ageostrophic

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. Compare the two equations
Thermodynamic
Vorticity/momentum
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. Add these equations to eliminate omega and we have a partial differential equation for geopotential tendency (assume J=0)
Vorticity Advection
Thickness Advection

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. What about the thickness advection?

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

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 midlatitudes the horizontal velocity is nearly in geostrophic balance. Recall: the geostrophic wind is nondivergent (for constant Coriolis parameter), that is Horizontal 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.

51. 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?

52. 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

53. 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)

54. 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.

55. 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

56. mean meridional circulation

57. 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

58. One more way for vertical velocity

59. Quasi-geostrophic equations cast in terms of geopotential and omega.
THERMODYNAMIC EQUATION
VORTICITY EQUATION
ELIMINATE THE GEOPOTENTIAL AND GET AN EQUATION FOR OMEGA

60. Quasi-Geostrophic Omega Equation
1.) Apply the horizontal Laplacian operator to the QG thermodynamic equation
2.) Differentiate the geopotential height tendency equation with respect to p
3.) Combine 1) and 2) and employ the chain rule of differentiation (chapter in Holton, note factor ‘2’ is missing in Holton Eq. (6.36), typo)
Advection of absolute vorticity by the thermal wind

61. Vertical Velocity Summary
Though small, vertical velocity is in some ways the key to weather and climate. It’s important to waves growing and decaying. It is how far away from “balance” the atmosphere is.
It is astoundingly difficult to calculate. If you use all of these methods, they should be equal. But using observations, they are NOT!
In fact, if you are not careful, you will not even to get them to balance in models, because of errors in the numerical approximation.

62. In Class Problem Solution

63. In Class Problem Soln (1)
Multiply the first equation through by H and subtract u-bar times the second equation (gets rid of one of the u’ terms)

64. In Class Problem Soln (2)
Now, take d/dx of this equation and subtract d/dt of the original second equation (gets rid of the other u’ term)
Collect terms together
(No loss of points if you did not do this)

65. In Class Problem Soln (3)
Now, plug in the wave form definition of h’
Remember the x and t derivatives of the exponential are:

66. In Class Problem Soln (4) (go back!)
h0 and exponential terms cancel, and we are left with
This is the dispersion relation; describes how the phase velocity (c) differs with respect to mean velocity, mean fluid depth, and density