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

2. Class News Ctools site (AOSS 401 001 F13)
Second Examination on December 10, 2013
Homework
Homework due November 26, 2013

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

4. Outline
Analysis of equations of atmospheric motion scaled for large-scale middle latitude dynamics / Quasi-geostrophic formalism
Review of large-scale formalism
Long and short waves / Barotropic waves
Examine ageostrophic wind
Cyclone development
Occlusion
Baroclinic  Barotropic /// energy conversion
Vertical motion

5. Vorticity Equation DIVERGENCE TILTING SOLENOIDAL or BAROCLINIC
Changes in relative vorticity are caused by:
Divergence
Tilting
Gradients in density on a pressure surface
Advection

6. QG Theory: Assumptions
Assume the horizontal wind is approximately geostrophic
Scale the material derivative
Neglect the vertical advection
Horizontal advection due to geostrophic wind
Assume the north-south variation of the coriolis parameter is constant
Divergence in the continuity equation only due to ageostrophic wind
Modify the thermodynamic equation
Advection by the geostrophic wind
Assume hydrostatic balance
Vertical velocity acts on a mean static stability profile

7. The quasi-geostrophic (QG) equations
momentum equation
geostrophic wind
continuity equation
thermodynamic equation

8. Geopotential tendency equation
Vorticity Advection
Thickness Advection
Linear partial differential equation for geopotential tendency.
Given a geopotential distribution at an initial time, can compute geopotential distribution at a later time.
The right hand side is like a forcing.

9. First major set of conclusions from the quasi-geostrophic system
We see that
Geostrophic advection of geostrophic vorticity causes waves to propagate
The vertical difference in temperature (thickness) advection causes waves to amplify

10. Remember our scaled vorticity equation?
We see that the QG vorticity equation is very similar to the scaled vorticity equation we developed before
…with a few additional assumptions

11. Long and Short Waves
In this discussion these are both “large-scale” meaning that rotation is important.
Therefore, this is a “short” large-scale wave versus a “long” large-scale wave.

12. Barotropic Wave Dispersion
Look at the barotropic wave equation

13. Consider a barotropic fluid
Think here
How do we get to this form of the equation with geopotential?
Barotropic: ageostrophic wind is zero, vertical velocity is zero, horizontal divergence is zero

14. Assume a “wave like solution” (get used to this…)
Dispersion relation. Relates frequency and wave number to flow. Must be true for waves.

15. Wind must be positive, from the west, for a wave.
Stationary wave?
U positive for wave to exist. Return to lecture where we considered eastward and westward flow over the mountain.
What is the wavelength for a stationary wave?  This leads to “planetary waves.” Scale order of 10,000 km. 1 – 2 – 3 of these around the planet, longitudinally.
Wind must be positive, from the west, for a wave.

16. Consider a more specific form of a wave solution

17. Assume that the geopotential takes the form of a wave
Mean
Wave
Mean Ф Gradient in y
Horizontal wavenumber

18. Remember the relationship between geostrophic wind and geopotential
Plug in the wave solution for the geopotential height

19. Divide the geostrophic wind into mean and perturbation
Divide wind into mean and perturbation, assume no mean north/south wind
Perturbation only
mean perturbation

20. Plug into the advection of relative vorticity

21. Plug into the advection of planetary vorticity

22. Compare advection of planetary and relative vorticity

23. 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

24. 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

25. More about the ageostrophic wind
Review ageostrophic wind and implications for vertical motion and cyclone development
Use all that we know to describe development of a mid-latitude cyclone

26. A closer look at the ageostrophic wind
Start with our momentum equation
Just for kicks, take
and see what happens

27. A closer look at the ageostrophic wind
Now, by the right hand rule:
and remember
so we can write

28. A closer look at the ageostrophic wind
We end up with
…the ageostrophic wind!
Knowing that the divergence of the ageostrophic wind leads to vertical motion, let’s explore the implications of this…

29. Where do we find acceleration?
Curvature
Acceleration
Ageostrophic wind
ΔΦ > 0
Φ0 - ΔΦ D C Φ0
y, north
Φ0 + ΔΦ
x, east
D = Divergence and C = Convergence

30. Where do we find acceleration?
Along-flow speed change

31. Where do we find acceleration?
Along-flow speed change
Acceleration
Ageostrophic wind D C J
Remember our example of a thermal circulation, where divergence and convergence at different levels are related.
D = Divergence and C = Convergence

32. Where do we find acceleration?
Along-flow speed change

33. Vorticity at upper and lower levels
Continue to examine the divergence of the wind  Which is a proxy for the vertical velocity

34. One more application… Start with the identity
Now, consider the divergence of the ageostrophic wind
Use the identity, and we have

35. What is this? Formally, this is
Which is how we derived the vorticity equation

36. What can we do with this? Plug in our QG assumptions
Let’s think about the difference in divergence (of the ageostrophic wind) between two levels
By the continuity equation, this means that mass is either increasing in the column (net convergence) or decreasing in the column (net divergence)
This should tell us whether low pressure or high pressure is developing at the surface…

37. Column Net Convergence/Divergence
Subtract vorticity equation at 1000 hPa from vorticity equation at 500 hPa
Gives us the net divergence between 1000 and 500 hPa
It can be shown that

38. Examine each term “Steering term”
Low-level centers of vorticity propagate in the direction of the thermal wind
(Along the gradient of thickness)

39. Examine each term “Development term” A bit complicated
Remember, thermal wind is the vertical change in the geostrophic wind
This term indicates the influence of a tilt with height of the location of the maximum (minimum) in vorticity
Fundamentally: if the location of the maximum in vorticity shifts westward with height, the low will develop.

40. Development term Combine terms Definition of thermal wind
Definition of geostrophic vorticity

41. Development term Remember the barotropic height tendency equation?
How about the
equation?

42. Development term
Development term is the vertical change in barotropic advection of vorticity
This is the same as the stretching term in the omega equation
If the upper-level wave propagates faster than the surface wave, the system decays
Otherwise, the system may develop…

43. Implications
Surface low and high pressure systems (centers of maximum/minimum vorticity) propagate along lines of constant thickness
If there is a vertical tilt westward with height of the vorticity, then a surface low pressure system can intensify (increase in low-level positive vorticity)

44. Look at Cyclones

45. Mid-latitude cyclone development

46. Mid-latitude cyclone development
From:

47. Mid-latitude cyclone development
From:

48. Mid-latitude cyclone development
From:

49. Mid-latitude cyclone development
From:

50. Mid-latitude cyclone development
From:

51. In the classic cyclone model
Occlusion describes the transition of a cyclone from baroclinic (west-ward tilt with height) to barotropic (“vertically stacked”)
Once there is no more westward vertical tilt with height, no further development can occur

52. Baroclinic  Barotropic

53. Idealized Development of a Baroclinic Wave
Start with a N-S temperature gradient and associated wind speed maximum in the upper troposphere
Introduce a low-level perturbation
Watch the conversion of PE to KE and the development of the wave…

54. Energetics of Midlatitude Cyclone Development
The jet stream is commonly associated with strong temperature gradients in the middle/lower troposphere (thermal wind relationship)
Mid-latitude cyclones develop along waves in the jet stream
By thermal wind balance, this means that mid-latitude cyclones develop along strong horizontal temperature gradients (fronts)
There is a link between the strength of the horizontal temperature gradient and cyclone development…

55. Idealized vertical cross section

56. Two important definitions
barotropic
density depends only on pressure.
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)

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

58. Barotropic/baroclinic atmosphere
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

59. Barotropic/baroclinic atmosphere
Energetics:
Baroclinic = temperature contrast = density contrast = available potential energy
Extratropical cyclones intensify through conversion of available potential energy to kinetic energy

60. Available Potential Energy
Defined as the difference in potential energy after an adiabatic redistribution of mass
COLD
WARM

61. Available Potential Energy
Defined as the difference in potential energy after an adiabatic redistribution of mass
WARM
COLD

62. Energetics in the atmosphere
Diabatic heating (primarily radiation) maintains the equator to pole temperature contrast
Strength of temperature contrast referred to as “baroclinicity”
Cyclones at mid-latitudes reduce this temperature contrast—adjust baroclinic atmosphere toward barotropic

63. Energetics in the atmosphere
Ability to convert potential energy to kinetic energy directly related to tilt with height (offset) of low/high pressure

64. Energetics in the atmosphere
Diabatic heating (primarily radiation) maintains the equator to pole temperature contrast
Strength of temperature contrast referred to as “baroclinicity”
Cyclones at mid-latitudes reduce this temperature contrast—adjust baroclinic atmosphere toward barotropic
We can quantify this, but first let’s go back to the ageostrophic wind…

65. Baroclinic  Barotropic Transition

66. 1200 UTC 31 March 1997

67. 1800 UTC 31 March 1997

68. 0000 UTC 1 April 1997

69. 0600 UTC 1 April 1997

70. 1200 UTC 1 April 1997

71. Vertical velocity: Omega equation
Kinematic method
Horizontal advection
Diabatic method
Omega equation
What are the ways that we think about vertical motion? Diabatic, Horizontal Advection, Kinematic Method.

72. Characteristics of large-scale vertical velocity
In all of the estimates for vertical velocity what is missing?
The answer is _______________
The vertical velocity in this large-scale, mid-latitude description of dynamics is exactly what is needed to maintain what balances ____________ ?

73. QG-omega equation Combine all QG equations
Link between vertical derivative of vorticity advection (divergence/stretching) and vertical motion

74. QG-omega equation Combine all QG equations
Link between vertical derivative of vorticity advection (divergence/stretching) and vertical motion
Link between temperature advection and vertical motion

75. QG-omega equation Combine all QG equations
Link between vertical derivative of vorticity advection (divergence/stretching) and vertical motion
Link between temperature advection and vertical motion
Link between diabatic heating and vertical motion

76. QG-omega equation (simplified)
Advection of absolute vorticity by the thermal wind

77. “Advection” by thermal wind?
How to analyze this on a map?
Thermal Wind is
Perpendicular to
Gradient of
Thickness
Look at contours of constant thickness

78. Vertical Motion on Weather Maps
Laplacian of omega is proportional to -ω
Omega can be analyzed as:
Remember, from definition of omega and scale analysis
Positive vorticity advection by the thermal wind indicates rising motion

79. Vertical Motion on Weather Maps
Positive vorticity advection by the thermal wind indicates rising motion
Descent
Ascent +
Lines of constant thickness

80. Omega Equation

81. 1200 UTC 31 March 1997

82. 1800 UTC 31 March 1997

83. 0000 UTC 1 April 1997

84. 0600 UTC 1 April 1997

85. 1200 UTC 1 April 1997

86. Take Away Messages Large-scale dynamics / quasi-geostrophic theory
Hydrostatic balance
Geostrophic balance
Vertical motion is important, but above balances are always maintained
Vertical motion is diagnosed / no time dependence (though it changes with time)
Vertical motion is linked to divergence of horizontal wind  ageostrophic wind
Vertical motion works against static stability, which is large. This is how baroclinicity is represented in the equations, despite baroclinic terms being scaled out
We have looked at dynamics of large-scale waves in different ways - heuristically, theoretically and on maps. We see consistency and different insights. The mathematics proves a way to both quantify and explore the behavior of the dynamics.