1. Richard B. Rood (Room 2525, SRB)
AOSS 401 Geophysical Fluid Dynamics: Atmospheric Dynamics Prepared: Vorticity / Flow
Richard B. Rood (Room 2525, SRB)
Cell:

2. Class News Ctools site (AOSS 401 001 F13)
Second Examination on December 10, 2013
Homework
Posted on Ctools / Due on Thursday 11/7/13

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

4. Outline
Vorticity and Flow

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

6. Scale Analysis of the Vorticity Equation
Changes in relative vorticity are caused by:
Divergence
Tilting
Gradients in density
Advection
Which of these are most important for large-scale flows?
Back to scale analysis…

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

8. Terms in Vorticity Equation
Time rate of change
Horizontal advection
Divergence
Planetary vorticity advection
Time rate of change
Horizontal advection
Vertical advection
Divergence
Tilting
Planetary vorticity advection
Solenoidal term

9. Assume balance among terms of 10-10s-2

10. 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.
baroclinic – density depends on pressure and temperature.

11. Barotropic Potential Vorticity
We can learn a lot about the atmosphere by considering the barotropic potential vorticity

12. Barotropic Potential Vorticity
Assume constant density
Integrate with height, z1  z2 over a layer of depth H.

13. Remember the Thermal Wind?
p is an independent variable, a coordinate. Hence, x and y derivatives are taken with p constant.

14. Implications of Thermal Wind for a Barotropic Fluid…
Barotropic: temperature is constant on a pressure surface
This means
Geostrophic wind is constant with height in pressure coordinates in a barotropic fluid

15. Barotropic Potential Vorticity

16. What happens if the depth (H) is constant?
Conservation of potential vorticity becomes conservation of absolute vorticity…

17. Barotropic Potential Vorticity
Potential vorticity is a measure of absolute vorticity relative to the depth of the vortex.
What happens if the depth (H) changes?

18. Relative vorticity with change of depth

19. The vortex went over the mountain
Surface with a hill.

20. Vorticity and depth
There is a relationship between depth and vorticity.
As the depth of the vortex changes, the relative vorticity has to change in order to conserve the potential vorticity.
We have now linked the rotational and irrotational components of the wind.
divergence and curl
vorticity and divergence
Potential vorticity indicates an interplay between relative and planetary vorticity through conservation of absolute angular momentum.

21. Let’s explicitly map these ideas to the Earth

22. Local vertical / planetary vorticity

23. relative vorticity/planetary vorticity

24. Compare relative vorticity to planetary vorticity
for large-scale and middle latitudes
planetary vorticity
is usually larger than relative vorticity

25. Relative and planetary vorticity
Planetary vorticity is cyclonic is positive vorticity
Planetary vorticity, in middle latitudes, is usually larger than relative vorticity
A growing cyclone “adds to” the planetary vorticity.
Lows are intense
A growing anticyclone “opposes” the planetary vorticity.
Highs are less intense

26. Compare relative vorticity to planetary vorticity and to divergence
Flow is rotationally dominated, but divergence is crucial to understanding the flow.

27. Return to our simple form of potential vorticity
From scaled equation, with assumption of constant density.

28. Fluid of changing depth
Stretching and shrinking of a column will change the relative vorticity.

29. Application to flow on the Earth

30. What might cause this wave-like flow?

31. Flow over a mountain
Mountain

32. Use our simple form of potential vorticity
From scaled equation, with assumption of constant density and temperature.

33. Flow over a mountain (long in the north-south) (can’t go around the mountain)
west
east

34. Flow over a mountain
Depth, H
Mountain
west
east

35. Flow over a mountain (assume flow is adiabatic)
θ + Δθ
Depth, H θ
Mountain
west
east

36. Flow over a mountain (far upstream constant zonal flow)
θ + Δθ
ζ=0
Depth, H θ
Mountain
west
east

37. Use the barotropic potential vorticity equation
From scaled equation, with assumption of constant density and temperature.

38. What happens as air gets to mountain?
θ + Δθ
ζ=0
Depth, H θ
Mountain
west
east

39. What happens as air gets to mountain?
Air is lifted.
Lifting higher at ground than upper air.
(pressure gradient force spreads it out)
θ + Δθ
ζ=0
Depth, H θ
Mountain
west
east

40. What happens as air gets to mountain?
Air is lifted.
Lifting higher at ground than upper air.
(pressure gradient force spreads it out)
θ + Δθ
ζ=0
Depth, H +ΔH θ
Mountain
west
east

41. What happens as air gets to mountain?
Air is lifted.
Lifting higher at ground than upper air.
(pressure gradient force spreads it out)
θ + Δθ
ζ must increase
Depth, H +ΔH θ
Mountain
west
east

42. What does it mean for the relative vorticity to increase?

43. What happens in these waves?
Loses cyclonic vorticity
Same as gains anticyclonic vorticity
Gains cyclonic vorticity

44. Or schematically
Rotational
Shear
Cyclonic
Anticyclonic

45. What happens as air gets to mountain?
Air turns cyclonically to increase vorticity.
In northern hemisphere turns north.
θ + Δθ
ζ must increase
Depth, H +ΔH θ
Mountain
west
east

46. In the (east-west, north-south) plane
MOUNTAINS
Depth, H
Depth, H +ΔH n s
west
east

47. What happens as air goes over mountain?
Air turns anti-cyclonically to decrease vorticity.
In northern hemisphere turns south.
θ + Δθ
ζ must decrease
Depth, H -ΔH θ
Mountain
west
east

48. In the (east-west, north-south) plane
MOUNTAINS
Depth, H
Depth, H +ΔH
Depth, H -ΔH n s
west
east

49. What happens as air goes down mountain?
Air turns cyclonically to increase vorticity.
In northern hemisphere turns north.
θ + Δθ
ζ must increase
Depth, H +ΔH θ
Mountain
west
east

50. In the (east-west, north-south) plane
MOUNTAINS
Depth, H
Depth, H +ΔH
Depth, H -ΔH
Depth, H +ΔH n
Arrives here with northward momentum “Overshoots”
Stretching here causes relative vorticity to increase; northward turning s
west
east

51. What is happening with planetary vorticity
What is happening with planetary vorticity? (In the (east-west, north-south) plane)
MOUNTAINS
Depth, H
Depth, H +ΔH
Depth, H -ΔH
Depth, H +ΔH n
f is greater for deflections to north
f is less for deflections to south s
west
east

52. What is happening with planetary vorticity
What is happening with planetary vorticity? (In the (east-west, north-south) plane)
MOUNTAINS
Depth, H
Depth, H +ΔH
Depth, H -ΔH
Depth, H +ΔH
Has an excess of potential vorticity; relative vorticity must decrease n
f = f1
ζ = ζ1
H = H1
f = f2 > f1
ζ = ζ2 = ζ1
H = H2 = H1 s
west
east

53. Excursion into the atmosphere

54. “Colorado Lows”

55. What happens if wind is from east?
θ + Δθ θ
Mountain
west
east

56. What is happening with planetary vorticity
What is happening with planetary vorticity? (In the (east-west, north-south) plane)
MOUNTAINS
Depth, H
Depth, H +ΔH
Depth, H -ΔH
Depth, H +ΔH n
Flow from east: planetary and relative vorticity offset each other; no overshoot or undershoot. s
west
east

57. Consider the vertical structure more

58. Where is this flow more barotropic?
10 m/s
5 m/s
20 m/s
30 m/s
B, cooler
A, warmer
- p, vertical
y, north

59. Idealized vertical cross section

60. Vorticity on Large Scales
Remember, vorticity is caused by
Wind shear
Rotation in the flow
Can we identify these on weather maps?
(The following maps come from

61. 300 mb Wind Speed

62. Where is there positive vorticity?

63. 500 mb Vorticity

64. Thermal Wind Remember, thermal wind relates
Vertical shear of geostrophic wind
Horizontal temperature gradients
Can we identify these on weather maps?

65. Where are the strongest ?

66. 850 mb Temperature

67. Convergence/Divergence
Remember, vertical motion on large scales directly related to
Convergence/divergence of ageostrophic wind
Curvature in the flow
Can we identify these on weather maps?

68. Where are surface lows/highs?

69. Surface Precipitation

70. 850 mb Temperature

71. Concepts Vorticity: shear and curvature
Why is curvature vorticity (as opposed to shear vorticity) usually associated with developing low pressure systems?
Divergence and convergence and location of surface high and low pressure systems
Thermal wind—vertical shear of the horizontal wind and horizontal temperature gradients

72. Concepts Features commonly found together Coincidence?
Jet stream
Upper level positive vorticity
Fronts
Midlatitude cyclones (low pressure systems)
Coincidence?
More on this later…

73. Mid-latitude cyclones
What we know:
Low pressure systems
Form through spinup of low-level positive vorticity
Divergence/convergence is key
This is just the beginning…
Always closely associated with fronts—why?
Sometimes develop rapidly, sometimes not at all—why?

74. The mid-latitude cyclone

75. Mid-latitude cyclones: Norwegian Cyclone Model

76. Fronts and Precipitation
Norwegian Cyclone Model
CloudSat Radar

77. Idealized vertical cross section

78. Cold and warm advection

79. Lifting and sinking

80. Increasing the pressure gradient force

81. Almost Weather

82. Mid-latitude cyclones: Norwegian Cyclone Model

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