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

2. Class News Ctools site (AOSS 401 001 F13)
First Examination on October 22, 2013
Second Examination on December 10, 2013
Homework posted:
Ctools Assignments tab
Due Thursday October 10, 2013
Derivations (using notes)

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

4. Outline Geostrophic Balance and the Real Wind Natural Coordinates
Balanced flows
Geostrophic
Cyclostrophic
Gradient

5. Atmosphere in Balance Hydrostatic balance (no vertical acceleration)
Geostrophic balance (no horizontal acceleration or divergence)
Adiabatic lapse rate (no clouds or precipitation)
Vertical motion is weak for large-scale flow.
A good approximation is that flow horizontal.
In the upper atmosphere the flow is often adiabatic.

6. Scale Analysis
Scale analysis is reliant on observations of the preferred motion of the fluid.
What are the size, spatial scale, of the motions?
What are the time scales of the motions?
How did we define large?
For our flow we compared f(rotation) to U/L

7. I present these equations:
Assume no viscosity and no vertical motion
What is vertical coordinate?
Pressure
What conservation law?
Momentum

8. Geostrophic balance Low Pressure High Pressure
Flow initiated by pressure gradient
Flow turned by Coriolis force

9. Geostrophic & observed wind 300 mb

10. Describe previous figure. What do we see?
At upper levels (where friction is negligible) the observed wind is parallel to geopotential height contours.
(On a constant pressure surface)
Wind is faster when height contours are close together.
Wind is slower when height contours are farther apart.
There is curvature in the flow
Hence acceleration

11. Consider this simple “map”
Contours of geopotential
Upper troposphere
Note coordinate system
What is the direction of the geostrophic wind?
Where is it weaker or stronger?

12. Geopotential (Φ) in upper troposphere
ΔΦ > 0
north
Φ0+ΔΦ Φ0
Φ0+2ΔΦ
Φ0+3ΔΦ
south
west
east

13. Geopotential (Φ) in upper troposphere
ΔΦ > 0
north
Φ0+ΔΦ Φ0 Δy
Φ0+2ΔΦ
Φ0+3ΔΦ
south
west
east

14. Geopotential (Φ) in upper troposphere
ΔΦ > 0
north
Φ0+ΔΦ
δΦ = Φ0 – (Φ0+2ΔΦ) Φ0 Δy
Φ0+2ΔΦ
Φ0+3ΔΦ
south
west
east

15. Geopotential (Φ) in upper troposphere
ΔΦ > 0
north
Φ0+ΔΦ Φ0 Δy
Φ0+2ΔΦ
Φ0+3ΔΦ
south
west
east

16. The horizontal momentum equation
Assume no viscosity

17. Geostrophic approximation

18. Geopotential (Φ) in upper troposphere
ΔΦ > 0
north
Φ0+ΔΦ Φ0 Δy
Φ0+2ΔΦ
Φ0+3ΔΦ
south
west
east

19. Geopotential (Φ) in upper troposphere
ΔΦ > 0
north
Φ0+ΔΦ Φ0 Δy
Φ0+2ΔΦ
Φ0+3ΔΦ
south
west
east

20. Think about this for a minute … Geopotential (Φ) in upper troposphere
ΔΦ > 0
north
T gradient
Φ0+3ΔΦ W cf
Φ0+2ΔΦ Δy
Φ0+ΔΦ
pgf Φ0 C
south
west
east
SOUTHERN HEMISPHERE

21. How did we get the wind?
This is the i (east-west, x) component of the geostrophic wind.
We have estimated the derivatives based on finite differences.
Does this seem like a reverse engineering of the methods we used to derive the equations?
There is a consistency
The direction comes out correctly! (towards east)
The strength is proportional to the gradient.

22. north Φ0+ΔΦ Φ0 Φ0+2ΔΦ Φ0+3ΔΦ south west east
The geostrophic wind can only be equal to the real wind if the height contours are straight.
north
Φ0+ΔΦ Φ0
Φ0+2ΔΦ
Φ0+3ΔΦ
south
west
east

23. Geostrophic & observed wind 300 mb

24. Geopotential (Φ) in upper troposphere
Think about the observed wind
Flow is parallel to geopotential height lines
There is curvature in the flow
Where is the effect of curvature here?

25. Geopotential (Φ) in upper troposphere
Think about the observed (upper level) wind
Flow is parallel to geopotential height lines
There is curvature in the flow
Geostrophic balance describes flow parallel to geopotential height lines
Geostrophic balance does not account for curvature
What’s one way to think about this?
Curvature means there is acceleration
How to best describe balanced flow with curvature?

26. Another Coordinate System?
We want to simplify the equations of motion
For horizontal motions on many scales, the atmosphere is in balance
Mass (p, Φ) fields in balance with wind (u)
It is easy to observe the pressure or geopotential height, much more difficult to observe the wind
Need to describe balance between pressure gradient, Coriolis and curvature

27. “Natural” Coordinate System
Follow the flow
From hydrodynamics—assumes no local changes (“steady state”)
No local change in geopotential height
No local change in wind speed or direction
Assume
Horizontal flow only (no vertical component)
No friction
This is like a Lagrangian parcel approach

28. Return to Geopotential (Φ) in upper troposphere
Define one component of the horizontal wind as tangent to the direction of the wind. t
north Φ0 t t t
Φ0+3ΔΦ
south
west
east
ΔΦ > 0

29. How do these natural coordinates relate to the tangential coordinates?Ω
They are still tangential, but the unit vectors do not point west to east and south to north.
The coordinate system turns with the wind.
And if it turns with the wind, what do we expect to happen to the forces? a
Φ = latitude
Earth

30. Looking down from above

31. Looking down from above

32. Looking down from above

33. Looking down from above

34. Looking down from above

35. Return to Geopotential (Φ) in upper troposphere
Define the other component of the horizontal wind as normal to the direction of the wind. n
north Φ0 n n n t t t
Φ0+3ΔΦ
south
west
east
ΔΦ > 0

36. “Natural” Coordinate System
Regardless of position (i,j)
t always points in the direction of flow
n always points perpendicular to the direction of the flow toward the left
Remember the “right hand rule” for vectors? Take k x t to get n
Assume
Pressure as a vertical coordinate
Flow parallel to contours of geopotential height

37. “Natural” Coordinate System
Advantage: We can look at a height (on a pressure surface) and pressure (on a height surface) and estimate the wind.
It is difficult to directly measure winds
We estimate winds from pressure (or hydrostatically equivalent height), a thermodynamic variable.
Natural coordinates are useful for diagnostics and interpretation.

38. “Natural” Coordinate System
For diagnostics and interpretation of flows, we need an equation…

39. Return to Geopotential (Φ) in upper troposphere
north
Low n n n t t t
Geostrophic assumption.
Do you notice that those n vectors point towards something out in the distance?
HIGH
south
west
east
ΔΦ > 0

40. Return to Geopotential (Φ) in upper troposphere
Do you see some notion of a radius of curvature? Sort of like a circle, but NOT a circle.
north
Low n n n t t
HIGH t
south
west
east

41. Time to look at the mathematics
One direction: no (u,v) components
Define velocity as:
Definition of magnitude:
First simplification: the velocity
Always positive
Always points in the positive t direction

42. Goal: Quantify Acceleration
acceleration is:
(Product Rule)
Change in speed
Change in Direction

43. How to get as a function of V, R ?

44. Remember our circle geometry…
Δs=RΔφ
this is not rotation of the Earth!
It is an element of curvature in the flow. Δφ Δt
t+Δt t R=
radius of curvature Δs t

45. Remember our circle geometry…
Δs=RΔφ
this is not rotation of the Earth!
It is an element of curvature in the flow. Δφ Δt
t+Δt n t R=
radius of curvature n Δs t

46. Remember our circle geometry…
Δs=RΔφ
If Δs is very small, Δt is parallel to n.
So, Δt points in the direction of n as Δs  0 Δφ Δt
t+Δt n t R=
radius of curvature n Δs t

47. Remember, we want an
expression for
From circle geometry
we have:
Rearrange and take
the limit
Use the chain rule
Remember the definition
of velocity

48. Goal: Quantify Acceleration
defined as:
(Product Rule)
We just derived:
So the total acceleration is

49. Acceleration in Natural Coordinates
Along-flow
speed change ?

50. Acceleration in Natural Coordinates
The total acceleration is
Definition of
wind speed
angle of rotation
Circle geometry
angular velocity
Plug in for Δs
Centrifugal force

51. Acceleration in Natural Coordinates
Along-flow
speed change
Centrifugal
Acceleration

52. Horizontal Momentum Equation?
Our horizontal momentum equation will consist of three terms
Acceleration
Coriolis force
Pressure gradient force

53. We have seen that Coriolis force is normal to the velocity.

54. Horizontal Pressure Gradient (by definition)

55. The horizontal momentum equation
Tangential Cartesian coordinate system / pressure in vertical
Natural coordinate system for horizontal flow in 2D

56. The horizontal momentum equation (in natural coordinates)
Along-flow
direction (t)
Across-flow
direction (n)

57. Simplification?
Which coordinate system is easier to interpret?
We are only looking at flow parallel to geopotential height contours

58. Simplification?
Which coordinate system is easier to interpret?
We are only looking at flow parallel to geopotential height contours

59. One Diagnostic Equation
Curved flow (Centrifugal Force)
Coriolis
Pressure Gradient

60. Uses of Natural Coordinates
Geostrophic balance
Definition: coriolis and pressure gradient in exact balance.
Parallel to contours  straight line  R becomes infinite

61. Geostrophic balance in natural coordinates

62. north Φ0+ΔΦ Φ0 Φ0+2ΔΦ Δn Φ0+3ΔΦ south west east
Which actually tells us the geostrophic wind can only be equal to the real wind if the height contours are straight.
north
Φ0+ΔΦ Φ0
Φ0+2ΔΦ Δn
Φ0+3ΔΦ
south
west
east

63. Therefore
If the contours are curved then the real wind is not geostrophic.

64. Cyclonic and Anticyclonic
What do these mean?
Cyclonic is flow around a low pressure system.
Anticyclonic is flow around a high pressure system.
These are words to describe rotational flow.

65. How does curvature affect the wind? (cyclonic flow/low pressure)Φ0
Φ0+ΔΦ
Φ0-ΔΦ
HIGH
Low R t n Δn

66. 1. Mathematical Perspective
Equation of motion
Split Coriolis into geostrophic and ageostrophic parts
Use definition of geostrophic wind

67. 1. Mathematical Perspective
Total centrifugal force balances ageostrophic part of coriolis
Look at sign of terms (R > 0)
Total wind is sum of its parts
Real wind speed is slower than geostrophic for cyclonic flow!

68. Add curvature (centrifugal force)
2. Physical Perspective
Geostrophic balance
Add curvature (centrifugal force) V
PGF
COR V
PGF
COR
CEN
>
Pressure gradient force is the same in each case. With curvature less coriolis force is needed to balance the pressure gradient.

69. Geostrophic & observed wind 300 hPa

70. Geostrophic & observed wind 300 hPa
Observed: 95 knots
Geostrophic: 140 knots

71. How does curvature affect the wind? (anticyclonic flow/high pressure)Φ0
Φ0+ΔΦ
Φ0-ΔΦ
HIGH
Low t n Δn R

72. 1. Mathematical Perspective
Total centrifugal force balances ageostrophic part of coriolis
Look at sign of terms (R > 0)
Total wind is sum of its parts
Real wind speed is faster than geostrophic for anticyclonic flow!

73. 2. Physical Perspective < Geostrophic balance Add curvature CEN V
PGF
COR
PGF V
COR
<
Pressure gradient force is the same in each case. With curvature more coriolis force is needed to balance the pressure gradient.

74. Geostrophic & observed wind 300 hPa

75. Geostrophic & observed wind 300 hPa
Observed:30 knots
Geostrophic: 25 knots

76. What did we just do?
Found a way to describe balances between pressure gradient, coriolis, and curvature
We assumed friction was unimportant and only looked at flow at a particular level
We assumed flow was on pressure surfaces
We saw that the simplified system can be used to describe real flows in the atmosphere
Can we describe other flow patterns? (Different scales? Different regions of the Earth?)

77. Summary:“Natural” Coordinates & Balanced Flows
Another perspective on flows
Within meteorological features
For horizontal motions on many scales, the atmosphere is in balance
Mass (p, Φ) fields in balance with wind (u)
It is easy to observe the pressure or geopotential height, much more difficult to observe the wind
Balance provides a way to infer the wind from the observed (p, Φ) / better estimate than geostrophic

78. Calculation / Interpretation: Gradient Wind Balance
For homework

79. Gradient Flow (Momentum equation in natural coordinates)
Balance in the normal, as opposed to tangential, component of the momentum equation
Balance between pressure gradient, coriolis, and centrifugal force

80. Gradient Flow (Momentum equation in natural coordinates)

81. Gradient Flow (Momentum equation in natural coordinates)
Look for real and non-negative solutions for V

82. Gradient Flow Solution (V) must be real and non-negative 8 Possible Solutions
Anomalous Low
Regular Low
Anomalous High
Regular High

83. Gradient Flow (Solutions for Lows, remember that square root.)
Pressure gradient force
NORMAL
ANOMALOUS
Low
Low V n n V
Coriolis Force
Centrifugal force

84. Gradient Flow (Solutions for Highs, remember that square root.)
Pressure gradient force
NORMAL
ANOMALOUS
High
High V n n V
Coriolis Force
Centrifugal force

85. Gradient Flow: Implications Solution (V) must be real

86. Gradient Flow Regular High and Low
Definition of normal, n, direction
Low
High
R > 0
R < 0 n n

87. Gradient Flow: Implications Solution must be real
Always satisfied
High
∂Φ/∂n < 0
R < 0
Trouble!
pressure gradient MUST go to zero faster than R goes to zero