1. AOSS 401, Fall 2013 Lecture 3 Coriolis Force September 10, 2013
Richard B. Rood (Room 2525, SRB)
Cell:

2. Class News Ctools site (AOSS 401 001 F13) New Postings to site
Syllabus
Lectures
Homework (and solutions)
New Postings to site
Homework 1: Due September 12
Read Greene et al. paper: Discussion Tuesday

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

4. Outline Conservation of Momentum Forces Greene et al. Paper Discussion
Coriolis Force
Centrifugal Force
Greene et al. Paper Discussion
Should be review. So we are going fast.
You have the power to slow me down.

5. Back to Basics: Newton’s Laws of Motion
Law 1: Bodies in motion remain in motion with the same velocity, and bodies at rest remain at rest, unless acted upon by unbalanced forces.
Law 2: The rate of change of momentum of a body with time is equal to the vector sum of all forces acting upon the body and is the same direction.
Law 3: For every action (force) there is and equal and opposite reaction.

6. Newton’s Law of Motion
Where i represents the different types of forces.

7. How do we express the forces?
In general, we assume the existence of an idealized parcel or “particle” of fluid.
We calculate the forces on this idealized parcel.
We take the limit of this parcel being infinitesimally small.
This yields a continuous, as opposed to discrete, expression of the force.
Use the concept of the continuum to extend this notion to the entire fluid domain.

8. Back to basics: A couple of definitions
Newton’s laws assume we have an “inertial” coordinate system; that is, and absolute frame of reference – fixed, absolutely, in space.
Velocity is the change in position of a particle (or parcel). It is a vector and can vary either by a change in magnitude (speed) or direction.

9. Apparent forces: A mathematical approach
Non-inertial, non-absolute coordinate system

10. One coordinate system related to another by:

11. Two coordinate systemsz’
z’ axis is the same as z, and there is rotation of the x’ and y’ axis z y’ y x x’

12. One coordinate system related to another by:
T is time needed to complete rotation.

13. Acceleration (force) in rotating coordinate system
The apparent forces that are proportional to rotation and the velocities in the inertial system (x, y, z) are called the Coriolis forces.
The apparent forces that are proportional to the square of the rotation and position are called centrifugal forces.

14. The importance of rotation
Non-rotating fluid
Rotating fluid

15. Apparent forces: A physical approach
Coriolis Force

16. Apparent forces
With one coordinate system moving relative to the other, we have the velocity of a particle relative to the coordinate system and the velocity of one coordinate system relative to the other.
This velocity of one coordinate system relative to the other leads to apparent forces. They are real, observable forces to the observer in the moving coordinate system.
The apparent forces that are proportional to rotation and the velocities in the inertial system (x,y,z) are called the Coriolis forces.
The apparent forces that are proportional to the square of the rotation and position are called centrifugal forces.

17. Consider angular momentum
We will consider the angular momentum of a “particle” of atmosphere to derive the Coriolis force.

18. Angular momentum?
Like momentum, angular momentum is conserved in the absence of torques (forces) which change the angular momentum.
This comes from considering the conservation of momentum of a body in constant body rotation in the polar coordinate system.
If this seems obscure or is cloudy, need to review a introductory physics text.

19. Angular speed ω
r (radius) v Δv Δθ v

20. What is the appropriate radius of rotation for a parcel on the surface of the Earth?Ω
Direction away from axis of rotation R
Earth

21. Magnitude of R the axis of rotation
R=acos(f) Ω R a
Φ = latitude
Earth

22. Tangential coordinate system
Place a coordinate system on the surface.
x = east – west (longitude)
y = north-south (latitude)
z = local vertical Ω R a Φ
Earth

23. Angle between R and axesΩ Φ R a
Φ = latitude
Earth

24. Assume magnitude of vector in direction RΩ
Vector of magnitude B R a
Φ = latitude
Earth

25. Vertical component Ω
z component = Bcos(f) R a
Φ = latitude
Earth

26. Meridional component Ω R
y component = Bsin(f) a
Φ = latitude
Earth

27. Earth’s angular momentum (1)
What is the speed of this point due only to the rotation of the Earth? Ω R a
Φ = latitude
Earth

28. Earth’s angular momentum (2)
Angular momentum is Ω R a
Φ = latitude
Earth

29. Earth’s angular momentum (3)
Angular momentum due only to rotation of Earth is Ω R a
Φ = latitude
Earth

30. Earth’s angular momentum (4)
Angular momentum due only to rotation of Earth is Ω R a
Φ = latitude
Earth

31. Angular momentum of parcel (1)
Assume there is some x velocity, u. Angular momentum associated with this velocity is Ω R a
Φ = latitude
Earth

32. Total angular momentum
Angular momentum due both to rotation of Earth and relative velocity u is Ω R a
Φ = latitude
Earth

33. Displace parcel south (1) (Conservation of angular momentum)
Let’s imagine we move our parcel of air south (or north). What happens? Δy Ω R a Φ
Earth

34. Displace parcel south (2) (Conservation of angular momentum)
We get some change ΔR Ω R a Φ
Earth

35. Displace parcel south (3) (Conservation of angular momentum)
But if angular momentum is conserved, then u must change. Ω R a Φ
Earth

36. Displace parcel south (4) (Conservation of angular momentum)
Expand right hand side, ignore squares and higher order difference terms.
Expected mathematical knowledge

37. Displace parcel south (5) (Conservation of angular momentum)
For our southward displacement

38. Displace parcel south (6) (Conservation of angular momentum)
Divide by Δt and take the limit
Coriolis term (check with previous mathematical derivation … what is the same? What is different?)

39. Displace parcel south (7) (Conservation of angular momentum)
What’s this? “Curvature or metric term.” It takes into account that y curves, it is defined on the surface of the Earth. More later.
Remember this is ONLY FOR a NORTH-SOUTH displacement.

40. Coriolis Force in Three Dimensions
Do a similar analysis displacing a parcel upwards and displacing a parcel east and west.
This approach of making a small displacement of a parcel, using conversation, and exploring the behavior of the parcel is a common method of analysis.
This usually relies on some sort of series approximation; hence, is implicitly linear. Works when we are looking at continuous limits.

41. Coriolis Force in 3-D

42. Definition of Coriolis parameter (f)
Consider only the horizontal equations (assume w small)
For synoptic-scale systems in middle latitudes (weather) first terms are much larger than the second terms and we have

43. Our momentum equation

44. Highs and Lows Motion initiated by pressure gradient
In Northern Hemisphere velocity is deflected to the right by the Coriolis force
Motion initiated by pressure gradient
Opposed by viscosity

45. Where’s the low pressure?

46. Geostrophic and observed wind 1000 mb (ocean)

47. Do you notice two “types” of motion?
Hurricane Charley
Do you notice two “types” of motion?

48. Our momentum equation Surface Body Apparent
This equation is a statement of conservation of momentum.
We are more than half-way to forming a set of equations that can be used to describe and predict the motion of the atmosphere!
Once we add conservation of mass and energy, we will spend the rest of the course studying what we can learn from these equations.

49. Our momentum equation Surface Body Apparent
Acceleration (change in momentum)
Pressure Gradient Force: Initiates Motion
Friction/Viscosity: Opposes Motion
Gravity: Stratification and buoyancy
Coriolis: Modifies Motion

50. The importance of rotation
Non-rotating fluid
Rotating fluid

51. Discussion of Greene et al. Paper

52. End

53. Apparent forces: A mathematical approach
Non-inertial, non-absolute coordinate system

54. Two coordinate systems
Can describe the velocity and forces (acceleration) in either coordinate system. x y z x’ y’ z’

55. One coordinate system related to another by:

56. Velocity (x’ direction)
So we have the velocity relative to the coordinate system and the velocity of one coordinate system relative to the other.
This velocity of one coordinate system relative to the other leads to apparent forces. They are real, observable forces to the observer in the moving coordinate system.

57. Two coordinate systemsz’
z’ axis is the same as z, and there is rotation of the x’ and y’ axis z y’ y x x’

58. One coordinate system related to another by:
T is time needed to complete rotation.

59. Acceleration (force) in rotating coordinate system
The apparent forces that are proportional to rotation and the velocities in the inertial system (x, y, z) are called the Coriolis forces.
The apparent forces that are proportional to the square of the rotation and position are called centrifugal forces.

60. Derivation of Centrifugal Force

61. Apparent forces: A physical approach

62. Circle Basics
Arc length ≡ s = rθ ω
r (radius) θ
Magnitude
s = rθ

63. Centrifugal force: Treatment from Holtonω
r (radius) Δv Δθ
Magnitude

64. Angular speed ω
r (radius) v Δv Δθ v

65. Centrifugal force: for our purposes

66. Now we are going to think about the Earth
The preceding was a schematic to think about the centrifugal acceleration problem. Note that the r vector above and below are not the same!

67. What direction does the Earth’s centrifugal force point?Ω
Ω2R R
Earth

68. What direction does gravity point?Ω R a
Earth

69. What direction does the Earth’s centrifugal force point?
So there is a component that is in the same coordinate direction as gravity (and local vertical).
And there is a component pointing towards the equator
We are now explicitly considering a coordinate system tangent to the Earth’s surface. Ω
Ω2R R
Earth

70. What direction does the Earth’s centrifugal force point?
So there is a component that is in the same coordinate direction as gravity:
~ aΩ2cos2(f)
And there is a component pointing towards the equator
~ - aΩ2cos(f)sin(f) Ω
Ω2R R
Φ = latitude
Earth

71. So we re-define gravity as

72. What direction does the Earth’s centrifugal force point?
And there is a component pointing towards the equator.
The Earth has bulged to compensate for the equatorward component.
Hence we don’t have to consider the horizontal component explicitly. Ω
Ω2R R
Earth

73. Centrifugal force of Earth
Vertical component incorporated into re-definition of gravity.
Horizontal component does not need to be considered when we consider a coordinate system tangent to the Earth’s surface, because the Earth has bulged to compensate for this force.
Hence, centrifugal force does not appear EXPLICITLY in the equations.

74. That’s all