AOSS 401, Fall 2013 Lecture 3 Coriolis Force September 10 , 2013

1. AOSS 401, Fall 2013Lecture 3Coriolis ForceSeptember 10, 2013 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Cell: 301-526-8572

2. Class News • Ctools site (AOSS 401 001 F13) • 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 • Model forecasts: • Weather Underground • Model forecasts: • NCAR Research Applications Program

4. Outline • Conservation of Momentum • Forces • 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 systems z’ axis is the same as z, and there is rotation of the x’ and y’ axis z’ 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 • http://climateknowledge.org/AOSS_401_Animations_figures/AOSS401_nonrot_MIT.mpg • Rotating fluid • http://climateknowledge.org/AOSS_401_Animations_figures/AOSS401_rotating_MIT.mpg

15. Apparent forces:A physical approach • Coriolis Force • http://climateknowledge.org/figures/AOSS401_coriolis.mov

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 v r (radius) 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 axes Ω Φ R a Φ = latitude Earth

24. Assume magnitude of vector in direction R Ω 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 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. 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. Surface Body Apparent Acceleration (change in momentum) Coriolis: Modifies Motion Friction/Viscosity: Opposes Motion Pressure Gradient Force: Initiates Motion Gravity: Stratification and buoyancy Our momentum equation

50. The importance of rotation • Non-rotating fluid • http://climateknowledge.org/AOSS_401_Animations_figures/AOSS401_nonrot_MIT.mpg • Rotating fluid • http://climateknowledge.org/AOSS_401_Animations_figures/AOSS401_rotating_MIT.mpg