AOSS 401, Fall 2007 Lecture 3 September 10 , 2007

1. AOSS 401, Fall 2007Lecture 3September 10, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530 Derek Posselt (Room 2517D, SRB) dposselt@umich.edu 734-936-0502

2. Class News • Ctools site (AOSS 401 001 F07) • A PDF of my Meeting Maker Calendar is Posted. • It has when I am in and out of town. • It has my cell phone number. • When I am out of town, I plan to be available for my Tues-Thursday office hours. • Write or call • Homework has been posted • Under “resources” in homework folder • Due Wednesday (September 12, 2007)

3. Weather • National Weather Service • http://www.nws.noaa.gov/ • Model forecasts: http://www.hpc.ncep.noaa.gov/basicwx/day0-7loop.html • Weather Underground • http://www.wunderground.com/cgi-bin/findweather/getForecast?query=ann+arbor • Model forecasts: http://www.wunderground.com/modelmaps/maps.asp?model=NAM&domain=US

4. Outline • Review • Coriolis Force • Vertical structure and vertical coordinate Should be review. So we are going fast. You have the power to slow us down.

5. From last time

6. Our momentum equation + other forces Now using the text’s convention that the velocity isu= (u, v, w).

7. Apparent forces

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

9. 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’

10. 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.

11. 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.

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

13. 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’

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

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

16. 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.

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

18. What direction does the Earth’s centrifugal force point? Ω Direction away from axis of rotation R Earth

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

20. 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

21. Angle between R and axes Ω Φ R a Φ = latitude Earth

22. Assume magnitude of vector in direction R Ω Vector of magnitude B R a Φ = latitude Earth

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

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

25. 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

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

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

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

29. 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

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

31. 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

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

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

34. Displace parcel south (4)(Conservation of angular momentum) Expand right hand side, ignore squares and higher of difference terms.

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

36. 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?

37. 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.

38. Coriolis Force in Three Dimensions(link to explicit derivation) • 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.

39. Coriolis Force in 3-D So let’s collect together today’s apparent forces.

40. 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

41. Our momentum equation + other forces that are, more often than not, ignored

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

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

44. Vertical StructurePressure as a vertical coordinate

45. Some basics of the atmosphere Troposphere ------------------ ~ 2 Mountain Troposphere ------------------ ~ 1.6 x 10-3 Earth radius Troposphere: depth ~ 1.0 x 104 m This scale analysis tells us that the troposphere is thin relative to the size of the Earth and that mountains extend half way through the troposphere.

46. Pressure altitude Under virtually all conditions pressure (and density) decreases with height. ∂p/∂z < 0. That’s why it is a good vertical coordinate. If ∂p/∂z = 0, then utility as a vertical coordinate falls apart.

47. Use pressure as a vertical coordinate? • What do we need. • Pressure gradient force in pressure coordinates. • Way to express derivatives in pressure coordinates. • Way to express vertical velocity in pressure coordinates.

48. Expressing pressure gradient force

49. Integrate in altitude Pressure at height z is force (weight) of air above height z.

50. Concept of geopotential Define a variable F such that the gradient of F is equal to g. This is called a potential function. We have assumed here that F is a function of only z.