AOSS 401 Geophysical Fluid Dynamics: Atmospheric Dynamics Prepared: 20130924 Scale Analysis

1. AOSS 401Geophysical Fluid Dynamics:Atmospheric DynamicsPrepared: 20130924Scale Analysis 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) • Disruption of schedule • Lectures for missed classes posted. • Class will be remote on Tuesday, September 24th • gotomeeting / • https://www3.gotomeeting.com/join/286347606 • Dial in 1 866 692 4541 // 3066034 • Homework posted on 18th due 26th • Paragraphs on paper due Tuesday Sept 24th

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

4. Outline • Scale analysis • Introduction to time scales • Scaling the horizontal momentum equations • Geostrophic balance

5. Time scales • Think about this: • Given: Rotation of Earth is important to our perception of motion • Given: We have an object moving through the atmosphere • Airplane • Baseball

6. Time scales • What is the natural time scale for the Earth’s rotation?

7. Time scales • Angular velocity of the Earth • Which says that the Earth’s surface moves through the circumference (at some latitude) in 24 hours.

8. 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’

9. Tangential coordinate system Ω Place a coordinate system on the surface. x = east – west (longitude) y = north-south (latitude) z = local vertical R a Φ The component of the angular velocity in the local vertical. Earth

10. Time scales • How do you figure out a time scale for the object in the atmosphere? • But first … what were the units of the angular velocity of the Earth? • What physical quantity does this represent?

11. Time scales • Back to the object in the atmosphere – • Airplane • How far from Denver to Detroit? • 1000 miles • How fast does an airplane fly? • 500 miles per hour • How long does it take to fly? • 2 hours • How do I compare this with the rotation of the Earth?

12. Time scales • Back to the object in the atmosphere – • Airplane • How do I compare this with the rotation of the Earth? • If this time scale is approximately the same size as the Earth’s rotation, then the rotation might be important to analyzing the motion. • What if something moves “fast?” - or “slow?” • Does the airplane pilot worry about the rotation of the Earth? • Does the baseball pitcher worry about the rotation of the Earth? (Why does the ball curve?)

13. Quantitative Comparison If something moves “fast?” The U/L is large. U/L large relative to f. Rotation is not very important. If something moves “slow,” then U/L is small and rotation might be very important.

14. Quantitative Comparison To make a comparison like this form a ratio and ask how that ratio compares to the number one.

15. Large scale If the Rossby number is comparable to or smaller than 1, then rotation is important and we say that the flow is “large-scale.”

16. Typical numbers In atmosphere a typical velocity above the ground might be 20 m/sec. The distance from the peak to a trough of a mid-latitude wave might be 1000 km. The angular velocity is 7.3 x 10-5 / sec. A representative velocity of the oceanic Gulf Stream might be 100 cm/sec, what is the length scale that would mean that rotation is important? A baseball pitcher throws the ball about 90 feet at about 90 miles / hour, is rotation important? For atmospheric example order 1/7. For oceanic example order 100/(2*7.3x10-5 L) ~ 1. Solve for L in cm

17. Some other numbers / redux

18. Some basics of Earth’s atmosphere atmosphere: depth ~ 1.0 x 105 m Mountain: height ~ 5.0 x 103 m Ocean Land Biosphere Earth: radius ≡ a = 6.37 x 106 m

19. Some basics of Earth’s atmosphere Troposphere ------------------ ~ 2 Mountain Troposphere ------------------ ~ 1.6 x 10-3 Earth radius Troposphere: depth ~ 1.0 x 104 m 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.

20. Take it the atmosphere

21. Atmospheric Momentum Equations

22. Scale Analysis • Scale analysis is reliant on observations of the preferred motion of the fluid. • What are the size, spatial scale, of the motions? • Intuitively, what might make something “large” on Earth? What are the special things about Earth that impact the motion? • What are the time scales of the motions?

23. Consider x and y components of the momentum equations Remember the units—each term must have units of acceleration m/sec2 or L/t2

24. How do we determine a time scale? Take a “typical” trajectory and ask “how far does a parcel go in a “characteristic time?”

25. Would like to define scales in terms of wind, pressure, and density distance = rate x time Estimate time as distance/(some average rate) L ≡ some characteristic distance U ≡ some characteristic speed Characteristic time ≡ L/U

26. Still, how to get a time scale? • But what is a characteristic time? • Look at the flow organization. • If a rotating cyclone, could be how long the parcel takes to go around the cyclone.

27. Hurricane Charley Do you notice two “types” of motion? Two time scales?

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

29. Would like to define scales in terms of wind, pressure, and density distance = rate x time Estimate time as distance/(some average rate) L ≡ some characteristic distance U ≡ some characteristic speed Characteristic time ≡ L/U

30. So for acceleration D ( )/Dt can be characterized by 1/(L/U)

31. Scale Analysis • Remember, we want to solve the system of equations that describes the atmosphere so that we can • understand how the atmosphere works • predict the motion and state of the atmosphere • Scale analysis  simplify the equations • Identify which processes are most important

32. Consider x and y components of the momentum equations Remember the units—each term must have units of acceleration m/sec2 or L/t2

33. acceleration Let us define:

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

35. Go to the board? • Write out the scales for the terms in the horizontal momentum equation.

36. U*W/a U*U/L Uf Wf U*U/a What are the scales of the terms?Horizontal momentum equations:

37. What are the scales of the terms? For “large-scale” mid-latitude

38. What are the scales of the terms? For “large-scale” mid-latitude Largest Terms

39. Consider only the largest terms • This is a dominant balance between the • pressure gradient force • Coriolis force (the dominant sin() components of the Coriolis force) This is the geostrophic balance.

40. Consider only the largest terms Note: There is no D( )/Dt term. Hence, no acceleration, no change with time. This is a balance. This is the geostrophic balance.

41. Geostrophic balance Check out: http://itg1.meteor.wisc.edu/wxwise/AckermanKnox/chap6/balanced_flow.html Pressure gradientforce (PGF) Low Pressure Coriolis force(Northern Hemisphere) High Pressure In Northern Hemisphere: Coriolis force points to the right(perpendicular) relative to the path of the air parcel

42. Geostrophic Wind Component form: Vector form: with the Coriolis parameter f = 2 sin Note: There is no D( )/Dt term. Hence, no acceleration, no change with time. The geostrophic wind describes the dominant balance between the pressure gradient force and the Coriois force.

43. Geostrophic wind 300 mb

44. Geostrophic & observed wind 300 mb

45. Geostrophic and observed wind 1000 mb (land)

46. Geostrophic and observed wind 1000 mb (ocean)

47. Geostrophic wind • The direction of the geostrophic wind is parallel to the isobars. • The geostrophic wind vg is a good approximation of the real horizontal wind vector, especially over oceans and at upper levels. Why? • The closer the isobars are together, the stronger the magnitude of the geostrophic wind | vg | (isotachs increase).

48. Next • Vertical Momentum Equation