AOSS 401, Fall 2006 Lecture 8 September 24 , 2007

1. AOSS 401, Fall 2006Lecture 8September 24, 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 • Contract with class. • First exam October 10. • Homework 3 is posted. • Due Friday • Solution sets for Homework 1 and 2 are posted.

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 • Vertical Structure Reset • Stability and Instability • Wave motion • Balances • Thermal Wind • Maps

5. Full equations of motion We saw that the first two equations were dominated by the geostrophic balance. What do we do for the vertical motion?

6. Thermodynamic equation(Use the equation of state)

7. Definition of potential temperature This is the temperature a parcel would have if it was moved from some pressure and temperature to the surface. This is Poisson’s equation.

8. This is a very important point. • Even in adiabatic motion, with no external source of heating, if a parcel moves up or down its temperature changes. • What if a parcel moves about a surface of constant pressure?

9. Adiabatic lapse rate. For an adiabatic, hydrostatic atmosphere the temperature decreases with height.

10. Another important point • If the atmosphere is in adiabatic balance, the temperature still changes with height. • Adiabatic does not mean isothermal. It means that there is no external heating or cooling.

11. The parcel method • We are going displace this parcel – move it up and down. • We are going to assume that the pressure adjusts instantaneously; that is, the parcel assumes the pressure of altitude to which it is displaced. • As the parcel is moved its temperature will change according to the adiabatic lapse rate. That is, the motion is without the addition or subtraction of energy. J is zero in the thermodynamic equation.

12. z Parcel cooler than environment Cooler If the parcel moves up and finds itself cooler than the environment then it will sink. (What is its density? larger or smaller?) Warmer

13. z Parcel cooler than environment Cooler If the parcel moves up and finds itself cooler than the environment, then it will sink. (What is its density? larger or smaller?) Warmer

14. z Parcel warmer than environment Cooler If the parcel moves up and finds itself warmer than the environment then it will go up some more. (What is its density? larger or smaller?) Warmer

15. z Parcel cooler than environment Cooler If the parcel moves up and finds itself cooler than the environment, then it will sink. (What is its density? larger or smaller?) This is our first example of “instability” – a perturbation that grows. Warmer

16. Let’s quantify this. Under consideration of T changing with a constant linear slope (or lapse rate).

17. Let’s quantify this. Under consideration of T of parcel changing with the dry adiabatic lapse rate

18. Stable: temperature of parcel cooler than environment.

19. Unstable: temperature of parcel greater than environment.

20. Stability criteria from physical argument

21. Let’s return to the vertical momentum equation

22. What are the scales of the terms? W*U/L Uf g 10-3 10-7 U*U/a 10 10 10-15 10-5

23. What are the scales of the terms? W*U/L Uf g 10-3 10-7 U*U/a 10 10 10-15 10-5

24. Vertical momentum equation  Hydrostatic balance

25. Hydrostatic balance

26. But our parcel experiences an acceleration Assumption of adjustment of pressure.

27. Solve for pressure gradient

28. But our parcel experiences an acceleration

29. Again, our pressure of parcel and environment are the same so

30. So go back to our definitions of temperature and temperature change above

31. Use binomial expansion

32. So go back to our definitions of temperature and temperature change above

33. Ignore terms in z2

34. For stable situation Seek solution of the form

35. For stable situation Seek solution of the form

36. z Parcel cooler than environment Cooler If the parcel moves up and finds itself cooler than the environment then it will sink. (What is its density? larger or smaller?) Warmer

37. Example of such an oscillation

38. For unstable situation Seek solution of the form

39. z Parcel cooler than environment Cooler If the parcel moves up and finds itself cooler than the environment, then it will sink. (What is its density? larger or smaller?) This is our first example of “instability” – a perturbation that grows. Warmer

40. This is our first explicit solution of the wave equation • These are called buoyancy waves or gravity gaves. • The restoring force is gravity, imbalance of density in the fluid. • We extracted an equation through scaling and use of balances. • This is but one type of wave that is supported by the equations of atmospheric dynamics. • Are gravity waves important in the atmosphere?

41. Near adiabatic lapse rate in the troposphere Troposphere ------------------ ~ 2 Mountain Troposphere ------------------ ~ 1.6 x 10-3 Earth radius Troposphere: depth ~ 1.0 x 104 m GTQ: What if we assumed that the atmosphere was constant density? Is there a depth the atmosphere cannot exceed?

42. Looking at the atmosphere • What does the following map tell you?

43. Cooling Warming Forced Ascent/Descent

44. An Eulerian Map

45. Let us return to the horizontal motions

46. Some meteorologist speak • Zonal = east-west • Meridional = north-south • Vertical = up and down

47. What are the scales of the terms? U*W/a U*U/L 10-8 10-4 Uf Wf U*U/a 10-3 10-3 10-6 10-12 10-5

48. What are the scales of the terms? U*W/a U*U/L 10-8 10-4 Uf Wf U*U/a 10-3 10-3 10-6 10-12 10-5 Largest Terms

49. Geostrophic balance Low Pressure High Pressure

50. Atmosphere in balance • Hydrostatic balance • Geostrophic balance • Adiabatic lapse rate • We can use this as a paradigm for thinking about many problems, other atmospheres. Suggests a set of questions for thinking about observations. What is the rotation? How does it compare to acceleration, represented by the spatial and temporal scales?