AOSS 401, Fall 2007 Lecture 13 October 5 , 2007

1. AOSS 401, Fall 2007Lecture 13October 5, 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 • Homework. • Homework due Monday • Answers to last homework are posted on ctools • We will go over the review questions on Friday • Think about them • Exam next Wednesday • Today we will talk about thermal wind and vertical velocity some more • Friday and Monday we will look at the material in different ways and more thoroughly • Also have your questions • Mid-term evaluation • “students will be notified soon thereafter that they can fill out the midterm evaluations between October 8 and October 14”

3. Some information about the exam

4. General Exam Instructions • Logistics • Write name on all pages. • Honor system signature required - see first page after intro material. • Turn in the exam sheets that are handed out. • No books, no calculators, no computers. • Personal notes allowed (~ 2 pages) • Mathematical formulas and identities will be provided. • General instructions. • You may use the provided equation numbers to refer to equations. • Draw pictures. • Show your work. A good start is worth points! • Work from fundamentals. • You have seen these ideas and techniques in your homework problems. • Don’t make the questions more complicated than they are. • Read all questions first. • There are N problems. • Point distribution. Problem#(Points) • P1(n1), P2(n2), ... • Total points for exam = 30

5. p is pressure ρ is density T is temperature α is specific volume u is velocity vector (ui, vj, wk) g is gravity force (assume constant) R is the gas constant for dry air cv is specific heat at constant volume cp is specific heat at constant pressure = (cv + R) a is radius of Earth Ω is angular velocity of Earth’s rotation n is kinematic viscosity coefficient Variables and constants

6. Units

7. Equations of motion(z, height, as vertical coordinate) tangential coordinate system on Earth’s surface (x, y, z) = (+ east, + north, + local vertical)

8. Equations of motion(p, pressure, as a vertical coordinate) tangential coordinate system on Earth’s surface (x, y, p) = (+ east, + north, pressure is vertical coordinate)

9. Mathematical Expressions(for arbitrary x)

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

11. Honor code NAME: ___________________________ signature "I have neither given nor received unauthorized aid on this examination, nor have I concealed any violations of the Honor Code."

12. Types of questions • Usually 6-7 questions • First two will be about definitions, what things are. • This equation represents what physical principle? • This term means? • Second two will be very similar to homework problems, but a little different • Pay attention to the definition of the problem • Last two will require putting together the concepts and tools in different ways.

13. Any questions about the test

14. Material from Chapter 3 • Thermal wind • Vertical Velocity • Review

15. Equations of motion in pressure coordinates(plus hydrostatic and equation of state)

16. Linking thermal field with wind field. • The Thermal Wind

17. Geostrophic wind

18. Hydrostatic Balance

19. Geostrophic wind Take derivative wrt p. Links horizontal temperature gradient with vertical wind gradient.

20. Thermal wind p is an independent variable, a coordinate. Hence, x and y derivatives are taken with p constant.

21. Relation between zonal mean temperature and wind is strong • This is a good diagnostic – an excellent check of consistency of temperature and winds observations. • We see the presence of jet streams in the east-west direction, which are persistent on seasonal time scales. • Is this true in the tropics?

22. Thermal wind

23. Thermal wind

24. Thermal wind

25. Thermal wind ?

26. From Previous LecturesThickness Note link of thermodynamic variables, and similarity to scale heights calculated in idealized atmospheres. Z2-Z1 = ZT ≡ Thickness - is proportional to temperature is often used in weather forecasting to determine, for instance, the rain-snow transition. (We will return to this.)

27. Similarity of the equations There is clearly a relationship between thermal wind and thickness.

28. Schematic of thermal wind. Thickness of layers related to temperature. Causing a tilt of the pressure surfaces. from Brad Muller

29. Another excursion into the atmosphere. X X X 850 hPa surface 300 hPa surface from Brad Muller

30. Another excursion into the atmosphere. X X X 850 hPa surface 300 hPa surface from Brad Muller

31. Another excursion into the atmosphere. 850 hPa surface 300 hPa surface from Brad Muller

32. Another excursion into the atmosphere. 850 hPa surface 300 hPa surface from Brad Muller

33. 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

34. A summary of ideas. • In general, these large-scale, middle latitude dynamical features tilt westward with height. • The way the wind changes direction with altitude is related to the advection of temperature, warming or cooling in the atmosphere below a level. • This is related to the growth and decay of these disturbances. • Lifting and sinking of geopotential surfaces.

35. Some simple links to vertical velocity

36. Equations of motion in pressure coordinates(plus hydrostatic and equation of state)

37. Let’s think about growing and decaying disturbances. Mass continuity equation.

38. Let’s think about growing and decaying disturbances. Formally links vertical wind and divergence.

39. Let’s think about growing and decaying disturbances. Convergence (divergence) of mass into (from) column above the surface will increase (decrease) surface pressure.

40. Another link to vertical velocity

41. Another link to vertical velocity

42. Estimating vertical velocity • First way: Mass continuity, the divergence of the horizontal wind • Errors in estimates of wind often dominate the calculation • Errors generally amplified when you take derivatives • Second way: Temperature field, advection of temperature • Generally easier to measure temperature and to get an estimate of temperature advection

43. Some take away messages

44. Some things that we learned (1) • Organizing structure provided by rotation. • Rotation is less important in the tropics, which is clearly observable in the atmosphere. • There is a theoretical limit on pressure gradients associated with high pressure systems. • Highs tend to be smeared out; they tend to have moderate wind speeds. • There is not such a limit for low pressure systems. • Lows can be very intense; The highest wind speeds are associated with lows.

45. Some things that we learned (2) • There is the possibility of “anomalous” circulations. • Possibility of cyclonic highs • Possibility of anti-cyclonic lows • We can estimate frictional dissipation based on the angle between lines of constant pressure, or height, and the observed wind.

46. Some things that learned (3) • Dynamical features can isolate air and allow the evolution of extraordinary chemical processes.

47. Dynamics is scale dependent • Planetary waves: 107 meters, 10,000 km • Have we seen one of these in our lectures? • Synoptic waves: Our large-scale, middle-latitude, 106 meters, 1000 km • What’s a synoptic wave? What does synoptic mean? • Hurricanes: 105 meters, 100 km • Fronts: 104 meters, 10 km • Cumulonimbus clouds: 103 meters, 1 km • Tornadoes: 102 meters, 0.1 km • Dust devils: 1 - 10 meters

48. Review Problems • The following three problems are good review problems for the test. • We will solve them in class on 10/5

49. 4.1 Review Problem The material derivative is written below in two different coordinate systems. One uses pressure as a vertical coordinate; the other uses height. What are the dependent and independent variables? When taking partial derivatives in one variable, what other parameters are held constant? Write down the definition of w and ω in terms of differentials of the independent variables. What are the units of ω?

50. 4.2 Review Problem Problem 3.3 from Text (Holton, 4th Edition) A tornado rotates with constant angular velocity ω. Show that the surface pressure at the center of the tornado is given by where p0 is the surface pressure at the distance r0 from the center and T is the temperature (assumed constant). If the temperature is 15o C and pressure and wind speed at 100 m from the center are 1000 hPa and 100 m s-1, respectively, what is the central pressure?