AOSS 401, Fall 2007 Lecture 4 September 12 , 2007

1. AOSS 401, Fall 2007Lecture 4September 12, 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 • Posselt office hours: Tues/Thurs AM and after class • If you are coming from outside the building for office hours (central or north campus), please email or call ahead • Class cancelled Friday 14 September • No office hours Thursday 13 September • I will be available during regular class time Friday • Homework 1 due today (Questions?) • Homework 2 posted by the end of the day • Under “resources” in homework folder • Due Monday (September 17, 2007)

3. Weather • NCAR Research Applications Program • http://www.rap.ucar.edu/weather/ • National Weather Service • http://www.nws.noaa.gov/dtx/ • 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 • Momentum equation(s) • Geopotential and atmospheric thickness • Transformation of vertical coordinates • Material Derivative • Lagrangian and Eulerian reference frames • Material (total, substantive) derivative • Mathematical tools needed for Homework 2

5. From last time

6. 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

7. 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.

8. Review: Vertical Structure and Pressure as Vertical Coordinate

9. Vertical Structure and Pressure as a Vertical Coordinate • Remember, we defined the geopotential as • And we were able to use hydrostatic balanceand the ideal gas law to show

10. Vertical Structure and Pressure as a Vertical Coordinate • Integrate from pressure p1 to p2 at heights z1 and z2 • From the definition of geopotential we get thickness and the fact that thickness is proportional to temperature • So, hydrostatic balance and the ideal gas law form the basis for the relationship between and

11. Pressure Gradient in Pressure Coordinates • Remember from Monday: horizontal pressure gradient force in pressure coordinates is the gradient of geopotential • Remember, if we have hydrostatic balance:

12. Pressure Coordinates: Why? • From Holton, p2: “The general set of … equations governing the motion of the atmosphere is extremely complex; no general solutions are known to exist. …it is necessary to develop models based on systematic simplification of the fundamental governing equations.” • Two goals of dynamic meteorology: • Understand atmospheric motions (diagnosis) • Predict future atmospheric motions (prognosis) • Use of pressure coordinates simplifies the equations of motion

13. Pressure Coordinates: Why? Horizontal momentum equations (u, v), no viscosity Height (z) coordinates Pressure (p) coordinates Density is no longer a part of the equations of motion Hidden inside the geopotential… We will see that this simplifies other relationships as well…

14. New Material: Holton Chapter 2 • Lagrangian and Eulerian Points of View • Material (total) derivatives • Review of key mathematical tools

15. Vector Momentum Equation(Conservation of Momentum)

16. Local vertical z k north y east j x i Vector Momentum Equation(Conservation of Momentum) Coordinate system is defined as tangent to the Earth’s surface Have entertained the possibility of several vertical coordinates z, p, … Velocity (u) = (ui + vj + wk)

17. Previously: Conservation of Momentum Now we are going to think about fluids. Consider a fluid parcel moving along some trajectory.

18. Consider a fluid parcel moving along some trajectory (What is the primary force for moving the parcel around?)

19. Consider several trajectories

20. How would we quantify this?

21. Use a position vector that changes in time Parcel position is a function of its starting point. The history of the parcel is known

22. Lagrangian Point of View • This parcel-trajectory point of view, which follows a parcel, is known as the Lagrangian point of view. • Benefits: • Useful for developing theory • Very powerful for visualizing fluid motion • The history of each fluid parcel is known • Problems: • Requires considering a coordinate system for each parcel • How do you account for interactions of parcels with each other? • How do you know about the fluid where there are no parcels? • How do you know about the fluid if all of the parcels bunch together?

23. Lagrangian Movie:Mt. Pinatubo, 1992

24. Consider a fluid parcel moving along some trajectory Could sit in one place and watch parcels go by.

25. How would we quantify this? • In this case: • Our coordinate system does not change • We keep track of information about the atmosphere at a number of (usually regularly spaced) points that are fixed relative to the Earth’s surface

26. Eulerian Point of View • This point of view, where is observer sits at a point and watches the fluid go by, is known as the Eulerian point of view. • Benefits: • Useful for developing theory • Requires considering only one coordinate system for all parcels • Easy to represent interactions of parcels through surface forces • Looks at the fluid as a field. • A value for each point in the field – no gaps or bundles of “information.” • Problems • More difficult to keep track of parcel history—not as useful for applications such as pollutant dispersion…

27. An Eulerian Map

28. Why Consider Two Frames of Reference? • Goal: understanding. Will allow us to derive simpler forms of the governing equations • Basic principles still hold: the fundamental laws of conservation • Momentum • Mass • Energy • are true no matter which reference frame we use

29. Movies Eulerian vs. Lagrangian Eulerian Lagrangian

30. Why Lagrangian? • Lagrangian reference frame leads to the material (total, substantive) derivative • Useful for understanding atmospheric motion and for deriving mass continuity…

31. On to the Material Derivative…

32. y Δy x Δx Material Derivative Consider a parcel with some property of the atmosphere, like temperature (T), that moves some distance in time Δt

33. Higher Order Terms Material Derivative We would like to calculate the change in temperature over time Δt, following the parcel. Expand the change in temperature in a Taylor series around the temperature at the initial position. Assume increments over Δt are small, and ignore Higher Order Terms

34. Material Derivative Divide through by Δt Take the limit for small Δt

35. Material Derivative Introduce the convention of d( )/dt ≡ D( )/Dt This is the material derivative: the rate of change of T following the motion

36. Lagrangian Eulerian Material Derivative Remember, by definition: and the material derivative becomes

37. Material Derivative (Lagrangian) Material derivative, T change following the parcel

38. Local Time Derivative (Eulerian) T change at a fixed point

39. COLD WARM Change Due to Advection Advection

40. A Closer Look at Advection Expanding advection into its components, we have

41. Change Due to Advection Advection

42. Class Exercise: Gradients and Advection • The temperature at a point 50 km north of a station is three degrees C cooler than at the station. • If the wind is blowing from the north at 50 km h-1 and the air is being heated by radiation at the rate of 1 degree C h-1, what is the local temperature change at the station? • Hints: • You should not need a calculator • Use the definition of the material derivative and of advection

43. Material Derivative Can be rewritten in terms of the local change We will use this again later…

44. 1100 UTC 1200 UTC 1300 UTC 1400 UTC 1500 UTC 1600 UTC Advection: A Recent Example Six-hour time temperature change at St. Cloud, MN

45. In the Eulerian reference frame, we have Non-linear This comes from Eulerian point of view Return to the Momentum Equation Remember, we derived from force balances This is in the Lagrangian reference frame

46. Homework 2:Mathematical Tools • Problem 2 in homework 2 asks you to expand various vector operators • A quick review of these follows

47. Gradient: Three-Dimensional Partial Spatial Derivative • A vector operator defined as • The gradient of a scalar (f) is a vector

48. Dot Product • The divergence is the dot product of the gradient with another vector • The dot product of two vectors A and B is

49. Laplacian: Divergence of a Gradient • Three-dimensional partial spatial second derivative. • Since it is a dot-product, it is NOT a vector itself… • The Laplacian of a scalar (f) is

50. Curl (Cross-Product) • The curl will be closely related to rotation—we will use this extensively when we cover vorticity • The result of taking the curl is a vector that is perpendicular (orthogonal) to both of the original vectors • The direction of the resulting vector depends on the order of operations… • We will return to this in more detail later…