1. AOSS 401, Fall 2013 Lecture 4 Material Derivative September 12, 2013
Richard B. Rood (Room 2525, SRB)
Cell:

2. Class News Ctools site (AOSS 401 001 F13) New Postings to site
Syllabus
Lectures
Homework (and solutions)
New Postings to site
Homework 1 due today
Greene et al. paper: Submit paragraph by Tuesday
Francis et al. paper added
Discuss on Thursday September 18
Paragraph due on Tuesday September 23

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

4. Outline Questions about homework? Material Derivative
Should be review. So we are going fast.
You have the power to slow me down.

5. Newton’s Law of Motion
Where i represents the different types of forces.

6. 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.

7. Our momentum equation Surface Body Apparent
Acceleration (change in momentum)
Pressure Gradient Force: Initiates Motion
Friction/Viscosity: Opposes Motion
Gravity: Stratification and buoyancy
Coriolis: Modifies Motion

8. The importance of rotation
Non-rotating fluid
Rotating fluid

9. Consider some parameter, like temperature, T
Total variations
Consider some parameter, like temperature, T Δx x y Δy
(x0,y0)

10. Taylor series expansion
It is sometimes convenient to estimate the value of a continuous function f(x) about a point x = x0 with a power series of the form:
In the last approximation, we neglected the higher order terms

11. If we move a parcel in time Δt
Using Taylor series expansion
Higher
Order
Terms
Assume increments over Δt are small, and
ignore Higher Order Terms

12. Assume increments over Δt are small
Total derivative
Total differential/derivative of the temperature T, T depends on t, x, y, z
Assume increments over Δt are small

13. Total Derivative
Divide by Δt
Take limit for small Δt

14. Total Derivative Introduction of convention of d( )/dt ≡ D( )/Dt
This is done for clarity.
By definition:
u,v,w: these are the velocities

15. Definition of the Total Derivative
The total derivative is also called material derivative.
describes a ‘Lagrangian viewpoint’
describes an ‘Eulerian viewpoint’

16. In class problem
In the absence of pressure forces and viscosity the equations of motion for purely horizontal motion are:
Assume that f, the Coriolis parameter, is constant. Calculate the velocity of an atmospheric parcel as a function of position (x,y).

17. Vector Momentum Equation (Conservation of Momentum)

18. Vector Momentum Equation (Conservation of Momentum)
Coordinate system is defined as tangent to the Earth’s surface x i y j z k
east
north
Local vertical
Velocity (u) = (ui + vj + wk)

19. Tangential coordinates

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

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

22. Consider several trajectories

23. How would we quantify this?

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

25. 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:
Need to keep track of a coordinate system for each parcel
How do you account for interactions between different parcels?
What can we say about the fluid where there are no parcels?
What can we say about the fluid as a whole if all of the parcels bunch together?

26. Lagrangian point of view: Eruption of Mount Pinatubo
Trajectories trace the motion of individual fluid parcels over a finite time interval
Volcanic eruption in 1991 injected particles into the tropical stratosphere (at N, E)
The particles got transported by the atmospheric flow, we can follow their trajectories
Mt. Pinatubo, NASA animation
Colors in animation reflect the atmospheric height of the particles. Red is high, blue closer to the surface.
This is a Lagrangian view of transport processes.

27. Lagrangian Movie: Mt. Pinatubo, 1992
Link to movie on server …. Clicking figure fails on Apple!

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

29. 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

30. Eulerian Point of View
This point of view, where an 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 (or clouds?)…

31. Zonally averaged circulation
Zonal-mean annual-mean zonal wind
Pressure (hPa)

32. An Eulerian Map

33. Consider some parameter, like temperature, Tx y
Material derivative, T change following the parcel

34. Consider some parameter, like temperature, Tx y
Local T change at a fixed point

35. Consider some parameter, like temperature, Tx y
Advection

36. Temperature advection term

37. Consider some parameter, like temperature, Tx y

38. Temperature advection term

39. Advection of cold or warm air
Temperature advection:
Imagine the isotherms are oriented in the E-W direction
Draw the horizontal temperature gradient vector!
pure west wind u > 0, v=0, w=0: Is there temperature advection? If yes, is it cold or warm air advection?
warm u y X
cold

40. Advection of cold or warm air
Temperature advection:
Imagine the isotherms are oriented in the E-W direction
Draw the gradient of the temperature (vector)!
pure south wind v > 0, u=0, w=0: Is there temperature advection? If yes, is it cold or warm air advection?
cold v y
warm X

41. Advection of cold or warm air
Temperature advection:
Imagine the isotherms are oriented as
Draw the horizontal temperature gradient!
pure west wind u > 0, v=0, w=0: Is there temperature advection? If yes, is it cold or warm air advection?
cold u
warm y X

42. Summary: Local Changes & Material Derivative
Local change at a fixed location
Advection term
Total change along a trajectory

43. Summary: For 2D horizontal flows
with horizontal wind vector and
horizontal gradient operator

44. Conservation and Steady-State

45. 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

46. Movies Eulerian vs. Lagrangian
Link to movie on server …. Clicking figure fails on Apple!

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

48. On to the Material Derivative…

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

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

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

52. Change Due to Advection
COLD
WARM

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

54. Change Due to Advection

55. 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

56. Class Exercise: Gradients and Advection
Material Derivative of T
Write in terms of local derivative
No east-west or vertical velocity
Plug in values

57. Class Exercise: Gradients and Advection
Material Derivative of T
Write in terms of local derivative
No east-west or vertical velocity
Plug in values

58. Class Exercise: Gradients and Advection
Material Derivative of T
Write in terms of local derivative
No east-west or vertical velocity
Plug in values

59. Class Exercise: Gradients and Advection
Material Derivative of T
Write in terms of local derivative
No east-west or vertical velocity
Plug in values

60. Written in terms of the local change leads to advection
Material Derivative
Written in terms of the local change leads to advection
We will use this again later…

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

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

63. That’s all