1. Richard B. Rood (Room 2525, SRB)
AOSS 401 Geophysical Fluid Dynamics: Atmospheric Dynamics Prepared: Conservation, Energy, Wave Equation
Richard B. Rood (Room 2525, SRB)
Cell:

2. Class News Ctools site (AOSS 401 001 F13)
First Examination on October 22, 2013
Second Examination on December 10, 2013
Homework will be posted later today.

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

4. Outline Conservation Equation Conservation of Energy
Full Equations of Motion
Buoyancy / Wave Equation

5. Simple Physics
Newtonian physics, Newton’s laws of motion applied to the atmosphere.
F = ma
a = dv/dt
v = dx/dt
Simple physics, but a complex problem.

6. Conservation (continuity) principle
There are certain parameters, for example, momentum, mass (air, water, ozone, number of atoms, … ), energy, that are conserved.
This is “classical” physics
Simple stuff, like billiard balls hitting each other, ice melting
Conserved? That means that in an isolated system that the parameter remains constant; it’s not created; it’s not destroyed.
Isolated system? A collection of things, described by the parameter, that might interact with each other, but does not interact with other things. That is, nothing new comes into the system – nothing leaves.
If you put a boundary around it then nothing crosses the boundary
Is the Earth an isolated system?
This is important to think about because it strongly constrains what can and cannot happen. There is just so much energy coming into the earth system, and it has to be conserved in the case of a stable climate. It will be slightly out of balance when the climate is warming (holding more energy) or cooling (losing energy). Ultimately, in the climate problem we will require a more detailed examination … namely we are talking about warming (or cooling) at the surface. Conservation will require cooling (or warming) elsewhere in the system, because ultimately, the earth needs to get rid of its energy.

7. Conservation principle
There are many other things in the world that we can think of as conserved. For example, money.
We have the money that we have.
If we don’t spend money or make money, then the money we have today is the same as the money we had yesterday.
Mtoday = Myesterday
That’s not very interesting, or realistic

8. Conservation principle (with “production” and “loss”)
Income
Mtoday = Myesterday + I - E
This is the balance equation, basic conservation. Changes come from production and loss.
Let’s get some money and buy stuff.
Expense

9. Conservation principle (with the notion of time)
Income
Mtoday = Myesterday + N(I – E)
Salary
Income per month = I
Rent
Expense per month = E
N = number of months
I = NxI and E= NxE
This is a meaningful representation of production and loss because it brings in the idea of there being a rate, a time dependency, to the production and loss. If it is just constant, it runs away with time.
Expense

10. Some algebra and some thinking
Mtoday = Myesterday + N(I – E)
Rewrite the equation to represent the difference in money
(Mtoday - Myesterday ) = N(I – E)
This difference will get more positive or more negative as time goes on.
Saving money or going into debt.
Divide by the number of months. Now we are headed to a rate in Money and a rate in Time (N). This is essentially a time derivative.
Divide both sides by N, to get some notion of how difference changes with time.
(Mtoday - Myesterday )/N = I – E

11. Conservation principle
dM/dt = Income (per time) – Expense (per time)
What we have done is a dimensional analysis
Start out with money (dollars)
Then we introduced the idea of time
Income per month
Expense per month
Then we divided by time to get some idea of how things change with time

12. Conservation principle (with “production” and “loss”)
Income
Mtoday = Myesterday + I - E
This is the balance equation, basic conservation. Changes come from production and loss.
Let’s get some money and buy stuff.
Expense

13. Conservation: Climate
Energy from the Sun
Stable Temperature of Earth could change from how much energy (production) comes from the sun, or by changing how we emit energy.
Earth at a certain temperature, T
Energy emitted by Earth
(proportional to T)

14. Let’s come back to our class

15. In our Eulerian point of view our system is our parcel, a fixed volume, and the fluid flows through it.
Production
“Income” Dz Dy
Loss
“Expense” Dx

16. Previously: Momentum & Mass
We have u, v, w, ρ, p which depend on (x, y, z, t). We need one more equation for the time rate of change of pressure…

17. Closing Our System of Equations
We have formed equations to predict changes in motion (conservation of momentum) and density (conservation of mass)
We need one more equation to describe either the time rate of change of pressure or temperature (they are linked through the ideal gas law)
Conservation of energy is the basic principle

18. Conservation of Energy: The thermodynamic equation
First law of thermodynamics:
Change in internal energy is equal to the difference between the heat added to the system and the work done by the system.
Internal energy is due to the kinetic energy of the molecules (temperature)
Total thermodynamic energy is the internal energy plus the energy due to the parcel moving

19. Mathematical Form
The mathematical representation follows from the first law of thermodynamics which is stated as: The increase in internal energy of a closed system is equal to the difference of the heat supplied to the system and the work done by it.
Many meteorological texts start with the enthalpy form of the first law of thermodynamics, where enthalpy is the total energy of a thermodynamic system. For a closed system this is written as
Where h is enthalpy, s is entropy and other variables as previously defined.

20. Thermodynamic Equation For a Moving Parcel
J represents sources or sinks of energy.
radiation
latent heat release (condensation/evaporation, etc)
thermal conductivity
frictional heating.
cv = 717 J K-1 kg-1, cvT = a measure of internal energy
specific heat of dry air at constant volume
amount of energy needed to raise one kg air one degree Kelvin if the volume stays constant.

21. Thermodynamic Equation
Involves specific heat at constant volume
Remember the material derivative form of the continuity equation
Following the motion, divergence leads to a change in volume
Reformulate the energy equation in terms of specific heat at constant pressure

22. Another form of the Thermodynamic Equation
Short derivation
Take the material derivative of the equation of state
Use the product rule and the fact that R=cp-cv
Substitute in from the thermodynamic energy equation in Holton
Leads to a prognostic equation for the material change in temperature at constant pressure

23. (Material derivative)
(Ideal gas law)
(Material derivative)
(Chain Rule)
(Use R=cp-cv)
Substitute in from the thermodynamic energy equation (Holton, pp )
(Cancel terms)

24. Thermodynamic equation
Prognostic equation that describes the change in temperature with time
In combination with the ideal gas law (equation of state) the set of predictive equations is complete

25. Atmospheric Predictive Equations
Momentum
Mass
Thermodynamic / Energy
Equation of State

26. Let’s do a problem Let’s explore the vertical velocity.
Solve for a class of motion.

27. Vertical momentum equation  scaling  Hydrostatic balance

28. Consider the vertical structure some more.
Hydrostatic
Eq. of State

29. Consider the vertical structure some more
Consider the vertical structure some more. Assume T as constant - (Isothermal)
Under special consideration of T as constant. (Isothermal)

30. Under special consideration of T as constant. (Isothermal)

31. Scale height for isothermal atmosphere
Units [R] = J/(kg*K)=kg*m*m/(s*s*kg*K), [T] = K
[RT]=m*m/(s*s) [RT/g]=m (unit of length)
This gives a number on order of 7-8 km.
Under special consideration of T as constant. (Isothermal)

32. Pressure altitude
Exponential decrease with height

33. Explore the vertical structure some more
Now let’s consider a non-isothermal atmosphere
What’s the next approximation to temperature that might make sense?
Linear decrease in temperature.

34. Consider a linear vertical temperature structure.
Under special consideration of T changing with a constant linear slope (or lapse rate).

35. Pressure for atmosphere with linear lapse rate
Under special consideration of T changing with a constant lapse rate. Or a linear slope.

36. Pressure for atmosphere with linear lapse rate
g positive, T decreases with height, pressure decreases with height.
g negative, T increases with height, pressure decreases with height.
Under special consideration of T changing with a constant lapse rate. Or a linear slope.

37. Consider a parcel in an atmosphere with a linear lapse rate

38. Let’s return to our linear lapse rate.
Under special consideration of T changing with a constant linear slope (or lapse rate).

39. Temperature as function of heightz
Cooler z
∂T/∂z is defined as lapse rate T
Warmer

40. 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.

41. Parcel cooler than environmentz
Cooler
If the parcel moves up and finds itself cooler than the environment, what happens?
What is its density? larger or smaller?
It will sink.
Warmer

42. Parcel warmer than environmentz
Cooler
If the parcel moves up and finds itself warmer than the environment, what happens.
What is its density? larger or smaller?
It rises.
Warmer

43. Parcel warmer than environmentz
Cooler
If the parcel moves up and finds itself warmer than the environment then it will go up some more.
This is our first example of “instability” – a perturbation that grows.
Warmer

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

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

46. Stable: temperature of parcel cooler than environment.

47. Unstable: temperature of parcel greater than environment.

48. Stability criteria from physical argument

49. Let’s return to the vertical momentum equation

50. We have an environment that looks like thisz
Cooler z
∂T/∂z is defined as lapse rate T
Warmer

51. Hydrostatic balance

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

53. Solve for pressure gradient

54. But our parcel experiences an acceleration

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

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

57. Use binomial expansion

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

59. Ignore terms in z2

60. For stable situation
Seek solution of the form

61. For stable situation
Seek solution of the form

62. Parcel cooler than environmentz
Cooler
If the parcel moves up and finds itself cooler than the environment then it will sink.
Warmer

63. Example of such an oscillation

64. Waves in my backyard (1)

65. Waves in my backyard (2)

66. Waves in my backyard (3)

67. For unstable situation
Seek solution of the form

68. Parcel warmer than environmentz
Cooler
If the parcel moves up and finds itself warmer than the environment then it will go up some more.
This is our first example of “instability” – a perturbation that grows.
Warmer

69. Motion in presence of balance
We scaled the momentum equations and found that vertical accelerations are in general very small.
Then we “allowed” a vertical acceleration in a hydrostatic atmosphere, and found the possibility of vertical motions.
Is this inconsistent in some way?

70. Skills we have used Scale analysis Drawing a picture
Decomposed into mean and deviation
Assumed functional form of temperature – a model
Series approximation
Wave-like solution
Substitution of assumed form of solution
Multiple perspectives on the same problem