1. Climate Modeling Theory - 1
Module 3
Climate Modeling Theory - 1
William J. Gutowski, Jr.
Iowa State University

2. Climate Modeling Theory - 1
Module 3
Climate Modeling Theory - 1
GOAL:
Understand basis for modeling climate from (almost) first principles

3. Climate Modeling Theory - 1
Module 3
Climate Modeling Theory - 1
OUTLINE (Part 1):
Symbolism
Conservation Laws
mass
thermodynamic energy
momentum
Equation of State
Water in the Atmosphere

4. Climate Modeling Theory - 1
Module 3
Climate Modeling Theory - 1
OUTLINE (Part 1):
Symbolism

5. Some Symbolism
t time
x west-east coordinate
y south-north coordinate
z vertical coordinate
f latitude
l longitude
horizontal wind
u west-east component of
v south-north component of
w vertical wind

6. Climate Modeling Theory - 1
Module 3
Climate Modeling Theory - 1
OUTLINE (Part 1):
Symbolism
Conservation Laws
mass
thermodynamic energy
momentum

7. Conservation
of “M”

8. Conservation
of “M”

9. Conservation
of “M”

10. Conservation
of “M”

11. Conservation
of “M”

12. (Continuity Equation)
Conservation of Mass
(Continuity Equation)
= density [kg/m3]
Source/sink = 0

13. Conservation
of Mass

14. Conservation of Water Mass
q = specific humidity
[kg-(H2O)v/kg-air]
s(q) = cond. - evap.

15. Conservation of Water Mass
(column integral)
E = sfc. evap.; P = precipitation

16. Conservation
of W
(Precipitable Water)

17. Conservation of General Constituent, i [kg-(constituent i)/kg-air]
qi = amount of i
[kg-(constituent i)/kg-air]
e.g., CO2, O3, etc.

18. = heating/mass [J-kg-1-s-1] Conservation of Thermodynamic Energy
~ First Law of Thermodynamics ~
Heat input = D (internal energy) + (work done)
= heating/mass
[J-kg-1-s-1]

19. Conservation of Thermodynamic Energy ~ First Law of Thermodynamics ~

20. Conservation of
Thermodynamic
Energy

21. RNET
FSH
RNET

22. Conservation of Momentum
~ Newton’s Second Law ~

23. Conservation of Momentum
~ Newton’s Second Law ~
Forces/mass:
gravity
pressure gradient
friction

24. Conservation of Momentum
~ Newton’s Second Law ~
Rotating Frame X

25. Conservation of Momentum
~ Newton’s Second Law ~
Rotating Frame

26. Conservation of Momentum
~ Newton’s Second Law ~
Sphere, Rotating Frame
rotation of direction

27. Conservation of Momentum Approximation: vertical
~ Newton’s Second Law ~
Approximation: vertical

28. Conservation of Momentum Approximation: vertical
~ Newton’s Second Law ~
Approximation: vertical

29. Conservation of Momentum Approximation: vertical
~ Newton’s Second Law ~
Approximation: vertical
Hydrostatic Approximation
Accurate to ~ 0.01% for weak vertical acceleration

30. Conservation of Momentum Approximation: horizontal, extratropical
~ Newton’s Second Law ~
Approximation: horizontal, extratropical

31. Conservation of Momentum Approximation: horizontal, extratropical
~ Newton’s Second Law ~
Approximation: horizontal, extratropical

32. Conservation of Momentum Approximation: horizontal, extratropical
~ Newton’s Second Law ~
Approximation: horizontal, extratropical
Geostrophic Approximation
Accurate to ~ %

33. Climate Modeling Theory - 1
Module 3
Climate Modeling Theory - 1
BREAK

34. Climate Modeling Theory - 1
Module 3
Climate Modeling Theory - 1
OUTLINE (Part 1):
Symbolism
Conservation Laws
mass
thermodynamic energy
momentum
Equation of State

35. Ideal Gas Law
R = gas constant
R = R(constituents)
Common practice:
R ≈ Rd = 287 J-kg-1-s-1
T = Tv

36. Climate Modeling Theory - 1
Module 3
Climate Modeling Theory - 1
OUTLINE (Part 1):
Symbolism
Conservation Laws
mass
thermodynamic energy
momentum
Equation of State
Water in the Atmosphere

37. q versus latitude & pressure
[g-kg-1]
Note: small part of atmosphere, but ...

38. … water
saturates
changes phase

39. Some Further Symbolism
q specific humidity [kg-kg-1]
mass (H2O)v/mass air
e vapor pressure [Pa]
partial pressure by water molecules
m mixing ratio [kg-kg-1]
mass (H2O)v/mass dry air
RH relative humidity [%]
ratio: m/msat

40. Water Cycle Q Q P P E E R

41. Water Cycle
Heat released E
Heat absorbed

42. Water is thus a primary form of heat transport
heat absorbed when evaporates
released when water condenses
largest individual source of energy
for the atmosphere

43. Water Cycle
Radiation absorbed by water & re-emitted

44. Water is thus a primary form of heat transport
heat absorbed when evaporates
released when water condenses
largest individual source of energy
for the atmosphere
and greenhouse gas
~ transparent to solar
absorbs/emits infrared

45. RH vs. latitude & pressure
[%] RH 70 70

46. precipitation vs. latitude & longitude
[dm-yr-1]
[dm-yr-1] = [100 mm-yr-1] =[0.27 mm-d-1]

47. Lift Moist Parcel z
9.8 K/km T

48. Lift Moist Parcel z T z RH

49. Lift Moist Parcel z
Lifting Condensation Level T z
LCL RH
100 %

50. Stable Precipitation
condensation
collision
coalescence

51. Stable Precipitation
condensation
collision
coalescence

52. Lift Further z
LCL T

53. Lift Further z
Environment’s T(z) T

54. Lift Further z
Environment’s T(z) T

55. Lift Further
Level of free convection z T

56. Convection
Level of free convection z T

57. Climate Modeling Theory - 1
Module 3
Climate Modeling Theory - 1
Final Question:
How much heating by condensation?

58. Use 1st Law of Thermodynamics
Assume: no work done
= heating/mass
[J-kg-1-s-1]

59. Apply to precipitating column

60. Apply to precipitating column
Heat released µ Mass condensed
µ Mass falling out
P [m/s]

61. Apply to precipitating column
Heat released µ Mass condensed
µ Mass falling out
Prw [kgw-m-2-s-1]

62. Apply to precipitating column
Heat released = LPrw [J-m-2-s-1]
Prw [kgw-m-2-s-1]

63. Apply to precipitating column
Heat released = LPrw [J -m-2-s-1]
Mass heated = ps/g [kgair-m-2]

64. Apply to precipitating column
Heat released = LPrw [J -m-3-s-1]
Mass heated = ps/g [kgair-m-2]
= Heating/mass
= LPrwg/ps [J -(kgair)-1-s-1]

65. How much heating by condensation?
P = 1000 mm-yr-1 = m-s-1
Ps = 1000 hPa =10+5 Pa
dT/dt = deg-s-1 = 0.67 deg-day-1
(Radiation ~ -1 deg-day-1)

66. Climate Modeling Theory - 2
Coming: Module 5
Climate Modeling Theory - 2
OUTLINE (Part 2):
Radiation
Surface Processes
Earth System