1. Water in the Atmosphere

2. Reading Hess Tsonis Wallace & Hobbs Bohren & Albrecht pp 43 - 44

3. Objectives Be able to define water vapor pressure
Be able to define virtual temperature
Be able to define specific humidity
Be able to define mixing ratio

4. Objectives Be able to calculate the water vapor pressure
Be able to calculate virtual temperature
Be able to calculate specific humidity
Be able to calculate mixing ratio

5. Water In the Atmosphere
Unique Substance
Occurs in Three Phases Under Normal Atmospheric Pressures and Temperatures
Gaseous State
Variable 0 – 4% H O

6. Water in the Atmosphere
Remember Dalton’s Law?
Law of Partial Pressures
Let’s look at the contribution of water
p = p1 + p2 + p3 + ….

7. Water Vapor Pressure (e)
Ideal Gas Law for Dry Air
Ideal Gas Law for Water Vapor
p = pressure of dry air
ad = specific volume of dry air
Rd = gas constant for dry air
e = vapor pressure of water vapor
av = specific volume of water vapor
Rv = gas constant for water vapor

8. Water Vapor Pressure (e)
Partial pressure that water vapor exerts
Total Pressure
p = pO2+pN2+pH2Ov
Water Vapor Pressure
e = pH2Ov

9. Water Vapor Pressure (e)
Gas Constant of Water Vapor H O
Molecular Weight (Mw )
Hydrogen = 1kg kmol-1 Oxygen = 16 kg kmol-1
Water = 18 kg kmol-1

10. Virtual Temperature (Tv)
The temperature dry air must have in order to have the same density as moist air at the same pressure
Fictitious temperature

11. Virtual Temperature (Tv)
Dry Air
Total Pressure = p
Volume = V
Temperature = T
Mass of Air = md

12. Virtual Temperature (Tv)
Moist Air (Mixture)
Total Pressure = p
Volume = V
Temperature = T
Mass of Air = md + mv

13. Virtual Temperature (Tv)
Density of mixture

14. Virtual Temperature (Tv)
Ideal Gas Law
For Dry Air
For Water Vapor Alone or or

15. Virtual Temperature (Tv)
Substitute into density expression

16. Virtual Temperature (Tv)
Dalton’s Law of Partial Pressure or

17. Virtual Temperature (Tv)
Substitute or

18. Virtual Temperature (Tv)
Remove Rd

19. Virtual Temperature (Tv)
Define e

20. Virtual Temperature (Tv)
Remove p

21. Virtual Temperature (Tv)
Rearrange terms

22. Virtual Temperature (Tv)
By definition, virtual temperature is the temperature dry air must have in order to have the same density as moist air (mixture) at the same pressure or
Instead of
p = total (mixture) pressure
r = mixture density
Use

23. Virtual Temperature (Tv)
Substitution of
Into
Produces

24. Virtual Temperature (Tv)
Rearrange

25. Virtual Temperature (Tv)
Start Canceling!

26. Virtual Temperature (Tv)
Still looks Ugly! Simplify!

27. Virtual Temperature (Tv)
p = total (atmospheric) pressure
e = water vapor pressure
T = temperature

28. Virtual Temperature (Tv)
Moist air (mixture) is less dense than dry air
Virtual temperature is greater than actual temperature
Small difference

29. Specific Humidity (q)
Ratio of the density of water vapor in the air to the (total) density of the air

30. Mixing Ratio (w) The mass of water vapor (mv) to the mass of dry air
Mass of Dry Air = md
Mass of Water Vapor = mv

31. Mixing Ratio (w) The mass of water vapor (mv) to the mass of dry air
Mass of Dry Air = md
Mass of Water Vapor = mv

32. Mixing Ratio (w) Expressed in g/kg Dry Air Tropical Air 1 to 2 g/kg

33. Mixing Ratio (w)
Can mixing ratio be expressed in terms of water vapor pressure?
Sure as it will rain on a meteorologist’s picnic!

34. Mixing Ratio (w)
By definition
Divide top and bottom by volume (V)

35. Mixing Ratio (w) But density is so..... w = mixing ratio
rv = density of water vapor in air
rd = density of dry air

36. Mixing Ratio (w)
Ideal Gas Law or or

37. Mixing Ratio (w)
Substitute

38. Mixing Ratio (w)
Simplify
Remember

39. Mixing Ratio (w) Substitute into But
p = total pressure of air (mixture)

40. Mixing Ratio (w)
Substitute into
Ta-Da!

41. Mixing Ratio (w) Expression for Mixing Ratio (w)
Water Vapor Pressure (e) in any units
Atmospheric Pressure (p) in any units

42. Mixing Ratio (w) Can be used to determine other water variables
Let’s look at
Specific Humidity
Water Vapor Pressure (e)
Virtual Temperature (Tv)

43. Specific Humidity (q) By definition But q = specific humidity
rv = density of water vapor in air
r = density of air
rd = density of dry air

44. Specific Humidity (q)
Substitute into
Results in
But

45. Specific Humidity (q)
Substitute into
Results in

46. Specific Humidity (q)
Eliminate V

47. Specific Humidity (q)
Divide top and bottom by md

48. Specific Humidity (q)
But so

49. Specific Humidity (q) Expression for specific humidity (q)
Mixing Ratio (w) in kg kg-1

50. Water Vapor Pressure (e)
Pressure exerted by water vapor is a fraction of total pressure of air
Fraction is proportional to # of moles in mixture
e = water vapor pressure
f = fractional amount of water vapor
p = total pressure of air

51. Water Vapor Pressure (e)
How many moles of water are in a sample of air?
Number of moles of water
nv = # of moles
mv = mass of water molecules
Mw = molecular weight of water

52. Water Vapor Pressure (e)
How many moles of dry air are in a sample of air?
Number of moles of dry air
nd = # of moles
md = mass of dry air
Md = mean molecular weight of dry air

53. Water Vapor Pressure (e)
How many moles of air are in a sample of air?
Number of moles of air

54. Water Vapor Pressure (e)
What is the molar fraction of water vapor in the air?
Substitute into

55. Water Vapor Pressure (e)
Yikes! Let’s make this more manageable!

56. Water Vapor Pressure (e)
Multiply top and bottowm by Mw/md

57. Water Vapor Pressure (e)
Canceling out

58. Water Vapor Pressure (e)
But
and
Mixing Ratio

59. Water Vapor Pressure (e)

60. Water Vapor Pressure (e)
Expression for water vapor pressure (e)
Mixing Ratio (w) in kg kg-1
Atmospheric Pressure (p)

61. Virtual Temperature (Tv)
Derive an expression for virtual temperature (Tv) using mixing ratio (w)

62. Virtual Temperature (Tv)
Expression for water vapor pressure or

63. Virtual Temperature (Tv)
Substituting

64. Virtual Temperature (Tv)
Expand

65. Virtual Temperature (Tv)
Common denominator w+e

66. Virtual Temperature (Tv)
Group

67. Virtual Temperature (Tv)
Simplify

68. Virtual Temperature (Tv)
Divide numerator by denominator (polynomial division) and eliminate w2 terms

69. Virtual Temperature (Tv)
Substitute e = .622

70. Virtual Temperature (Tv)
Expression for virtual temperature
Mixing Ratio (w) in kg kg-1

71. Review of Water Variables
Water Vapor Pressure

72. Review of Water Variables
Virtual Temperature

73. Review of Water Variables
Mixing Ratio

74. Review of Water Variables
Specific Humidity

75. Water in the Atmosphere
Moisture Variables
Water Vapor Pressure
Virtual Temperature
Mixing Ratio
Specific Humidity
Amount of Moisture in the Atmosphere

76. Water in the Atmosphere
Unanswered Questions
How much water vapor can the air hold?
When will condensation form?
Is the air saturated?
The Beer Analogy

77. The Beer Analogy You are thirsty! You would like a beer.
Obey your thirst!

78. The Beer Analogy
Pour a glass but watch the foam

79. The Beer Analogy
Wait!
Some joker put a hole in the bottom of your Styrofoam cup!
It is leaking!

80. The Beer Analogy
Having had many beers already, you are intrigued by the phenomena!

81. The Beer Analogy
Rate at beer flows from keg is constant

82. The Beer Analogy Rate at beer flows from keg is constant
Rate at beer flows from cup depends on height

83. The Beer Analogy
The higher the level of beer in the cup, the faster it leaks!

84. The Beer Analogy The cup fills up Height becomes constant
Equilibrium Reached
Inflow
(Constant)
Leakage
(Varies with Height)

85. The Beer Analogy What do you do? Inflow (Constant) Leakage
(Varies with Height)

86. The Beer Analogy
Get a new cup!

87. Overview
Similar to what happens to water in the atmosphere

88. Overview
Molecules in liquid water attract each other
In motion

89. Overview
Collisions
Molecules near surface gain velocity by collisions

90. Overview
Fast moving molecules leave the surface
Evaporation

91. Overview
Soon, there are many water molecules in the air

92. Overview
Slower molecules return to water surface
Condensation

93. Overview Net Evaporation
Number leaving water surface is greater than the number returning
Evaporation greater than condensation

94. Overview Molecules leave the water surface at a constant rate
Depends on temperature of liquid

95. Overview Molecules return to the surface at a variable rate
Depends on mass of water molecules in air

96. Overview Rate at which molecule return increases with time
Evaporation continues to pump moisture into air
Water vapor increases with time

97. Overview Eventually, equal rates of condensation and evaporation
“Air is saturated”
Equilibrium

98. Overview
Derive a relationship that describes this equilibrium

99. Clausius-Clapeyron Equation