1 Water in the Atmosphere
2 Reading Hess Tsonis Wallace & Hobbs Bohren & Albrecht pp 43 - 44
3 Objectives Be able to define water vapor pressureBe 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 pressureBe able to calculate virtual temperature Be able to calculate specific humidity Be able to calculate mixing ratio
5 Water In the AtmosphereUnique Substance Occurs in Three Phases Under Normal Atmospheric Pressures and Temperatures Gaseous State Variable 0 – 4% H O
6 Water in the AtmosphereRemember 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 airMass of Dry Air = md Mass of Water Vapor = mv
31 Mixing Ratio (w) The mass of water vapor (mv) to the mass of dry airMass 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 ratiorv = 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 Butp = 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 variablesLet’s look at Specific Humidity Water Vapor Pressure (e) Virtual Temperature (Tv)
43 Specific Humidity (q) By definition But q = specific humidityrv = 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 VariablesWater Vapor Pressure
72 Review of Water VariablesVirtual Temperature
73 Review of Water VariablesMixing Ratio
74 Review of Water VariablesSpecific Humidity
75 Water in the AtmosphereMoisture Variables Water Vapor Pressure Virtual Temperature Mixing Ratio Specific Humidity Amount of Moisture in the Atmosphere
76 Water in the AtmosphereUnanswered 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 constantRate 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 constantEquilibrium 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 EvaporationNumber leaving water surface is greater than the number returning Evaporation greater than condensation
94 Overview Molecules leave the water surface at a constant rateDepends on temperature of liquid
95 Overview Molecules return to the surface at a variable rateDepends on mass of water molecules in air
96 Overview Rate at which molecule return increases with timeEvaporation 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