Water in the Atmosphere

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

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

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

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

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

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

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

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

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

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

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

Virtual Temperature (Tv) Density of mixture

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

Virtual Temperature (Tv) Substitute into density expression

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

Virtual Temperature (Tv) Substitute or

Virtual Temperature (Tv) Remove Rd

Virtual Temperature (Tv) Define e

Virtual Temperature (Tv) Remove p

Virtual Temperature (Tv) Rearrange terms

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

Virtual Temperature (Tv) Substitution of Into Produces

Virtual Temperature (Tv) Rearrange

Virtual Temperature (Tv) Start Canceling!

Virtual Temperature (Tv) Still looks Ugly! Simplify!

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

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

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

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

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

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

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

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

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

Mixing Ratio (w) Ideal Gas Law or or

Mixing Ratio (w) Substitute

Mixing Ratio (w) Simplify Remember

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

Mixing Ratio (w) Substitute into Ta-Da!

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

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

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

Specific Humidity (q) Substitute into Results in But

Specific Humidity (q) Substitute into Results in

Specific Humidity (q) Eliminate V

Specific Humidity (q) Divide top and bottom by md

Specific Humidity (q) But so

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

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

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

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

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

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

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

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

Water Vapor Pressure (e) Canceling out

Water Vapor Pressure (e) But and Mixing Ratio

Water Vapor Pressure (e)

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

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

Virtual Temperature (Tv) Expression for water vapor pressure or

Virtual Temperature (Tv) Substituting

Virtual Temperature (Tv) Expand

Virtual Temperature (Tv) Common denominator w+e

Virtual Temperature (Tv) Group

Virtual Temperature (Tv) Simplify

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

Virtual Temperature (Tv) Substitute e = .622

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

Review of Water Variables Water Vapor Pressure

Review of Water Variables Virtual Temperature

Review of Water Variables Mixing Ratio

Review of Water Variables Specific Humidity

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

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

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

The Beer Analogy Pour a glass but watch the foam

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

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

The Beer Analogy Rate at beer flows from keg is constant

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

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

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

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

The Beer Analogy Get a new cup!

Overview Similar to what happens to water in the atmosphere

Overview Molecules in liquid water attract each other In motion

Overview Collisions Molecules near surface gain velocity by collisions

Overview Fast moving molecules leave the surface Evaporation

Overview Soon, there are many water molecules in the air

Overview Slower molecules return to water surface Condensation

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

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

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

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

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

Overview Derive a relationship that describes this equilibrium

Clausius-Clapeyron Equation