prof. fred remer university of north dakota water in the atmosphere

Prof. Fred RemerUniversity of North Dakota

Water in the AtmosphereWater in the Atmosphere

ReadingReading

Hess– pp 43 - 44– pp 58 – 60

Tsonis– pp 93 – 97

Wallace & Hobbs– pp 66 – 67– pp 79 – 84

Bohren & Albrecht– pp 181-188

ObjectivesObjectives

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

ObjectivesObjectives

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 AtmosphereWater In the Atmosphere

Unique Substance Occurs in Three Phases Under

Normal Atmospheric Pressures and Temperatures

Gaseous State– Variable 0 – 4%

HH HHOO

Remember Dalton’s Law?– Law of Partial Pressures

– Let’s look at the contribution of water

p = p1 + p2 + p3 + ….p = p1 + p2 + p3 + ….

Water Vapor Pressure (e)Water Vapor Pressure (e)

Ideal Gas Law for Dry Air

Ideal Gas Law for Water Vapor

TRp dd

TRe vv

p = pressure of dry airp = pressure of dry airdd = specific volume of dry air = specific volume of dry air

RRdd = gas constant for dry air = gas constant for dry air

e = vapor pressure of water vapore = vapor pressure of water vaporvv = specific volume of water vapor = specific volume of water vapor

RRvv = gas constant for water vapor = gas constant for water vapor

Partial pressure that water vapor exerts

Total PressureTotal Pressurep = pp = pOO22

+p+pNN22+p+pHH22OOvv

Water Vapor PressureWater Vapor Pressuree = pe = pHH22OOvv

Gas Constant of Water Vapor

HH HHOO

Molecular Weight Molecular Weight (M(Mw w ))

Hydrogen = 1kg kmolHydrogen = 1kg kmol-1 -1

Oxygen = 16 kg kmolOxygen = 16 kg kmol-1-1

Water = 18 kg kmolWater = 18 kg kmol-1-1

v kmolkg18

kmolKJ8314R

11v kgKJ461R

Virtual Temperature (TVirtual Temperature (Tvv))

The temperature dry air must have in order to have the same density as moist air at the same pressure

Fictitious temperature

Dry Air

Total Pressure = pTotal Pressure = p

Volume = VVolume = V

Temperature = TTemperature = T

Mass of Air = mMass of Air = mdd

Moist Air (Mixture)

Total Pressure = pTotal Pressure = p

Volume = VVolume = V

Temperature = TTemperature = T

Mass of Air = mMass of Air = md d ++ mmvv

Density of mixture

Ideal Gas Law

– For Dry Air

– For Water Vapor Alone

TRp ddd

TRe vv

ororTR

vv oror

Substitute into density expression

Dalton’s Law of Partial Pressure

epp d oror eppd

Substitute eppd

Remove Rd

Re)ep(

Define 622.

Remove p

Rearrange terms

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

TRp ddd TRe vvInstead ofInstead of oror

UseUsep = total (mixture) pressurep = total (mixture) pressure

= mixture density= mixture density

Substitution of

vdTRp Into

Produces

Rearrange

Start Canceling!

Still looks Ugly! Simplify!

)622.1)(p/e(1

)p/e378(.1

p = total (atmospheric) pressurep = total (atmospheric) pressure

e = water vapor pressuree = water vapor pressure

T = temperatureT = temperature

Moist air (mixture) is less dense than dry air

Virtual temperature is greater than actual temperature

Small difference

)p/e378(.1

Specific Humidity (q)Specific Humidity (q)

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

Mixing Ratio (w)Mixing Ratio (w)

The mass of water vapor (mv) to the mass of dry air

Mass of Dry Air = mMass of Dry Air = mdd

Mass of Water Vapor = mMass of Water Vapor = mvv

The mass of water vapor (mv) to the mass of dry air

Mass of Dry Air = mMass of Dry Air = mdd

Mass of Water Vapor = mMass of Water Vapor = mvv

Expressed in g/kg– Dry Air

1 to 2 g/kg

– Tropical Air 20 g/kg

Can mixing ratio be expressed in terms of water vapor pressure?

Sure as it will rain on a meteorologist’s picnic!

By definition

Divide top and bottom by volume (V)

But density is so.....

w = mixing ratiow = mixing ratio

vv = density of water vapor in air = density of water vapor in air

dd = density of dry air = density of dry air

Ideal Gas Law

TRp ddd

TRe vv

Substitute

Simplify

Remember

Substitute into

eppd p = total pressure of air (mixture)p = total pressure of air (mixture)

Substitute into

Ta-Da!

ew eppd

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

e622.w

Can be used to determine other water variables

Let’s look at– Specific Humidity – Water Vapor Pressure (e)

– Virtual Temperature (Tv)

By definition

vqq = specific humidityq = specific humidity

vv = density of water vapor in air = density of water vapor in air

= density of air = density of air

dv dd = density of dry air = density of dry air

Substitute into

Results in

Substitute into

Results in

V/mV/m

Eliminate V

V/mV/m

Divide top and bottom by md

m/mm/m

But so

m/mm/m

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

Pressure exerted by water vapor is a fraction of total pressure of air

Fraction is proportional to # of moles in mixture

pfe e = water vapor pressuree = water vapor pressure

f = fractional amount of water vapor f = fractional amount of water vapor

p = total pressure of airp = total pressure of air

How many moles of water are in a sample of air?

Number of moles of water

nnvv = # of moles = # of moles

mmvv = mass of water molecules = mass of water molecules

MMww = molecular weight of water = molecular weight of water

How many moles of dry air are in a sample of air?

Number of moles of dry air

nndd = # of moles = # of moles

mmdd = mass of dry air = mass of dry air

MMdd = mean molecular weight of dry air = mean molecular weight of dry air

How many moles of air are in a sample of air?

Number of moles of air

What is the molar fraction of water vapor in the air?

Substitute into

M/mM/m

Yikes! Let’s make this more manageable!

M/mM/m

pM/mM/m

Multiply top and bottowm by Mw/md

M/mM/m

Canceling out

M/mM/m

pM/Mm/m

Mixing RatioMixing Ratio

andand 622.M

pM/Mm/m

Expression for water vapor pressure (e)– Mixing Ratio (w) in kg kg-1

– Atmospheric Pressure (p)

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

Expression for water vapor pressure

Substituting

Expand

Common denominator w+

Simplify

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

Substitute = .622

w61.1TTv

Virtual Temperature (Tv)Virtual Temperature (Tv)

Expression for virtual temperature– Mixing Ratio (w) in kg kg-1

w61.1TTv

Review of Water VariablesReview of Water Variables

Water Vapor Pressure

TRe vv

Virtual Temperature

w61.1TTv

)p/e378(.1

Mixing Ratio

e622.w

Specific Humidity

Moisture Variables– Water Vapor Pressure– Virtual Temperature– Mixing Ratio– Specific Humidity

Amount of Moisture 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 AnalogyThe Beer Analogy

You are thirsty! You would like a

beer. Obey your thirst!

Pour a glass but watch the foam

Wait! Some joker put

a hole in the bottom of your Styrofoam cup!

It is leaking!

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

Rate at beer flows from keg is constant

Rate at beer flows from cup depends on height

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

The cup fills up Height

becomes constant

Equilibrium Reached

Inflow(Constant)

Leakage(Varies with

Height)

What do you do?

Inflow(Constant)

Leakage(Varies with

Height)

Get a new cup!

OverviewOverview

Similar to what happens to water in the atmosphere

OverviewOverview

Molecules in liquid water attract each other

In motion

OverviewOverview

Collisions Molecules near

surface gain velocity by collisions

OverviewOverview

Fast moving molecules leave the surface

Evaporation

OverviewOverview

Soon, there are many water molecules in the air

OverviewOverview

Slower molecules return to water surface

Condensation

OverviewOverview

Net Evaporation– Number leaving

water surface is greater than the number returning

– Evaporation greater than condensation

OverviewOverview

Molecules leave the water surface at a constant rate

Depends on temperature of liquid

OverviewOverview

Molecules return to the surface at a variable rate

Depends on mass of water molecules in air

OverviewOverview

Rate at which molecule return increases with time– Evaporation

continues to pump moisture into air

– Water vapor increases with time

OverviewOverview

Eventually, equal rates of condensation and evaporation

“Air is saturated” Equilibrium

OverviewOverview

Derive a relationship that describes this equilibrium

Clausius-Clapeyron Clausius-Clapeyron EquationEquation

prof. fred remer university of north dakota water in the atmosphere

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