Prof. Fred Remer University of North Dakota Water in the Atmosphere

Prof. Fred Remer University of North Dakota Reading l Hess –pp –pp 58 – 60 l Tsonis –pp 93 – 97 l Wallace & Hobbs –pp 66 – 67 –pp 79 – 84 l Bohren & Albrecht –pp

Prof. Fred Remer University of North Dakota Objectives l Be able to define water vapor pressure l Be able to define virtual temperature l Be able to define specific humidity l Be able to define mixing ratio

Prof. Fred Remer University of North Dakota Objectives l Be able to calculate the water vapor pressure l Be able to calculate virtual temperature l Be able to calculate specific humidity l Be able to calculate mixing ratio

Prof. Fred Remer University of North Dakota Water In the Atmosphere l Unique Substance l Occurs in Three Phases Under Normal Atmospheric Pressures and Temperatures l Gaseous State –Variable 0 – 4% H H O

Prof. Fred Remer University of North Dakota Water in the Atmosphere l Remember Dalton’s Law? –Law of Partial Pressures –Let’s look at the contribution of water p = p1 + p2 + p3 + ….

Prof. Fred Remer University of North Dakota Water Vapor Pressure (e) l Ideal Gas Law for Dry Air l Ideal Gas Law for Water Vapor p = pressure of dry air  d = specific volume of dry air R d = gas constant for dry air e = vapor pressure of water vapor  v = specific volume of water vapor R v = gas constant for water vapor

Prof. Fred Remer University of North Dakota Water Vapor Pressure (e) l Partial pressure that water vapor exerts Total Pressure p = p O 2 +p N 2 +p H 2 O v Water Vapor Pressure e = p H 2 O v

Prof. Fred Remer University of North Dakota Water Vapor Pressure (e) l Gas Constant of Water Vapor H H O Molecular Weight (M w ) Hydrogen = 1kg kmol -1 Oxygen = 16 kg kmol -1 Hydrogen = 1kg kmol -1 Oxygen = 16 kg kmol -1 Water = 18 kg kmol -1

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l The temperature dry air must have in order to have the same density as moist air at the same pressure l Fictitious temperature

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Dry Air Total Pressure = p Total Pressure = p Volume = V Temperature = T Mass of Air = m d

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Moist Air (Mixture) Total Pressure = p Total Pressure = p Volume = V Temperature = T Mass of Air = m d + m v

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Density of mixture

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Ideal Gas Law –For Dry Air –For Water Vapor Alone or or

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Substitute into density expression

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Dalton’s Law of Partial Pressure or

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Substitute or

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Remove R d

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) Define 

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Remove p

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Rearrange terms

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l 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 Instead of or Use p = total (mixture) pressure  = mixture density

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Substitution of l Into l Produces

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Rearrange

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Start Canceling!

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Still looks Ugly! Simplify!

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) Virtual Temperature (T v ) p = total (atmospheric) pressure e = water vapor pressure T = temperature

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Moist air (mixture) is less dense than dry air l Virtual temperature is greater than actual temperature l Small difference

Prof. Fred Remer University of North Dakota Specific Humidity (q) l Ratio of the density of water vapor in the air to the (total) density of the air

Prof. Fred Remer University of North Dakota Mixing Ratio (w) l The mass of water vapor (m v ) to the mass of dry air Mass of Dry Air = m d Mass of Water Vapor = m v

Prof. Fred Remer University of North Dakota Mixing Ratio (w) l The mass of water vapor (m v ) to the mass of dry air Mass of Dry Air = m d Mass of Water Vapor = m v

Prof. Fred Remer University of North Dakota Mixing Ratio (w) l Expressed in g/kg –Dry Air l 1 to 2 g/kg –Tropical Air l 20 g/kg

Prof. Fred Remer University of North Dakota Mixing Ratio (w) l Can mixing ratio be expressed in terms of water vapor pressure? l Sure as it will rain on a meteorologist’s picnic!

Prof. Fred Remer University of North Dakota Mixing Ratio (w) l By definition l Divide top and bottom by volume (V)

Prof. Fred Remer University of North Dakota Mixing Ratio (w) l But density is so..... w = mixing ratio  v = density of water vapor in air  d = density of dry air

Prof. Fred Remer University of North Dakota Mixing Ratio (w) l Ideal Gas Law or or

Prof. Fred Remer University of North Dakota Mixing Ratio (w) l Substitute

Prof. Fred Remer University of North Dakota Mixing Ratio (w) l Simplify l Remember

Prof. Fred Remer University of North Dakota Mixing Ratio (w) l Substitute into l But p = total pressure of air (mixture)

Prof. Fred Remer University of North Dakota Mixing Ratio (w) l Substitute into l Ta-Da!

Prof. Fred Remer University of North Dakota Mixing Ratio (w) l Expression for Mixing Ratio (w) –Water Vapor Pressure (e) in any units –Atmospheric Pressure (p) in any units

Prof. Fred Remer University of North Dakota Mixing Ratio (w) l Can be used to determine other water variables l Let’s look at –Specific Humidity –Water Vapor Pressure (e) –Virtual Temperature (T v )

Prof. Fred Remer University of North Dakota Specific Humidity (q) l By definition l But q = specific humidity  v = density of water vapor in air  = density of air  d = density of dry air

Prof. Fred Remer University of North Dakota Specific Humidity (q) l Substitute into l Results in l But

Prof. Fred Remer University of North Dakota Specific Humidity (q) l Substitute into l Results in

Prof. Fred Remer University of North Dakota Specific Humidity (q) l Eliminate V

Prof. Fred Remer University of North Dakota Specific Humidity (q) l Divide top and bottom by m d

Prof. Fred Remer University of North Dakota Specific Humidity (q) l But so

Prof. Fred Remer University of North Dakota Specific Humidity (q) l Expression for specific humidity (q) –Mixing Ratio (w) in kg kg -1

Prof. Fred Remer University of North Dakota Water Vapor Pressure (e) l Pressure exerted by water vapor is a fraction of total pressure of air l Fraction is proportional to # of moles in mixture e = water vapor pressure f = fractional amount of water vapor p = total pressure of air

Prof. Fred Remer University of North Dakota Water Vapor Pressure (e) l How many moles of water are in a sample of air? l Number of moles of water n v = # of moles m v = mass of water molecules M w = molecular weight of water

Prof. Fred Remer University of North Dakota Water Vapor Pressure (e) l How many moles of dry air are in a sample of air? l Number of moles of dry air n d = # of moles m d = mass of dry air M d = mean molecular weight of dry air

Prof. Fred Remer University of North Dakota Water Vapor Pressure (e) l How many moles of air are in a sample of air? l Number of moles of air

Prof. Fred Remer University of North Dakota Water Vapor Pressure (e) l What is the molar fraction of water vapor in the air? l Substitute into

Prof. Fred Remer University of North Dakota Water Vapor Pressure (e) l Yikes! Let’s make this more manageable!

Prof. Fred Remer University of North Dakota Water Vapor Pressure (e) l Multiply top and bottowm by M w /m d

Prof. Fred Remer University of North Dakota Water Vapor Pressure (e) l Canceling out

Prof. Fred Remer University of North Dakota Water Vapor Pressure (e) l But Mixing Ratio and

Prof. Fred Remer University of North Dakota Water Vapor Pressure (e)

Prof. Fred Remer University of North Dakota Water Vapor Pressure (e) l Expression for water vapor pressure (e) –Mixing Ratio (w) in kg kg -1 –Atmospheric Pressure (p)

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Derive an expression for virtual temperature (T v ) using mixing ratio (w)

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Expression for water vapor pressure or

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Substituting

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Expand

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) Common denominator w+ 

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Group

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Simplify

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) l Divide numerator by denominator (polynomial division) and eliminate w 2 terms

Prof. Fred Remer University of North Dakota Virtual Temperature (T v ) Substitute  =.622

Prof. Fred Remer University of North Dakota Virtual Temperature (Tv) l Expression for virtual temperature –Mixing Ratio (w) in kg kg -1

Prof. Fred Remer University of North Dakota Review of Water Variables l Water Vapor Pressure

Prof. Fred Remer University of North Dakota Review of Water Variables l Virtual Temperature

Prof. Fred Remer University of North Dakota Review of Water Variables l Mixing Ratio

Prof. Fred Remer University of North Dakota Review of Water Variables l Specific Humidity

Prof. Fred Remer University of North Dakota Water in the Atmosphere l Moisture Variables –Water Vapor Pressure –Virtual Temperature –Mixing Ratio –Specific Humidity l Amount of Moisture in the Atmosphere

Prof. Fred Remer University of North Dakota Water in the Atmosphere l Unanswered Questions –How much water vapor can the air hold? –When will condensation form? –Is the air saturated? l The Beer Analogy

Prof. Fred Remer University of North Dakota The Beer Analogy l You are thirsty! l You would like a beer. l Obey your thirst!

Prof. Fred Remer University of North Dakota The Beer Analogy l Pour a glass but watch the foam

Prof. Fred Remer University of North Dakota The Beer Analogy l Wait! l Some joker put a hole in the bottom of your Styrofoam cup! l It is leaking!

Prof. Fred Remer University of North Dakota The Beer Analogy l Having had many beers already, you are intrigued by the phenomena!

Prof. Fred Remer University of North Dakota The Beer Analogy Rate at beer flows from keg is constant

Prof. Fred Remer University of North Dakota The Beer Analogy Rate at beer flows from keg is constant Rate at beer flows from cup depends on height

Prof. Fred Remer University of North Dakota The Beer Analogy The higher the level of beer in the cup, the faster it leaks!

Prof. Fred Remer University of North Dakota The Beer Analogy l The cup fills up l Height becomes constant l Equilibrium Reached Inflow (Constant) Leakage (Varies with Height)

Prof. Fred Remer University of North Dakota The Beer Analogy l What do you do? Inflow (Constant) Leakage (Varies with Height)

Prof. Fred Remer University of North Dakota The Beer Analogy l Get a new cup!

Prof. Fred Remer University of North Dakota Overview l Similar to what happens to water in the atmosphere

Prof. Fred Remer University of North Dakota Overview l Molecules in liquid water attract each other l In motion

Prof. Fred Remer University of North Dakota Overview l Collisions l Molecules near surface gain velocity by collisions

Prof. Fred Remer University of North Dakota Overview l Fast moving molecules leave the surface l Evaporation

Prof. Fred Remer University of North Dakota Overview l Soon, there are many water molecules in the air

Prof. Fred Remer University of North Dakota Overview l Slower molecules return to water surface l Condensation

Prof. Fred Remer University of North Dakota Overview l Net Evaporation –Number leaving water surface is greater than the number returning –Evaporation greater than condensation

Prof. Fred Remer University of North Dakota Overview l Molecules leave the water surface at a constant rate l Depends on temperature of liquid

Prof. Fred Remer University of North Dakota Overview l Molecules return to the surface at a variable rate l Depends on mass of water molecules in air

Prof. Fred Remer University of North Dakota Overview l Rate at which molecule return increases with time –Evaporation continues to pump moisture into air –Water vapor increases with time

Prof. Fred Remer University of North Dakota Overview l Eventually, equal rates of condensation and evaporation l “Air is saturated” l Equilibrium

Prof. Fred Remer University of North Dakota Overview l Derive a relationship that describes this equilibrium

Prof. Fred Remer University of North Dakota Clausius-Clapeyron Equation