Lecture 7
Water Vapor
Water Vapor amount in the air is variable.
Concentration of water vapor can be
quantified by:
Vapor pressure
Mixing ratio
Specific humidity
Absolute humidity
Relative humidity
Dew point depression
Wet-bulb temperature
Warmer air can hold more water vapor at
equilibrium than colder air.
Air that holds this equilibrium amount is
saturated.
If air is cooled below the saturation
temperature, some of the water vapor
condenses into liquid, which releases latent
heat and warms the air.
Thus, temperature and water vapor interact in
a way that cannot be neglected.
Saturation Vapor Pressure
Vapor pressure:
Air is a mixture of gases. All of the gases
contribute to the total pressure. The pressure
associated with any one gas in a mixture is
called the partial pressure.
Water vapor is a gas, and its partial pressure in
air is called the vapor pressure.
Symbol e is used for vapor pressure. Units are
pressure units: kPa.
Saturation
Air can hole any proportion of water vapor.
For humidities greater than a threshold called the
saturation humidity, water vapor tends to condense
into liquid faster than it re-evaporates.
This condensation process lowers the humidity
toward the equilibrium (saturation) value.
The process is so fast that humidities rarely exceed
the equilibrium value.
Saturation
Thus, while air can hold any portion of water vapor,
the threshold is rarely exceeded by more than 1% in
the real atmosphere.
Air that contains this threshold amount of water vapor
is saturated.
Air that holds less than that amount is unsaturated.
Saturation
The equilibrium (saturation) value of vapor
pressure over a flat surface of pure water is
given the symbol:
es
For unsaturated air, e < es
Air can be slightly supersaturated (e > es).
When there are no surfaces upon which water
vapor can condense.
Saturation – Technical Definition
Water Vapor
Liquid Water
Sealed
Container
Water Vapor Fluxes
Flux of water
molecules from
liquid to vapor
Flux of water
molecules from
vapor to liquid
Saturation
Saturation exists when these two fluxes of
water vapor are equal
Flux of water
molecules from
liquid to vapor
Flux of water
molecules from
vapor to liquid
Saturation Vapor Pressure
Formula for es(T) called the
Clausius-Clapeyron Equation
Approximation:
 L
es  e0  exp
R
 v
 1
1 




T

 0 T 
Where e0 = 0.611 kPa, T = 273 K,
Rv = 461 J K-1 Kg-1 is the gas constant for water vapor.
Absolute temperature in Kelvins must be used for T.
Clausius-Clapeyron Equation
This equation describes the relationship
between temperature and saturation vapor
pressure.
Because clouds can consist of liquid droplets
and ice crystals suspended in air, we must
consider saturations with respect to water
and ice.
Teten’s Formula
Is an empirical expression for saturation vapor
pressure with respect to liquid water that
includes the variation of latent heat with
temperature.
 b  (T  T1 ) 
es  e0  exp

 T  T2 
B = 17.2694, T1 = 276.16 K, T2 = 35.86 K
Exercise
Calculate es(T) for T = 0C, 10C, 20C,
30C, 40C
Graph of Clausius-Clapeyron Equation
System of Dry Air + Water Vapor
Assume system is closed

i.e., no exchange of mass with environment
Dry air + water vapor
Saturation, Sub-Saturation, Super-Saturation
Super-saturated air
Saturated air
Sub-saturated air
Super-Saturation and Condensation
Suppose air becomes
super-saturated
“Excess” water vapor
will condense
Supersaturation
Supersaturation occurs when e > es
Supersaturation is a temporary state
Water vapor condenses until state of
supersaturation is relieved
Humidity Variables
Mixing Ratio
the ratio of mass of water vapor to mass of dry
air is called the mixing ratio, r or w. It is given
by:
mv
w
md
(W&H 3.57)
If neither condensation or evaporation take place, w of a parcel
remains constant. Therefore, it is a conserved quantity.
Units are g/g but is usually presented as g/kg, but when solving
numerical problems, must be expressed as a dimensionless
quantity: kg/kg or g/g.
Humidity Variables
Mixing Ratio
the ratio of mass of water vapor to mass of dry
air is called the mixing ratio, r or w. It is given
by:
r w
 e
Pe
(Stull 5.3)
Where ε = rd/rv = 0.622 g vapor/g dry air is the ratio of gas
constants for dry air to that for water vapor.
r is proportional to the ratio of partial pressure of water vapor
(e) to partial pressure of the remaining gases in the air (P-e).
Humidity Variables
The saturated mixing ratio, rs,
is where es is used in place of e.
Units are g/g but is usually presented as g/kg:
= grams of water vapor per kilogram of dry air.
Humidity Variables
Specific Humidity
The ratio of mass of water vapor to mass of
total (moist+ dry) air, q, to a good
approximation is given by:
mv
w
 e
q


mv  md 1  w
P
(W&H)
(Stull)
Humidity Variables
Absolute Humidity
The concentration of water vapor in air is
called the absolute humidity, and has units
of grams of water vapor per cubic meter
(g/ m3).
Because absolute humidity is essentially a
partial density, it can be found from the
partial pressure using the ideal gas law for
water vapor:
e
e
v 
    d
Rv  T P
Humidity Variables
Relative Humidity
The ratio of actual amount of water vapor in
the air compared to the equilibrium
(saturation) amount at that temperature is
called the relative humidity.
RH
e
q
 r
  

100% es qs  s rs
Cooling a Parcel -- Constant
Pressure
Reminder
Recall
e nv

p n
where nv = number of moles of water vapor
and n = total number of moles
 nv
e 
n

p

i.e., e is proportional to p.
Cool the system at constant pressure
Closed system  nv and n remain
constant
 e remains constant
Start with Sub-Saturated Air
Cool air at
constant pressure
e
Cool at Constant Pressure
e
Cool at Constant Pressure
e
Cool at Constant Pressure
e
Cool at Constant Pressure
e
Cool at Constant Pressure
e
Saturation Achieved
Continue to cool air
e
Super-Saturation!
e
Dew
Dew forms when super-saturation occurs
near a surface, e.g., a blade of grass
DEW
Dew Point (Td)
Definition: The temperature at which
saturation would first be achieved if the air
were cooled at constant pressure
Temperature and Dew Point
So, Td is the temperature that
satisfies es(Td) = e.
e
Td
T
Note
If the Td < 0C and super-saturation
occurs, frost forms

Water vapor turns directly to ice
Note: Frost is not frozen dew!
Frost
Relative Humidity (RH)
w
RH  100
ws
where w is the mixing ratio and ws is
the saturation mixing ratio
Relative Humidity
Approximation
e
RH  100
es
Simpler, as es is a function of T only.
Exercise
Let T = 20.0C and e = 12.0 hPa
Calculate RH using the approximate form

First, calculate es(T)
 17.67  20.0C 
e s (20C )  6.112hPa  exp

 20.0C  243.5C 
 23.4 hPa
12.0 hPa
RH  100
 100 0.51  51%
23.4 hPa
Increased Accuracy
For greatest accuracy, use the exact form of RH
and use tabulated values of ws
Best source: Smithsonian Meteorological
Tables (SMT)
Supersaturation
When condensation is occurring on a
surface, a thin layer of air next to the
surface is supersaturated

i.e., RH > 100%
Technically, Td > T where condensation is
occurring
However, Td – T is quite small and cannot
be measured by standard instruments
So, for practical purposes, Td  T
Adiabatic Cooling
(Adiabatic expansion due to falling
pressure)
nv
Again, e 
p
n
Closed system  nv/n is constant
But, p is decreasing
Therefore, e is decreasing
RH of Expanding Parcel
e
Recall, RH  100
es
e is decreasing due to expansion
But, parcel is cooling
 es is also decreasing
It turns out that es decreases faster than e
e and es for a rising parcel
z
e decreases as
parcel rises
es decreases faster than e
e
es
e and es for a sinking parcel
z
es
e
RH and Adiabatic Processes
RH of a rising parcel increases
 condensation can occur if parcel can be
lifted sufficiently
RH of a sinking parcel decreases
 condensation will not occur if air is
sinking
Lifting Condensation Level (LCL)
Definition: Level at which saturation is first
achieved if an air parcel is lifted
adiabatically
The LCL is usually an accurate indication
of the height of the cloud base
LCL
z
LCL
e
es