AOSC 620
PHYSICS AND CHEMISTRY
OF THE ATMOSPHERE, I
Lecture 5A, Moist Air
Professor Russell Dickerson
Room 2413, Computer & Space Sciences Building
Phone(301) 405-5364
russ@atmos.umd.edu
web site www.meto.umd.edu/~russ
Copyright © 2013 R. R. Dickerson
1
Skew-T Practice
If a parcel of air with T = 15 oC and Td = 2oC rises
adiabatically form 1000 hPa to the LCL find the alt,
press, temp and sat mix ratio at the LCL.
If the parcel continues to rise (pseudo-adiabatically)
another 200 hPa how much water can be condensed
out? How much rain could this produce?
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
2
Dew Point Temperature Td
Temperature to which moist air may be cooled with
pressure and mixing ratio held constant to just reach
saturation with respect to H2O.
The “frost point” is the saturation temperature with
respect to ice.
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
3
At the dewpoint w = ws(Td, P)
w
es (Td )
P  es (Td )
As in Henry’s Law, but L is enthalpy,
at a given temperature:
 Lv  1 1 
es (T )  es (T0 ) exp    
 Rv  T0 T 
 Lv  1 1 
es (Td )  es (T0 ) exp    
 Rv  T0 Td 
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
4
 Lv
es (T )
 exp 
es (T0 )
 Rv
 1 1 
Lv  1 1 
    1     for T  T0
Rv  T0 T 
 T0 T 
liquid
e
solid
gas
T0
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
T
5
Phase Diagram of Water
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
6
Relating relative
humidity to dew point.
Copyright © 2014 R. R. Dickerson
w
ws (Td , P) es (Td )
f =
=
»
ws (T ) ws (T, P)
es (T )
æ es (Td ) ö
ln f = ln ç
÷
è es (T ) ø
From Clausius - Clapeyron :
æ es (Td ) ö Lv æ 1 1 ö
ln ç
÷@
ç - ÷
è es (T ) ø Rv è T Td ø
T
or
Td =
æ
ö
Rv
ln f ÷
ç1- T
Lv
è
ø 7
Wet-Bulb Temperature Tw
Temperature to which air may be cooled by
evaporating water into it at constant pressure. When
water is evaporated into air, energy is added to the
water. This energy comes at the expense of the dry
air, which is cooled. How do we get from Tw to w
and RH?
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
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Consider
1. Isobaric process
2. Mixing ratio increased by evaporating
water into air: w => ws(Tw,p)
The heat necessary to evaporate dw grams of
water per kilogram of dry air is:
dq = Lvdw
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
9
To find the heat lost to dry air alone due to evaporation
of water, we must correct for the mass of the water that
the dry air now contains:
(1  w)dq   Lv dw'
Lv dw
or dq  
 c pm dT
1 w
Lv dw
Lv
dT  
  dw
c pm 1  w
cp
Integrate from T to Tw
w => ws(Tw,p)
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
10
Lv
(Tw  T )   ( ws (Tw )  w)
cp
T  Tw
Lv
or

ws (Tw )  w c p
Useful for isobaric condensation.
Measure using a Sling Pychrometer or aspirated wet
and dry bulbs:
We measure T and Tw. Since ws is a known function
of Tw and p, you can determine w from ws and the
above equation.
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
11
Alternatively, if w and T are known one can
calculate the wet bulb temperature, Tw.
Example:
ws (Tw )   e(Tw ) / p
we p
We may now apply the Clausius Clapeyron
equation.
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
12
{ (
Lv 1 1
es (Tw ) @ es (T) exp R
v T Tw
)}
æ
ö
L
es (Tw )
1
1 ÷
v
ç
or
» 1+
es (T)
R çè T T ÷ø
v
w
ö
L æ1
1 ÷
v
ç
es (Tw ) - es (T) = es (T)
R çè T T ÷ø
v
w
multiply by e
p
:
ö
L æ1
1 ÷
v
ç
ws (Tw ) - ws (T) = ws (T)
R çè T T ÷ø
w
Copyright © 2013 R. R. Dickerson v
& Z.Q. Li
13
From Isobaric Condensation:
Lv
(T  Tw )  ws (Tw )  w
cp
The Clausius Clapeyron Equation gives:
ws (Tw )  ws (T ) 
Lv ws (T )  1 1 
 

Rv  T Tw 
Solve for Tw:
(f = w/ws(T))
ws (Tw ) - w / f
=
Lvw/ f
Rv
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
æ1 1 ö
ç - ÷
è T Tw ø
14
After extensive algebra:
(1  f )
Tw  T 
w
f
cp
Lv w

Lv
fRvT 2
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
15
Also note:
since ws (Tw )  w;
then Tw  Td
or Td  Tw  T
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
16
Equivalent Temperature Te
Temperature a sample of moist air would reach
if all the moisture were condensed out at
constant pressure (i.e., latent heat converted to
sensible heat).
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
17
1st Law :
d q  c p dT  dp
but d q   Lv dw  c p dT for this problem.
0
Te
w
T
  Lv dw   c p dT
c p (Te  T )  Lv w
Lv w
Te  T 
cp
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
18
Isentropic Condensation
Temperature Tc
Tc is the temperature at which saturation is reached
when moist air is cooled adiabatically with w held
constant. See R&Y Figure 2.3 or W&H Figure
3.10.
Tc can be determined by the intersection of the
adiabatic equation (Poisson’s) and the Clausius
Clapeyron equation and found on a SkewT.
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
19
( p)
p
Tc = T s
k
(Poisson)
o
ì L æ 1 1 öü
es (Tc ) = es (T ) exp í v ç - ÷ý
î Rv è T Tc øþ
p
Constant H2O mixing ratio
Dry adiabat
SkewT
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
20
For completeness
Absolute humidity, rv, density of water vapor.
Specific humidity, q, g H2O /kg air
(not dry air). Same as [H2O].
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
21
Conservative Properties of Air Parcels
Variable
dry adiabatic
saturated/pseudo adiabatic
q
C
NC
qe
C
C
qw
C
C
Td
NC
NC
Tw
NC
NC
w
C
NC
T*
NC
NC
Te
NC
NC
Tc
C
NC
f
NC
C
C
C
qq
Copyright © 2013 R. R. Dickerson
& Z.Q. Li
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