EAS 6140 Thermodynamics of Atmospheres and Oceans
Worksheet – Humidity Variables
Humidity variables
1. The water vapor mixing ratio, wv, is defined as the ratio of the mass of water vapor to
the mass of dry air. Write an expression for wv in terms of the partial pressure of water
vapor (e) and the total atmospheric pressure (p). (see p 114)
wv 
mv  v
e


md  d
pe
2. Relative humidity is frequently defined as H=w/ws. Is this expression equivalent to
H=e/es (4.34a)? H=e/es is the actual definition of relative humidity. However if we look
at the water vapor missing ratio wv  
ws  
e
and the saturation mixing ratio
pe
es
, because p is so much greater than e or es we essentially have H=w/ws
p  es
giving us a reasonable approximation of relative humidity.
3. Relative humidity with respect to liquid water saturation is always (less than, greater
than, equal to) the relative humidity with respect to ice saturation
greater than *Note you cannot have a saturation vapor pressure with respect to ice
@ temps above 0°C. Also refer to table 4.4
4. The total mass of water vapor in a column of unit cross-sectional area extending from the
surface to the top of the atmosphere is called the precipitable water, Wv:

W =
 dz
v
0
v
(4.40)
Incorporate the hydrostatic equation in (4.40) to derive an expression for Wv in terms of
specific humidity, qv.

Wv    v dz
0

Wv    v
0
Wv  
dz 
p
g
 dp
g
1 p v
dp
g po 
1 po  v
dp
  a a  d  v
g p 
mv
v
1 po  v
Wv  
dp
wv  qv 

g p d  v
mv  md  v   d
Wv 
Wv 
1 po
qv dp
g p
5. Write the SI units for each of the following parameters
relative humidity____no units_________
water vapor mixing ratio___no units_________
specific humidity ____no units_______
precipitable water ____kg/m2________
6. Air at 50oC with RH=10% contains (more, less) water vapor than air at -10oC with
RH=100%. (Hint: es=123.39 hPa at 50oC; es=2.8622 hPa at -10oC.)
50oC
10% * 123.39 = 12.3
12.3 > 2.86
-10oC
1005 * 2.8622 = 2.86
7. An air mass has a temperature of 30°C and a relative humidity of 50% at a pressure of
1000 hPa. Determine the following: (note: you may use Appendix E for values of es.
In Appendix E, ew corresponds to es)
a) water vapor partial pressure
H
e
es
.5 
e
42.427hPa
b) mixing ratio
e  21.21hPa
wv  
m
e
  v  0.622
pe
md
21.21hPa


wv  0.622 
  0.0135
 1000hPa  21.21hPa 
c) specific humidity
qv 
wv
0.0135

 0.0133
1  wv 1  0.0135
d) virtual temperature
Tv  (1  0.608qv )T
Tv  (1  0.608*0.0133)303K  305.45 K or 32.45C
Thermodynamic Charts
Using the Stuve (or Skew-T) diagram, read off the temperature and determine the
potential temperature at each of the following levels
Level
MAF sounding
T

MFL sounding
T

27°C 300.15K
900 mb
20°C 302.12 K
20°C 302.12K
700 mb
10°C 313.6K
6.5°C 309.7K
500 mb
-6°C
-7°C
1000 mb
325.7 K
324.5K
Estimate dT/dz and d/dz for the following layers
Level
MAF sounding
T

1000-900 mb
MFL sounding
T

-10.2°C/km 2.87°C/km
700-500 mb
-4.98°C/km
-5.9°C/km
-4.47°C/km
5.46°C/km
Determine the change in T and  of an air parcel initially at 900 mb if subjected to
MAF sounding
MFL sounding
T

T

a. adiabatic lifting of 100 mb
-10
0
-10
0
b. adiabatic lowering of 100 mb
10
0
10
0
c. radiative heating of 10oC
10
10
10
10
d. radiative cooling of 10oC
-10
-10
-10
-10
Sketch the following paths on the MFL diagram
a) A parcel of air is lifted dry adiabatically from an initial height of 900 mb to a height of
700 mb. The parcel is then lowered back down to the initial pressure. Label this path A.
Is this path reversible? Yes this is a reversible process. Although latent heat will
positively change the temperature of the parcel the parcel does not loose any condensate
or heat to the environment. This is essentially why this process is reversible. As long as
we are not losing mass or heat you should consider the process to be reversible. Other
processes that are irreversible include radiative heating and cooling, and mixing. This
answer should also help for part b.
b) A parcel of air is lifted dry adiabatically from an initial height of 900 mb to a height of
700 mb. The parcel is then heated radiatively by 10oC at constant pressure. The parcel is
then lowered back down to the initial pressure. Label this path B. Is this path reversible?
No.
Use the Stuve (or Skew-T) diagram to answer the following questions. Lines of constant
water vapor mixing ratio are given by the red dashed line (with scale on the x axis). The
vertical profile indicated by the curve left of the temperature curve represents the profile of dew
point temperatures. The dew-point temperature, denoted by TD, is defined by wv =ws TD (6.15)
So the dew point temperature is a humidity variable that is directly related to the mixing ratio.
The dewpoint depression is defined as T-TD.
Fill in the table below for the MAF sounding, by answering the following questions for
each of the indicated levels
Level
w
1000
TD
T-TD
ws
RH
(Air parcel begins @ a pressure < 1000mb)
850
12.5g/kg
14.5
20-14.5
19g/kg
12.5/19
700
7.5g/kg
4.5
10-4.5
12g/kg
7.5/12
4 00
(Not able to determine exact values with skew t)
Which layers are most likely to have clouds?
None have 100% humidity; we shouldn’t expect any clouds. However the 900-850 mb
and @ 650mb are the layers with the highest RH. You can determine this by the spacing
between the dewpoint curve and the temperature curve.
Fill in the table below for the MFL sounding, by answering the following questions for
each of the indicated levels
Level
w
TD
T-TD
ws
RH
1000
18g/kg
23
27-23
24g/kg
18/24
850
14g/kg
16
16-16
14g/kg
14/14
700
9.5g/kg
6.5
6.5-6.5
9.5g/kg
9.5/9.5
4 00
Which layers are most likely to have clouds?
Layers @ 850 and 700 mb