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H
Appendix H The adiabatic atmosphere
Let us consider an adiabatic atmosphere in which movements are fast enough for heat
transfer between different parts to be negligible.
=
=−
d U
d Q
P d V
P d V
(H.53)
We also assume that it is made of perfect gases for which the internal energy is only a
function of the temperature and use C V as the heat capacity at constant volume:
d U
=
C V d T
(H.54)
Combining the two equations and using the law of perfect gases to eliminate P , we get
d T
T +
R d V
C V
V =
0
(H.55)
The gas density
ρ
relates to the volume through
ρ
V
=
M (molar mass), and therefore:
d
ρ
ρ
C V
R
d T
T
=
(H.56)
which we can be integrated as:
T
T 0
C R
ρ
ρ 0 =
(H.57)
where the subscript 0 refers to surface conditions (elevation z
0). The next task is to find
a relationship between temperature and elevation in the atmosphere. To this end, we will
use the law of perfect gases one more time to eliminate V from (H.55) :
R d P
=
d T
T
P = (
+
R
C V )
(H.58)
and combine it with the hydrostatic condition:
M
V g d z
d P
=− ρ
g d z
=−
(H.59)
to obtain
M RT
PV g d z
= (
R
+
C V )
d T
(H.60)
or, using R = C P
C V :
d T
d z =−
Mg
C P
(H.61)
 
 
 
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