Geoscience Reference
In-Depth Information
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)