Geography Reference
In-Depth Information
purposes, the total energy of the atmosphere is the sum of internal energy, gravita-
tional potential energy, and kinetic energy. However, it is not necessary to consider
separately the variations of internal and gravitational potential energy because in
a hydrostatic atmosphere these two forms of energy are proportional and may be
combined into a single term called the
total potential energy
. The proportionality
of internal and gravitational potential energy can be demonstrated by considering
these forms of energy for a column of air of unit horizontal cross section extending
from the surface to the top of the atmosphere.
If we let dE
I
be the internal energy in a vertical section of the column of height
dz, then from the definition of internal energy [see (2.4)]
dE
I
=
ρc
v
Tdz
so that the internal energy for the entire column is
∞
E
I
=
c
v
ρT dz
(8.29)
0
However, the gravitational potential energy for a slab of thickness dz at a height z
is just
ρgzdz
so that the gravitational potential energy in the entire column is
dE
P
=
∞
0
E
P
=
ρgzdz
=−
zdp
(8.30)
p
0
0
where we have substituted from the hydrostatic equation to obtain the last integral
in (8.30). Integrating (8.30) by parts and using the ideal gas law we obtain
∞
∞
E
P
=
pdz
=
R
ρT dz
(8.31)
0
0
Comparing (8.29) and (8.31) we see that c
v
E
P
=
RE
I
. Thus, the total potential
energy may be expressed as
=
c
p
/c
v
E
I
=
c
p
/R
E
P
E
P
+
E
I
(8.32)
Therefore, in a hydrostatic atmosphere the total potential energy can be obtained
by computing either E
I
or E
P
alone.
