Geoscience Reference
In-Depth Information
of (2.8) can also be written as
p
=
R
m
T
/α
(2.16)
relates the three variables,
, the temperature
T
and the pressure
p
; thus only two of the
three are needed to define the state. If
α
α
and
T
are chosen as these independent variables,
Equation (2.15) becomes
∂
∂
p
d
u
u
∂α
dh
=
dT
+
T
+
α
(2.17)
∂
T
α
Since by definition the specific heat capacity for constant volume is
c
v
=
(
∂
u
/∂
T
)
α
and since it can be shown that (
∂
u
/∂α
)
T
=
0, combination of the differential form of
Equations (2.16) with (2.17) produces
=
(
c
v
+
−
α
dh
R
m
)
dT
dp
(2.18)
or
dh
=
c
p
dT
−
α
dp
(2.19)
where by definition
c
p
=
∂
/∂
T
)
p
is the specific heat for constant pressure. With the
hydrostatic law, giving the pressure change with height in a fluid at rest, i.e.
(
h
dp
=−
ρ
gdz
(2.20)
one finally obtains from Equation (2.19)
dh
=
c
p
dT
+
gdz
(2.21)
Equation (2.21) is derived here by a combination of the principle of conservation of
energy with the equation of state and the hydrostatic equation. This result was obtained
for air containing water vapor; however, the moisture content dependency of the specific
heat at constant pressure, namely
c
p
=
q
)
c
pd
, is very weak and therefore
Equation (2.21) is usually applied with the specific heat for dry air, i.e.
c
pd
.
The criterion for the stability of an atmosphere at rest can be obtained by the following
thought experiment. Consider a small parcel of air with a temperature
T
1
that undergoes
a small vertical displacement without mixing with the surrounding body of air; this
displacement is sufficiently small and fast, so that the pressure of the particle adjusts to
its new environment in an adiabatic fashion, that is as a reversible process without heat
exchange with its surroundings. Two cases are of interest, depending on the degree of
saturation of the air.
qc
pw
+
(1
−
2.2.1
Stability of a partly saturated atmosphere
Dry adiabatic lapse rate
If the atmosphere can be assumed to remain partly saturated during this process, there
is no vaporization or condensation, and the change in heat of the parcel is
dh
=
0. With
