Geoscience Reference
In-Depth Information
the temperature used in heat transfer formulations should be corrected to account for
that fact. This can be done by using the potential temperature
instead of the actual
temperature
T
. The potential temperature is the temperature that would result if air were
brought adiabatically to a standard pressure level
p
0
=
θ
1000 hPa; for such a process
dh
=
0, and Equation (2.19), after substitution of
α
with (2.16), can be integrated to
yield Poisson's equation
p
)
R
d
/
c
p
θ
=
T
(
p
0
/
(2.23)
which can serve to define the potential temperature
and also to calculate it for a given
pressure
p
and temperature
T
; note that in Equation (2.23)
R
m
is replaced by
R
d
for
convenience. During an adiabatic process the potential temperature is conserved and
therefore under perfectly neutral conditions in a dry atmosphere, or when the specific
humidity is constant with height, the potential temperature should be a constant. A dry
atmosphere is unstable when
θ
decreases, and stable when it increases with height. Nev-
ertheless, the difference between
T
and
θ
θ
is usually rather small, especially in the lower
layers of the air near the ground surface, where most measurements are made. Therefore
in many situations, when the height difference of the temperature measurements is only
a few meters, the use of the actual temperature
T
is allowed; otherwise
θ
must be used
in heat transfer formulations.
Density stratification due to water vapor
In the above considerations of atmospheric stability, the density stratification due to vertical
humidity gradient
∂
q
/∂
z
was not taken into account. Under some conditions this can be an
important factor, but it can be readily shown (see, for example, Brutsaert, 1982) that this
may be incorporated in the analysis by means of the virtual potential temperature, defined as
θ
V
=
(1
+
0
.
61
q
)
θ
; the virtual potential temperature is related to the potential temperature,
in the same way the virtual temperature is related to the actual temperature, as indicated
in Equation (2.10). Thus, strictly speaking, in the presence of humidity stratification, the
atmosphere can be considered statically neutral, not when
θ
is constant, but only when
θ
V
is constant; it is unstable when
θ
V
decreases, and stable when it increases with height.
Put differently, under such conditions the stability criterion for an atmosphere is not the lapse
rate of the temperature, but the lapse rate of the virtual temperature; in practice, however,
this difference is often ignored.
2.2.2
Stability of a saturated atmosphere
When the air is saturated, any increase in heat content
dh
of a parcel of air during an
adiabatic process can only be the result of condensation, that is a decrease in the water
vapor content of the air; this can be written as
dh
=−
L
e
dq
, in which
L
e
is the latent heat
of vaporization and
q
, defined in Equation (2.2), is the specific humidity. With Equation
(2.21) one obtains now
dT
1
dz
=
d
+
L
e
c
p
dq
dz
−
(2.24)
