Geoscience Reference
In-Depth Information
8.3.2 The extension to cloud air
The density
ρ
cl
of cloud air is
M
d
+
M
v
+
M
l
ρ
cl
=
=
ρ
d
+
ρ
v
+
ρ
l
.
(8.46)
volume
Rewriting in terms of the density of moist air gives
ρ
cl
1
ρ
l
ρ
cl
ρ
d
+
ρ
v
=
ρ
cl
−
ρ
l
=
−
=
ρ
cl
(
1
−
q
l
) ,
(8.47)
where
q
l
ρ
l
/ρ
cl
is the
specific liquid water content
. Solving for the cloud air
density yields
=
ρ
d
+
ρ
v
p
R
d
T
v
(
1
p
R
d
T
vcl
,
ρ
cl
=
q
l
)
=
q
l
)
=
(8.48)
(
1
−
−
with
T
vcl
the virtual temperature of cloud air,
T
vcl
=
T
v
(
1
−
q
l
)
T (
1
+
0
.
61
q
−
q
l
) ,
(8.49)
neglecting the higher-order term. Like
T
v
,
T
vcl
uses the gas constant for dry air in
its equation of state:
p
=
ρ
cl
R
d
T
vcl
.
(8.50)
The extension to potential temperature is
θ
vcl
(z)
=
θ
v
(
1
−
q
l
)
θ(
1
+
0
.
61
q
−
q
l
).
(8.51)
8.3.3 A conserved temperature in cloud air
We saw that the application of entropy conservation in dry air leads straightfor-
wardly to the concept of potential temperature, which is conserved in isentropic
(reversible and adiabatic) processes. But as
Bohren and Albrecht
(
1998
) lament, its
application to cloud air is much more complicated:
[Accounting for entropy conservation in cloud air] is tedious and you are almost certain to
make errors along the way. We derived the lapse rate for a saturated parcel several times,
each time obtaining different results. Indeed, you are likely to find expressions similar but
not identical to ours. Nevertheless, we think that what follows is correct.
We shall summarize the Bohren-Albrecht result.
A saturated cloud parcel, of uniform temperature
T,
contains mass
M
d
of dry
air, whose partial pressure, gas constant, and specific heat at constant pressure are
p
d
,
R
d
,
c
pd
. The total water (vapor plus liquid) mixing ratio of the cloud parcel is
w
t
; the specific heat of its liquid water is
c
w
; its water vapor mixing ratio is the
