Geoscience Reference
In-Depth Information
saturation value,
w
s
; and the latent heat of vaporization is
v
. The statement of total
entropy conservation, Eq. (6.113) of
Bohren and Albrecht
(
1998
), is
v
w
s
T
.
1
M
d
DS
Dt
c
pd
)
1
DT
Dt
R
d
p
d
Dp
d
Dt
D
Dt
=
(w
t
c
w
+
−
+
(8.52)
T
With the weighted specific heat
w
t
c
w
+
c
pd
written simply as
c
p
,thisis
v
w
s
T
v
w
s
T
,
(8.53)
1
M
d
DS
Dt
=
c
p
T
DT
Dt
−
R
d
p
d
Dp
d
D
Dt
c
p
θ
d
Dθ
d
D
Dt
Dt
+
=
Dt
+
with
θ
d
the “dry-air” potential temperature
T
d
(z)
p
d
(
0
)
p
d
(z)
R
d
c
p
θ
d
(z)
=
.
(8.54)
The
equivalent potential temperature θ
e
is defined through
Eq. (8.53)
:
v
w
s
T
1
M
d
DS
Dt
=
c
p
θ
d
Dθ
d
Dt
+
D
Dt
c
p
θ
e
Dθ
e
Dt
.
=
(8.55)
It is conserved in isentropic processes in cloud air, including processes with
condensation and evaporation. It follows that
θ
e
is (
Betts
,
1973
)
θ
d
exp
v
w
s
c
p
T
T
p
d
(
0
)
p
d
exp
v
w
s
c
p
T
.
R
d
c
p
θ
e
=
=
(8.56)
Physically,
θ
e
is approximately the potential temperature a cloud parcel would have
if it were moved upwards, reversibly and adiabatically, to a level where the pressure
is low enough to allow all its water vapor to condense.
8.4 The averaged equations for moist air
We'll discuss the averaged equations for moist air, considering first the general case
where the average can be over space or over an ensemble of realizations. We'll show
that the averaging produces some new Reynolds terms involving thermodynamic
properties. Then we'll generalize the ensemble-averaged Reynolds-flux equations
in
Chapter 5
to include buoyancy and Coriolis effects.
