Geoscience Reference
In-Depth Information
R
∗
is the universal gas constant and
m
d
and
m
v
are the molecular weights of dry
air and water vapor. By introducing the
specific humidity q
,definedas
M
v
M
d
+
ρ
v
ρ
,
q
=
M
v
=
(8.39)
we can write
Eq. (8.37)
as
q
m
d
m
v
R
d
=
R
=
(
1
−
q)R
d
+
qR
v
=
(
1
−
q)R
d
+
(
1
+
0
.
61
q)R
d
.
(8.40)
Thus in turbulent air containing water vapor the gas constant
R
has turbulent
fluctuations and the air-density deviation
ρ
( T)
of
Section 8.2
becomes
ρ
=˜
ρ
=
˜
˜
ρ
( T, R)
. If we again linearize about a dry base state,
R
R
, from
Eq. (8.40)
˜
=
R
d
+
R
=
we have
0
.
61
qR
d
and we can write
T
T
0
−
ρ
ρ
0
−
0
.
61
q,
(8.41)
which quantifies how both warmer air and moister air are buoyant.
Instead of carrying two contributions to buoyancy in moist air, as in
Eq. (8.41)
,
we can write the gas law for moist air in terms of the gas constant for dry air, using
Eq. (8.40)
:
p
=
ρRT
=
ρR
d
T(
1
+
0
.
61
q).
(8.42)
Then if we define a
virtual temperature T
v
for moist air as
T
v
=
T(
1
+
0
.
61
q),
(8.43)
the gas law
(8.42)
for moist air is
p
=
ρR
d
T
v
.
(8.44)
Physically,
T
v
is the temperature of dry air having the same pressure and density as
moist air at temperature
T
.
It
is also traditional
to generalize the definition of potential
temperature,
Eq. (8.21)
,to
virtual
potential temperature:
T
v
(z)
p(
0
)
p(z)
0
.
61
q)
p(
0
)
p(z)
R
c
p
R
c
p
θ
v
(z)
=
=
T(
1
+
=
θ(
1
+
0
.
61
q).
(8.45)
It is a conserved variable
(Problem 8.16)
. Thus, in moist air we simply use the
deviation of
virtual
potential temperature, rather than potential temperature, as the
buoyancy variable in the equation of motion
(8.34)
.
