Geoscience Reference
In-Depth Information
From the ideal gas law for dry air ( p
¼
R
d
T, where R
d
is the gas constant
for dry air) and (2.3) and (2.4), it is seen that
=
0
¼
p
0
p
T
0
=
=
T
ð
2
:
11
Þ
where T
¼
T
ð
z
Þþ
T
0
ð
x
;
y
;
z
;
t
Þ
. Thus,
B
¼ð
p
0
p
T
0
=
=
T
Þg
ð
2
:
12
Þ
In other words, buoyancy depends not only on the temperature excess or deficit
over that of the environment, but also on the pressure excess or deficit over that
of the environment. It turns out, however, that p
0
=
p can be neglected in compar-
=
ison with T
0
T for substantially subsonic flow. To see this, consider the horizontal
equation of motion (2.1) expressed in terms of the base state and perturbations
(2.3) and (2.4)
r
p
0
Dv
h
=
Dt
¼ @
v
h
=@
t
þ
v
EJ
v
h
¼
1
=
ð
2
:
13
Þ
Suppose that each component of
the wind u
; v;
w
U, which is
typically
10m s
1
. Then it follows from (2.13) that
=r
p
0
jj
v
j
1
v
h
j
ð
2
:
14
Þ
EJ
and therefore that
p
0
U
2
=
ð
2
:
15
Þ
From the ideal gas law it follows that
p
0
=ð
R
d
p
U
2
T
Þ
U
2
c
2
=
=
ð
2
:
16
Þ
where the speed of sound c in dry air is given by
R
d
1
=
2
c
¼ð
T
Þ
ð
2
:
17
Þ
and
¼
C
p
=
C
v
¼
1
:
4
1
ð
2
:
18
Þ
where C
p
and C
v
are the specific heat of air at constant pressure and volume,
respectively. Thus, with the exception of perhaps the most intense tornadoes,
U
c
300m s
1
the Mach number U
2
c
2
such that
the
square of
=
2
10
3
T
0
300
10
2
, so that to a good approximation
B
¼ð
T
0
100
=ð
300
Þ
=
T
5
=
=
T
Þg
ð
2
:
19
Þ
It will be shown later that, if buoyancy is expressed in terms of potential
temperature rather than temperature, we do not have to be concerned with the
Mach number in the computation of buoyancy.
Suppose now that air includes both water vapor and condensate; the pressure
of the air is the sum of the partial pressure of the dry air and that of the water
vapor, while the condensate does not contribute to the total pressure. Suppose
also that we treat the air containing both water vapor and condensate as if it were
dry (i.e., with no water vapor or condensate) and note that the density is modified
by the less dense water vapor and the more dense condensate, so that
p
¼
d
R
d
T
þ
v
R
v
T
¼ð
d
þ
v
Þ
R
d
T
v
¼ð
d
þ
v
þ
l
þ
i
Þ
R
d
T
cv
ð
2
:
20
Þ
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