Geoscience Reference
In-Depth Information
the scale height of the atmosphere, which is
10 km. We would like to avoid
making an assumption about the time scale, so we estimate the magnitude of the
time derivative term—the first on the RHS of (2.29)—in terms of the horizontal
advection term—the second on the RHS of (2.29)—as
v
p
j
J
z
p
j
,
where U is the scale of the horizontal wind speed. We are thus assuming that the
time derivative term is the same order of magnitude as the horizontal advection
term, but not necessarily the same magnitude as the vertical advection term. We
found earlier (2.15) that p
0
=
p
EJ
z
p
U
=
U
2
, so that
j
v
EJ
z
p
j
U
2
R
T
j
J
z
j
v
jj
U
3
R
=
=
p
=
=
TL,
where L is the horizontal scale. So, in summary
R
t
ð
ln p
Þ
U
3
@=@
=
ð
2
:
30
Þ
TL
J
z
ð
ln p
Þ
U
3
R
=
TL
ð
2
:
31
Þ
@=@
z
ð
ln p
Þ
1
=
H
ð
2
:
32
Þ
The LHS of (2.29) is
U
L and the scale of vertical velocity w
W, so that the
scale of (2.29) may be expressed as follows:
=
R
R
C
p
ð
U
3
TL
þ
U
3
U
=
L
C
v
=
=
=
TL
þ
W
=
H
Þ
ð
2
:
33
Þ
From the continuity equation (2.28) it is seen that U
D, where D is the
vertical scale and not necessarily the same scale as the scale height H. Substituting
for W in the third term on the RHS of (2.33) and making use of the formula for
the speed of sound (2.17), it follows that:
=
L
W
=
1
U
2
c
2
=
þ
C
v
=
C
p
D
=
H
ð
2
:
34
Þ
where we have combined the first and second terms because they are the same.
The first term on the RHS of (2.34) can therefore be neglected when U
c; that
is, when wind speeds are much less than the speed of sound, which is
300m s
1
(i.e., for low Mach numbers). This condition holds for most winds in convective
storms
30m s
1
or less, but definitely not for strong tornadoes, in which wind
speeds are
100m s
1
. We conclude then that the time derivative term (or the
horizontal advection term) may be neglected for typical non-tornadic, subsonic
wind speeds, but not for strong tornadoes. If sound waves must be taken into
account when considering the dynamics of strong tornadoes, then the conse-
quences of a relatively high ratio between the wind speed and the speed of sound
(Mach number) must be taken into account and the first term on the RHS of
(2.34) must be included.
Under some conditions, the second term on the RHS of (2.34), which
represents the vertical advection of pressure (cf. (2.29)), can also be neglected.
This term may be neglected when D
H (i.e., when the vertical scale is much less
than the scale height or, in other words, when convection is ''shallow'').
Search WWH ::
Custom Search