Geoscience Reference
In-Depth Information
We can justify this approximation qualitatively if the horizontal momentum of air
flowing into the storm's updraft is approximately conserved as it rises up into the
storm without mixing with environmental air (and neglecting water vapor and
water substance loading). Consider a simple case when the vertical shear is
unidirectional; in the example shown in
Figure 4.3
the shear is westerly. So in
mid-levels air approaches the storm from the west, but slows down as it
encounters easterly momentum brought up from below; to conserve momentum, it
must be diverted around the updraft and, furthermore, speeds up as it passes
through the channel to the left of the updraft. Then, from (4.16) and (4.17) it
follows that
0
p
0
f
¼
2
1
2
j
v
i
j
<
0
ð
4
:
18
Þ
so that there is a negative perturbation pressure at mid-levels on the left side of
the updraft. Now, consider another air parcel, but this one begins from the
boundary layer, far ahead of the storm, so once again p
0
i
¼
0. Imagine that this air
parcel, however, ascends in the updraft and ends up on the north side of the
updraft, just where the previous air parcel had ended its journey. Using (4.16) and
(4.18) we see that
ð
mid
-
level
2
w
2
m
¼
2
2
1
1
1
2
j
v
0
j
þ
2
j
v
m
j
þ
Bdz
ð
4
:
19
Þ
LFC
where w
m
is the updraft in the storm at mid-levels;
j
v
0
j
is the storm-relative wind
speed in the boundary layer upstream from the storm;
j
v
m
j
is the storm-relative
wind speed at mid-levels on the left side of the updraft, and CAPE at mid-levels is
ð
mid
-
level
CAPE
m
¼
Bdz
ð
4
:
20
Þ
LFC
It follows that
2
2
1
=
2
w
m
¼ð
2 CAPE
m
þj
v
0
j
þj
v
m
j
Þ
ð
4
:
21
Þ
So, when there is no vertical shear, both the storm-relative winds in the boundary
layer and at mid-levels are zero, and (4.21) is equivalent to that predicted by
parcel theory from buoyancy alone (3.7). However, when there is vertical shear,
there is storm-relative wind in the boundary layer or at mid-levels or both. The
stronger the vertical shear, the stronger the updraft (4.21).
Finally, consider the special case in which the storm moves along with the
pressure-weighted mean wind from the surface to 6 km and that the storm-relative
boundary-layer wind is from the east at U ms
1
. Then the vertical shear in the
lowest 6 km is U/6 km. From (4.21), we see that in the reference frame of the
storm
j
v
0
j
2
U
2
2
¼
0. Therefore, the dynamic contribution to vertical
velocity (w
m
) is equivalent to that from buoyancy
½ð
2CAPE
Þ
and
j
v
m
j
1
=
2
when
2CAPE
¼
U
2
ð
4
:
22
Þ
(i.e., when CAPE
¼
U
2
1, U
2
=
2, or when R
¼
1). For R
<
=
>
CAPE, and the
contribution from dynamic perturbation pressure gradients dominates the
buoyancy contributions. We will shortly analyze the details of what contributes to
the dynamic vertical perturbation pressure gradient force. We now take a step
backward and look at a brief history of supercell research.
2
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