Geoscience Reference
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Figure 3.42. Demonstration of ''optimum'' orientation of flow normal to a cold outflow (gust
front, density current) from a precipitating convective cloud when environmental shear is as
indicated by the vertical profile of gust front relative winds below z ¼ h at the lower right.
Vertical cross section across the leading edge of the cold pool. The red line encloses the cold
outflow; the green streamline indicates the vertically upright flow along the leading edge of the
outflow. Other symbols are explained in the text.
Under what circumstance(s) is this possible? To find out, we integrate the steady-
state, inviscid, flux form of the horizonal vorticity equation as we did for (3.36),
but this time only over that part of the domain that extends up to a height that is
higher than the top of the cold pool, but not higher than the top of the previous
domain; we integrate only up to z ¼ d
<
H, but z
>
h. It follows, then, from
(3.37), (3.38), and (3.39) ( Figure 3.42 ) that
u R ; d u R ; 0 ¼ D
u u R ; h u R ; 0
ð 3
:
40 Þ
and that
u L ; d ¼ 0
ð 3
:
41 Þ
We first show that the second term on the LHS of (3.36) vanishes. Integrating
with respect to height first, we find that
ð R
L ð w
ð R
L ð w
d
Þ
dx ¼
Þ d dx
ð 3
:
42 Þ
0
because w ð z ¼ 0 Þ¼ 0. Now,
d ¼ð@
u
=@
z Þ d ð@
w
=@
x Þ d ¼ð@
w
=@
x Þ d
ð 3
:
43 Þ
because it is seen in Figure 3.42 that
ð@
u
=@
z Þ d ¼ 0
ð 3
:
44 Þ
If there is an erect updraft at the leading edge of the density current and w ¼ 0at
z ¼ 0atx ¼ R and x ¼ L, then we may assume that the updraft is (locally)
 
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