Geoscience Reference
In-Depth Information
So, the divergence equation may be written as
2
p
0
¼½ð@
2
2
2
2
2
1
0
r
u
=@
x
Þ
þð@v=@
y
Þ
þð@
w
=@
z
Þ
2
½j
D
j
j!j
þ@
B
=@
z
ð
4
:
39
Þ
To isolate the effects of the dynamic part of the pressure perturbation field, we
consider only that part of the divergence equation associated with the dynamic
perturbation pressure (cf. 2.70)
2
p
0
d
¼½ð@
2
2
2
2
2
1
0
r
u
=@
x
Þ
þð@v=@
y
Þ
þð@
w
=@
z
Þ
2
½j
D
j
j!j
ð
4
:
40
Þ
and do not consider
2
p
0
b
¼ @
0
r
B
=@
z
ð
4
:
41
Þ
where p
0
¼
p
0
d
þ
p
0
b
(cf. (2.63)).
According to (4.40), the shape of the three-dimensional pressure field is
determined by terms involving the vertical and horizontal shears of each com-
ponent of the wind, including deformation and vorticity. With proper boundary
conditions, the pressure field can be determined. Since the operator on the LHS of
(4.40) is a (three-dimensional) Laplacian, the sign of each forcing function on the
RHS of (4.40) is of the opposite sign of its contribution to dynamic perturbation
pressure.
To isolate the effects of the updraft/downdraft on its environment, each
variable is expressed in terms of
the environmental
(mean) value and the
perturbation (primed) storm value. Thus,
u
¼
U
ð
z
Þþ
u
0
ð
x
;
y
;
z
;
t
Þ
ð
4
:
42
Þ
v ¼
V
ð
z
Þþv
0
ð
x
;
y
;
z
;
t
Þ
ð
4
:
43
Þ
w
¼
w
0
ð
x
;
y
;
z
;
t
Þ
ð
4
:
44
Þ
The reader is cautioned that the perturbation terms in (4.42)-(4.44) are not
necessarily small compared with the respective mean terms (in (4.42)-(4.44)), as
was the case in our earlier linear stability analyses; in fact, they are typically on
the same order of magnitude as the mean terms. In (4.42)-(4.44) it is seen that the
environmental horizontal wind field V is chosen, for simplicity, to be horizontally
homogeneous, varying only as a function of height, and the vertical environmental
wind field (vertical velocity in the environment) is zero (i.e., the vertical motion
field is ''resting''). This environmental wind field may be represented by a hodo-
graph (
Figure 4.32
), a plot of wind vs. height that is represented by the locus of
points marked by the tip of the wind vectors at each height, where each wind
vector is plotted at a common origin.
The storm-related wind field, however, varies as a function of three-
dimensional space and time. In nature, there are inhomogeneities in the
environmental wind field, but they are neglected and usually are considered to be
second-order effects, which is not necessarily always the case (e.g., consider what
happens when a convective storm approaches, straddles, or crosses a surface front
or outflow boundary or other surface boundary). Also, there is often a band of
mesoscale ascent where convective storms are triggered, so the average value of
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