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Table 2.1  Vertical profile of eddy vertical velocity perturbation W
Height above surface R
Length scale ratio z = R / r *
Vertical velocity W = w * fz cm s −1
1 cm
1 (  r * = 1 cm)
30 × 10 −7 (= w * )
100 cm
100
1.10 × 10 −4
100 m
100 × 100
2.20 × 10 −3
1 km
1000 × 100
8.71 × 10 −3 ≈ 0.01
10 km
10000 × 100
3.31 × 10 −2
The values of large eddy vertical velocity perturbation W produced by the process
of MFC at normalized height z computed from Eq. 1.6 are given in Table 2.1 . The
turbulence length scale r * is equal to 1 cm, and the related vertical velocity perturba-
tion w * is equal to 30 × 10 −7 cm/s for the height interval 1 cm to 1000 m (cloud-base
level) for the computations shown in Table 2.1 . Progressive growth of successively
larger eddies generates a continuous spectrum of semipermanent eddies anchored to
the surface and with increasing circulation speed W .
The above values of vertical velocity, although small in magnitude, are present
for long enough time period in the lower levels and contribute to the formation and
development of clouds as explained in the next section.
2.2.2
Large Eddy-Growth Time
The time T required for the large eddy of radius R to grow from the primary turbu-
lence scale radius r * is computed from Eq. (1.36) as follows:
x
2
r
w
π
2
*
*
T
=
li
( .
z
x
1
x
=
z
and
x
=
z
.
1
1
2
2
In the above equation, z 1 and z 2 refer, respectively, to the lower and upper limits of
integration and li is the Soldner's integral or the logarithm integral. The large eddy-
growth time T can be computed from Eq. (1.36) as follows.
As explained earlier, a continuous spectrum of eddies with progressively increas-
ing speed (Table 2.1 ) anchored to the surface grows by MFC originating in turbulent
fluctuations at the planetary surface. The eddy of radius 1000 m has a circulation
speed equal to 0.01 cm/s (Table 2.1 ). The time T seconds taken for the evolution of
the 1000-m (10 5 cm) eddy from 1 cm height at the surface can be computed from
the above equation by substituting for z 1 = 1 cm and z 2 = 10 5 cm such that x 1 = 1 and
x 2 ≈ 317.
317
1
0012 1
π
T
=
li
().
z
.
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