Cumulus Cloud Model - Rain Formation in Warm Clouds: General Systems Theory - page 39

∫ 1 317

li () The value of is equal to 71.3.

Hence, T ≈ 8938, s ≈ 2 h 30 min.
Thus, starting from the surface level cloud growth begins after 2 h 30 min. This
is consistent with the observations that under favourable synoptic conditions solar
surface heating during the afternoon hours gives rise to cloud formation.
The dominant turbulent eddy radius at 1000 m in the sub-cloud layer is 100 m
starting from the 1-cm-radius dominant turbulent eddy at surface and the formation
of successively larger dominant eddies at decadic length scale intervals as explained
in Sect. 1.5.1. Also, it has been shown in Sect. 2.1.1 that the radius of the dominant
turbulent eddy ( r * ) inside the cloud is 1 m. These features suggest that the scale
ratio is 100 times larger inside the cloud than below the cloud. The 1000-m (1 km)
eddy at cloud-base level forms the internal circulation for the next stage of eddy
growth, namely 10 km eddy radius with circulation speed equal to 0.03 cm/s. Cloud
growth begins at 1 km above the surface and inside this 10-km eddy, with dominant
turbulent eddy radius 1 m as shown above. The circulation speed of this 1-m-radius
eddy inside cloud is equal to 3 m/s as shown in the following. Since the eddy con-
tinuum ranging from 1 cm to 10 km radius grows from the surface starting from the
same primary eddy of radius r * and the perturbation speed w * (cm/s), the circulation
speeds of any two eddies of radii R 1 , R 2 with corresponding circulation speeds W 1
and W 2 are related to each other as follows from Eq. (1.1):

r
R w

2

2 2

*

W =

* , 1

π

1

2 r

R

2 * 2

W = w

* , 2

π

2

2

W
W

R
R

2 1

= ,

2

2

1

W
W

R
R

2
1

1
2

= .

As mentioned earlier, cloud growth with dominant turbulent eddy radius 1 m begins
at 1 km above surface and forms the internal circulation to the 10-km eddy. The
circulation speed of the in-cloud dominant turbulent eddy is computed as equal to
3 m/s from the above equation where the subscripts 1 and 2 refer, respectively, to
the outer 10-km eddy and the internal 1-m eddy.
The value of vertical velocity perturbation W at cloud base is then equal to 100
times the vertical velocity perturbation just below the cloud base. Vertical velocity
perturbation just below the cloud base is equal to 0.03 cm/s from Table 2.1 . There-
fore, the vertical velocity perturbation at cloud base is equal to 0.03 × 100 cm/s, i.e.
3 cm/s and is consistent with airborne observations over the Indian region during
the monsoon season (Selvam et al. 1976 ; Pandithurai et al. 2011).