Cumulus Cloud Model - Rain Formation in Warm Clouds: General Systems Theory - page 42

2.2.3 In-Cloud Temperature Lapse Rate

The in-cloud lapse rate Γ s is computed using the following expression (Eq. 2.2):

=− θ *
*

fz

ΓΓ
s

.

r

The primary eddy radius length r * at cloud base is equal to 1 m as shown earlier
(Sect. 2.1). The model computations for in-cloud vertical profile of vertical velocity
W, temperature perturbation θ and lapse rate Γ c at 1 km height intervals above the
cloud base are given in Table 2.2 .
The predicted temperature lapse rate decreases with height and becomes less sat-
urated than adiabatic lapse rate near the cloud top, the in-cloud temperatures being
warmer than the environment. These results are in agreement with the observations.

2.2.4 Cloud-Growth Time

The large eddy-growth time (Eq. 1.36) can be used to compute cloud-growth time
T c (Eq. 2.4):

r
w

π
2

* z

T = li (), z

2

c

z

1

*

where li is the Soldner's integral or the logarithm integral. The vertical profile of
cloud-growth time T c is a function of the cloud-base primary turbulent eddy fluctua-
tions of radius r * and perturbation speed w * alone. The cloud-growth time T c using
Eq. (2.4) is shown in Fig. 2.5 for the two different cloud-base CCN spectra, with
mean volume radii equal to 2.2 and 2.5 μm, respectively. The cloud-growth time
remains the same since the primary trigger for cloud growth is the persistent turbu-
lent energy generation by condensation at the cloud base in primary turbulent eddy
fluctuations of radius r * and perturbation speed w * .
Let us consider r * is equal to 100 cm and w * is equal to 1 cm s −1 the time taken
for the cloud to grow (see Sect. 2.1.6) to a height of, e.g. 1600 m above cloud base
can be computed as shown below. The normalized height z is equal to 1600 since
dominant turbulent eddy radius is equal to 1 m:

100
100 0012 1600
100 1 2536 15 84
30

π li

T

c = ×
=× ×
≈

.

. . s

min .

The above value is consistent with cloud-growth time observed in practice.