Cumulus Cloud Model - Rain Formation in Warm Clouds: General Systems Theory - page 46

rain formation. According to the general systems theory for suspended particulate
size spectrum in turbulent fluid flows (Sect. 1.6), larger suspended particulates are
associated with the turbulent regions (smaller scale length with larger fluctuation
speed) of the vertical velocity spectrum. Spontaneous formation of larger cloud/
raindrops may occur by collision and coalescence of smaller drops in these regions
of enhanced turbulence.

2.2.7 Rainfall Commencement

Rainfall sets in at the height at which the terminal velocity w T of the raindrop be-
comes equal to the mean cloud updraft W . Let the mean volume radius R m be rep-
resentative of the precipitation drop at level z above cloud base. In the intermediate
range (40 μm < R m < 0.6 mm), an approximated formula for the fall speed is given
by Yau and Rogers (1989) as wKR m

T = 3 , where K 3 = 8 × 10 3 s −1 .
For droplets with radii < 30 μm, the terminal velocity is given as wKR
T

2 ,

= 1

m

where K 1 ~ 1.19 × 10 6 cm −1 s −1 .
For large drops, w

T = 2 . K 2 can be approximated as K 2 = 2010 cm 1/2 s −1 for
a raindrop size between 0.6 and 2 mm (Yau and Rogers 1989).

R m

2.2.8 Rainfall Rate

The in-cloud rainfall rate ( R t ) can be computed as shown below.
Rain water production rate over unit area (rainfall rate mm/s/unit area) Q rt in the
cloud at any level z above the cloud base (Eq. 2.5) is given by

4
3

3

RWWRN
tz

=− ( ) π .

T m

The in-cloud rainfall rate R tz is derived from the normalized flux value for height z,
i.e. eddy wavelength z . Therefore, for unit-normalized height interval z, the rainfall
rate R t is equal to R tz / z cm/s.

RRz = ( / ) × 3600 × 10 mm/h.

t tz

In the above equation, W T in cm/s is the terminal velocity of the raindrop of mean
volume radius R m at level z . Rainfall rate ( R t ) derived above is representative of the
mean cloud scale rain intensity.
Rain formation computed for two different cloud-base CCN mean volume ra-
dius, namely 2.2 and 2.5 μm are shown in Fig. 2.7 . Rain with larger rain rate forms
at a lower level and extends up to a higher level for the larger size (mean volume
radius 2.5 μm) cloud-base CCN spectrum.