Cumulus Cloud Model - Rain Formation in Warm Clouds: General Systems Theory - page 37

2.2.1 Vertical Velocity Profile in the Atmospheric

Boundary Layer

The MFC-generated values of vertical velocity have been calculated for different
heights above the surface for clear air conditions and above the cloud base for in-
cloud conditions for a representative tropical environment with favourable moisture
supply. A representative cloud-base height is considered to be 1000 m above sea
level (a.s.l), and the corresponding meteorological parameters are surface pressure
1000 mb, surface temperature 30 °C, relative humidity at the surface 80 % and tur-
bulent length scale 1 cm. The values of the latent heat of vaporisation L V and the
specific heat of air at constant pressure C p are 600 and 0.24 cal gm −1 , respectively.
The ratio values of m w / m 0 , where m 0 is the mass of the hygroscopic nuclei per unit
volume of air and m w is the mass of water condensed on m 0 , at various relative
humidities as given by Winkler and Junge ( 1971 , 1972 ) have been adopted and the
value of m w / m 0 is equal to about 3 for relative humidity of 80 %. For a representa-
tive value of m 0 equal to 100 µg m −3 , the temperature perturbation θ ′ is equal to
0.00065 °C, and the corresponding vertical velocity perturbation (turbulent) w * is
computed and is equal to 21.1 × 10 −4 cm s −1 from the following relationship between
the corresponding virtual potential temperature θ v , and the acceleration due to grav-
ity g , which is equal to 980.6 cm s −2 :

g

′

w =

θ θ
v

.

*

Heat generated by the condensation of water equal to 300 μg on 100 μg of hygroscopic
nuclei per metre cube, say in 1 s, generates vertical velocity perturbation w * (cm s −2 )
equal to 21.1 × 10 −4 cm s −2 at surface levels. Since the time duration for water vapour
condensation by deliquescence is not known, in the following it is shown that a value
of w * equal to 30 × 10 −7 cm s −2 , i.e. about three orders of magnitude less than that
shown in the above example is sufficient to generate clouds as observed in practice.
From the logarithmic wind-profile relationship (Eq. 1.4) and the steady state
fractional upward mass flux f of surface air at any height z (Eq. 1.8), the correspond-
ing vertical velocity perturbation W can be expressed in terms of the primary verti-
cal velocity perturbation w * as (Eq. 1.6):

Wwfz = ,

*

W may be expressed in terms of the scale ratio z as follows:
From Eq. (1.8),

2
π

f = ln . z

z

Therefore,

2

z zw z

2

Wwz = ln = ln . z