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2
π
f
=
ln .
z
z
In the above equation,
f
represents the fraction of the air mass of the surface origin
which reaches the height
z
after dilution by vertical mixing caused by the turbulent
eddy fluctuations.
Considering that the cloud-base level is 1000 m, the value of
R
= 1000 m and the
value of turbulence length scale
r
below cloud base is equal to 100 m so that the nor-
malized length scale
z
=
R
/
r
= 1000 m/100 m = 10, and the corresponding fractional
volume dilution
f
= 0.6.
The value of
q
/
q
a
at the cloud-base level is also found to be about 0.6 by several
observers (Warner
1970
).
The fractional volume dilution
f
will also represent the ratio
q
/
q
a
inside the
cloud. The observed (Warner
1970
)
q
/
q
a
profile inside the cloud is seen (closely) to
follow the profile obtained by the model for dominant eddy radius
r
= 1 m (Fig. 1.4).
It is, therefore, inferred that, inside the cloud, the dominant turbulent eddy radius
is 1 m, while below the cloud base, the dominant turbulent eddy radius is 100 m.
2.1.2
In-Cloud Vertical Velocity Profile
The logarithmic wind-profile relationship (Eq. 1.4) derived for the PBL in Sect. 1.5.2
holds good for conditions inside a cloud because the same basic physical process,
namely MFC, operates in both the cases. The value of vertical velocity inside the
cloud will, however, be much higher than in cloud-free air.
From Eq. (1.6), the in-cloud vertical velocity profile can be expressed as
Wwfz
=
,
*
where
W
is the vertical velocity at height
z, w
∗
is the production of vertical veloc-
ity per second by the MFC at the reference level, i.e. cloud-base level, and
f
is the
fractional upward mass flux of air at level
z
originating from the cloud-base level.
The
f
profile is shown in Fig. 1.4. The vertical velocity profile will follow the
fz
profile assuming
w
∗
is the constant at the cloud-base level during the cloud-growth
period.
2.1.3
In-Cloud Excess Temperature Perturbation Profile
The relationship between temperature perturbation
θ
and the corresponding vertical
velocity perturbation is given as follows:
g
=
θ
θ
0
W
,
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