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cumuliform cloud curling around and trapping pockets of unsaturated air inside
the main body of the cloud. In 1947, Henry Stommel published a classic paper in
which he suggested that entrainment into clouds occurs and described a method
for calculating its effects. If the air outside the buoyant cloud is cooler or drier
than it is inside the cloud (by definition the air outside a cloud must be drier),
then the cloud's buoyancy is reduced.
Consider the vertical mass flux in a steady-state plume:
R
2
M
¼
w
ð
3
:
1
Þ
where R is the radius of the plume; and w is the mean updraft velocity. It is seen
from (2.189) that the fractional change of mass flux with height is
=ð
wR
2
ð
1
=
M
Þ
dM
=
dz
¼½
1
Þ
2R
w
¼
2
=
R
ð
3
:
2
Þ
So, the entrainment rate, which is defined by the fractional rate of change of
vertical mass flux (1
Dt), is inversely proportional to the width of the
buoyant air parcel in a steady-state plume.
In an entraining thermal bubble, entrainment is given by the fractional rate of
increase in volume V as the thermal bubble rises. So, from (2.200) it follows that
=
MDM
=
R
2
R
3
4
1
=
MdM
=
dz
¼ð
1
=
w
Þð
1
=
V
Þ
DV
=
Dt
¼ð
1
=
w
Þð
4
w
Þ=ð
3
Þ¼
3
=
R
ð
3
:
3
Þ
This entrainment rate for a thermal is also inversely proportional to the width of
the buoyant air parcel. Wide air parcels are protected from the potentially deleter-
ious effects of a cooler environment because mixing at the outer edges is felt less
and less the farther inward you go; narrow parcels are not protected, because
mixing at the outer edges easily reaches the center of the parcel. At the top of the
cloud, where vertical velocity decreases to zero, the parcel spreads out into an
anvil (
Figure 3.5
) as an artificial wall of sorts (a wall of non-buoyant or negatively
buoyant air) prevents any air from rising any further, so that air is forced to
spread laterally as a consequence of continuity. This wall may be at
the
tropopause, or below at stable layers.
The entrainment rate (1
dz) is a function of vertical velocity (cf. (3.1)).
For the plume and thermal models, however, vertical velocity does not explicitly
appear in the expression for the entrainment rate because the vertical derivative of
the mass flux is proportional to vertical velocity. In general (i.e., not in idealized,
steady-state plumes or thermals), however, the faster an air parcel moves upward,
the less time it has to be diluted by environmental air. So, the lapse rate of the
cloud-bearing layer can have an effect on whether or not a cloud can be initiated:
if the rate of increase in buoyancy as an air parcel ascends upward is greater than
the rate of reduction in buoyancy owing to the entrainment of environmental air,
then the cloud has a good chance of reaching upper levels in the troposphere if
there are no intervening stable layers. On the other hand, if the rate of increase in
buoyancy is less than the reduction of buoyancy owing to entrainment, then the
cloud will more likely not reach the upper troposphere if there are no intervening
stable layers. The process of formation of a convective cloud such that it is able to
=
MdM
=
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