Universal Spectrum for Atmospheric Suspended Particulates: Comparison with Observations: Data Set I - Rain Formation in Warm Clouds: General Systems Theory - page 57

available. Many microphysical mechanisms are still not understood in quantitative
detail (Pruppacher and Klett 1997 ).
Although the relative humidity of clouds and fogs usually remains close to 100 %,
considerable departures from this value have been observed. The spatial and tempo-
ral nonuniformity of the humidity inside clouds and fogs results in a corresponding
rapid spatial variation of the concentration of cloud drops and the cloud liquid water
content. Based on his observations, Warner ( 1969 ) suggested that bimodal DSDs
are the result of a mixing process between the cloud (cumulus) and the environ-
ment. Warner proposed that the mixing process producing the bimodality is due
mostly to entrainment of drier air at the growing cloud top, and to a lesser degree,
to entrainment at the cloud edges. The size distribution experiences a broadening
effect with increasing distance from cloud base. Spectra with double maxima have
also been observed by others in other regions. If we consider the spatial distribution
of the drop size, number concentration, and liquid water content, we find strongly
inhomogeneous conditions. The cloud liquid water content w L varies rapidly over
short distances along a horizontal flight path in a manner which is closely related
to the variation of the vertical velocity in the cloud and also w L varies essentially as
the total number concentration of drops. Vulfson et al. ( 1973 ) demonstrate that the
cloud water content typically increases with height above the cloud base, assumes
a maximum somewhere in the upper half of the cloud, and then decreases again
toward the cloud top. In most cases, a comparison between the observed cloud
water content w L and adiabatic liquid water content (w L ) ad computed on the basis
of a saturated adiabatic ascent of moist air shows that generally w L < (w L ) ad . In most
cases, w L /(w L ) ad is found to decrease with increasing height above cloud base but
to increase with cloud width. This implies that the entrainment is especially pro-
nounced near the cloud top, while the net dilution effect by entrainment is less in
wider clouds than narrower ones.

3.3.2 Formulations for DSDs in Clouds and Fog

For many fog and cloud modeling purposes, it is necessary to be able to approxi-
mate the observed DSD by an analytical expression. Fortunately, DSDs measured
in many different types of clouds and fogs under a variety of meteorological condi-
tions often exhibit a characteristic shape. Generally, the concentration rises sharply
from a low value to a maximum, and then decreases gently towards larger sizes.
Such a characteristic shape can be approximated reasonably well by either a gamma
distribution or a lognormal distribution. In order to describe a DSD with two or
more maxima, one or more unimodal distributions may be superposed. As an ex-
ample, according to Khrgian and Mazin ( 1952 )(in Borovikov et al. ( 1963 ), many
DSDs with a single maximum may also be quite well be represented by a gamma
distribution. Another convenient representation of the cloud DSD is the empirical
formula developed by Best ( 1951a , 1951b ). These various analytical expressions
only represent average distributions. Individual drop size spectra may be signifi-
cantly different (Pruppacher and Klett 1997 ). A wealth of aircraft measurements in