Colin McKellar
MET 215
Midterm
10/27/10
The Effects of Fluctuations in Liquid Water Content on the Evolution of Large Drops by Coalescence
Previous research into the evolution of large drops by coalescence in explaining rain
development assumed that the liquid water content was uniformly distributed within a cloud. This
assumption is used in the continuous growth equation and the Bowen Model to describe the formation
of large drops. Despite the advancements in describing large drop growth in these methods, the
assumption made concerning a uniform drop distribution within a cloud leads to inaccuracies in
explaining the time it takes for large drops to form. Thus, the main propose of this paper is to describe
first; the sensitivity of the coagulation process to the liquid water content within a cloud and two, to
describe the coagulation process in a cloud with spatially varying liquid water content.
The coagulation equation, also known as the stochastic coagulation equation, is used to describe
the rate of change of droplet concentration in a size interval dv. Also contained in this equation is the
coagulation coefficient K. This coefficient can be described as the probability of a drop with volume v,
collecting a drop of volume u. The first term in this equation describes
the total number of coalescences in a unit time that is experienced by the drops in the size interval. In
effect, this will account for all the possible coalescences of the drops in dv by the contribution of the
large and small droplets. The net effect of this term is to reduce the number of drops within the volume
since more coalescence will lead to fewer drops. The second term of the equation accounts for the
increase in the number of drops within the volume by the coalescences of all pairs of smaller drops. The
individual volumes of the smaller drops then sum to the total volume.
Contained within the coagulation equation is the expression for liquid water content (w). This
describes how the number of droplets in the volume will vary with a change in w. Thus, an increase in w
is due to either an increase in the number or size of the droplets. This makes sense since w is a function
of droplet distribution and concentration. This relationship will lead to two possibilities that can occur.
The first possibility is that changes in w are produced by changes in the number of drops. When this
assumption is used in the coagulation equation, the result is a dependence between w and time where a
change in the liquid water content is proportional to a change in time. As a result, any increase in the
liquid water content within a given volume will lead to a faster growth rate of the droplets.
A second possibility for w is to allow for the concentration and shape of the droplet distribution
to be independent of w. However, any changes in the drop volume will remain proportional to a change
in w. When this assumption is used in the coagulation equation, the first effect is on the coagulation
coefficient K. The coefficient is now computed as the product of the liquid water content with both of
the two individual droplet volumes. In this process, the size of the kernel is increased since the cross
section, falling speed, and collection efficiency will also have to increase since these are functions of
droplet size. However, if the collection efficiency is assumed to be constant, then w will be valid in the
Stokes law region. As a consequence of this relationship, the kernel will be proportional to the fourth
power of the radius and to the four thirds power of the liquid water content. In addition, this allows for
an examination into the effect of different values of the kernel size and its effect on coalescence. The
end result of scaling the coagulation coefficient with w is that as the liquid water content increases, the
time needed for the growth of the droplet decreases.
The main conclusion for both possibilities is when the distribution is changed statistically by the
stochastic coagulation equation; the time it takes for a droplet to evolve is faster as opposed to when
the droplet distribution is held constant. This process occurs in a highly non-linear way since
coalescence of the smaller drops will take a longer period of time as opposed to large drops as described
in figure 1. This makes sense since a larger drop will “sweep out” a wider area and thus have a higher
probability of collecting droplets through coalescence. If there is a higher probability of coalescence
occurring, then the time it takes for the droplet to grow is decreased.
Previous research into liquid water content distribution from the Telford model has shown that
liquid water concentration varies greatly within a cloud. This variance can be described by Poisson`s
probability law. It is also understood that the liquid water content in much less than the given adiabatic
value through the ascent in the cloud due to mixing of dry air. Despite the low frequency at which high
levels of the liquid water content occur within a cloud, the impact of such a distribution is profound in
large drop evolution. Figure 2 shows that as the liquid water content increases, the total number of
large drops produced will increase and will occur at a faster pace. The primary conclusion of this result
is that growth of large drops is dominated by this process of higher than average liquid water content.
In conclusion, it has been shown that regions of higher than average liquid water content likely
exist within clouds. The reason for this is most likely due to the non-linearity of the mixing process. The
production of large droplets within high areas of liquid water is greater owing to higher probability of
coagulation occurring within such areas. The impact of this is to produce a high level of large drops in a
very quick manner and thus account for the total amount of large drops produced in short time scales.