A Theoretical Power-Law for the Size Distribution of Small Particles or Drops falling through the Atmosphere

A theoretical study is made of the size distribution of small particles or drops in the atmosphere as produced by vertical motion and coalescence. It is demonstrated that a power-law distribution is a possible solution both in the time-dependent case, when conditions are assumed to be spatially homogeneous, and in the steady state case. The theory is in special applied to cloud droplets falling and coalescing under the effect of gravity and Stokes friction. In this case it is found that the frequency distribution falls off as r⁻⁵ in the time-dependent case and as r⁻³ in the steady state case, r being the radius. A comparison with observed size distributions of droplets in the range 20–80 microns given by Weickmann and aufm Kampe indicate that, on an average, the r⁻⁵-distribution and the r⁻³-distribution are reasonable approximations for fair weather cumulus clouds and for cumulus congestus- and cumulonimbus clouds, respectively. In the time-dependent case the theory predicts that the concentration of droplets of a given size should change inversely to a linear function of time. After a definite time T, infinite concentrations are reached and the model breaks down. This characteristic growth time is inversely proportional to the initial concentration of particles. For the mean concentration of cloud droplets given by Weickmann and aufm Kampe, T takes on values of the order 150 seconds, assuming that only droplets which differ in radii less than 10 times coalesce.