Here’s a paper by Liu et al. deriving a theoretical relative dispersion based on the aerosol concentration and updraft velocity.

Analytical expression for the relative dispersion of the cloud droplet size distribution (Subscription required)

Abstract: An analytical expression that relates the relative dispersion (ratio of standard deviation to mean radius) of the cloud droplet size distribution to CCN spectra and updraft velocity is derived from adiabatic growth theory of cloud droplets. Coupled with the Twomey expression for droplet concentration, the analytical expression is used to examine the relationship of relative dispersion to droplet concentration under different combinations of CCN spectra and updraft velocities. These analytical results compare favorably with the corresponding simulations of an adiabatic parcel model. The analytical expression theoretically demonstrates that an increase in aerosol loading (CCN concentration) leads to concurrent increases in the droplet concentration and relative dispersion whereas a larger updraft velocity leads to a higher droplet concentration but a smaller relative dispersion.

Liu, Y., P. H. Daum, and S. S. Yum (2006), Analytical expression for the relative dispersion of the cloud droplet size distribution, Geophys. Res. Lett., 33, L02810, doi:10.1029/2005GL024052.

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