The influence of entrainment on the evolution of cloud droplet spectra: I. A model of inhomogeneous mixing

In this, the first of two related papers, we present calculations of the growth of a population of condensate droplets rising above cloud base within small cumuli which are entraining undersaturated environmental air. It is assumed, on the basis of dimensional arguments and laboratory experiments on entrainment, conducted within a cloud droplet evolution tunnel, that this mixing process is inhomogeneous. In the extreme situation to which the calculations apply undersaturated air is entrained in a stream, or in blobs, and some droplets of all sizes are completely removed from the condensate spectrum by evaporation, while others do not change in size. This is equivalent to assuming that the time constant for turbulent mixing (τ τ ) is large relative to that for droplet evaporation (τ r ), and is thus the antithesis to the homogeneous model utilized by other workers, which assumes implicitly that τ τ /τ r = 0. The calculations based on the extreme inhomogeneous model produce spectral shapes which agree well with those reported in cumulus by Warner (1969a) and indicate that a small proportion of the droplets can grow several times faster through the condensate spectrum than classical theory predicts.