We present a mean-field model that describes droplet growth due to condensation and collisions and droplet loss due to fallout. The model allows for an effective numerical simulation. We study how the rain initiation time depends on different parameters. We also present a simple model that allows one to estimate the rain initiation time for turbulent clouds with an inhomogeneous concentration of cloud condensation nuclei. In particular, we show that over-seeding even a part of a cloud by small hygroscopic nuclei one can substantially delay the onset of precipitation.
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