Linear relation between convective cloud drop number concentration and depth for rain initiation

Coalescence of cloud droplets is essential for the production of small raindrops at a given vertical distance above the cloud base (D-p). The rate of droplet coalescence is determined mainly by droplet size, spectrum width and concentrations. The droplet condensational growth is determined by the number of activated CCN (N-a) and height above cloud base. Here we show that when the droplet mean volume radius, r(v), exceeds similar to 13 mu m, or when droplet effective radius (r(e)) exceeds similar to 14 mu m, considerable precipitation mass (>0.03 g kg(-1)) is likely to be present in growing convective clouds. This is because the rate of droplet coalescence is proportional to similar to r(v)(5) which practically implies the existence of a threshold r(v) above which efficient warm rain formation can occur, and also because the vertical profile of r(v), even in diluted clouds, nearly follows the theoretical adiabatic condensational growth curve. The small observed deviations are mainly caused by deviations from purely inhomogeneous mixing which cause partial droplet evaporation. Consequently, D-p must theoretically change nearly linearly with N-a. This is confirmed here observationally, where increasing N-a by 100 per milligram (approximate to cm(3) at cloud base) of air, resulted in an increase of similar to 280 m in D-p for both Israeli and Indian deep convective clouds. This means that in highly polluted clouds or where strong cloud-base updrafts occur, clouds have to grow well above the freezing level, even in tropical atmosphere, before precipitation forms either by warm or by mixed-phase processes.