We use a very simple model with no adjustable parameters to simulate the increase in relative humidity between 11 and 14 km in the tropics. The rate of increase in relative humidity, in this interval, appears to be largely determined by the shape of the convective detrainment profile, the temperature profile, and the level of zero radiative heating. It appears to be unaffected by evaporative moistening or the complex three-dimensional structure of convective systems. We also show that the rapid increase in the fraction of supersaturated air parcels above 150 mb (~14.5 km) is associated with a transition from large scale descent to ascent.