The region of the tropical atmosphere between the main convective outflow level around 200--150 hPa and the cold point, the substratosphere, shares properties of the troposphere and the stratosphere. An analogous region exists in single-column radiative-convective models between the top of the convectively adjusted region and the cold point, so a radiative-convective model has been used to investigate the radiative and dynamical factors that allow a substratosphere to exist and to which it is sensitive. A key result is that localized heating in the 15 ¿m CO2 band, owing to the sharp curvature in the temperature profile near the convection top, forces apart the cold point and convection top creating the substratosphere. A possible radiation-convective-transport feedback involving O3 is identified that could amplify the response of the substratosphere and cold point to changes in forcing such as sea surface temperature or stratospheric meridional circulation. A comparison of timescales suggests that the substratosphere is characterized by (1) a radiative timescale shorter than the convective timescale, so that radiation dominates in setting the temperature structure, but (2) a convective transport timescale shorter than the timescales for other processes affecting O3, so that the import of air from the boundary layer keeps O3 mixing ratios low there.