The mechanisms responsible for the extreme dryness of the stratosphere have been debated for decades. A key difficulty has been the lack of models which are able to reproduce the observations. Here we examine results from a new atmospheric chemistry general circulation model (ECHAM5/MESSy1) together with satellite observations. Our model results match observed temperatures in the tropical lower stratosphere and realistically represent recurrent features such as the semi-annual oscillation (SAO) and the quasi-biennual oscillation (QBO), indicating that dynamical and radiation processes are simulated accurately. The model reproduces the very low water vapor mixing ratios (1–2 ppmv) periodically observed at the tropical tropopause near 100 hPa, as well as the characteristic tape recorder signal up to about 10 hPa, providing evidence that the dehydration mechanism is well-captured, albeit that the model underestimates convective overshooting and consequent moistening events. Our results show that the entry of tropospheric air into the stratosphere at low latitudes is forced by large-scale wave dynamics; however, radiative cooling can regionally limit the upwelling or even cause downwelling. In the cold air above cumulonimbus anvils thin cirrus desiccates the air through the sedimentation of ice particles, similar to polar stratospheric clouds. Transport deeper into the stratosphere occurs in regions where radiative heating becomes dominant, to a large extent in the subtropics. During summer the stratosphere is moistened by the monsoon, most strongly over Southeast Asia.