Sprite discharges above thunderclouds at altitudes of 40-90 km (refs1-5) are usually created by a strong positive cloud-to-ground lightning flash6. Sometimes these sprite discharges emerge from a visible halo 5,7-9, and during the first stage they always propagate downwards and branch on their way 5,7,9-11. Modelling efforts have been restricted to conditions of non-ionized air of constant density and show double-headed sprites12 or sprites starting from metal electrodes, but they do not explain why observations exclusively record sprites that propagate downwards. Here we present simulations with a numerical discharge model on a non-uniform, dynamically adapted computational grid13 to capture the wide range of emerging spatial scales, and we use realistic air and electron densities that vary with altitude. Our model shows a downward-propagating screening-ionization wave in the lower ionosphere that sharpens and collapses into a sprite streamer as it propagates farther down. Streamer velocity, diameter and length until branching agree with observations9 within measuring accuracy. We speculate that sprites generically emerge through the collapse of a wide screening-ionization wave into a sprite streamer, although this wave is only sometimes visible as a luminous halo. © 2009 Macmillan Publishers Limited. All rights reserved.