Top: position (relative to the top of the box) of the mean ionization front as a function of time for the 3D turbulent simulations (at Mach 1, 2, and 4 in solid blue) of Tremblin et al. (2012) and the corresponding analytical solutions (dotted red) and 1D simulations (dashed green) in which the turbulence is taken into account by an effective temperature/sound speed. Bottom: snapshots of the column density of the 3D simulations at a time of 1 Myr after the ionization is switched on at the top of the box. The blue-dashed lines show the mean position of the ionization front.