Relations have been developed for the entrainment of a turbulent layer into a stratified medium under the action of a surface stress. A constant gradient Richardson number J is assumed at the interface, and the solutions are assumed to be self-similar. It has been shown that the requirements of similarity are that the scales of the (mean and turbulent) velocity and temperature should all be powers of time t. The cases of linear and two-layer density distributions have been given special attention, and a constant of proportionality in each case is determined by comparison with numerical solutions. The results are identical to those obtained by Pollard et al.--type slab models if Rv? 2J, where Rv = buoyance/(mean velocity)2. The nondimensional entrainment rate for the two-layer case is nearly twice that for linear stratification, as observed. The density gradient in the two-layer case is proportional to t-2, which explains the observed gradual disappearance of the density step. Numerical solutions show that the self-similarity assumption is quite good with two layers and is approximately valid with linear stratification if 100500, nonlinear mechanisms create a sharp interface with linear stratification. Thus whereas a two-layer experiment destroys the initial sharp interface, an experiment with initially linear stratification creates a sharp interface.