A wind-tunnel simulation of an atmospheric boundary layer, artificially thickened as is often used in neutral flow wind-loading studies, has been investigated for weakly unstable stratification, including the effect of an overlying inversion. Rather than using a uniform inlet temperature profile, the inlet profile was adjusted iteratively by using measured downstream profiles. It was found that three cycles are sufficient for there to be no significant further change in profiles of temperature and other quantities. Development to nearly horizontally-homogeneous flow took a longer distance than in the neutral case because the simulated layer was deeper and therefore the length scales larger. Comparisons show first-order and second-order moments quantities are substantially larger than given by 'standard forms' in the mixed layer but are close in the surface layer. Modified functions, obtained by matching one to the other, are suggested that amount to an interpolation in the mixed layer between the strongly unstable and the weakly unstable cases.