Theoretical models are constructed with the aim of relating, explaining and predicting features of a radiatively active turbulent cloud layer over the sea and under a strong subsidence inversion. Both dry aerosol clouds (no phase change) and wet clouds (with a phase change and latent heat exchanges) are considered. For the wet cloud case an important element of the theory is the requirement that the wet-bulb potential temperature must increase upwards in the inversion. For both cases entrainment of the upper warm air is hypothesized to lie between upper and lower limits determined from the turbulent energy budget. The dry cloud case is solved for both steady state and transient results, with only the transient behaviour depending on the entrainment hypothesis. Only steady state solutions are presented for the more complex wet cloud case and these differ somewhat for the maximum and minimum entrainment limits. Observational data from Oakland, California are used for comparison with those steady state solutions, with results indicating the essential validity of the approach. Detailed comparisons, especially for determination of the most correct entrainment rate, are hampered both by inadequate measurement of the inversion properties and by uncertainties in the net radiation flux leaving the cloud top. Computations of the latter suggest that several presently used radiation models are still in serious disagreement, at least for application to downward flux under an inversion. It is suggested that the present theory provides a partial explanation of the origin of the trade wind inversion.