Assuming local thermodynamic equilibrium in the fluid, an expression is derived for the rate of destruction of the mean square of the temperature fluctuations by radiative transfer of heat. This takes a particularly simple form (a) if the fluid is effectively transparent over distances equal to the scale of the turbulent motion, when the effect appears as a decay time for temperature fluctuations from the mean, and (b) if the fluid is effectively opaque, when the effect is of an increased conductivity due to radiation. A theory of the interaction of the temperature and velocity fields developed in a previous paper shows that, if the radiative effects are relatively weak, a sudden collapse of the turbulent motion occurs while the flux Richardson number is still less than one. If the radiative effects are strong, the turbulent intensity approaches zero as the flux Richardson number approaches one. The effects of radiation are always to increase the critical value of the ordinary Richardson number. Criteria for fully turbulent motion of an unrestricted flow are given in terms of the gradients of mean velocity and mean temperature and of the rate of radiative cooling. The relevance of these calculations to motions of the atmosphere is briefly discussed.