A simple numerical model is used to investigate the warming of the mixed layer during the early summer. The structure consists of two layers separated by a thermocline, across which there is both downward diffusion of heat and upward entrainment of the lower layer by the warmer upper layer. Assuming that the diffusion rate in the thermocline varies with Richardson number (Ri) according to Munk and Anderson's (1948) formulation, it follows that the heat flux through the thermocline is small for either very small or very large gradients of temperature (i.e., density). The heat flux is maximum at a temperature gradient corresponding to a 'limiting Richardson number' Ri*=0.6. This maximum value of heat flux is proportional to the vertical velocity shear squared. Model results for a shallow area show that, for low surface heat fluxes throughout the early summer, weak warming occurs uniformly at all depths in proportion to the heat flux. For high surface heat fluxes, much stronger warming occurs, but it is confined to the surface layer. As a consequence the early summer warming of the cold bottom water actually will be decreased if the surface heat flux is increased beyond the maximum heat flux in the thermocline. If a 6-day period of strong surface heating were to occur in the early spring, the model would predict that the resulting strong thermocline would persist for the remainder of the summer, allowing only 27% of the entire spring surface heat flux to penetrate to the lower layer. If the 6-day period of strong surface heating were postponed until the end of the early summer, 52% more heat would be able to penetrate the thermocline during the entire spring. It is concluded that conditions favorable for the formation of a cold pool can prevail if the early summer surface heating is stronger or if a brief strong heating event occurs soon after the spring surface heating begins.