Using successive versions of a global climate model, we show how convective transport to the free troposphere of the humidity evaporated at the surface or, reciprocally, entrainment of dry air from the free troposphere into the mixed layer, controls surface evaporative cooling and then sea surface temperature. This control is as important as the radiative effect of boundary layer clouds on radiation. Those aspects are shown to be improved when activating a mass flux representation of the organized structures of the convective boundary layer coupled to eddy diffusion, the so-called "thermal plume model," leading to an increased near-surface drying compared to the use of turbulent diffusion alone. Controlling detrainment by air properties from just above the boundary layer allows the thermal plume model to be valid for both cumulus and stratocumulus regimes, improving the contrast in near-surface humidity between the trade winds region and East Tropical oceans. Using pairs of stand-alone atmospheric simulations forced by sea surface temperature and of coupled atmosphere-ocean simulations, we show how the improvement of the surface fluxes that arise from this improved physics projects into an improvement of the representation of sea surface temperature patterns in the coupled model, and in particular into a reduction of the East Tropical Ocean warm bias. The work presented here led to the bias reduction in sea surface temperature in the Institute Pierre Simon Laplace coupled model, IPSL-CM6A, developed recently for the 6th phase of the Coupled Model Intercomparison Project, CMIP6. Plain Language Summary Global numerical models used to anticipate the future of our climate under global warming still suffer from significant errors, some of which are shared among all models. Among those shared errors is the tendency to predict too warm sea surface temperature over the east part of the tropical ocean in the tropics. We show how a better representation of the vertical convective transport in the first km above sea surface improves the representation of transport of dry air from the free troposphere. The drying of the near surface increases evaporation at the surface, that in turn contributes to significantly cool the sea surface in those regions.