Absorption of sunlight forms a diurnal warm layer (DWL) in the afternoon at the surface and upper meters of the ocean when wind is weak. Analyses using models and remote sensing data disagree on the frequency of DWLs stronger than 1 °C. In situ time series in the central Indian Ocean showed the DWL exceeded 1 °C for 24% of days in October-December 2011 during the Dynamics of the Madden Julian Oscillation (DYNAMO) experiment. For 4 days with mean wind of 1.4 m/s, the DWL was 1.5-2.5 °C. We observed atmospheric turbulence over the DWL using Doppler lidar. Mixed layer turbulence is convective, generated mostly by surface buoyancy flux. The turbulent kinetic energy dissipation of the convective marine atmospheric mixed layer scales with surface buoyancy flux like diurnal convective boundary layers over land and convective mixed layers in the ocean and lakes. This convection in the afternoon is out of phase with buoyancy flux from nocturnal atmospheric net radiative cooling. The afternoon atmospheric convective turbulence over the tropical ocean mixes heat and humidity from the ocean to the lifted condensation level of clouds.