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Turbulent mixing and energy dissipation have important roles in the global circulation but are not resolved by ocean models. We use direct numerical simulations of a geostrophic circulation, resolving turbulence and convection, to examine the rates of dissipation and mixing. As a starting point, we focus on circulation in a rotating rectangular basin forced by a surface temperature difference but no wind stress. Emphasis is on the geostrophic regime for the horizontal circulation, but also on the case of strong buoyancy forcing (large Rayleigh number), which implies a turbulent convective boundary layer. The computed results are consistent with existing scaling theory that predicts dynamics and heat transport dependent on the relative thicknesses of thermal and Ekman boundary layers, hence on the relative roles of buoyancy and rotation. Scaling theory is extended to describe the volume‐integrated rate of mixing …