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The mean state of the atmosphere and ocean is set through a balance between external forcing (winds, radiation, heat and freshwater fluxes) and the emergent turbulence, which transfers energy to dissipative structures. The forcing gives rise to jets in the atmosphere and currents in the ocean, which spontaneously develop turbulent eddies through the baroclinic instability. A critical step in the development of a theory of climate is to properly include the resulting eddy-induced turbulent transport of properties like heat, moisture, and carbon. The baroclinic instability generates flow structures at the Rossby deformation radius, a length scale of order 1000 km in the atmosphere and 100 km in the ocean, smaller than the planetary scale and the typical extent of ocean basins respectively. There is therefore a separation of scales between the large-scale temperature gradient and the smaller eddies that advect it randomly, inducing effective diffusion. Numerical solutions of the two-layer quasi-geostrophic model, the standard model for studies of eddy motions in the atmosphere and ocean, show that such scale separation remains in the strongly nonlinear turbulent regime, provided there is sufficient bottom drag. We compute the scaling-laws governing the eddy-driven transport associated with baroclinic turbulence. First, we provide a theoretical underpinning for empirical scaling-laws reported in previous studies, for different formulations of the bottom drag law. Second, these scaling-laws are shown to provide an important first step toward an accurate local closure to predict the impact of baroclinic turbulence in setting the large-scale temperature profiles in the atmosphere and ocean.