In the simplest case - quasi-geostrophic turbulence in two layers - theory suggests that a diffusive theory may approximately hold, with a diffusivity that increases rapidly with supercriticality so resembling in some ways an adjustment. However, in the earth's atmosphere the eddy scale is barely larger than the deformation scale and an inverse cascade, if it exists at all, is evidently of limited extent. It is not clear whether this is a general result, or is particular to a certain set of parameters, including those of the Earth's atmosphere. We have investigated this problem in perhaps the simplest framework that allows changes in stratification, namely a two-level primitive-equation model on the beta plane. Our results suggest that quasigeostrophic theory works reasonably well, at least in some parameter regimes, provided that the stratification is diagnosed from the primitive-equation model. In particular, for some parameter regimes the mean flow is found to be supercritical and eddies are created that are considerably larger than the deformation radius, suggesting a vigorous inverse cascade. To go beyond quasi-geostrophic theory one also needs a theory for the stratification and we propose a closure for the eddy vertical heat flux assuming that the eddy mixing slope scales with the isentropic slope. This captures reasonably well the empirical dependency of the flux, and provides full closure to the non-geostrophic problem.