A general strategy is presented for the study of convection in a turbulent-fluid system, such as the Earth’s core, in which the adiabatic density differences across the system are much larger than the density differences that drive the convection. This situation is drastically different from the laboratory, where the density differences due to convection are the greater, and where the Boussinesq approximation is valid. In the case considered here, the anelastic approximation deals satisfactorily with the large basic density differences across the system. Turbulent transport of large scale fields such as entropy is evaluated through the application of a local description of turbulence. The resulting theory is applied to the Earth’s core, and the system of equations obtained by Braginsky and Roberts (Geophys. Astrophys. Fluid Dynam. 79, 1, 1995) is recovered; insights that have emerged since that paper was written are added.