Gross thermodynamics of two-component core convection

We model the inner core by an alloy of iron and 8 per cent sulphur or silicon and the outer core by the same mix with an additional 8 per cent oxygen. This composition matches the densities of seismic model, Preliminary Reference Earth Model (PR-EM). When the liquid core freezes S and Si remain with the Fe to form the solid and excess 0 is ejected into the liquid. Properties of Fe, diffusion constants for S, Si, 0 and chemical potentials are calculated by first-principles methods under the assumption that S, 0, and Si react with the Fe and themselves, however, not with each other. This gives the parameters required to calculate the power supply to the geodynamo as the Earth's core cools. Compositional convection, driven by light O released at the inner-core boundary on freezing, accounts for half the entropy balance and 15 per cent of the heat balance. This means the same magnetic field can be generated with approximately half the heat throughput needed if the geodynamo were driven by heat alone. Chemical effects are significant: heat absorbed by disassociation of Fe and 0 almost nullify the effect of latent heat of freezing in driving the dynamo. Cooling rates below 69 K Gyr(-1) are too low to maintain thermal convection everywhere; when the cooling rate lies between 35 and 69 K Gyr(-1) convection at the top of the core is maintained compositionally against a stabilizing temperature gradient; below 35 K Gyr(-1) the dynamo fails completely. All cooling rates freeze the inner core in less than 1.2 Gyr, in agreement with other recent calculations. The presence of radioactive heating will extend the life of the inner core, however, it requires a high heat flux across the core-mantle boundary. Heating is dominated by radioactivity when the inner core age is 3.5 Gyr. We, also, give calculations for larger concentrations of O in the outer core suggested by a recent estimation of the density jump at the inner-core boundary, which is larger than that of PREM. Compositional convection is enhanced for the higher density jumps and overall heat flux is reduced for the same dynamo dissipation, however, not by enough to alter the qualitative conclusions based on PREM. Our preferred model has the core convecting near the limit of thermal stability, an inner-core age of 3.5 Gyr and a core heat flux of 9 TW or 20 per cent of the Earth's surface heat flux, 80 per cent of which originates from radioactive heating.