Modelling the isotopic evolution of the Earth

We present a flexible multi-reservoir (primitive lower mantle, depleted upper mantle, upper continental crust, lower continental crust and atmosphere) forward-transport model of the Earth, incorporating the Sm-Nd, Rb-Sr, U-Th-Pb-He and K-Ar isotope-decay systematics. Mathematically, the model consists of a series of differential equations, describing the changing abundance of each nuclide in each reservoir, which are solved repeatedly over the history of the Earth. Fluxes between reservoirs are keyed to heat production and further constrained by estimates of present-day fluxes (e.g. subduction, plume flux) and current sizes of reservoirs. Elemental transport is tied to these fluxes through 'enrichment factors', which allow for fractionation between species. A principal goal of the model is to reproduce the Pb-isotope systematics of the depleted upper mantle, which has not been done in earlier models. At present, the depleted upper mantle has low (238)U/(204)Pb (mu) and (232)Th/(238)U (kappa) ratios, but Pb-isotope ratios reflect high time-integrated values of these ratios. These features are reproduced in the model and are a consequence of preferential subduction of U and of radiogenic Pb from the upper continental crust into the depleted upper mantle. At the same time, the model reproduces the observed Sr-, Nd-, Ar- and He-isotope ratios of the atmosphere, continental crust and mantle. We show that both steady-state and time-variant concentrations of incompatible-element concentrations and ratios in the continental crust and upper mantle are possible. Indeed, in some cases, incompatible-element concentrations and ratios increase with time in the depleted mantle. Hence, assumptions of a progressively depleting or steady-state upper mantle are not justified. A ubiquitous feature of this model, as well as other evolutionary models, is early rapid depletion of the upper mantle in highly incompatible elements; hence, a near-chondritic Th/U ratio in the upper mantle throughout the Archean is unlikely. The model also suggests that the optimal value of the bulk silicate Earth's K/U ratio is close to 10000; lower values suggested recently seem unlikely.