Mixing of a passive heterogeneity by mantle convection

Accretion and early differentiation processes have left Earth's mantle in a chemically heterogeneous state at the end of the Hadean. Since then, these primordial heterogeneities have been progressively erased by mantle convection stirring. This is well-illustrated by short lived isotopic systems such as ¹⁴⁶Sm-¹⁴²Nd: mantle-derived rocks 2.7 to 4.0 Gy old have been found with measurable anomalies in ¹⁴²Nd/¹⁴⁴Nd, while younger rocks show no detectable deviations from the mantle average. This indicates that convective stirring within the mantle has reduced the level of heterogeneities below the instrumental detection limit in ~1.8 Gy since Earth's formation. These observations have the potential of giving constraints on the mantle stirring rate in the Archean, and therefore on the mantle's dynamical state. However, the survival time of an heterogeneity depends not only on the mixing rate, but also on the initial level of heterogeneity and instrumental detection limit. For these reasons, and also because of the relative scarcity of available data, the observed survival time cannot be simply translated into a mantle stirring time. A quantitative interpretation of the geochemical data in terms of stirring rate requires comparison with a model that can predict the evolution of the probability density function (PDF) of the abundance of a geochemical tracer (or, equivalently, histograms of concentration), as a function of the convective regime and characteristics of the initial heterogeneity. We present here an analytical model for the time evolution of the PDF of a chemical tracer that is initially heterogeneously distributed. The model predictions compare very well with results from numerical simulations. This provides a solid physical basis for interpreting ¹⁴²Nd/¹⁴⁴Nd variations in terms of mantle dynamical state.