Inverse problem of thermal convection: numerical approach and application to mantle plume restoration

Modern seismic tomographic images of the Earth’s interior facilitate the inference of the complex trajectories of present-day convective flow in the upper mantle. Quantitative reconstruction of both the observed mantle structure and temperature field backwards in time requires a numerical tool for solving the inverse problem of thermal convection at infinite Prandtl number. In this paper we present a variational approach to three-dimensional numerical restoration of thermoconvective mantle flow with temperature-dependent viscosity. This approach is based on a search for the mantle temperature and flow in the geological past by minimizing differences between present-day mantle temperature derived from seismic velocities (or their anomalies) and that predicted by forward models of mantle flow for an initial temperature guess. The past mantle temperatures so obtained can be employed as constraints on forward models of mantle dynamics. To demonstrate the applicability of this technique, we restore numerically a fluid dynamic model of the evolution of upper mantle plumes and show that the initial shape of the plumes can be accurately reconstructed. We then model the evolution of the plumes forward in time (plume upbuilding) starting from the restored state to the state they were before the restoration and demonstrate the high accuracy of the model predictions. We also show that the neglect of thermal diffusion in the backward modeling of thermal plumes (in order to simplify the numerical procedure) results in erroneous restorations of the plumes.