Mixing by time-dependent convection

Abstract

Two-dimensional numerical flow models are employed to study the mixing of passive heterogeneities in the earth's mantle. Both simple kinematic models of time-dependent flow and dynamic convection models are studied. Initially small blocks of tracers are inserted into the flow and their progressive dispersal is monitored through time. Some of the tracers also serve as infinitesimal strain markers. Three different regimes of mixing are observed: (A) all heterogeneities are rapidly strained and stirred according to a turbulent mixing law, where the size of a blob decreases exponentially with time; (B) most heterogeneities are only slowly strained and possibly follow a laminar mixing law with a linear decrease with time of the size of a heterogeneity; (C) the size reduction proceeds rapidly in part of the fluid, but some poorly stirred “islands” persist for a long time. All three regimes are found in various kinematic flow models, while in the dynamic convection models only regime A is observed. The effective strain rate in the exponential (turbulent) mixing law is more than an order of magnitude smaller than the average flow strain rate. When the results from the dynamic convection models with variable viscosity are scaled to the upper mantle, mixing over small length scales would be achieved after a few hundred million years. For whole-mantle convection with a significant increase of viscosity with depth, mixing times could be in the order of 1–2 Ga. In the latter case it appears marginally possible that part of the isotopic heterogeneity observed in oceanic basalts has evolved within the convecting mantle rather than in physically separated reservoirs.