Three-dimensional numerical calculations of mantle convection with the two major phase transitions have been carried out in a 5 × 5 × 1 configuration to study the effects of increasing vigor of convection, internal heating and extremely negative Clapeyron slope of the spinel-perovskite phase transition. Depth-dependent properties of thermal expansivity, viscosity and thermal conductivity have been incorporated. Three-dimensional solutions for surface Rayleigh number (Ras) between 2 × 106 and 4 × 108 show that there is a distinct transition between Ras = 4 × 107 and 108 in which the system changes from single-layered to layered convection with the mass flux decreasing to below 10% at Ras = 108. Surface heat flux does not decrease with increasing Ras and with the accompanying decrease in the mass flux at the transition zone. The effects of internal heating are to reduce the wavelengths of the planforms which cause greater degree of layering in the system. Comparison between 2D and 3D results shows that there is a greater mass flux passing through the transition zone in the 2D models for the 5 × 5 × 1 box, but for larger aspect ratio (8 × 8 × 1) the 3D flows become more layered than the corresponding 2D solution. Increasing the magnitude of the negative Clapeyron slope by three times the experimental value can bring about a dramatic reduction in the amount of mass flux across the 670 km discontinuity. Results from a 8 × 8 × 1 box show that a greater amount of layering is produced in the larger aspect-ratio configuration because of the shorter wavelengths of the developed planforms. Increasing the degrees of freedom in a 3D system by either greater amounts of convective vigor or larger domains may give rise to a greater tendency for layered convection. Three-dimensional spherical shell models may produce a greater degree of layering than the corresponding axisymmetric models.
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We thank Satoru Honda and Yu.Yu. Pod- lachikov for the stimulating discussions, M. Lund-gren and T. Bui for preparation of the manuscript, and Bobby Bolshoi for encouragement. We thank Uli Christensen for a constructive criticism of this manuscript. This research has been supported by the Geochemistry program at NSF, NASA, the Von Humboldt Stiftung of Germany and the Deutsche Forschungsgemeinschaft. V. Steinbach acknowledges support by the Minnesota Supercomputer Institute as a visiting scholar.