On scaling relations in time-dependent mantle convection and the heat transfer constraint on layering

We present a series of simulations of the mantle convection process based u pon an axisymmetric numerical model and highlight a wide range of results i n which scaling emerges. For the more challenging simulations it was found necessary to employ a finite difference mesh with uneven grid spacing in th e radial coordinate, and we present the appropriate transformed field equat ions that are required to implement a model of this kind. The statistics of mass flux events transiting the 660-km phase transition are calculated for a large number of high-resolution calculations, and some of these are show n to display scale invariance properties in the high Rayleigh number regime . We also present a new parameterized model of convection and demonstrate i ts success in predicting the manner in which many of the bulk properties of the convection process scale with convection control parameters. Results a re also presented which demonstrate that quantities such as heat flow, char acteristic velocity, and thermal boundary layer thickness scale with the me an viscosity even in time-dependent simulations in which the effects of pha se transitions, depth-varying viscosity, and internal heating are active. T he heat flow scaling exponent is seen to decrease in magnitude with increas ing internal heating rate and Clapeyron slope of the 660-km phase transitio n, but it is shown to be insensitive to depth variation of viscosity. Heat flow is seen to be reduced only modestly as the degree of layering increase s unless layering is extreme. These calculations clearly demonstrate that i n order for the surface heat flow predicted by the model to equal that char acteristic of Earth, the mean viscosity of the mantle that controls the con vection process must be considerably higher than the viscosity inferred on the basis of postglacial rebound and/or the flow must be significantly laye red by the endothermic phase transition at 660 km depth. If mantle viscosit y may be assumed to be Newtonian, in which case the creep resistance that c ontrols rebound and convection must be the same, this constitutes a strong argument for the importance of layering. The force of this argument depends upon the existence of an accurate estimate of the temperature at the core- mantle boundary which has only recently become available.