We present a series of simulations of the mantle convection process based upon an axisymmetric numerical model and highlight a wide range of results in which scaling emerges. For the more challenging simulations it was found necessary to employ a finite difference mesh with uneven grid spacing in the radial coordinate, and we present the appropriate transformed field equations 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 shown to display scale invariance properties in the high Rayleigh number regime. We also present a new parameterized model of convection and demonstrate its success in predicting the manner in which many of the bulk properties of the convection process scale with convection control parameters. Results are also presented which demonstrate that quantities such as heat flow, characteristic velocity, and thermal boundary layer thickness scale with the mean viscosity even in time-dependent simulations in which the effects of phase transitions, depth-varying viscosity, and internal heating are active. The heat flow scaling exponent is seen to decrease in magnitude with increasing internal heating rate and Clapeyron slope of the 660-km phase transition, 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 increases unless layering is extreme. These calculations clearly demonstrate that in order for the surface heat flow predicted by the model to equal that characteristic of Earth, the mean viscosity of the mantle that controls the convection process must be considerably higher than the viscosity inferred on the basis of postglacial rebound and/or the flow must be significantly layered by the endothermic phase transition at 660 km depth. If mantle viscosity may be assumed to be Newtonian, in which case the creep resistance that controls 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. ¿ 2000 American Geophysical Union