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

Abstract

Abstract. 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.

Layered convection and the impacts of the perovskite‐postperovskite phase transition on mantle dynamics under isochemical conditions

“…Although it will require detailed further analyses of the thermal history of the planet so as to extend the results of Butler and Peltier [2000, 2002] in order to make close contact with these recent results of Condie and others, an initial analysis based on results from the current model is worthwhile. To this end, we show in Figure 11 the cooling history result from the model C00D00VM3 (CMB temperature held fixed at 4000 K) in which the initial mean temperature is ∼280 K higher.…”

“…The database produced by these analyses will nevertheless prove to be extremely useful in the next phase of this new series of analyses of mantle convection in which the new control volume‐based model will be deployed in both fully 3‐D analyses and in the development of complete thermal histories of Earth evolution. In thermal history mode, the new numerical structure will enable us to considerably refine the recent results of Butler and Peltier [2000] that were obtained using the original model of Solheim and Peltier [1994a, 1994b]. Our goal will be to more fully explore a possible connection between the episodically layered model of the convective circulation and the so‐called “supercontinent cycle” in which continents at Earth's surface are on occasion brought together to form a single “super” continent and then later dispersed.…”

Global glacial isostatic adjustment: palaeogeodetic and space‐geodetic tests of the ICE‐4G (VM2) model

“…Butler and Peltier (2000), for example, demonstrate that when a viscosity structure such as VM3 is directly inserted into an apriori model of the mantle convection process then the convection model tends to significantly overpredict the surface heat flow. Taken at face value this has been construed to suggest either that the viscosity which controls the process of mantle convection is significantly higher than that which controls rebound (implying that the rheology of the mantle is significantly non-Newtonian), or that the extent to which the convective circulation is layered is much higher than is often assumed.…”

Goldschmidt Conference Abstracts 2005

“…Fei et al, 2004) have led to a claim that the Clapeyron slope for this transition was initially overestimated. The model that we have elected to employ for the purpose of providing an initial investigation of this effect is that previously developed by the Toronto group ( Solheim and Peltier, 1994;Butler and Peltier, 2000). It needs to be understaood, however, that there exists a pressure coincident additional phase transition, namely that from Ilmenite to Perovskite, which also has a very negative Clapeyron slope.…”