Viscosity Stratification and a 3-D Compressible Spherical Shell Model of Mantle Evolution

Summary The viscosity stratification has a strong influence on the thermal evolution of a compressible Earth’s mantle with time-dependent internal heating. The differential equations for infinite Prandtl-number convection are solved using a three-dimensional finite-element spherical-shell method on a computational mesh derived from a regular icosahedron with 1.351.746 or, alternatively, 10.649.730 nodes. We formulate a radial viscosity profile from solid-state physics considerations using the seismic model PREM. New features of this viscosity profile are a high-viscosity transition layer beneath the usual asthenosphere, a second low-viscosity layer below the 660-km endothermic phase boundary and a considerable viscosity increase in the lower 80 % of the lower mantle. To be independent of the special assumptions of this derivation, we vary the level and the form of this profile as well as the other physical parameters in order to study their consequences on the planforms and on the convection mechanism. The effects of the two mineral phase boundaries at 410 and 660 km depth proved to be smaller than effects of the strong variation of viscosity with radius. The latter had more influence on the convective style than all other parameters. Values of our material parameters are time independent and constant in the lateral directions, except for viscosity. The focus of this paper is a variation of non-dimensional numbers as Rayleigh number, Ra, Nusselt number, Nu, the reciprocal value, Ror, of the Urey number, viscosity-level parameter, r n , etc. We explored the parameter range for special solutions. For the wide parameter range −0.5 ≤ r n ≤ +0.3, that includes our preferred viscosity profile, we obtain solutions characterized by reticular connected thin cold sheet-like downwellings. The downwellings are thinner than similar features in previous publications. They bear a resemblance to observed subducting slabs but are purely vertical. We find it remarkable that the downwellings penetrate the high-viscosity transition layer. They remain sheet-like to 1350 km depth. Below this depth they begin to lose definition but their locations are still visible at 1550 km depth. Such thin subducting sheets are notable since the viscosity is Newtonian. On the other hand, it is not surprising there are no transform-like features at the surface of the model. We compute laterally averaged heat flow at the Earth’s surface, the ratio of heat output to radiogenic heat production, Ror, the Rayleigh number and the Nusselt number as a function of time. Nu(2) denotes the temporal average of the Nusselt number of a run for the last 2000 Ma of the evolution, Ra(2) is the temporal average of the Rayleigh number, respectively. For a wide parameter range, we obtain Nu(2) = 0.120 Ra (2) 0.295 in this model.