Toward a Thermochemical Model of the Evolution of the Earth’s Mantle

This is a report on first steps for a combination of two numerical models of the evolution of the Earth’s mantle: The first one, K3, is a new 2-D convection-fractionation model that simulates the growth of continents and of the geochemically complementary depleted mantle reservoir. The second model shows the 3-D generation of oceanic lithospheric plates and subducting sheet-like downwellings in a spherical-shell mantle. Based on the abundances of the present-day geochemical reservoirs of Hofmann (1988) we developed a numerical dynamical model of convection and of chemical differentiation in the Earth’s mantle. It is shown that a growing and additionally laterally moving continent and a growing depleted mantle evolved from an initially homogeneous primordial mantle. The internal heat production density of the evolving mantle depends on the redistribution of the radioactive elements by fractionation and convection. The fractionation generates separate geochemical reservoirs. However, the convection blurs the reservoirs by mixing. Although we take into account also the effects of the two phase transitions in 410 and 660 km depth, it is essentially the dependence of the viscosity on radius which guarantees the conservation of the major geochemical reservoirs. This model has no internal compulsory conditions. The principal idea of this first model is to compute the relative viscosity variations as a function of depth from observable quantities. We develop a self-consistent theory using the Helmholtz free energy, the Ullmann-Pan’kov equation of state, the free volume Grüneisen parameter and Gilvarry’s formulation of Lindemann’s law. In order to receive the relative variations of the radial factor of the viscosity, we insert the pressure, P, the bulk modulus, K, and ∂K/∂P from PREM. For mantle layers deeper than 771 km we used the perovskite melting curve by Zerr and Boehler (1993, 1994) in order to estimate the relative viscosity. For the calibration of the viscosity we have chosen the standard postglacial-uplift viscosity beneath the continental lithosphere. Furthermore, we took into account the dependence of the viscosity on temperature and on the degree of depletion of volatiles. An essential first new result of this paper is a high-viscosity transition layer and a second low-viscosity layer below it. Although our model mantle is essentially heated from within, we assume additionally a small heat flow at the CMB. This is necessary because of the dynamo theory of the outer core. The second main result of this first model is a more distinct bipartition of the mantle in a depleted upper part and a lower part rich in incompatible elements, yet. This result is rather insensitive to variations of the Rayleigh number and of the thermal boundary condition at CMB. The different parts of this paper are closely connected by the algorithm. The continuation of the first finding leads to a 3-D, up to now purely thermal model of mantle evolution and plate generation. This second model was used to carry out a series of three-dimensional compressible spherical-shell convection calculations with another new, but related viscosity profile, called eta3, that is derived from PREM and mineral physics, only. Here, the Birch-Murnaghan equation was used to derive the Grüneisen parameter as a function of depth. Adding the pressure dependence of the thermal expansion coefficient of mantle minerals, we derived the specific heats, c p and c v, too. Using the Gilvarry formulation, we found a new melting temperature of the mantle and the new viscosity profile, eta3. The features of eta3 are a high-viscosity transition layer, a second low-viscosity layer beginning under the 660-km discontinuity, and a strong viscosity maximum in the central parts of the lower mantle. The rheology is Newtonian but it is supplemented by a viscoplastic yield stress, ∂ y. A viscosity-level parameter, r n, and ∂ y have been varied. For a medium-sized Rayleigh-number-yield-stress area, eta3 generates a stable, plate-tectonic behavior near the surface and simultaneously thin sheet-like downwellings in the depth. Outside this area three other types of solution were found. The presence of two internal low-viscosity layers and of ∂y is obviously conducive for plateness and thin sheet-like downwellings. The distribution of the downwellings is more Earth-like if the yield stress is added. The outlines of a combination of the two models have been discussed.