3. Mantle Viscosity and the Thickness of the Convective Downwellings

This paper is an investigation into the thermal evolution of the Earth´s mantle with variable viscosity and time-dependent internal heating. The differential equations of the infinite Prandtl-number convection are solved using a three-dimensional finite-element spherical-shell method with a computational mesh derived from a regular icosahedron with 1.351746 × 10⁶ or, alternatively, 1.0649730 × 10⁷ nodes. The radial factor of the viscosity was derived from solid-state physical considerations using the seismic model PREM. The new features of this viscosity profile are a high-viscosity transition layer (Fig. 3) beneath the usual asthenosphere, a second low-viscosity layer below the 660-km endothermic phase boundary and a considerable viscosity hill in the lower 80 % of the lower mantle. In spite of this deduction, a strong variation of the viscosity profile (see, e.g., Fig. 6) and of other physical parameters was carried out in order to study the consequences for the planforms and for the convection mechanism. The effects of the two mineral phase boundaries in 410 and 660 km depth proved to be smaller than the effects of the strong dependence of the viscosity upon the radius. The latter had more influence on the mechanism than all other parameters. The Grüneisen parameter, , the coefficient of thermal expansion, , the specific heat at constant pressure, c_p, and the specific heat at constant volume, c_v, were derived as functions of the radius using geophysical and physical observables (Fig. 2). Among the physical parameters, only for the viscosity a temporally and laterally dependent factor was additionally used in the calculations. Using our most favored viscosity profile (Fig. 3), reticular connected thin cold sheet-like downwellings are a result of the convection calculations. The downwellings bear a resemblance to the observed subducting slabs but they are perpendicular. It is remarkable that the slabs penetrate the high-viscosity transition layer. They are sheet-like in 1350 km depth, yet. Underneath, the slabs begin to dissolve but their locations are visible in 1550 km depth, yet. The thin subducting sheets are remarkable since the viscosity is Newtonian. It is, however, not surprising that there are no transform fault zones at the surface of the model. The laterally averaged heat flow at the Earth´s surface, qob, the ratio of heat output to radiogenic heat production, Ror, the Rayleigh number and the Nusselt number have been calculated as a function of time (Fig. 4). The temperature spectra and the spectral heterogeneity maps of the temperature show a strong reddening for all depths.

The first reason for introducing a high-viscosity transition layer was discussed by Karato et al. (1995). The second reason is based on a paper by Walzer and Hendel (2002). In that paper, the authors derived the viscosity as a function of depth. The method is independent of the geophysical methods used up to now and independent of ideas on the distribution of minerals in the mantle. The authors started from a self-consistent theory using the Helmholtz free energy, an equation of state (EoS), the free-volume Grüneisen parameter and Lindemann´s law. The viscosity has been determined as a function of the melting temperature which stems from Lindemann´s law. Walzer and Hendel (2002) applied the Ullmann-Pan´kov EoS whereas in the present paper the Birch-Murnaghan EoS is used which is a special case of the Ullmann-Pan'kov EoS for = 4 where is the pressure derivative of the bulk modulus at vanishing pressure. In order to receive the relative variations of the radial factor of the viscosity, Walzer and Hendel (2002) inserted the pressure, P, the bulk modulus, K, and from PREM (Dziewonski and Anderson, 1981) into the above mentioned equations. PREM stems from seismic observations. For the calibration of the viscosity profile, Walzer and Hendel (2002) used the standard postglacial-uplift viscosity of the asthenosphere beneath the continental lithosphere. Two important conclusions of Walzer and Hendel (2002) are that not only the asthenosphere, but also the upper part of the lower mantle is a low-viscosity layer and that a high-viscosity transition layer tends to divide the mantle into two principle reservoirs regarding the incompatible elements and regarding the volatiles. The transition layer acts as a barrier, but a permeable one, to flow across the mantle transition zone.

In the present approach, the authors did not directly apply the values of P, K and from PREM but values which have been smoothed by means of the Birch-Murnaghan EoS (cf. Fig. 1). Of course, we did not smooth across the phase boundaries. So, in the present paper the radial factor of the viscosity is derived for 0 to 1129.9 km depth using the method of Walzer and Hendel (2002). This procedure avoids complex assumptions on the depth distribution of mineral phases and nevertheless allows a derivation of the viscosity from observables. For depths between 1129.9 km and 2891 km, the radial factor of the viscosity was determined from the melting curve of perovskite (Zerr and Boehler, 1993, 1994; Boehler 1997). The result of this computations is given in Fig. 3 where the viscosity jumps at the phase boundaries were replaced by strong viscosity gradients in order to avoid numerical difficulties with the convection code. The general form of the P, T-dependence of the viscosity, , is given by

with where r is the radius, the colatitude, the longitude and t the time. Therefore, the viscosity depends on these four independent variables, too.