Variation of Non-Dimensional Numbers and a Thermal Evolution Model of the Earth’s Mantle

A 3-D compressible spherical-shell model of the thermal convection in the Earth’s mantle has been investigated with respect to its long-range behavior. In this way, it is possible to describe the thermal evolution of the Earth more realistically than by parameterized convection models. The model is heated mainly from within by a temporally declining heat generation rate per volume and, to a minor degree, from below. The volumetrically averaged temperature, T a , diminishes as a function of time, as in the real Earth. Therefore, the temperature at the core-mantle boundary, T CMB,av , has not been kept constant but the heat flow, in accord with Stacey (1992). Therefore,T CMB,av decreases like T a . This procedure seems to be reasonable since evidently nobody is able to propose a comprehensible thermostatic mechanism for CMB. First of all, a radial distribution of the starting viscosity has been derived using PREM and solid-state physics. The time dependence of the viscosity is essential for the evolution of the Earth since the viscosity rises with declining temperature. For numerical reasons, the temperature-dependent factor of the model viscosity is limited to four orders of magnitude. The focus of this paper is an investigation of the variation of parameters, especially of the non-dimensional numbers as the Rayleigh number, Ra, the Nusselt number, Nu the reciprocal value of the Urey number, Ror, the viscosity level, r n , etc. For 0.0 ≤ r n ≤ +0.3, the authors arrived at Earth-like models. This interval contains the starting model. The quantification of the essential features of the model is provided by the eight plots. Numerical procedure: The differential equations are solved using a fast multigrid solver and a second-order Runge-Kutta procedure with a FE method. On 128 processors, runs with 10649730 grid points need about 50 hours. Figure 11 shows the scalling degree of our code. If the temperature dependence of the viscosity, Eq. (4), is replaced by Eq. (10) then, in the interval 0.0 ≤ r n ≤ +0.3, reticularly conected thin cold sheet-like downwellings are found from the surface down to 1350km depth. However, the movements along the upper surface are not plate-like.