Comparison of the dynamic and parameterized models of mantle convection including core cooling

The two-dimensional cooling model of mantle convection, including core cooling was studied. The core temperature is assumed to be constant in space, but it changes with time because of the convective retrieval of heat. The viscosity is also constant in space but it is a function of the area-averaged (mean) temperature in order to check the possible effects of temperature-dependent viscosity on the cooling. The rate of cooling is controlled by three factors: the temperature change of the core by a unit change in the core heat flux; the viscosity law, governed by the mean temperature; and the contribution of the internal heating. Several different types of temperature field were used as the initial conditions. Some results showed considerable fluctuations in the relation between the temporal Rayleigh number and the Nusselt numbers, which are appropriately defined for both the upper and lower thermal boundary layers. A parameterized model of the cooling history of the mean and bottom temperatures was calculated assuming the power-law relation between the Rayleigh and the Nusselt numbers, which is determined by the least-square fit of the results of the dynamic calculations. For the parameterized models we integrated backward in time. The agreement between the dynamic and parameterized calculations is generally good, except for the cases when the rate of cooling is large or the decay in internal heating is small. When the rate of the cooling is large the lower thermal boundary layer disappears. This results in unpredictability of the bottom temperature. However, this case is not applicable for the Earth. When the decay in internal heating is small, we find that the agreement between the dynamic and parameterized models is sensitive to the change in the amount of internal heating assumed in parameterized models. This is because the internal heat source directly affects the rate of cooling. The beta value, which is the power index of the Nu-Ra relation, varies from 0.3 to 0.4. When the bottom thermal boundary layer becomes weak, because of the fast cooling of the core, beta deviates from 0.3 considerably.