Numerical models of the Earth's thermal history: Effects of inner-core solidification and core potassium

Recently there has been renewed interest in the evolution of the inner core and in the possibility that radioactive potassium might be found in significant quantities in the core. The arguments for core potassium come from considerations of the age of the inner core and the energy required to sustain the geodynamo [Nimmo, F., Price, G.D., Brodholt, J., Gubbins, D., 2004. The influence of potassium on core and geodynamo evolution. Geophys. J. Int. 156, 363-376; Labrosse, S., Poirier, J.-P., Le Mouël, J.-L., 2001. The age of the inner core. Earth Planet Sci. Lett. 190, 111-123; Labrosse, S., 2003. Thermal and magnetic evolution of the Earth's core. Phys. Earth Planet Int. 140, 127-143; Buffett, B.A., 2003. The thermal state of Earth's core. Science 299, 1675-1677] and from new high pressure physics analyses [Lee, K., Jeanloz, R., 2003. High-pressure alloying of potassium and iron: radioactivity in the Earth's core? Geophys. Res. Lett. 30 (23); Murthy, V.M., van Westrenen, W., Fei, Y.W., 2003. Experimental evidence that potassium is a substantial radioactive heat source in planetary cores. Nature 423, 163-165; Gessmann, C.K., Wood, B.J., 2002. Potassium in the Earth's core? Earth Planet Sci. Lett. 200, 63-78]. The Earth's core is also located at the lower boundary of the convecting mantle and the presence of radioactive heat sources in the core will affect the flux of heat between these two regions and will, as a result, have a significant impact on the Earth's thermal history. In this paper, we present Earth thermal history simulations in which we calculate fluid flow in a spherical shell representing the mantle, coupled with a core of a given heat capacity with varying degrees of internal heating in the form of K40 and varying initial core temperatures. The mantle model includes the effects of the temperature dependence of viscosity, decaying radioactive heat sources, and mantle phase transitions. The core model includes the thermal effects of inner core solidification and we present models for which the final size of the inner core is the same that for the present-day Earth. We compare the results of simulations with and without the effects of inner core solidification and we compare the results of the numerical model with those of a parameterized model. Models with concentrations of potassium in the core of roughly 600 ppm best satisfy the present-day surface heat flow constraint; however, the core temperatures in these models are somewhat high. In addition, we find that models with lesser degrees of heating in the core can also satisfy the surface heat flow constraint provided that the mantle is in a particularly active state. Our models predict a relatively young inner core with the greatest age being 1756 Ma. We demonstrate that models with high core temperatures in the latter part of simulations result in high CMB heat flows which lead to predictions of young inner cores. For fixed initial core temperatures, this leads to a slight decrease in the predicted age of the inner core with increasing concentration of radioactive elements in the core.