Time-dependent heat transfer in the spherical Earth: Implications on the power and thermal evolution of the core

Abstract:

We quantitatively investigate the time-dependence of heat conduction for a post-core, spherical Earth that is not convecting, due to compositional layering, based on hundreds of measurements of thermal diffusivity (D) for insulators and metals. Consistency of our solutions for widely ranging input parameters indicates how additional heat transfer mechanisms (mantle magmatism and convection) affect thermal evolution of the core. We consider 1) interior starting temperatures (T) of 273-5000 K, which represent variations in primordial heat, 2) different distributions and decay of long-lived radioactive isotopes, 3) additional heat sources in the core (primordial or latent heat), and 4) variable depth-T dependence of D. Our new analytical solution for cooling of a constant D sphere validates our numerical results. The bottom line is that the thermally insulating nature of minerals, combined with constraints of spherical geometry, limits steep thermal gradients to the upper mantle, consistent with the short length scale (x ~700 km) of cooling over t = 4.5 Ga indicated by dimensional analysis [x² ~ 4Dt], and with plate tectonics. Consequently, interior temperatures vary little so the core has remained hot and is possibly warming.

Findings include: 1) Constant vs. variable D affects thermal profiles only in detail, with D for the metallic core being inconsequential. 2) The hottest zone in Earth may lie in the uppermost lower mantle; 3) Most radiogenic heat is released in Earth’s outermost 1000 km thereby driving an active outer shell; 4) Earth’s core is essentially isothermal and is thus best described by the liquid-solid phase boundary; 5) Deeply sequestered radioactivity or other heat will melt the core rather than by run the dynamo (note that the heat needed to have melted the outer core is 10% of radiogenic heat generated over Earth’s history); 6) Inefficient cooling of an Earth-sized mass means that heat essentially remains where it is generated, until it is removed by magmatism; 7) Importantly, the observed plate velocities are consistent with a Nusselt number of 1, i.e. the present day cooling is essentially conductive. Conductive cooling plus magmatism largely governs Earth’s thermal structure and dynamics, below a unicellular upper mantle. Core dynamics and magnetism are likely driven by rotational effects.