Non-linear effects from variable thermal conductivity and mantle internal heating: implications for massive melting and secular cooling of the mantle

The temperature-dependence of the phonon portion of the thermal conductivity k(T,P) devised by Hofmeister [Science 283 1699-1706] decreases with temperature, the same as in the dependence of mantle viscosity. Such a functional relationship of ∂k/∂T<0, when coupled with internal heating would present a situation very conducive for positive feedback action. On the other hand, the photon dependence of the conductivity has a functional relationship of ∂k/∂T>0. Thus, there can be a tradeoff between the phonon and photon contributions in the conductivity in the presence of internal heating. We have conducted two-dimensional calculations of mantle convection up to a surface Rayleigh number of around five million and an internal heating of chondritic abundance, with the extended-Boussinesq approximation in which the dissipation number has been set to 0.47 and depth-dependent thermal expansivity, decreasing by a factor of 5 across the mantle. The value of the constant mantle viscosity and the amount of internal heating are varied. For an enhanced radiative contribution [J. Geophys. Res. 84 (B4) 1603-1610] the radiative component of the thermal conductivity can exceed the phonon contribution in the upper mantle. Our results show that in all cases with basal heating the average mantle temperature of the variable conductivity models are higher than those of the corresponding constant conductivity models. But the interior thermal difference between the two conductivity models decreases (1) with greater vigor of convection, (2) an increase of internal heating and (3) an increase in the radiative contribution to the conductivity. The interior mantle temperature is significantly hotter, more than 500 °C, than the constant conductivity model, for the k(T,P) model with the less enhanced radiative component [Science 283 1699–1706]. These results would suggest that some sort of massive melting in the young earth might have occurred with k(T,P) and that there should not be so much radioactivity in the lower mantle today without incurring the wrath of some melting. We have also studied the effects of k(T,P) on slowing down the mantle secular cooling process by monitoring the gradual decrease in mantle temperature following an imposition of an adiabatic boundary condition at the core-mantle boundary. A decay time of 3.6 Gy has been taken for the mantle radioactivity and we have varied the initial amount of radioactive heating from chondritic value to four times the chondritic value. A significant delay in the cooling process of at least 1–2 Gy is found for a surface Rayleigh number of between 5×106 to 5×107. The mantle temperature can be heated up by 300–400 °C for initial radiogenic heating value characteristic of the Archean. We find the strongest deviations from the constant conductivity case for a silicate model by Hofmeister [Science 283 1699-1706] and intermediate values for an enhanced radiative conductivity model comparable to the model of Shankland et al. [J. Geophys. Res. 84 (B4) 1603-1610]. Such high mantle temperatures maintained for a long time by variable thermal conductivity would have important consequences on the thermal and petrological evolution of the mantle.