Convective heat transfer as a function of wavelength: Implications for the cooling of the Earth

Attempting to reconstruct the thermal history of the Earth from a geophysical point of view has for a long time been in disagreement with geochemical data. The geophysical approach uses parameterized models of mantle cooling. The rate of cooling of the Earth at the beginning of its history obtained in these models is generally too rapid to allow a sufficient present‐day secular cooling rate. Geochemical estimates of radioactive element concentrations in the mantle then appear too low to explain the observed present mantle heat loss. Cooling models use scaling laws for the mean heat flux out of the mantle as a function of its Rayleigh number of the form Q ∝ Ra β . Recent studies have introduced very low values of the exponent β, which can help reduce the cooling rate of the mantle. The present study instead focuses on the coefficient C in the relation Q = C Ra β and, in particular, on its variation with the wavelength of convection. The heat transfer strongly depends on the wavelength of convection. The length scale of convection in Earth's mantle is that of plate tectonics, implying convective cells of wide aspect ratio. Taking into account the long wavelength of convection in Earth's mantle can significantly reduce the efficiency of heat transfer. The likely variations of this wavelength with the Wilson cycle thus imply important variations of the heat flow out of the Earth on a intermediate timescale of 100 Ma, which renders parameterized models of thermal evolution inaccurate for quantitative predictions.