Nusselt–Rayleigh number scaling for spherical shell Earth mantle simulation up to a Rayleigh number of 109109

An investigation of the power law relationship between Nusselt number (Nu) and Rayleigh number (Ra) for Earth’s convecting mantle is presented. The Nu(Ra) relationship was calculated from the results of a model with three dimensional spherical geometry and free slip boundary conditions. Both basally and internally heated convection has been examined. For Nu(Ra)=aRaβNu(Ra)=aRaβ, ββ was found to be 0.294±0.0040.294±0.004 for basally heated systems, which is lower than the value of 1/3 suggested by conventional boundary layer theory. The exponent β=0.337±0.009β=0.337±0.009 for internally heated systems, when the internally heated Ra is converted to a basally heated equivalent for comparison. The influence of the method used to calculate ββ was also considered, with particular attention paid to high Ra. As an example of the significance of β=0.29β=0.29 rather than 1/3, a Ra of 109109 results in a surface heat flux which is ≈≈32% lower. Within the range of Ra used in this study, there is no evidence that ββ changes at high Ra. Therefore, that mechanism cannot be used to moderate mantle temperature when projecting back to early Earth conditions. The differing planform of basally and internally heated models was shown to result in different scaling relationships between root mean square surface velocity and Ra for the two modes of heating, in particular, a much lower surface velocity for internally heated cases relative to equivalent Ra basally heated cases.