A comparison of heat transfer laws for mantle convection at very high Rayleigh numbers

Abstract

The Nusselt number ( Nu)-Rayleigh number ( Ra) law for heat transfer in convection at very high Rayleigh numbers serves as the basis for modeling the temperature distribution in the mantle and the Earth's thermal history. Three different approaches are commonly used to construct the Nu Ra law; two-dimensional grid calculations, one-dimensional calculations using the mean field approximation, and boundary layer theory. The three approaches give significantly different results. A comparison is made of Nu Ra laws derived from these methods for the classical problem of steady isoviscous convection in a layer with stress-free, isothermal boundaries. The comparison indicates the following: mean field calculations become exact as Ra → Ra(critical), but systematically overestimate Nu at large Ra by assigning too large a correlation coefficient to temperature and vertical velocity. Boundary layer theory underestimates Nu, but becomes exact as Ra → ∞. Nusselt numbers from finite difference, uniform grid calculations by Jarvis and finite element, variable grid calculations by Schubert and Anderson in the range 10 5 < Ra < 10 9 converge slowly to the Nusselt number predicted by boundary layer theory. The discrepancy is 14% at Ra = 10 6 and 3–7% at Ra = 10 8. Mean field computations by Quareni et al. over the same range give Nusselt numbers 50–60% higher. All three methods predict maximum heat transfer for cells with aspect ratio ∼ 0.8. The heat transfer falls off rapidly with increasingly elongate cells.