Anelastic convection in the mantle with variable properties

The convective motion and the thermal state of the Earth's mantle are investigated through the mean-field approximation, deriving the steady-state solutions of the set of conservation equations in cartesian geometry for a compressible fluid with variable viscosity, bulk modulus, thermal expansion and thermal conductivity. The density and bulk modulus are taken from the Preliminary Reference Earth Model (PREM) and the Grüneisen parameter is evaluated through the Debye-Brillouin formulation, which has been proven to be an effective approximation. Thermal expansion is derived from the thermodynamical definition of the Grüneisen parameter, and thermal conductivity and the other thermodynamic parameters are written in terms of the elastic constants according to the quasi-harmonic theory. Comparison of the solutions obtained with an exponential density profile following the Adams-Williamson equation of state and the polynomial profile of the PREM shows that the detail of the radial behaviour of density has a minor influence. The radial variation of thermal expansivity is found to produce a lower thermal gradient at high Rayleigh numbers through a complex feedback mechanism involving buoyancy and adiabatic heating. The radial dependence of thermal conductivity strongly affects both the mechanical and thermal state of the convective cells and inhibits the formation of thermal boundary layers at the bottom of the cells. In this case, the temperature dependence of the viscosity determines a 'cold' and a 'hot' branch, which may be related to the composition of the outer core. © 1991.