Analyses of marginal stability, heat transfer and boundary layer properties for thermal convection in a compressible fluid with infinite Prandtl number

Thermal and dynamical evolution of planets is controlled by thermal convection in planetary mantles. Mantle compressibility, which measures volume change due to pressure change and its associated energetic effects, can have important effects on planetary mantle convection. However, key issues including marginal stability analysis, thermal boundary properties and heat transfer in compressible mantle convection are not well understood. This paper studies the influence of mantle compressibility on thermal convection in an isoviscous and compressible fluid with infinite Prandtl number, using both marginal stability analysis and numerical modelling. For the marginal stability analysis, a new formulation of the propagator matrix method is implemented to compute the critical Rayleigh number Ra-c and the corresponding eigenfunctions for compressible convection at different wavelengths (i.e. wavenumber k(x)) and dissipation number Di which measures the compressibility. Ra-c from the analysis is in a good agreement with that determined from the numerical experiment using the eigenfunctions as initial perturbations. Our study suggests that if Ra is defined by the surface density, the minimum Ra-c may occur at non-zero Di. Finite element models are computed for compressible mantle convection at different Ra and Di. Heat flux and thermal boundary layer (TBL) properties including boundary layer thickness and temperature difference are quantified and analysed from the numerical results. Scaling laws of temperature differences across TBLs and of the heat flux are derived analytically for compressible mantle convection and are verified by the numerical results. This study shows that while TBL thicknesses and the heat flux are still scaled with Ra to the -1/3 and 1/3 power, respectively, as those for incompressible convection, they also strongly depend on Di. In particular, compressibility breaks the symmetry for the top and bottom TBLs, and the ratios of thickness and temperature difference for the top TBL to those for the bottom TBL are exp(Di/2). These results have important implications for compressible mantle convection.