Numerical solutions of compressible convection with an infinite Prandtl number: comparison of the anelastic and anelastic liquid models with the exact equations

We developed a numerical method for the set of equations governing fully compressible convection in the limit of infinite Prandtl numbers. Reduced models have also been analysed, such as the anelastic approximation and the anelastic liquid approximation. The tests of our numerical schemes against self-consistent criteria have shown that our numerical simulations are consistent from the point of view of energy dissipation, heat transfer and entropy budget. The equation of state of an ideal gas has been considered in this work. Specific effects arising because of the compressibility of the fluid are studied, like the scaling of viscous dissipation and the scaling of the heat flux contribution due to the mechanical power exerted by viscous forces. We analysed the solutions obtained with each model (fully compressible model, anelastic and anelastic liquid approximations) in a wide range of dimensionless parameters and determined the errors induced by each approximation with respect to the fully compressible solutions. Based on a rationale on the development of the thermal boundary layers, we can explain reasonably well the differences between the fully compressible and anelastic models, in terms of both the heat transfer and viscous dissipation dependence on compressibility. This could be mostly an effect of density variations on thermal diffusivity. Based on the different forms of entropy balance between exact and anelastic models, we find that a necessary condition for convergence of the anelastic results to the exact solutions is that the product $\unicode[STIX]{x1D716}q$ must be small compared to unity, where $\unicode[STIX]{x1D716}$ is the ratio of the superadiabatic temperature difference to the adiabatic difference, and $q$ is the ratio of the superadiabatic heat flux to the heat flux conducted along the adiabat. The same condition seems also to be associated with a convergence of the computed heat fluxes. Concerning the anelastic liquid approximation, we confirm previous estimates by Anufriev et al. (Phys. Earth Planet. Inter., vol. 152, 2005, pp. 163–190) and find that its results become generally close to those of the fully compressible model when $\unicode[STIX]{x1D6FC}T{\mathcal{D}}$ is small compared to unity, where $\unicode[STIX]{x1D6FC}$ is the isobaric thermal expansion coefficient, $T$ is the temperature (here $\unicode[STIX]{x1D6FC}T=1$ for an ideal gas) and ${\mathcal{D}}$ is the dissipation number.