A benchmark for finite Prandtl number convection: comparison of Boltzmann and Navier-Stokes solutions

While modern thermal convection in rocky planets is controlled by a slow solid-state creep flow, the earliest stages of terrestrial planets likely experienced turbulent flow during which their silicate envelope was fully molten, usually called magma ocean. The main parameter separating the two regimes is the Prandtl number (Pr), which is so high for mantle convection to be usually assumed infinite, whereas magma oceans are characterized by on the order of 1. We compared the results of isoviscous convection simulations performed with three codes: (GAIA, TLBM, StreamV). These codes are based on different numerical formulations and were used for modelling convection with ranging from 1 to 1000, while exploring different convection intensity by varying the Rayleigh number (Pr) from to. GAIA (Generic Automaton for planetary Interior Analysis) is a Finite Volume fluid flow and energy solver for the Navier-Stokes equations across arbitrary geometries. TLBM (Thermal Lattice Boltzmann Method) solves the mesocale momentum and energy distribution densities for colliding particles on a discrete lattice. StreamV is a Eulerian-Lagrangian Finite Volume code that solves the Navier-Stokes equations under the Boussinesq approximation. The codes are compared over 24 different simulation setups, analogue to the classical Blankenbach infinite benchmark, but extending it to finite and to two types of boundary conditions, free-slip and no-slip. We show that the results of the three codes are generally in good agreement, and discuss differences. Finite solutions show a much richer dynamics varying from stable steady-state solutions, to oscillatory and chaotic ones, and converging to infinite Prandtl number solution for increasing values of for larger: is sufficient for but is required for. Our results offer a robust set of solutions useful for testing future finite Prandtl number convection codes.