Does the Boussinesq approximation correctly describe statistically stationary states of Rayleigh-Bénard convection?

RESUMO

It is generally believed that statistically stationary states of Rayleigh-Bénard convection (RBC) can be described by statistically stationary solutions to the Boussinesq approximation. Based on an analysis of the mechanical energy equation, we argue that this belief is not justified, because the expression for mean conversion of internal energy into kinetic energy by compressions and expansions, which is implicitly given by the approximation, does not qualify as an approximation. In consequence, it is questionable whether the exact Boussinesq expression relating the mean kinetic energy dissipation in a stationary state to the Rayleigh and Nusselt numbers is valid. Using assumptions that partly lie outside the Boussinesq approximation, we derive scaling relations for the mean kinetic energy dissipation in a stationary state in three and two dimensions. The relation for the three-dimensional system is similar to the Boussinesq expression, but includes an unknown prefactor, while the relation for the two-dimensional system is completely different. Our arguments suggest that statistically stationary solutions to the Boussinesq approximation in three dimensions, reproduce scaling relations between various statistical quantities quite well, although they should not be regarded as approximations in a strict sense, while solutions in two dimensions completely fail. We suggest that this should be investigated by performing DNS of weakly compressible RBC and compare with DNS of Boussinesq RBC.