Stationary statistical properties of Rayleigh-Bénard convection at large Prandtl number

Xiaoming Wang

This is the third in a sequel in our study of Rayleigh-B\'enard convection at large Prandtl number. More specifically we investigate if stationary statistical properties of the Boussinesq system for Rayleigh-B\'enard convection at large Prandtl number are related to that of the infinite Prandtl number model for convection which is formally derived from the Boussinesq system via setting the Prandtl number to infinity. We study asymptotic behavior of stationary statistical solutions, or invariant measures, to the Boussinesq system for Rayleigh-B\'enard convection at large Prandtl number. In particular, we show that the invariant measures of the Boussinesq system for Rayleigh-B\'enard convection converge to that of the infinite Prandtl number model for convection as the Prandtl number approaches infinity. We also show that specific statistical properties such as the Nusselt number for the Boussinesq system is asymptotically bounded by the Nusselt number of the infinite Prandtl number model for convection at large Prandtl number. We discover that the Nusselt numbers are saturated by ergodic invariant measures. Moreover, we derive a new upper bound on the Nusselt number for the Boussinesq system at large Prandtl number of the form $Ra^{\frac13}(\ln Ra)^{\frac13} + c \frac{Ra^{\frac72}\ln Ra}{Pr^2}$ which asymptotically agrees with the (optimal) upper bound on Nusselt number for the infinite Prandtl number model for convection.