We discuss the scaling of the long time-averaged rate of heat transport in the vertical direction, quantified as the Nusselt number, within the Rayleigh-Benard model for convection. How the Nusselt number scales in terms of the parameters of the system, i.e., the Rayleigh and Prandtl numbers, is an outstanding open problem in classical physics. We will focus on the case of large but finite Prandtl number with the goal of deriving bounds on the Nusselt number for the system that are consistent with the best-known bounds on the Nusselt number for the infinite Prandtl number model. Both the case of classical no-slip boundary condition as well as the geophysically relevant free-slip boundary condition will be discussed. Numerical methods that are able to preserve the Nusselt number asymptotically will be presented as well.
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