Bounds on vertical heat transport for infinite-Prandtl-number Rayleigh–Bénard convection

For the infinite-Prandtl-number limit of the Boussinesq equations, the enhancement of vertical heat transport in Rayleigh–Bénard convection, the Nusselt number Nu , is bounded above in terms of the Rayleigh number Ra according to Nu\,≤\,0.644 \,× Ra^1/3 [log Ra]^1/3 as Ra\,→\,∞ . This result follows from the utilization of a novel logarithmic profile in the background method for producing bounds on bulk transport, together with new estimates for the bi-Laplacian in a weighted L² space. It is a quantitative improvement of the best currently available analytic result, and it comes within the logarithmic factor of the pure 1/3 scaling anticipated by both the classical marginally stable boundary layer argument and the most recent high-resolution numerical computations of the optimal bound on Nu using the background method.