Bound on vertical heat transport at large Prandtl number

We prove a new upper bound on the vertical heat transport in Rayleigh–Bénard convection of the form c Ra 1 3 ( ln Ra ) 2 3 under the assumption that the ratio of Prandtl number over Rayleigh number satisfies Pr Ra ≥ c 0 where the non-dimensional constant c 0 depends on the aspect ratio of the domain only. This new rigorous bound agrees with the (optimal) Ra 1 3 bound (modulo logarithmic correction) on vertical heat transport for the infinite Prandtl number model for convection due to Constantin and Doering [P. Constantin, C.R. Doering, Infinite Prandtl number convection, J. Stat. Phys. 94 (1) (1999) 159–172] and Doering, Otto and Reznikoff [C.R. Doering, F. Otto, M.G. Reznikoff, Bounds on vertical heat transport for infinite Prandtl number Rayleigh–Bénard convection, J. Fluid Mech. 560 (2006) 229–241]. It also improves a uniform (in Prandtl number) Ra 1 2 bound for the Nusselt number [P. Constantin, C.R. Doering, Heat transfer in convective turbulence, Nonlinearity 9 (1996) 1049–1060] in the case of large Prandtl number.