Bound on vertical heat transport at large Prandtl number

We prove a new upper bound on the vertical heat transport in Rayleigh-Benard convection of the form c Ra-1/3(InRa)(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 Prandl 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 I infinite Prandtl number Rayleigh-Benard 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) 1049-10601 in the case of large Prandtl number. (C) 2007 Elsevier B.V. All rights reserved.