A bound on the vertical transport of heat in the ‘ultimate’ state of slippery convection at large Prandtl numbers

Abstract

AbstractAn upper bound on the rate of vertical heat transport is established in three dimensions for stress-free velocity boundary conditions on horizontally periodic plates. A variation of the background method is implemented that allows negative values of the quadratic form to yield ‘small’ ($O\left(1/ \mathit{Pr}\right)$) corrections to the subsequent bound. For large (but finite) Prandtl numbers this bound is an improvement over the ‘ultimate’$R{a}^{1/ 2} $scaling and, in the limit of infinite$Pr$, agrees with the bound of$R{a}^{5/ 12} $recently derived in that limit for stress-free boundaries.