Ultimate State of Two-Dimensional Rayleigh-B\'enard Convection between Free-Slip Fixed-Temperature Boundaries*

Rigorous upper limits on the vertical heat transport in two dimensional Rayleigh-B\'enard convection between stress-free isothermal boundaries are derived from the Boussinesq approximation of the Navier-Stokes equations. The Nusselt number $Nu$ is bounded in terms of the Rayleigh number $Ra$ according to $Nu \leq 0.2891 Ra^{5/12}$ uniformly in the Prandtl number $Pr$. This Nusselt number scaling challenges some theoretical arguments regarding the asymptotic high Rayleigh number heat transport by turbulent convection.