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We revisit the optimal heat transport problem for Rayleigh-B\'enardconvection in which a rigorous upper bound on the Nusselt number, $Nu$, issought as a function of the Rayleigh number $Ra$. Concentrating on the2-dimensional problem with stress-free boundary conditions, we impose the fullheat equation as a constraint for the bound using a novel 2-dimensionalbackground approach thereby complementing the `wall-to-wall' approach ofHassanzadeh \etal \,(\emph{J. Fluid Mech.} \textbf{751}, 627-662, 2014).Imposing the same symmetry on the problem, we find correspondence with theirresult for $Ra \leq Ra_c:=4468.8$ but, beyond that, the optimal fieldscomplexify to produce a higher bound. This bound approaches that by a1-dimensional background field as the length of computational domain$L\rightarrow\infty$. On lifting the imposed symmetry, the optimal2-dimensional temperature background field reverts back to being 1-dimensionalgiving the best bound $Nu\le 0.055Ra^{1/2}$ compared to $Nu \le 0.026Ra^{1/2}$in the non-slip case. % We then show via an inductive bifurcation analysis thatimposing the full time-averaged Boussinesq equations as constraints (byintroducing 2-dimensional temperature {\em and} velocity background fields) isalso unable to lower this bound. This then exhausts the background approach forthe 2-dimensional (and by extension 3-dimensional) Rayleigh-Benard problem withthe bound remaining stubbornly $Ra^{1/2}$ while data seems more to scale like$Ra^{1/3}$ for large $Ra$. % Finally, we show that adding a velocity backgroundfield to the formulation of Wen \etal\, (\emph{Phys. Rev. E.} \textbf{92},043012, 2015), which is able to use an extra vorticity constraint due to thestress-free condition to lower the bound to $ Nu \le O(Ra^{5/12})$, also failsto improve the bound.

Exhausting the background approach for bounding the heat transport in Rayleigh-B\'enard convection