Limitations of the background field method applied to Rayleigh-B\'enard convection

We consider Rayleigh-B\'enard convection as modeled by the Boussinesq equations, in case of infinite Prandtl number. There is a broad interest in bounds of the upwards heat flux, as given by the Nusselt number ${\rm Nu}$, in terms of the forcing via the imposed temperature difference, as given by the Rayleigh number in the turbulent regime ${\rm Ra}\gg 1$. In several works, the background field method applied to the temperature field has been used to provide upper bounds on ${\rm Nu}$ in terms of ${\rm Ra}$. In these applications, the background field method comes in form of a variational problem where one optimizes a stratified temperature profile subject to a certain stability condition; the method is believed to capture marginal stability of the boundary layer. The best available upper bound via this method is ${\rm Nu}$ $\lesssim {\rm Ra}^\frac{1}{3}(\ln {\rm Ra})^\frac{1}{15}$; it proceeds via the construction of a stable temperature background profile that increases logarithmically in the bulk. In this paper, we show that the background temperature field method cannot provide a tighter upper bound in terms of the power of the logarithm. However, by another method one does obtain the tighter upper bound ${\rm Nu}\lesssim {\rm Ra}^\frac{1}{3}(\ln\ln {\rm Ra})^\frac{1}{3}$, so that the result of this paper implies that the background temperature field method is unphysical in the sense that it cannot provide the optimal bound.