Limitations of the background field method applied to Rayleigh-Bénard convection

We consider Rayleigh-Bénard convection as modeled by the Boussinesq equations, in case of infinite Prandtl number and with no-slip boundary condition. There is a broad interest in bounds of the upwards heat flux, as given by the Nusselt number $Nu$, in terms of the forcing via the imposed temperature difference, as given by the Rayleigh number in the turbulent regime $Ra \gg 1$. In several works, the background field method applied to the temperature field has been used to provide upper bounds on $Nu$ in terms of $Ra$. 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 $Nu \lesssim Ra^{1/3}(\ln Ra)^{1/15}$. 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 $Nu \lesssim Ra^{1/3}(\ln \ln Ra)^{1/3}$, 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.