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

We consider Rayleigh-Bénard convection as modeled by the Boussinesq equations, in the case of infinite Prandtl numbers 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 $Ra ≫1$⁠. In several studies, 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 the form of a variational problem where one optimizes a stratified temperature profile subject to a certain stability condition; the method is believed to capture the marginal stability of the boundary layer. The best available upper bound via this method is $Nu ≲ R a 1 3 ( ln R a ) 1 15$⁠; 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 $Nu ≲ Ra 1 3 ( ln ln Ra ) 1 3$ 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.