MiS Preprint Repository

We consider Rayleigh-B\'enard 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 $\mathrm{N}\mathrm{u}$, in terms of the forcing via the imposed temperature difference, as given by the Rayleigh number in the turbulent regime $\mathrm{R}\mathrm{a}\gg 1$. In several works, the background field method applied to the temperature field has been used to provide upper bounds on $\mathrm{N}\mathrm{u}$ in terms of $\mathrm{R}\mathrm{a}$. 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 $\mathrm{N}\mathrm{u}\lesssim {\mathrm{R}\mathrm{a}}^{\frac{1}{3}}(\ln \mathrm{R}\mathrm{a}{)}^{\frac{1}{15}}$; 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 $\mathrm{N}\mathrm{u}\lesssim {\mathrm{R}\mathrm{a}}^{\frac{1}{3}}(\ln \ln \mathrm{R}\mathrm{a}{)}^{\frac{1}{3}}$, 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.