Bounds for convection between rough boundaries

We consider Rayleigh-B'enard convection in a layer of fluid between rough no-slip boundaries where the top and bottom boundary heights are functions of the horizontal coordinates with square-integrable gradients. We use the background method to derive an upper bound on mean heat flux across the layer for all admissible boundary geometries. This flux, normalized by the temperature difference between the boundaries, can grow with the Rayleigh number ($Ra$) no faster than $\mathcal{O}(R{a}^{1\mathrm{/}2})$ as $Ra\to \mathrm{\infty }$. Our analysis yields a family of similar bounds, depending on how various estimates are tuned, but every version depends explicitly on the boundary geometry. In one version the coefficient of the $\mathcal{O}(R{a}^{1\mathrm{/}2})$ leading term is $0.242+2.925\mathrm{\parallel }\mathrm{\nabla }h{\mathrm{\parallel }}^2$, where $\mathrm{\parallel }\mathrm{\nabla }h{\mathrm{\parallel }}^2$ is the mean squared magnitude of the boundary height gradients. Application to a particular geometry is illustrated for sinusoidal boundaries.