LAUSR

Abstract We prove a new rigorous bound for the mean convective heat transport $\langle w T \rangle$, where $w$ and $T$ are the non-dimensional vertical velocity and temperature, in internally heated convection between an insulating lower boundary and an upper boundary with a fixed heat flux. The quantity $\langle wT \rangle$ is equal to half the ratio of convective to conductive vertical heat transport, and also to $\frac 12$ plus the mean temperature difference between the top and bottom boundaries. An analytical application of the background method based on the construction of a quadratic auxiliary function yields $\langle w T \rangle \leq \frac {1}{2}(\frac {1}{2}+ \frac {1}{\sqrt {3}} ) - 1.6552\, {\textit {R}}^{-(1/3)}$ uniformly in the Prandtl number, where R is the non-dimensional control parameter measuring the strength of the internal heating. Numerical optimisation of the auxiliary function suggests that the asymptotic value of this bound and the $-1/3$ exponent are optimal within our bounding framework. This new result halves the best existing (uniform in $ {\textit {R}}$) bound (Goluskin, Internally Heated Convection and Rayleigh–Bénard Convection, Springer, 2016, table 1.2), and its dependence on $ {\textit {R}}$ is consistent with previous conjectures and heuristic scaling arguments. Contrary to physical intuition, however, it does not rule out a mean heat transport larger than $\frac 12$ at high $ {\textit {R}}$, which corresponds to the top boundary being hotter than the bottom one on average.