Bounds for internally heated convection with fixed boundary heat flux

TL;DR

This paper establishes a new rigorous upper bound on mean convective heat transport in internally heated convection with fixed boundary heat flux, improving previous bounds and aligning with theoretical conjectures.

Contribution

It introduces an improved analytical bound on heat transport using the background method, which is optimal within its framework and advances understanding of internal heating convection.

Findings

New bound halves previous maximum estimates.

Bound's asymptotic behavior matches conjectured scaling.

Results suggest mean heat transport can exceed half at high heating rates.

Abstract

We prove a new rigorous bound for the mean convective heat transport $\langle w T \rangle$, where $w$ and $T$ are the nondimensional 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 $\frac12$ 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 $\langle w T \rangle \leq \tfrac{1}{2}\big(\tfrac{1}{2}+ \tfrac{1}{\sqrt{3}} \big) - 1.6552\, R^{-\frac13}$ uniformly in the Prandtl number, where $R$ is the nondimensional 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 $R$) bound (Goluskin 2016, Springer, Table 1.2) and its dependence on $R$ 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 $\frac12$ at high $R$, which corresponds to the top boundary being hotter than the bottom one on average.