Reply to He et al.: The dependence of heat transport law on aspect ratio is still unclear

In their comment on Iyer et al. (1), He et al. (2) appear to address two putative issues: 1) that Iyer et al. neglect to point out that the small aspect ratio of their simulation could have an impact on the outcome; 2) He et al. (2) use their own data on the Nusselt–Rayleigh scaling to attempt to quantify its dependence on the aspect ratio $\mathrm{\Gamma }=d/H$, where $d$ is the diameter and $H$ is height of the convection cell.

On issue 1, we point out that, in the Abstract, Introduction, and final summary, Iyer et al. (1) have stated that their aspect ratio of $\mathrm{\Gamma }=1/10$ is indeed small. The assertion of He et al. (2) that Iyer et al. have neglected to mention the smallness of aspect ratio, or were unaware of its potential impact, is very misleading.

On issue 2, He at al. (2) make two points. First, they use their data (figure 1 in ref. 2) to prove a point about the ultimate state, but neglect to note that their fits to those data have been questioned (3); see also the rebuttal in ref. 4. Second, in figure 2 of ref. 2, they fit a power law to just two data points on the aspect ratio dependence and extrapolate substantially to arrive at their conclusion that the aspect ratio used in Iyer et al. (1) would imply a transition to the ultimate state at a Rayleigh number of about $10^{17}$. They then use data from Roche et al. (5) to lend support for their two-data-point fits, but these data lower the transition Rayleigh number by three orders of magnitude into the range of the simulations of ref. 1. Reasons for the differences between experiments are not discussed. While we agree that the aspect ratio dependence is not well understood, we regret to have to note the weakness of the arguments used by He et al. to impute that effect.

We take this opportunity to point out that no high Rayleigh number data have been obtained for large aspect ratios; neither have the thin boundary regions on the heated bottom and cooled top plate been explored well to know the nature of boundary layer transition. Except when the aspect ratios are very large, invoking classical boundary layers and their characteristic transition scenarios is very rough, at best. What may matter more for turbulent convection are local thermal instabilities which are associated with ongoing plume detachments, as discussed by Howard (6). If we insist on a boundary layer description, which may be useful for some purposes, what may matter more is the ratio of the boundary layer thickness to the cell diameter (which is a large value of $\sim \mathrm{1,000}$ for the highest $\mathrm{R}\mathrm{a}$ of ref. 1) and not $\mathrm{\Gamma }$, per se. The direct numerical simulations of ref. 1 show similar magnitudes and intermittent statistics of velocity derivatives as in fully developed turbulence; they are present even when the classical $1/3$ scaling continues to hold for the global heat transport law.