Steady Rayleigh--B\'enard convection: strongly nonlinear high-wavenumber rolls

Abstract

In Rayleigh--B\'{e}nard convection, two-dimensional steady rolls bifurcate supercritically at a Rayleigh number $Ra$ that depends on their horizontal-to-vertical aspect ratio $\Gamma$, and they exist at all larger $Ra$ despite being unstable. Heat transport by certain rolls---quantified by the Nusselt number $Nu$---closely resembles turbulent transport, yet $Nu$ scalings of rolls are understood only for specific boundary conditions and $\Gamma$--$Ra$ limits. Here we investigate the high-wavenumber limit $\Gamma = O(Ra^{-1/4})$ as $Ra \to \infty$, using numerics and matched asymptotic analysis. We compute steady rolls between stress-free boundaries for Prandtl numbers $10^{-1} \leq Pr \leq 10^{3/2}$ and $Ra$ reaching $10^{19}$. While the $\Gamma = O(Ra^{-1/4})$ limit gives smaller $Nu$ than when $\Gamma = O(1)$, we identify prefactors $c$ in $\Gamma = c\,Ra^{-1/4}$ that locally maximize $Nu$. These locally $Nu$-maximizing rolls display approximate scalings $Nu \propto Ra^{0.29}$ and $Re \propto Ra^{0.40}$, with the Reynolds number $Re$ defined using root-mean-square velocity. Our asymptotic analysis reveals a vertically stacked four-layer structure near each boundary, predicting $Nu = O(Ra^{3/10})$ and $Re = O(Ra^{2/5})$. This asymptotic construction largely follows that of Taylor vortices by \cite{Deguchi2023}, but we identify a thin plume region within the middle boundary layer whose inclusion eliminates the logarithmic factors in Deguchi's predictions. Asymptotic arguments and numerics suggest the same scalings for stress-free or no-slip boundaries, unlike in other $\Gamma$--$Ra$ limits. Our asymptotics extend the weakly nonlinear analysis of Blennerhassett \& Bassom (1994) into the strongly nonlinear regime and complement the asymptotics of Chini \& Cox (2009) for $\Gamma = O(1)$ rolls.