Steady Rayleigh--Bénard convection: strongly nonlinear high-wavenumber rolls

Abstract

In Rayleigh--Bénard 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 $\Gamma =O(R{a}^{-1/4})$ as $Ra\to \infty$, using numerics and matched asymptotic analysis. We compute steady rolls between stress-free boundaries for Prandtl numbers $10^{-1}\le Pr\le 10^{3/2}$ and $Ra$ reaching $10^{19}$. While the $\Gamma =O(R{a}^{-1/4})$ limit gives smaller $Nu$ than when $\Gamma =O(1)$, we identify prefactors $c$ in $\Gamma =cR{a}^{-1/4}$ that locally maximize $Nu$. These locally $Nu$-maximizing rolls display approximate scalings $Nu\propto R{a}^{0.29}$ and $Re\propto R{a}^{0.40}$, 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 $Nu=O(R{a}^{3/10})$ and $Re=O(R{a}^{2/5})$. 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.