Asymptotic structure of strongly nonlinear large-wavenumber steady states in Rayleigh-Benard convection

Recent numerical studies (e.g., Wen et al. J. Fluid Mech. 2020, 2022) have revealed the existence of highly nonlinear steady states in two-dimensional (2D) Rayleigh-Benard convection (RBC) having a horizontal wavenumber k that increases in proportion to Ra^1/4, where Ra is the Rayleigh number. At O(1) Prandtl number Pr, these large-wavenumber states are local (in k) maximizers of the heat flux in steady 2D convection for both stress-free and no-slip walls. Here, we elucidate the asymptotic structure of these large-wavenumber equilibria as Ra→∞. Our construction thus complements the analysis of Blennherhassett & Bassom (IMA J. Appl. Math. 1994), who analyzed less strongly supercritical steady convective states having the same wavenumber scaling, and that by Chini & Cox (Phys. Fluids 2009), who analyzed steady cellular flows arising in stress-free RBC having O(1) wavenumber. We demonstrate the emergence of an intricate four-region wall-normal asymptotic structure and show that the Nusselt number Nu=O(Ra^3/10). These results are corroborated by numerical computations of coherent convective states in 2D RBC at extreme values of Ra and consistent with recent work by Deguchi (J. Fluid Mech. 2023) on steady axisymmetric Taylor-Couette flow.