Asymptotic transport of steady roll solutions in Rayleigh–Bénard convection at infinite Prandtl number

Rayleigh–Bénard convection at infinite Prandtl number (Pr) provides a simplified yet physically relevant framework for understanding heat transport in highly viscous fluids like the Earth's mantle. This work explores the asymptotic behavior of strongly nonlinear convection between no-slip boundaries in this regime. Instead of examining chaotic turbulence from DNS at large Rayleigh numbers (Ra), our study focuses on identifying and analyzing dynamically unstable, steady roll solutions to uncover the fundamental transport mechanisms at large Ra. We compute these solutions over various aspect ratios (Γ) at fixed Ra and extend the computations to large Ra. Our results show that for fixed Γ, the Nusselt number (Nu) scales as Ra^1/5, in agreement with the asymptotic prediction by Vynnycky & Masuda (2013). At fixed large Ra, a local maximum in Nu emerges at a specific Γ that scales as Ra^-1/4 as Ra→∞. Guided by these numerical findings, we perform matched asymptotic analysis of roll solutions with Γ = L Ra^-1/4 in the limit Ra→∞, for arbitrary constant L. Our analysis reveals that these steady rolls develop an asymptotic structure consisting of four vertically stacked regions near each boundary—similar to the structure found in locally Nu-maximizing rolls at finite Pr. The corresponding Nusselt number satisfies Nu = c_n Ra^3/10, where the prefactor c_n depends on L. We derive the relevant asymptotic scalings in each region, validate them against numerical data, and propose an iterative numerical scheme for computing c_n.