Asymptotic scalings of steady rolls in Rayleigh-Bénard convection

In Rayleigh-Bénard convection, steady two-dimensional rolls are the simplest nonlinear states. They exist for all Rayleigh numbers (Ra) above the primary instability but are unstable at large Ra. The large-Ra asymptotic scalings of heat transport and other mean quantities are more accessible for steady rolls than for turbulence. Nonetheless, the possible scalings of steady rolls are surprisingly rich and not fully understood. Various scaling exponents are possible in the large-Ra limit, depending on the boundary conditions and on which simultaneous limit (if any) is taken in the aspect ratio of the rolls and/or the Prandtl number of the fluid. This talk will survey what is known about large-Ra limits of steady rolls at finite Prandtl numbers. For various asymptotic limits, we will show numerical computations of steady rolls with Ra as large as 10¹⁹, as well as matched asymptotic equations that are consistent with the numerical results. Depending on boundary conditions and on how the aspect ratio varies with Ra, the Nusselt number of steady rolls is predicted to scale like Ra^α with at least three different exponents that include α=1/3, 1/4, and 25/84.