Heat transport in Rayleigh–Bénard convection with linear marginality

Recent direct numerical simulations (DNS) and computations of exact steady solutions suggest that the heat transport in Rayleigh–Bénard convection (RBC) exhibits the classical 1/3 scaling as the Rayleigh number Ra→∞ with Prandtl number unity, consistent with Malkus–Howard’s marginally stable boundary layer theory. Here, we construct conditional upper and lower bounds for heat transport in two-dimensional RBC subject to a physically motivated marginal linear-stability constraint. The upper estimate is derived using the Constantin–Doering–Hopf (CDH) variational framework for RBC with stress-free boundary conditions, while the lower estimate is developed for both stress-free and no-slip boundary conditions. The resulting optimization problems are solved numerically using a time-stepping algorithm. Our results indicate that the upper heat-flux estimate follows the same 5/12 scaling as the rigorous CDH upper bound for the two-dimensional stress-free case, indicating that the linear-stability constraint fails to modify the boundary-layer thickness of the mean temperature profile. By contrast, the lower estimate successfully captures the 1/3 scaling for both the stress-free and no-slip cases. These estimates are tested using marginally-stable equilibrium solutions obtained under the quasi-linear approximation, steady roll solutions and DNS data. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.