Transition to turbulent thermal convection beyond [Formula Presented] detected in numerical simulations

We have conducted high-resolution two-dimensional calculations for a Boussinesq convection model with a Prandtl number of unity in an aspect-ratio 3 box, going from Rayleigh numbers between [Formula Presented] to [Formula Presented] A grid of [Formula Presented] grid points consisting of a cosine-sine basis set has been employed for free-slip boundary conditions. We have found evidence for a transition involving the branching of plumes at a Rayleigh number of [Formula Presented] Inside the core of these “superplumes,” the structure is extremely complex. There may be another transition at Ra of [Formula Presented] where a secondary instability may develop in regions of the local Rayleigh number which becomes supercritical inside the core of the complex “superplumes.” For Ra of [Formula Presented] to [Formula Presented] Ra follows a [Formula Presented] power law in the Nusselt-Rayleigh number relationship. From Ra of [Formula Presented] to [Formula Presented] Ra follows a [Formula Presented] power law. Above this value the Nusselt number becomes insensitive to the variation in the global Rayleigh number and this is due to the development of small-scale convection cells vertically aligned in the interior of the extremely high Ra number flow. The global Reynolds number scales as [Formula Presented] up to Ra of [Formula Presented] Scaling relationships based on global properties would not work in extremely high Ra situations beyond Ra of [Formula Presented] because of the complex turbulent layered convection in the core of the flow and the severe degradation of the boundary layers.