Multi-scale steady solution for Rayleigh–Bénard convection

We found a multi-scale steady solution of the Boussinesq equations for Rayleigh–Benard convection in a three-dimensional periodic domain between horizontal plates with a constant temperature difference. This was realised using a homotopy from the wall-to-wall optimal transport solution provided by Motoki et al. (J. Fluid Mech., vol. 851, 2018, R4). A connected steady solution, which is a consequence of bifurcation from a thermal conduction state at Rayleigh number scaling law with a consistent prefactor, and the energy transfer to small scales with a nearly constant flux in the wavenumber space is in accordance with the turbulent energy transfer.