Marginally stable thermal equilibria of Rayleigh-Bénard convection

Abstract

We study Boussinesq convection by computing marginally-stable mean temperature profiles for various Ra which are thermal equilibria of the quasilinear equations. We find these marginally-stable thermal equilibria by solving one-dimensional eigenvalue problems and allowing the mean temperature to evolve according to diffusion and advection by the eigenmodes. The mode amplitudes are chosen such that the mean temperature maintains marginally stability. We find that multiple marginally-stable modes become important for Ra > 10${}^{6}$, and Nu ~ Ra${}^{1/3}$ up to our maximum Ra = 10${}^{9}$.