An Analysis of Stable Mode Contributions to Rayleigh-B\'{e}nard Convection

Abstract

In this work, we investigate the presence and impact of stable eigenmodes in Rayleigh-B\'{e}nard convection that arises due to a temperature gradient within a fluid system. The nonlinear evolution of the canonical convection system is cast in terms of eigenmode amplitudes. The linear modes that play a significant role in the dynamics across a range of Rayleigh numbers are identified. We find that while unstable eigenmodes are a significant contributor, a small number of linearly stable modes grow to large amplitude via nonlinear interactions and are essential in modeling system dynamics. Importantly, the stable eigenmodes are seen to contain the majority of the boundary layer structure that forms in the nonlinear state. We also demonstrate that the scaling of convective heat transport (as quantified by the Nusselt Number) with Rayleigh number can be effectively quantified with only a small fraction of the total number of eigenmodes in the system.