Triply resonant penetrative convection

A thermal convection model is considered that consists of a layer of viscous incompressible fluid contained between two horizontal planes. Gravity is acting vertically downward, and the fluid has a density maximum in the active temperature range. A heat source/sink that varies with vertical height is imposed. It is shown that in this situation there are three possible (different) sub-layers that may induce convective overturning instability. The possibility of resonance between the motion in these layers is investigated. A region is discovered where a very sharp increase in Rayleigh number is observed. In addition to a linearized instability analysis, two global (unconditional) nonlinear stability thresholds are derived.