Weakly non-linear stability of stratified natural convection in a vertical cavity with lateral temperature gradient

Stability of free convection is examined in a vertical cavity with a fixed lateral temperature difference on the walls, which is also heated from the bottom by a constant heat flux causing vertical stratification of the base flow. Weakly nonlinear stability equations are derived, under the assumption of fully developed flow conditions. Critical Rayleigh numbers and Landau coefficients are determined in terms of the Prandtl number and stratification parameter. It is found that critical disturbances are always two-dimensional, for stationary and oscillating instabilities as well. Alternatives are considered for the calculation of the Landau coefficient. Analysis reveals that unstationary instability is the preferred mode at the codimension point.