Finite-amplitude instability of the buoyancy boundary layer in a thermally stratified medium

The finite-amplitude instability of the buoyancy-driven boundary layer is considered on a vertical plate immersed in thermally stratified ambient medium, where wall and surrounding fluid have different temperature gradients. Although linear stability this configuration has been investigated, solution arising from critical studied only for specific parameter values. We extend by using amplitude expansion method. primary bifurcations to two-dimensional least unstable mode gradient ratios ( $0 \leqslant \lambda 10$ ) Prandtl numbers $10^{-1} Pr 10^{4}$ are investigated. Only supercritical found occur when < 2$ $Pr 2800$ , while subcritical also larger values ratio number. Analysis contribution nonlinear terms Landau coefficient reveals that interaction modification mean flow second harmonic velocity with fundamental plays an important role bifurcation. Based equation, threshold equilibrium discussed as well. These encouraging results should be helpful understanding such system.