Penetrative convection induced by a statically unstable density distribution in a very thin central layer

Several models of penetrative convection have been studied (Gribov $\&$ Gurevich, 1957; Veronis, 1963; Matthews, 1988; Batchelor $\&$ Nitsche, 1990; Simitev $\&$ Busse, 2010). We consider Rayleigh-B\'{e}nard convection with a static density distribution that has a piecewise linear dependence on the vertical coordinate and whose unstably stratified part occupies a central layer of thickness $\epsilon \ll 1$. Some limiting cases corresponding to the linear eigenvalue problem are treated analytically and the results confirmed by a detailed numerical investigation. Steady two-dimensional flow patterns are determined numerically for supercritical Rayleigh numbers in the range $\epsilon \ge 0.06$. For $0.2 \le \epsilon \le 0.5$, an analytical nonlinear stability three-dimensional study is undertaken in the case of poorly conducting boundaries. A weakly nonlinear evolution equation for the leading order temperature perturbation is also derived and solved numerically as function of $\epsilon$ and Prandtl number. The effect of the boundaries on the flow characteristics diminishes as $\epsilon \rightarrow 0$, leading us to study the stability of an unbounded stratified fluid for which similarity type solutions are obtained. Our findings are compared to those of the models mentioned above.