Stability of an Initially, Stably Stratified Fluid Subjected to a Step Change in Temperature

The onset of convective instability in an initially quiescent, stably stratified fluid layer between two horizontal plates is analyzed with linear theory. The bottom boundary is heated suddenly from below, subjected to a step change in surface temperature. The critical time tc to mark the onset of Rayleigh-Bénard convection is predicted by propagation theory. This theory uses the length scaled by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sqrt{\alpha_t}$\end{document}, where α denotes thermal diffusivity. Under the normal mode analysis the dimensionless disturbance equations are obtained as a function of τ(=αt/d2) and ζ(=Z/\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sqrt{\alpha_t}$\end{document}), where d is the fluid layer depth and Z is the vertical distance. The resulting equations are transformed to self-similar ones by using scaling and finally fixing τ as τc under the frame of coordinates τ and ζ. For a given γ, Pr and τc, the minimum value of Ra is obtained from the marginal stability curve. Here γ denotes the temperature ratio to represent the degree of stabilizing effect, Pr is the Prandtl number and Ra is the Rayleigh number. With γ=0, the minimum Ra value approaches the well-known value of 1708 as τc increases. However, it is inversely proportional to τc3/2 as τc decreases. With increasing γ, the system becomes more stable. It is interesting that in the present system, propagation theory produces the stability criteria to bound the available experimental data over the whole domain of time.