Scaling Laws for the Wavelength and the Appearance Time of Cells Resulting from Double-Diffusive Convection

We investigate the solutions of partial differential equations describing a two-layered fluid system. The lower layer contains a fastly diffusing, stabilizing substance T and a slowly diffusing, destabilizing substance S. The upper layer contains only the solvent. Double-diffusive convection yields cells in agreement with experiments performed with a tenside (S) and glycerine (T) dissolved in water. Approximation of the PDEs renders scaling laws for the appearance time Tem of the cells and their wavelength λ, as functions of the initial concentrations S0 and T0, as well as λ as a function of time. These approximations agree well with the solutions of the PDEs.