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Scaling Laws for the Wavelength and the Appearance Time of Cells Resulting from Double-Diff.pdf

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

Scaling Laws for the Wavelength and the Appearance Time of Cells Resulting from Double-Diffusive Convection Karsten K?tter a and Mario Markus a,* Max-Planck-Institut für molekulare Physiologie Postfach 500247, D-44202 Dortmund, Germany. * Corresponding author, E-mail: mario.markus@mpi-dortmund.mpg.de Received 25 May 2001 Accepted 28 Sep 2001 ABSTRACT 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, desta- bilizing 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 S 0 and T 0 , as well as λ as a function of time. These approximations agree well with the solutions of the PDEs. KEYWORDS: double-diffusive convection, salt fingers, self-organization ScienceAsia 28 (2002) : 49-54 INTRODUCTION Double-diffusive convection results from an instability at an interface between two fluid layers. 1- 3 Such layers contain different amounts of a slowly diffusing, gravitationally destabilizing dissolved substance S and a fastly diffusing, stabilizing dis- solved substance T. The instability is counterintuitive at first glance; in fact, if one considers only the total densities and not their composition, the layers would be stable. A fluctuation at the interface causing a protrusion P from layer L 1 to layer L 2 , may be depleted of T fastly enough by diffusion, so that S in P can abandon L 1 and P grows. As examples for two horizontal layers: i) S (heavier than water, but diffusing slowly) and T (lighter than water, but diffusing fastly) are layered above water; ii) S (lighter than water, but diffusing slowly) and T (heavier than water, b