The effect of variable stratification on linear doubly diffusive stability

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Abstract
The effect of variable stratification on the linear bifurcations of a doubly diffusive plane parallel layer is examined numerically by expanding in a Fourier series. Because the motivation is analysis of solar-pond stability, a Prandtl number of 7 and ratio of diffusivities of is used in the study, with (large) solute Rayleigh numbers Rs ranging from 104 to 1012. Stratification of solute is cubic antisymmetric about midlayer; because temperature has a higher diffusivity, it is given a linear stratification. Exchange of stabilities results also solve the ‘fingering’ and thermal problems with cubic stratification. For the overstable case, the numerical results approach Walton's perturbation solution at large Rs, but differ significantly at smaller Rs (< 108). While both exchange of stabilities and overstable modes display an expected tendency to localize about the point of minimum solute gradient, the overstable modes behave in other, non-intuitive ways. Sublayers of reversed salinity gradient, if small enough, can be stable. Above Rs = 1012 computations become prohibitively expensive as a continuous spectrum is approached. A simple sublayer scaling rule defines an infinite family of Rs and stratification parameters on which the localized eigensolution is nearly invariant.