The effect of variable stratification on linear doubly diffusive stability

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 $\frac{1}{80}$ is used in the study, with (large) solute Rayleigh numbers Rs ranging from 10 4 to 10 12 . 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 (< 10 8 ). 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 = 10 12 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.