The role of the buoyancy layer in determining the structure of salt fingers

Abstract

An initial state consisting of sugar solution lying above a denser salt solution in a Hele-Shaw cell is unstable to disturbances that evolve into long, slender fingers. An analysis of the structure of fully evolved (infinitely long) fingers that are independent of the vertical coordinate concludes that fingers with a width of the order of the buoyancy-layer thickness have maximum growth rate. Since effective gravity can be altered by inclining the Hele-Shaw cell toward the horizontal, fingers of different preferred widths can be established. An abrupt change of the angle of inclination changes the preferred width. A stability analysis of the resulting initial-value problem shows that perturbations with a vertical scale of the order of the buoyancy-layer thickness grow, and fluid from each finger penetrates laterally into the two adjacent fingers. The unstable modes resemble those observed experimentally by Taylor & Veronis (1986). It turns out that all vertically uniform fingers, even ones with the preferred width of the basic state, are unstable to a non-oscillatory peturbation that changes straight fingers to ones that have a vertically wavy structure. In all cases the vertical scale of the most unstable disturbance is of the order of the buoyancy-layer thickness. Also included is a discussion of the need for a model describing the transient evolution of fingers and particularly one that contains an analysis of the role of the transition region between the salt-finger zone and the reservoirs above and below.