Salt fingers in three dimensions

Three dimensional (3D) numerical calculations are made for a vertically unb ounded fluid with initially uniform vertical gradients of sugar (S) and sal t (T), where tau = kappa(S)/kappa(T) = 1/3 is the diffusivity ratio, and th e molecular viscosity is v much greater than kappa(T). The latter inequalit y allows us to neglect the nonlinear term in the momentum equation, while r etaining such terms in the T-S equations. The discrete 3D Fourier spectrum resolves the fastest growing horizontal wavelength, as well as the depth in dependent Fourier component. Unlike previous calculations for the pure 2D c ase the finite amplitude equilibration in 3D is primarily due to the instab ility of the lateral S-gradients in the fingers, and the consequent transfe r of energy to vertical scales comparable with the finger width. It is show n that finite amplitude two-dimensional disturbances are unstable and give way to three dimensional fingers with much larger fluxes. Calculations are also made for rigid boundary conditions at z = (0, L) in order to make a ro ugh quantitative comparison with previous lab experiments wherein a finger layer of finite thickness is sandwiched between two well-mixed (T, S) reser voirs. The flux ratio is in good agreement, and the fluxes agree within a f actor of two even though the thin interfacial boundary layer between the re servoir and the fingers is not quite rigid because sheared fingers pass thr ough it. It is suggested that future experiments be directed toward the muc h simpler unbounded gradient model, for which flux and variance laws are gi ven herein.