Three dimensional (3D) numerical calculations are made for a vertically unbounded fluid with initially uniform vertical gradients of sugar ( S ) and salt ( T ), where τ = κS/κT = 1/3 is the diffusivity ratio, and the molecular viscosity is ν >> κT. The latter inequality allows us to neglect the nonlinear term in the momentum equation, while retaining such terms in the T-S equations. The discrete 3D Fourier spectrum resolves the fastest growing horizontal wavelength, as well as the depth independent Fourier component. Unlike previous calculations for the pure 2D case the finite amplitude equilibration in 3D is primarily due to the instability of the lateral S-gradients in the fingers, and the consequent transfer of energy to vertical scales comparable with the finger width. It is shown 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 rough quantitative comparison with previous lab experiments wherein a finger layer of finite thickness is sandwiched between two well-mixed (T,S) reservoirs. The flux ratio is in good agreement, and the fluxes agree within a factor of two even though the thin interfacial boundary layer between the reservoir and the fingers is not quite rigid because sheared fingers pass through it. It is suggested that future experiments be directed toward the much simpler unbounded gradient model, for which flux and variance laws are given herein.
Radko, Timour, and Melvin E. Stern. 1999. "Salt fingers in three dimensions." Journal of Marine Research 57, (3). https://elischolar.library.yale.edu/journal_of_marine_research/2325