#### Abstract

Numerical solutions for salt fingers in an unbounded thermocline with uniform overall vertical temperature-salinity gradients are obtained from the Navier-Stokes-Boussinesq equations in a finite computational domain with periodic boundary conditions on the velocity. First we extend previous two-dimensional (2D) heat-salt calculations [Prandtl number *Pr* = *ν/k _{T}* = 7 and molecular diffusivity ratio τ =

*k*/

_{S}*k*= 0.01] for density ratio

_{T}*R*= 2; as

*R*decreases we show that the average heat and salt fluxes increase rapidly. Then three-dimensional (3D) calculations for

*R*= 2.0,

*Pr*= 7, and the numerically "accessible" values of τ = 1/6, 1/12 show that the ratio of these 3D fluxes to the corresponding 2D values [at the same (τ,

*R, Pr*)] is approximately two. This ratio is then extrapolated to τ = 0.01 and multiplied by the directly computed 2D fluxes to obtain a first estimate for the 3D heat-salt fluxes, and for the eddy salt diffusivity (defined in terms of the overall vertical salinity gradient). Since these calculations are for relatively "small domains" [

*O*(10) finger pairs], we then consider much larger scales, such as will include a slowly varying internal gravity wave. An analytic theory which assumes that the finger flux is given parametrically by the small domain flux laws shows that if a critical number

*A*is exceeded, the wave-strain modulates the finger flux divergence in a way which amplifies the wave. This linear theoretical result is confirmed, and the finite amplitude of the wave is obtained, in a 2D numerical calculation which resolves both waves and fingers. For highly supercritical

*A*(small

*R*) it is shown that the temporally increasing wave shear does

*not*reduce the fluxes until the wave Richardson number drops to ~0.5, whereupon the wave starts to overturn. The onset of density inversions suggests that at later time (not calculated), and in a sufficiently large 3D domain, strong convective turbulence will occur in patches.

#### Recommended Citation

Stern, Melvin E., Timour Radko, and Julian Simeonov.
2001.
"Salt fingers in an unbounded thermocline."
*Journal of Marine Research*
59,
(3).
https://elischolar.library.yale.edu/journal_of_marine_research/2395