Salt fingers in an unbounded thermocline

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 = nu /k(T) = 7 and molecular diffusivity ratio tau = k(S)/k(T) = 0.01] for density ratio 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 tau = 1/6, 1/12 show that the ratio of these 3D fluxes to the corresponding 2D values [at the same (tau, R, Pr)] is approximately two. This ratio is then extrapolated to tau = 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 similar to0.5, 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.