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 = ν/kT = 7 and molecular diffusivity ratio τ = kS/kT = 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 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