Finite-amplitude salt fingers in a vertically bounded layer

Abstract

We compute numerically the amplitude of long thin fingers that form in a liquid stratified with sugar S* and salt T* (measured in buoyancy units), for which τ = kS/kT = 1/3 is the ratio of the two diffusivities and the Prandtl number is Pr = v/kT ∼ 103, where v is the viscosity. The finger layer in our model is bounded by rigid and slippery horizontal surfaces with constant T*, S* (the setup is similar to the classical Rayleigh convection problem). The numerically computed steady fluxes compare well with laboratory experiments in which the fingers are sandwiched between two deep (convectively mixed) reservoirs with given concentration differences ΔT*, ΔS*. The model results, discussed in terms of a combination of asymptotic analysis and numerical simulations over a range of density ratio R = ΔT*/ΔS*, are consistent with the (ΔS*)4/3 similarity law for the fluxes. The dimensional interfacial height (H*) in the reservoir experiments (unlike that in our rigid lid model) is not an independent parameter, but it adjusts to a statistically steady value proportional to (ΔS*)−1/3. This similarity law is also explained by our model when it is supplemented by a consideration of the stability of the very thin horizontal boundary layers with large gradients (∂S*/∂z) which form near the rigid surfaces. The preference for three-dimensional salt fingers is also explained by a combination of analytical and numerical considerations.