Estimating salt and heat fluxes and melt rate at a vertical ice-ocean interface using an eddy solute diffusivity model

The melt rate of a vertical ice face in oceans depends on the salt concentration and salt flux at the ice-ocean interface with the salt concentration at the interface further determined by the ratio R of salt and heat fluxes at the interface and the temperature and salt concentration in the ambient seawater. Motivated by this, we study turbulent double-diffusive convection in a fluid confined between two infinite vertical walls at different temperatures and salt concentrations with the flow driven by buoyancy due to salt concentration difference as in realistic ice-ocean situations. The salt flux, as measured by the dimensionless saline Nusselt numbers Nu_S, can be written as an integral in terms of the eddy solute diffusivity K_s(x) and the diffusion coefficient κ_S of salt in water, where x is the distance along the normal direction to the walls with x=0 at the colder wall. We show that the eddy thermal diffusivity K_T(x) is approximately equal to K_s(x) and thus the heat flux, as measured by the heat Nusselt number Nu_T, can also be expressed as an integral in terms of K_s(x) and the thermal diffusivity κ_T. We propose a three-layer analytical model for K_s(x) with two parameters that depend on the salinity Rayleigh number and the Schmidt number (Sc). Using this model, we can obtain estimates of Nu_S and Nu_T, and thus R, and the melt rate of a vertical ice face at realistic values of Sc ≈ 2500 and Lewis number (Le) ≈ 180.