Structure of mushy layers grown from perfectly and imperfectly conducting boundaries. Part 1. Diffusive solidification

Abstract

We model transient mushy-layer growth for a binary alloy solidifying from a cooled boundary, characterising the impact of liquid composition and thermal growth conditions on the mush porosity and growth rate. We consider cooling from a perfectly conducting isothermal boundary, and from an imperfectly conducting boundary governed by a linearised thermal boundary condition. For an isothermal boundary we characterise different growth regimes depending on a concentration ratio, which can also be viewed as characterising the ratio of composition-dependent freezing point depression versus the temperature difference across the mushy layer. Large concentration ratio leads to high porosity throughout the mushy layer and an asymptotically simplified model for growth with an effective thermal diffusivity accounting for latent heat release from internal solidification. Low concentration ratio leads to low porosity throughout most of the mushy layer, except for a high-porosity boundary layer localised near the mush–liquid interface. We identify scalings for the boundary-layer thickness and mush growth rate. An imperfectly conducting boundary leads to an initial lag in the onset of solidification, followed by an adjustment period, before asymptoting to the perfectly conducting state at large time. We develop asymptotic solutions for large concentration ratio and large effective heat capacity, and characterise the mush structure, growth rate and transition times between the regimes. For low concentration ratio the high porosity zone spans the full mush depth at early times, before localising near the mush–liquid interface at later times. Such variation of porosity has important implications for the properties and biological habitability of mushy sea ice.