Convective Phenomena in Mushy Layers

Since the Annual Review of Fluid Mechanics review of mushy layers by Worster (1997) , there have been significant advances in the understanding of convective processes in mushy layers. These advances have come in the areas of ( a) more detailed analysis, computation, and understanding of convective instabilities and chimney convection in binary alloys; ( b) investigations of diffusive and convective transport processes in ternary alloys; and ( c) applications of mushy layer theory in materials science, sea ice, and polar climate modeling, as well as other geophysical applications such as the convective dynamics of the Earth's core. Our objective for this review is to provide an updated account of the understanding of mushy layer convection and related applications and, in doing so, to provide a new resource to the fluid dynamics research community interested in these complex systems.

Solidification of binary aqueous solutions under periodic cooling. Part 2. Distribution of solid fraction

We report an experimental study of the distributions of temperature and solid fraction of growing $\text{NH}_{4}\text{Cl}$–$\text{H}_{2}\text{O}$ mushy layers that are subjected to periodical cooling from below, focusing on late-time dynamics where the mushy layer oscillates about an approximate steady state. Temporal evolution of the local temperature $T(z,t)$ at various heights in the mush demonstrates that the temperature oscillations of the bottom cooling boundary propagate through the mushy layer with phase delays and substantial decay in the amplitude. As the initial concentration $C_{0}$ increases, we show that the decay rate of the thermal oscillation with height also decreases, and the propagation speed of the oscillation phase increases. We interpret this as a result of the solid fraction increasing with $C_{0}$, which enhances the thermal conductivity but reduces the specific heat of the mushy layer. We present a new methodology to determine the distribution of solid fraction $\unicode[STIX]{x1D719}(z)$ in mushy layers for various $C_{0}$, using only measurements of the temperature $T(z,t)$. The method is based on the phase behaviour during thermal modulation, and opens up a new approach for inferring mushy-layer properties in geophysical and engineering settings, where direct measurements are challenging. In our experiments, profiles of the solid fraction $\unicode[STIX]{x1D719}(z)$ exhibit a cliff–ramp–cliff structure with large vertical gradients of $\unicode[STIX]{x1D719}$ near the mush–liquid interface and also near the bottom boundary, but much more gradual variation in the interior of the mushy layer. Such a profile structure is more pronounced for higher initial concentration $C_{0}$. For very low concentration, the solid fraction appears to be linearly dependent on the height within the mush. The volume-average of the solid fraction, and the local fluctuations in $\unicode[STIX]{x1D719}(z)$ both increase as $C_{0}$ increases. We suggest that the fast increase of $\unicode[STIX]{x1D719}(z)$ near the bottom boundary is possibly due to diffusive transport of solute away from the bottom boundary and the depletion of solute content near the basal region.

Mushy-layer growth and convection, with application to sea ice

Sea ice is a reactive porous medium of ice crystals and liquid brine, which is an example of a mushy layer. The phase behaviour of sea ice controls the evolving material properties and fluid transport through the porous ice, with consequences for ice growth, brine drainage from the ice to provide buoyancy fluxes for the polar oceans, and sea-ice biogeochemistry. We review work on the growth of mushy layers and convective flows driven by density gradients in the interstitial fluid. After introducing the fundamentals of mushy-layer theory, we discuss the effective thermal properties, including the impact of salt transport on mushy-layer growth. We present a simplified model for diffusively controlled growth of mushy layers with modest cooling versus the solutal freezing-point depression. For growth from a cold isothermal boundary, salt diffusion modifies mushy-layer growth by around 5–20% depending on the far-field temperature and salinity. We also review work on the onset, spatial localization and nonlinear development of convective flows in mushy layers, highlighting recent work on transient solidification and models of nonlinear convection with dissolved solid-free brine channels. Finally, future research opportunities are identified, motivated by geophysical observations of ice growth. This article is part of the theme issue ‘The physics and chemistry of ice: scaffolding across scales, from the viability of life to the formation of planets’.

Solid–mush interface conditions for mushy layers

A subtle issue in the study of mushy zones which form during the solidification of binary alloys is that there are two distinct types of solid–mush interfaces which may occur. One of these is a eutectic front and the other is a front which separates the mushy layer and, assuming complete solute rejection, a layer of pure solid. For semi-infinite-domain configurations that admit similarity solutions, such as those at a uniform initial temperature and concentration with an imposed cold temperature at the lower boundary, only one of the two types appears, and the type of front is determined by the various parameters of the system. In a finite domain, it is possible for each type of front to appear at different times. Specifically, the advance of the eutectic front is restricted by the isotherm associated with the eutectic temperature, and the other front type will appear over a longer time scale. Leading up to the time when the front changes type, the concentration being frozen into the solid decreases. This process writes a history of the system into the solid.

Localisation of convection in mushy layers by weak background flow

It is known that freckles form at the sidewalls of directionally solidified materials. We present a weakly nonlinear analysis of the effects of a weak and slowly varying background flow formed by non-axial thermal gradients on convection near onset in a mushy layer. We find that in the two-dimensional case, the onset of mush convection occurs away from the walls. However if three-dimensional disturbances are allowed, the onset occurs near the walls of the container confining the mush. We derive amplitude equations governing this behaviour and simulate their evolution numerically.