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Certain geological features have been interpreted as evidence of channelized magma flow in the mantle, which is a compacting porous medium. Aharonov et al. (J. Geophys. Res., vol. 100 (B10), 1995, pp. 20433–20450) developed a simple model of reactive porous flow and numerically analysed its instability to channels. The instability relies on magma advection against a chemical solubility gradient and the porosity-dependent permeability of the porous host rock. We extend the previous analysis by systematically mapping out the parameter space. Crucially, we augment numerical solutions with asymptotic analysis to better understand the physical controls on the instability. We derive scalings for the critical conditions of the instability and analyse the associated bifurcation structure. We also determine scalings for the wavelengths and growth rates of the channel structures that emerge. We obtain quantitative theories for and a physical understanding of, first, how advection or diffusion over the reactive time scale sets the horizontal length scale of channels and, second, the role of viscous compaction of the host rock, which also affects the vertical extent of channelized flow. These scalings allow us to derive estimates of the dimensions of emergent channels that are consistent with the geologic record.
The free-boundary problem between a liquid region and a mushy layer (a reactive porous medium) must respect both thermodynamic and fluid dynamical considerations. We develop a steady two-dimensional forced-flow configuration to investigate the thermodynamic condition of marginal equilibrium that applies to a solidifying mushy layer with outflow and requires that streamlines are tangent to isotherms at the interface. We show that a ‘two-domain’ approach in which the mushy layer and liquid region are distinct domains is consistent with marginal equilibrium by extending the Stokes equations in a narrow transition region within the mushy layer. We show that the tangential fluid velocity changes rapidly in the transition region to satisfy marginal equilibrium. In convecting mushy layers with liquid channels, a buoyancy gradient can drive this tangential flow. We use asymptotic analysis in the limit of small Darcy number to derive a regime diagram for the existence of steady solutions. Thus we show that marginal equilibrium is a robust boundary condition and can be used without precise knowledge of the fluid flow near the interface.
Solute transport within solidifying binary alloys occurs predominantly by convection from narrow liquid chimneys within a porous mushy layer. We develop a simple model that elucidates the dominant structure and driving forces of the flow, which could be applied to modelling brine fluxes from sea ice, where a cheaply implementable approach is essential. A horizontal density gradient within the mushy layer in the vicinity of the chimneys leads to baroclinic torque which sustains the convective flow. In the bulk of the mushy layer, the isotherms are essentially horizontal. In this region, we impose a vertically linear temperature field and immediately find that the flow field is a simple corner flow. We determine the strength of this flow by finding a similarity solution to the governing mushy-layer equations in an active region near the chimney. We also determine the corresponding shape of the chimney, the vertical structure of the solid fraction and the interstitial flow field. We apply this model first to a periodic, planar array of chimneys and show analytically that the solute flux through the chimneys is proportional to a mush Rayleigh number. Secondly we extend the model to three dimensions and find that an array of chimneys can be characterized by the average drainage area alone. Therefore we solve the model in an axisymmetric geometry and find new, sometimes nonlinear, relationships between the solute flux, the Rayleigh number and the other dimensionless parameters of the system.
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