Steady, compositional convection in a mushy layer, cooled from below, is analysed for cases in which the Rayleigh number is very large. Solutions are found in which the flow is downwards everywhere except through narrow, vertical chimneys in the layer. The fluxes of mass, heat and solute from the mushy layer into the overlying melt are determined in terms of a single parameter that is proportional to the number density of chimneys. The model solutions can be used to interpret the results of laboratory experiments.

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