Drainage through subglacial water sheets

Subglacial drainage plays an important role in controlling coupling between glacial ice and underlying bed. Here, we study the flow of water in thin, macroporous sheets between ice and bed. Previous work shows that small perturbations in depth of a nearly parallel-sided water film grow unstably because these areas have enhanced viscous dissipation that leads to enhanced melting of an ice roof. We argue that in the presence of bed protrusions bridging a water sheet, downward motion of the ice roof can stabilize this sheet. Stability results when the rate of roof closure increases faster with water depth than the rate of viscous dissipation. We therefore modify existing theory to include protrusions that partially support the overlying glacier. Differences in the pressure on protrusions relative to water pressure drive roof closure. The mechanisms of both regelation and creep normal to the bed close the overlying ice roof and decrease the ice-bed gap. In order to account for multiple protrusion sizes along the bed (for instance, resulting from an assortment of various-sized sediment grains), we incorporate a method of partitioning overburden pressure among different protrusion size classes and the water sheet. Partitioning is dependent on the amount of ice protrusion contact and, therefore, water depth. This method allows prediction of roof closure rates. We then investigate stable, steady sheet configurations for reasonable parameter choices and find that these steady states can occur for modest water depths at very low effective pressures, as is appropriate for ice streams. Moreover, we find that multiple steady sheet thicknesses exist, raising the possibility of switches between low and high hydraulic conductivity regimes for the subglacial water system.