Subglacial water sheets: Depth and stability are compatible

There exists a broad spectrum of subglacial hydraulic configurations. These are typically categorized based on parameters such as cross-sectional area, along-flow morphology, and water discharge. Common examples of categories include cavities linked through small orifices and subglacial hydraulic sheets that are much wider than deep. Sheets have often been discounted in the glaciological literature because they decay to channels or other drainage configurations above a critical water depth (≤ 4 mm) (Walder, 1982). We show that under reasonable conditions much thicker sheets can remain stable. We present a new formulation of water sheets based on insight gained from glacier sliding. In this case, we consider an ice ceiling that intrudes into a water sheet because viscous creep and regelation act normal to the ice--bed interface. We modify a stress renormalization technique originally formulated for the study of linked cavities. In our renormalization, obstacles, such as sediment grains of different sizes, support some of the weight of overlying ice with water pressure supporting the remainder. Obstacles are divided into size classes with each different class bearing a different percentage of the weight of the overlying ice. This formulation yields a number of simultaneous equations: one velocity equation for each size class, and one total stress balance. Intrusion velocities are dependent on the effective pressure, which is the ice overburden pressure less the water pressure. For zero effective pressure, no closure occurs. When effective pressure is high, results show that instantaneous closure velocities can range up to approximately 0.4 m h-1. However, these velocities are not sustained indefinitely. As ice intrudes farther into the sheet (i.e., water depth decreases), the ice rests on more obstacles, stress concentrations disperse, and velocities decrease. Stress solution distributions can vary considerably for each obstacle size class. Thus, a true controlling obstacle size is ill-defined and not necessarily justifiable. In addition, results suggest that water sheets can be stable to much greater depths because intrusion velocities are coupled to effective pressure. This conclusion results from the inclusion of both regelation and viscous creep into the governing equations. Creep acts rapidly and is the controlling mechanism for high effective pressures. Regelation is the controlling mechanism when effective pressures are low. For these low stress cases, sheets close slowly and are more stable than a simple viscous closure model suggests.