Stability of quasi-steady flow in an englacial conduit

Abstract

The stability of water flow in an englacial conduit is examined with particular reference to catastrophic outbursts of water. Quasi-steady flow of water in a conduit is considered, the conduit being simultaneously enlarged by frictional heating and compressed by plastic deformation in response to the pressure difference across the tunnel wall. The conduit is fed by an ice-dammed reservoir. With the aid of simplifying assumptions, we have devised a mathematical model consisting of two time-dependent, non-linear, dimensionless ordinary differential equations, which describe the time evolution of the conduit cross-section and the water depth in the reservoir. The conditions leading to different types of time-dependent flow behaviour are examined. Regions of the parameter space where the water flow is stable and unstable have been identified. In the unstable regime, the process of emptying the reservoir has either an oscillatory or an exponential character. In the stable regime, the system's return to equilibrium, following a perturbation, also exhibits an oscillatory or exponential character. Examples of this time-dependent behaviour are presented. The model has also been used to study the influence of the glacier, conduit and reservoir geometries on the system's stability. The results show that an increase in the horizontal area of the water reservoir or an increase in the slope of the conduit enhance the likelihood of a sudden outburst. However, an increase in the glacier thickness or the conduit length stabilizes the equilibria.