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Supraglacial lakes play a central role in storing melt water, enhancingsurface melt, and ultimately in driving ice flow and ice shelf melt throughinjecting water into the subglacial environment and facilitating fracturing.Here, we develop a model for the drainage of supraglacial lakes through thedissipation-driven incision of a surface channel. The model consists of the StVenant equations for flow in the channel, fed by an upstream lake reservoir,coupled with an equation for the evolution of channel elevation due toadvection, uplift, and downward melting. After reduction to a `streampower'-type hyperbolic model, we show that lake drainage occurs above acritical rate of water supply to the lake due to the backward migration of ashock that incises the lake seal. The critical water supply rate depends onadvection velocity and uplift (or more precisely, drawdown downstream of thelake) as well as model parameters such as channel wall roughness and theparameters defining the relationship between channel cross-section and wettedperimeter. Once lake drainage does occur, it can either continue until the lakeis empty, or terminate early, leading to oscillatory cycles of lake filling anddraining, with the latter favoured by large lake volumes and relatively smallwater supply rates.

The drainage of glacier and ice sheet surface lakes