  Supraglacial lakes play a central role in storing melt water, enhancing
surface melt, and ultimately in driving ice flow and ice shelf melt through
injecting water into the subglacial environment and facilitating fracturing.
Here, we develop a model for the drainage of supraglacial lakes through the
dissipation-driven incision of a surface channel. The model consists of the St
Venant equations for flow in the channel, fed by an upstream lake reservoir,
coupled with an equation for the evolution of channel elevation due to
advection, uplift, and downward melting. After reduction to a `stream
power'-type hyperbolic model, we show that lake drainage occurs above a
critical rate of water supply to the lake due to the backward migration of a
shock that incises the lake seal. The critical water supply rate depends on
advection velocity and uplift (or more precisely, drawdown downstream of the
lake) as well as model parameters such as channel wall roughness and the
parameters defining the relationship between channel cross-section and wetted
perimeter. Once lake drainage does occur, it can either continue until the lake
is empty, or terminate early, leading to oscillatory cycles of lake filling and
draining, with the latter favoured by large lake volumes and relatively small
water supply rates.
