Mathematical Models of ice stream dynamics and supraglacial drainage

Patterning is a recurrent feature of glacial systems, which characterizes as much subglacial and supraglacial environments as the flow of ice itself. Some examples include bedforms developing at the contact between ice and bed, spatial organization in subglacial and supraglacial drainage networks, the narrow corridors of fast flowing ice known as ice streams that form the arterial drainage system of large ice sheets, and temporal switches between slow and fast flow regimes in glacier and ice stream flow. This thesis focusses on two types of glacial patterns, namely ice streams and channelization in supraglacial drainage networks. Ice flow within ice sheets is far from uniform, with the narrow bands known as ice streams flowing at velocity two order of magnitude larger than the rest of the ice sheet. In the Siple Coast region of West Antarctica ice streams experiance weak topographic confinement, thus suggesting that they may originate spontaneously from an otherwise uniform flow as a fingering instability. Motivated by observations suggesting that the marked contrast in velocity between ice streams and surrounding ice is due to a transition from frozen, thus sticky bed underneath slow flowing regions, to molten, thus well lubricated bed under ice streams, we investigate the role of basal thermal transitions in relation to the onset of ice streams. Our findings suggest that basal transitions from frozen to molten bed (or vice versa) can undergo an instability potentially leading to the onset of streaming. An asymptotic analysis for short wavelenght perturbations shows that, at wavelengths of few ice thicknesses, such instability is controlled by the interplay between strain heating and heat advection from the region upstream of the transition. We also find that the background structure of the ice sheet is key to pattern formation. In particular, in the case of ice flowing from molten to frozen regions we find an instability at the ice sheet thickness scale or smaller, which is not resolved by most ice sheet models. Observations reveal that ice streams experience significant temporal variability on a variety of time scales, ranging from decadal to multi-millennial ones. As much as spatial patterning, such variability holds implications for the future of ice sheets, sea level change, and the interpretation of geological records. Recent work \citep{robel} shows that the switch between steady streaming conditions and self-sustained oscillations with multi-millennial periodicity can be understood as a Hopf bifurcation. Little is presently known about shorter scale variability, which however appears more likely to originate from external forcing. In chapter \ref{ch:stoch} we explore the effects of a specific type of forcing, i.e. stochastically-varying climatic conditions, on the temporal dynamics of ice stream flow. We find that data-based climate fluctuations alter the deterministic dynamics substantially, and are capable of introducing widespread, short-scale oscillations even in ranges of the parametric regime where the deterministic dynamics predict steady streaming. We thus conclude that noise-induced transitions may play a role in the observed temporal dynamics of ice stream flow. In part \ref{drain} we turn to patterning in drainage networks on the surface of glaciers. Supraglacial drainage networks route meltwater originating on the surface of glaciers towards moulins and crevasses, through which it eventually reaches the base of the ice. Therefore, understanding the physical controls on the structure of the drainage network has implications for how surface melt influences the motion of ice. Here we focus on the physical controls on the formation of evenly spaced channels on the surface of glaciers. In particular, we find that the flow of meltwater on bare ice is capable of carving evenly spaced channels as a result of a morphological instability. We show that in certain conditions the network is shaped solely by the hydrodynamics of meltwater regardless of ice thermal conditions, which justifies widely-observed regular patterns in drainage networks. Finally, comparison of our results with the geometrical feature of supraglacial networks reported in the literature shows good agreement between model's predictions and observations.