A theory of glacier surges

We propose a model of glacier flow that is capable of explaining temperate glacier surges. The laws of conservation of mass and momentum are supplemented by the prescription of a sliding law that gives the basal shear stress τ as a function of the basal velocity u and the effective pressure N. The effective drainage pressure N is determined by a simple study of the subglacial hydraulic system. Following Rothlisberger, we determine N = NR for the case of drainage through a single subglacial tunnel. Alternatively, following Kamb, we find that the corresponding theory for a linked-cavity drainage system yields N = NK < NR. Furthermore, the stability of each drainage system depends on the velocity u, such that for large enough u, there is a transition from tunnel to cavity drainage. Consequently, one can write N = N(u). We then find that the sliding law τ = τ(u) is multivalued, and hence so also is the flux/depth relation Q = Q(H). An analysis of the resulting system of equations is sketched. For large enough accumulation rates, a glacier will undergo regular relaxation oscillations, resembling a surge. The surge is triggered at the point of maximum stress; from this point two hydraulic transition fronts travel up and down glacier to calculable boundary points. The speed of propagation is the order of 50 metres an hour. At these fronts, the tunnel drainage system collapses, and a high water pressure cavity drainage system is installed. This activated zone has high velocities and quickly relaxes (surges) to a quasi-equilibrium state. This relaxation is much like opening a sluice gate, in that a large wave front propagates forward. Behind this wave front, the velocity can decay oscillatorily, and thus the flow can be compressive. We conclude with some discussion of the effects of seasonal variation and of prospects for the current theory's applicability to soft-bedded glaciers.