A minimal model of a tidewater glacier

We propose a simple, highly parameterized model of a tidewater glacier. The mean ice thickness and the ice thickness at the glacier front are parameterized in terms of glacier length and, when the glacier is calving, water depth. We use a linear relation between calving rate and water depth. The change in glacier length is determined by the total change in the mass budget (surface balance and calving flux), but not by the details of the glacier profile and the related velocity field. We show that this may still yield relatively rapid rates of retreat for an idealized bed geometry with a smooth overdeepening. The model is able to simulate the full cycle of ice-free conditions, glacier terminus on land, tidewater glaciers terminus, and backwards. We study two cases: (i) a glacier with a specific balance (accumulation) that is spatially uniform, and (ii) a glacier in a warmer climate with the specific balance being a linear function of altitude. Equilibrium states exhibit a double branching with respect to the climatic forcing (equilibrium-line altitude). One bifurcation is related to the dependence of the calving process on the bed profile; the other bifurcation is due to the height-mass-balance feedback. We discuss the structure of the solution diagram for different values of the calving-rate parameter. The model results are similar to those of Vieli and others (2001), who combined a fairly sophisticated two-dimensional (vertical plane) numerical ice-flow model with the modified flotation criterion suggested by Van der Veen (1996). With regard to the global dynamics of a tidewater glacier, we conclude that the details of the glacier profile or velocity field are less significant than the bed profile and the relation between the water depth and the calving rate.