The flow of glaciers and ice-sheets as a problem in plasticity

Abstract A calculation is made of the distribution of stress and velocity in an ideal glacier and in an ideal ice-sheet. The ice is assumed to have a constant yield stress and to obey, like other polycrystalline plastic aggregates, the Levy-Mises equations of flow and either the Mises or the Tresca criterion of yielding. The solution obtained for an ideal glacier represents the two-dimensional flow of a long slab of ice down a gently undulating rough slope. The addition of ice to the upper surface by snowfall and the removal of ice by ablation are allowed for, but the frictional resistance of the sides of the glacier valley is neglected. Two states of flow are possible, ‘active’ and ‘passive’, corresponding to the active and passive Rankine states in soil mechanics. Which of these states occurs at a given place depends upon the relative magnitudes of the curvature of the bed and the rate of snowfall or ablation; a simple algebraic expression of this dependence is obtained. In both states of flow the velocity is greatest at the surface and decreases with depth according to an elliptical law. It is shown that, in accordance with observation, crevasses of limited depth can open in active flow but not in passive flow. The slip-line field for the problem has a close connexion with the directions and positions of shear faults (although the laminated structure of a glacier is doubtless also an important factor here). In passive flow the faults to be expected are similar to the ‘thrust planes’ often seen on glaciers. The theory suggests that in active flow a complementary sort of shear fault with the opposite direction of movement may occur—and there is some observational evidence for this. The tendency of glaciers to accentuate hollows in their beds is connected with the suggestion that erosion should proceed faster under passive flow than under active flow. The second solution obtained is formally similar but represents the two-dimensional flow of a large ice-sheet, such as the Greenland ice-cap. If a horizontal bed is assumed the profile is calculated to be formed from parts of two parabolas, the maximum height being given by the yield stress of ice. In the accumulation area flow is active. The maximum velocity is every­ where at the surface while the maximum shear rate is on the bed. The solution thus gives no support to the belief that the weight of ice above squeezes out the underlying ice at a faster rate.