As we discussed in Chapter 3, a great deal of ice is lost from the Greenland and Antarctic Ice Sheets and from tidewater glaciers by calving. However, the calving process is poorly understood. In the case of tidewater glaciers, extending longitudinal strain in the last several kilometers of the glacier usually results in extensive crevassing, so the ice arrives at the calving face in a weakened condition. At the calving face, blocks ranging in size from fractions of a cubic meter to 104 m3 break off and fall into the water. Other blocks break off below the water surface and float upward. Finally, there is a rain of smaller fragments, most of which are probably released by melting along grain boundaries. In Antarctica, in contrast, glaciers reaching the sea tend to form floating ice shelves. Any crevasses that were present near the grounding line are largely healed. Calving from ice shelves commonly involves blocks from 105 to 1011 m3. While the processes of calving from grounded tidewater glaciers and floating ice shelves both involve propagation of fractures (Chapter 4), it seems likely that the origin of the stresses is substantively different in the two cases.
It is widely believed that the demise of the Late Pleistocene ice sheets was facilitated by loss of ice in calving bays that formed at the ends of ice streams and migrated rapidly headward. Such calving would resemble that in grounded tidewater glaciers. For this reason, the process of calving of such glaciers has attracted considerable interest over the past decade. One of the first efforts to tackle this problem was by Brown etal. (1982). Using the method described in Chapter 3 (Equation 3.14), they found that calving speeds, uc, were proportional to mean water depth, hw, thus:
However, as we noted, the physical reasons for this relation are unclear. Analytical efforts to describe the static stress distribution in a calving ice tongue, involving both longitudinal stresses and torques due to the imbalance between hydrostatic stresses in the ice and in the water at the calving face, failed to detect any stresses that might vary with water depth and hence be responsible for an empirical relation like Equation (11.14).
To study this problem further, Hanson and Hooke (2000) resorted to a plain-strain steady-state finite-element model. The model domain was 2000 m long in order to buffer the area of interest, the last ~500 m, from a poorly constrained ice flux into the upglacier end. The reference model had a calving face 200 m high and contained 16 000 elements and 16 440 nodes. (This necessitated solution of 48 440 simultaneous equations!) The lower 140 m of the face were submerged, so the subaerial partwas 60 m high, a typical average height (Brown etal., 1982). Sliding was allowed along the bed.
Figure 11.6 shows calculated distributions of horizontal velocity, u, and longitudinal stress deviator, a 'xx, in the reference model. A zone of high u and a 'xx is present just below the water line near the calving face. We hypothesized that this would tend to produce an overhang in the face, and that this might facilitate calving. As a measure of the rate of overhang development, we calculated the velocity gradient, du/dz, between this point of maximum velocity and the bed (where the glacier was sliding). Comparison of models with total calving-face heights ranging from 100 to 300 m, all of which had subaerial heights of 60 m, suggested that, in nature, du/dz probably increases nearly linearly with water depth (Figure 11.7). In addition, the rate of stretching along the bed just upglacier from the calving face, a'xx bed, increases with water depth (Figure 11.7). The latter may facilitate the formation of bottom crevasses, and hence submarine calving. The two effects, combined, provide a plausible physical explanation for the empirical relation, Equation (11.14).
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