Theoretical Glaciology

In Chapter 1 the constitutive theory was treated from a rather general point of view and no specific details were given about ice as a material. Here, we give a brief summary of its constitutive response from a view-point of material science. No claim of completeness will be made, however.

The previous chapters were devoted to the derivation of the basic principles and to the material properties of ice with no particular intention of applying the collected information to a physical problem. Here, in this chapter, attention will be focussed on a first simple application of the knowledge gained so far. In particular, an attempt is made to shed some light on the deformation mechanism and thermal conditions of glaciers and ice sheets.

In Chapter 3, the strictly parallel-sided ice slab was analysed. Solutions to the flow and temperature problem were constructed for which stresses did not vary with the coordinate along the glacier axis. Two solutions were found which satisfied these conditions. In the first, apart from stresses, velocities and temperature are also independent of the coordinate along the ice slab. For this solution to be consistent with the boundary conditions of the general model, both accumulation and ablation had to be neglected. To account for the inclusion of the latter, and for floating ice shelves, in order to properly allow for mass flux from the grounded portion of the ice sheet, the velocity field had to be assumed to vary with the length coordinate. On the basis that stress and temperature are still independent of this coordinate, it was found that the longitudinal velocity components vary linearly along the ice sheet axis, whereas transverse velocities remain independent of the length coordinate and vary linearly with depth. Longitudinal and transverse stretchings are equal and constant in this case, and the constant transverse surface velocity gives the possibility of including a constant accumulation rate. For a grounded ice slab, it was shown that this solution is in violation with a stress-dependent sliding law. For an ice shelf, on the other hand, the solution is meaningful, provided that the drag from the ocean current is neglected; in this case, the flow parameters can be related to the thermal conditions in the atmosphere. It is clear from the above that for ice slabs resting on a rockbed, a more general solution to the flow problem should be sought which is free from the defects mentioned above. This is one reason why one should search for a solution, in which longitudinal stretching effects are consistently taken into account. A second reason is the fact that glaciers and ice sheets are only very nearly parallel-sided. Generally, the rockbed and free surface undulate about a mean plane surface. Rockbed protuberances force the ice to flow around them. Together with the spatially-dependent accumulation rate, they cause (also in the steady state) surface undulations and are thus responsible for the longitudinal stretching effects occurring in ice slopes. It is evident in these instances that stresses can no longer be independent of the longitudinal coordinate.

In the previous chapter, the nearly parallel-sided ice slab was thoroughly analysed; in particular, its steady-state response was treated, and it was shown how small-amplitude bottom protuberances affect surface topography, basal stresses and surface velocities. Attention was restricted to plane motion and to ice slabs for which the mean bed and mean surface were plane and strictly parallel so that the mean ice thickness was constant. In reality, this is only approximately correct as it is well known that ice-sheet thicknesses may vary from up to 3000 m inland or more to zero thickness at the snout. On length scales which are comparable with the extension of the ice sheet, the assumption of small variations about a parallel-sided ice slab is thus unrealistic. On the other hand, calculations, as those performed in Chapter 4, must be limited to length scales for which ice-sheet thicknesses do not deviate appreciably from a layer with a constant thickness.

The investigations of ice-flow problems which have been undertaken in this book so far, have been rather qualitative and thus theoretical because only plane flow was considered and, furthermore, very simple glacier geometries were assumed. Either conditions near the strictly parallel-sided ice slab were analysed or the top and bottom surfaces were assumed to vary slowly in the horizontal directions. Three important problems, however, remained untouched; they are: