Can we understand how microphysics determine the phenome-nological constitutive relation for ice? At present, this is only partly possible (Duval etal., 1983; Weertman, 1983; Budd & Jacka, 1989; Alley, 1992; Goldsby & Kohlstedt, 2001).
The bulk constitutive properties are manifestations of the underlying deformation mechanisms. These will vary with conditions of stress, temperature and grain size (Duval et al., 1983; Goldsby & Kohlstedt, 2001). It is first essential to recognize that the constitutive relation for ice analysed and supported in this article (power-law creep, with n ~ 3) is applicable specifically to those parts of the ice sheets most important for their large-scale flow and structure: the rapidly deforming layers near the glacier bed on the ice sheet flanks and in ice-stream shear margins. A modified constitutive rule will be needed elsewhere if conditions favour different deformation mechanisms, such as in the low-stress environment of the ice divide region (see Pettit, this volume, Chapter 58).
One important deformation mechanism relevant here is dislocation glide, the motion of crystal lattice defects along the basal planes of the hexagonal ice crystals. Individual ice crystals are like little decks of cards, with easy shear deformation in one plane only; these are the basal planes. This deformation mechanism is certainly dominant in ice, as it quantitatively explains the observed development of crystal c-axis fabrics (the statistics of orientations) and explains the observed dependence of fluidity enhancement on these fabrics (Li et al., 1996; Azuma & Goto-Azuma, 1996; De La Chapelle et al., 1998). It is unlikely to be the rate-limiting mechanism, however. Laboratory measurements of single ice crystals deforming by dislocation glide show the stress exponent is n ~ 2-3 for this mechanism in isolation (data are reviewed by Goldsby & Kohlstedt, 2001). In a polycrystal, ice grains cannot deform only by this mechanism because adjacent grains with different shapes and orientations interfere. Other mechanisms must accommodate such incompatible deformations, and these are probably rate-limiting. These may include dislocation motion in stiff directions, diffusion along grain boundaries and grain-boundary sliding (which may also be a consequence of dislocation motion). Rate limitation may also arise from a need to recrystallize or move grain boundaries to eliminate dislocation tangles (Montagnat & Duval, 2000). Uncertainty about the rate-limiting mechanism through the full range of icesheet stresses is a major gap in our current knowledge.
Generally, the rate of deformation arising from dislocation mechanisms is proportional to the product of dislocation speed and dislocation density (Weertman, 1983). Both speed and density depend on deviatoric stress, and it is the product of these dependences that determines the exponent n for these mechanisms. The speed of dislocation motion is linear with stress. Weertman (1983) has argued that n = 3 occurs when the deformation attains a steady-state, requiring that the external applied stress is balanced by internal stresses associated with lattice distortions around dislocations; in such a condition, theory suggests that dislocation density will be proportional to the square of stress, yielding a cubic product. This sort of steady-state is likely to occur in the rapidly straining deep layers of the ice sheets and smaller glaciers (Alley, 1992), where recrystallization is known to occur.
Clouding of this simple picture (Weertman, 1983) arises from clear data showing a transition to n ~ 4 at high stresses in excess of 1 MPa (data reviewed by Goldsby & Kohlstedt, 2001). This is the dislocation creep regime, where the rate-limiting process is dislocation motion in stiff directions. This has led to the idea that the phenomenological result n ~ 3 near 1 bar stress is describing a transition regime, where dislocation creep with high n is comparable in magnitude to some other mechanism with lower n that is rate-limiting at lower stress.
Molecular-scale properties that control the ice softness A will also be different depending on which deformation mechanisms are active and rate-limiting. For dislocation-motion mechanisms (Weertman, 1983), the molecular spacing in the lattice is important. Each dislocation jump will move a distance similar to this spacing (the Burgers vector), so the dislocation speed should be proportional to it. Dislocation density, if controlled by the balance of internal and external stresses, will depend inversely on the second power of both the molecular spacing and the shear modulus for the ice lattice. There is presently no calculation of polycrystalline ice viscosity from first principles and known lattice properties.
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