Profile results

Using a standard value of n = 3, the calculated elevation profiles match the topographies very well (Fig. 57.1). To interpret the optimized values, I use the fact that these are essentially thermometers (though imprecise ones). I calculate the equivalent effective temperature T* as the temperature corresponding to the A value given by

The inverse of A gives effective temperature via Equation (4).

For Greenland, E is known from borehole studies to be ca. 2 to 4 due to the ice-age ices being soft, and I use the value E = 2.5 as suggested by Paterson (1991). For the Antarctic cases, I use E = 1. Much of Greenland is frozen at its bed, so the appropriate flux partitioning qd = 1. For East Antarctica and inland West Antarctica I will use both qd = 1 and qd = 0.5, the latter meaning equal partitioning of the flux between deformation and basal motion. Optimized values are shown in Table 57.1.

In column 5 T* is effective temperature calculated independently from direct temperature estimates from boreholes and models (Huybrechts & Oerlemans, 1988; Engelhardt & Kamb, 1993; Paterson, 1994, p. 222; Huybrechts, 1996; Engelhardt, 2005), and using Equation (10). The agreement is very strong between these independent estimates. Figure 57.1 (a) Calculated elevation profiles compared with measured elevations for the three representative transects described in the text. (b) Root-mean-square mismatch of calculated and measured elevations, averaged for the three profiles, as a function of stress exponent. For each n value, each profile is least-squares optimized by adjusting softness parameters as described in the text. distance from margin or grounding line (km) stress exponent n

Figure 57.1 (a) Calculated elevation profiles compared with measured elevations for the three representative transects described in the text. (b) Root-mean-square mismatch of calculated and measured elevations, averaged for the three profiles, as a function of stress exponent. For each n value, each profile is least-squares optimized by adjusting softness parameters as described in the text.

Table 57.1 Optimized values used in profile calculations

is some suggestion here that n ~ 4-5 is more appropriate than n = 3, but the approximations used here must be remembered. Note that (Equation 16) the sensitivity of profile shape to n becomes progressively smaller as n becomes larger, toward the plastic limit; the broad upper half of the mismatch curve (Fig. 57.1) is inherent in the method.

The topographic profiles of the modern ice sheets thus provide direct confirmation that, for the deep, rapidly deforming ices most relevant here, ice properties have been accurately characterized phenomenologically using laboratory experiments and local in situ measurements in active glaciers and ice shelves (Weertman, 1983). This is a major accomplishment.

There is a bit of deception hidden in Fig. 57.1, which is that for Greenland the use of a flat basal topography, as shown, reproduces the elevation profile better than does a more realistic topography, given a spatially uniform S. The second tabulated value for Greenland softness (So(G)) included here is a more realistic case for which the western margin mountain range has been included as basal topography in the profile calculation (abstracted from Bamber et al., 2001b). In this case the profile matches the measured elevations as well as shown in Fig. 57.1 only if S increases toward the ice margin, which is expected to be the case, because the bed becomes warmer toward the margin and some sliding occurs.

The profile shapes depend also on the stress exponent n, and this permits evaluation of which n values are plausible. I specify n, find best-fit values for the four softness parameters, and then calculate the root-mean-square mismatch of the model profile elevations from the known topography. The result of this exercise (Fig. 57.1) clearly indicates that 'average' ice is neither linear viscous nor plastic, and that the non-linearity is pronounced (certainly n > 2). This, of course, matches expectation from laboratory experiments, and in situ analyses of ice deformation. There