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Figure 57.3 Dependence of WAIS-type volume per length on ice stream width (2w), at three different effective temperatures for the shear margins.

above the flotation level as the shear margins become stronger. This type of ice sheet could, in principle, convert from being concave-upward on its flank to being convex-upward, as ice streams stiffen or are replaced by strong-bed regions.

Given the measured activation energies for ice, the full range of effective temperatures for terrestrial glaciers could induce a nearly factor-of-two variation in ice volumes. However, because the thick ice sheets are thermal insulators, it is unlikely that effective temperatures for the ice sheets would vary outside the range of approximately -1 to —15°C, corresponding to a ca. 50% volume variation. The capacity for enhancement variations to induce volume variation is even smaller, but still significant. An enhancement variation of a factor of three, as observed in some of the Arctic ice-cap boreholes, would induce a ca. 15% variation (as can be seen directly from Equation 18).

The ability of temperature decrease to stiffen the margins of ice streams implies that the ice-stream width versus ice-sheet volume relation for a WAIS-type ice sheet is temperature dependent (Fig. 57.3). Thus, despite the vast complexity of the ice-stream-dominated system, it is clearly important to account for the activation energy of ice creep when analysing, for example, the configuration of the ice streams during the Last Glacial Maximum.

57.4.2 Response time

An ice sheet has many different response times, corresponding to different types of external and internal forcings. Most important of the external forcings are changes of interior accumulation rate, changes of marginal position induced by ablation-zone processes or sea-level change, and changes in temperature. These response times decrease as the stress exponent n increases (for reasons explained in section 57.2.2.1). They are also functions of ice thickness through the characteristic time H/b, which is essentially a

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(d) East Antarctic n = 1 n = 3

14 15 16 13 14 15 Log1 0 effective viscosity (Pa s)

14 15 16 13 14 15 Log1 0 effective viscosity (Pa s)

Figure 57.4 Dependence of approximate characteristic response-times on effective viscosity, stress exponent, and one control on viscosity (temperature), shown in panel e. Black diamonds (in a, b, c, and d) are estimates for modern EAIS and GIS. The response-time to accumulation-rate change is that for which ice surface slope does not change, corresponding to a situation wherein the ice margin is free to expand. The comparable response times for fixed margins are smaller, by a factor of (n + 2)/(2n + 2). These response times neglect covari-ant change in temperature, which will slow all responses.

effective temperature (°C)

Figure 57.4 Dependence of approximate characteristic response-times on effective viscosity, stress exponent, and one control on viscosity (temperature), shown in panel e. Black diamonds (in a, b, c, and d) are estimates for modern EAIS and GIS. The response-time to accumulation-rate change is that for which ice surface slope does not change, corresponding to a situation wherein the ice margin is free to expand. The comparable response times for fixed margins are smaller, by a factor of (n + 2)/(2n + 2). These response times neglect covari-ant change in temperature, which will slow all responses.

residence time and is related to how rapidly mass can be redistributed to approach a new equilibrium configuration.

Two examples of how response times might increase with effective viscosity and with decreased stress exponent are shown here (Fig. 57.4): (i) the linearized estimate for response time of ice thickness to a change in accumulation rate (H/(n + 2)b; see Whillans, 1981); and (ii) the diffusive-limit estimate for response time of interior thinning to retraction of the ice margin (which depends inversely on the diffusivity y,; see Cuffey & Clow, 1997). Figure 57.4 also shows the dependence of the response times on effective ice temperature; cold ice sheets respond more slowly. Because this effect arises entirely from the dependence of

n = 3

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13 14 15 16 -35 -20 -5 Log1 0 effective viscosity (Pa s) effective temperature (°C)

Figure 57.5 Relations between steady-state ice volume per divide length and other variables for a Greenlandic-type ice sheet with a flat bed, allowing for feed-back between accumulation rate and topography. Black squares indicate points for which the calculation was performed. In panel (b), enhancement values are indicated corresponding to the equivalent softness change induced by the temperature change as shown.

ice-sheet geometry on temperature, it is in fact identical to the volume-temperature relation.

57.4.3 Ice volume, given climate feed-back

In the preceding calculations, climatic forcing and ice-sheet span were fixed. In reality, there will be complex feed-backs between ice-sheet topography and the climatic forcing, with melt rates generally being reduced as elevation increases, and with ice-sheet interiors becoming very dry as they are elevated and isolated from atmospheric moisture transported to the margins (Weertman, 1976; Oerlemans, 1980). In some cases, the margins (and hence L) will indeed be fixed, by sea-level or topography. In other cases, as with the GIS and the Pleistocene Laurentide Ice Sheet, the ice sheet is free to expand across the land surface until a warmer climate is reached or a calving margin established.

The effect of such feed-backs is to greatly amplify the dependence of ice volume on viscosity and on the stress exponent. Consider one example (Fig. 57.5), a Greenland-type ice sheet for which the accumulation rate above the equilibrium line is fixed as a function of elevation (above equilibrium line) rather than as a function of position. Further, let the equilibrium line altitude rise linearly with distance from the ice-sheet centre, with ablation rate at zero elevation increasing similarly. This situation corresponds to an ice sheet that is centred in a climatically cold region and that encounters an increasingly hostile warm climate as it expands outward (L increasing). Here the ice-sheet profiles are calculated using a finite-difference solution to Equation (13), and this profile calculation iterated with increasing L values until the steady-state profile is found. Note that I do not include isostatic effects here (see Oerlemans, 1980).

The sensitivity of the steady-state ice volume to effective viscosity is much greater in this case than in those of Fig. 57.2 because the thicker ice also implies a cooler surface climate and therefore a narrower ablation zone. Further, as the span L increases, the total flux through the equilibrium line becomes larger so the ablation needs to be more intense to balance it. Reducing the stress exponent n causes a particularly large increase in L and volume before a new steady configuration is attained because the ice-sheet margin is steeper, as discussed in section 57.4.1.

The exact magnitude of volume change per viscosity change shown in Fig. 57.5 is, of course, dependent on the choice of horizontal ablation-rate gradient (-0.01 myr-1km-1 used here), and so only applies for this one case.

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