Figure 57.2 Relations between steady-state ice volume per divide length (i.e. area of cross-sectional profiles), effective viscosity, stress exponent and two specific controls on viscosity (temperature and enhancement). (a) Black diamonds indicate estimated positions for the three modern ice sheets, according to representative profiles: EA, East Antarctic-type; WA, West Antarctic-type; G, Greenland-type. West Antarctic volumes are those above the flotation level (hence sea-level equivalent). (b and c) Volumes have been normalized to values at 0°C and at E = 1. For 'WA, basal' the effective temperature or enhancement of basal shear has been varied. For 'WA, margins' only the effective temperature or enhancement of the ice-stream shear margins has been varied.

and a similar relation for the ice-stream case. Effective viscosities shown here are averages of h for each profile.

57.4.1 Ice volume, given fixed climatic forcing and span

Ice volumes (per length of ice divide) are calculated by integrating profile solutions. In the case for which both L and b(x) are fixed, ice volume depends on the microphysical parameters as shown in Fig. 57.2. Effective viscosity exerts primary control, with ice volume increasing by approximately a factor of two per order-of-magnitude increase in viscosity. This sublinear (roughly logarithmic) dependence of volume on h arises because the flux is fixed by climate, whereas the accommodation of the flux by ice flow relies on ice thickness in at least three multiplicative ways: one through the magnitude of stress, one through the integration to convert strain rate to velocity, and one through the integration to convert velocity to flux.

The consequence of having a stress exponent n > 1 is to reduce ice volume. Compared with an ice sheet with n = 3, a linear-viscous one would have a flatter surface in the ice-sheet interior, a thicker mid-flank, and a much steeper marginal zone. At the low stresses in the interior of the ice sheet, the linear-viscous ice would be more readily deformed, allowing a smaller surface slope to accommodate the ice flux for a given thickness.

The more pronounced dependence of the WAIS-type ice-sheet volume on viscosity results from steepening of the weak-bed flank

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