Englacial and basal temperatures along a flowline calculated using the Column model

Let us now consider the temperature distribution along a flowline calculated with the use of the Column model (Figure 6.12). The original objective of the modeling shown in Figure 6.12 was to investigate the possibility that, along the margin of the Laurentide Ice Sheet in North Dakota, there could have been a ~2 km-wide zone in which the ice was frozen to the bed. Such a temperature distribution is implied by glacial landforms, as discussed further below (Moran et al., 1980). Thus, the flowline modeled was assumed to extend from Hudson Bay to North Dakota.

In the model, the accumulation rate was assumed to be 0.20 m a-1 65 km upglacier from the equilibrium line, and to decrease linearly to 0.05 m a-1 at the divide, and to 0 at the equilibrium line. The decrease in bn toward the divide is consistent with the present accumulation pattern in Antarctica (Figure 3.9) and northern Greenland, although not southern Greenland (Zwally and Giovinetto, 2000). In the ablation area, the ablation rate increased linearly downglacier from the equilibrium line, and the rate of increase was adjusted to provide a balanced mass budget. The horizontal velocity was approximated by the balance velocity (Equation (5.1) modified to allow for divergence of the flowlines). The ice sheet profile was adjusted to provide the shear stress necessary to yield this horizontal velocity, using a relation similar to the first of Equations (5.19) with a sliding law to estimate ub. Isostatic depression of the earth's crust was included. The vertical velocity was calculated from the submergence or emergence velocity relation (Equation (5.26)), and was assumed to decrease linearly with depth (Equation 6.15). The temperature at the margin was -7.5 °C. The temperature along the surface was calculated assuming a lapse rate of -0.01 Km-1, and making an empirical correction for warming effects of percolating melt water. The geothermal fluxes used were appropriate to the geologic terrane along the flowline. To circumvent certain problems, discussed later, it was assumed that the warming rate was 1 uak instead of uak.

Several features of the temperature distribution in Figure 6.12 merit comment.

• The downward and outward advection of cold ice is represented by the reversal in slope of the —20 °C and —25 ° C isotherms ~900 km from the divide.

• The progressive compression of the isotherms near the bed downglacier from the divide reflects the outward increase in basal temperature gradient as strain heating increases.

• Basal melting occurs over the first 250 km of the flowline because the accumulation rate here is low, and downward advection of cold ice is, thus, less important than it is further downglacier. A more realistic vertical velocity distribution would lead to more melting here, while a higher accumulation rate would lead to less melting.

• Between ~250 and ~420 km from the divide, half of the meltwater formed in the first 250 km is refrozen to the base. This keeps the bed at the pressure melting temperature. The rest of the water was assumed to have drained away into the bedrock. (Had it been assumed, instead, that more of the meltwater stayed at the ice/bed interface, the zone of subfreezing temperatures between ~420 and ~840 km from the divide would be smaller or absent.)

• The zone of subfreezing basal temperatures between ~420 and ~840 km owes its existence to increased downward advection of cold ice as the accumulation rate increases outward.

• Basal melting resumes downglacier from ~840 km as strain heating warms the basal ice. It becomes particularly important in the ablation area where upward vertical velocities decrease the basal temperature gradient, thus trapping more heat at the bed.

• The basal frozen zone at the margin, barely visible at the scale of the figure, is a result of cold atmospheric temperatures at the margin and decreasing vertical velocity as the margin is approached. The vertical velocity decreases because, as the surface slope steepens, a greater fraction of the ablation rate is balanced by the us tan a term in the emergence velocity. In addition, it is assumed that the meltwater formed in the outer ~500 km of the glacier leaves the system as groundwater or by way of localized subglacial conduits.

Even though the Column model is relatively crude in comparison with numerical models being used today, it does reproduce what are probably the essential features of the temperature distribution at the base of a continental-scale ice sheet with an ablation zone of significant width,

Basal melt rate

1300

^ Basal temperature Pressure-^ '

melting 500

Basal melt rate

10 a

1000

1300

Distance from divide, km

Figure 6.12. Temperature distribution along a flowline calculated with the use of the Column model. The bed is at the pressure melting temperature except in the section labeled "Basal temperature". (From Moran et al., 1980, Figure 6. Reproduced with permission of the International Glaciological Society.)

namely: (1) melting beneath the divide if the accumulation rate is sufficiently low and freezing otherwise; (2) in the former case, a zone of freeze-on in the lower part of the accumulation area followed by a possible zone in which the ice is frozen to the bed; (3) melting beneath the ablation area; and (4) a possible frozen toe in areas where marginal temperatures were relatively cold. The distribution of these zones depends on bn, Pg, and 0s. Temporal changes in bn and 0s due to climate change will alter the basal temperature distribution, but there will be a lag of order 103 years between any change in climate and a response at the bed.

The fact that water from melting basal ice flows downglacier along the bed and refreezes is consistent with observations of layers of dirty ice, several meters thick, that were encountered at the bottoms of both the Byrd Station, Antarctica, and the Camp Century, Greenland, ice cores. In both cases, the dirt was dispersed throughout the ice, and the dirty ice had fewer air bubbles than the overlying clean ice. In the Camp Century core, the oxygen isotope ratios indicated that the basal dirty ice was formed from water that originally condensed at lower temperatures than the overlying ice. All of these observations are consistent with melting of ice that originally formed at a higher altitude than the overlying ice, downglacier flow of that water along the bed, and refreezing of the water incorporating dispersed dirt in the process. It is difficult to account for meters-thick layers of basal ice with dispersed dirt in any other way, although regelation of ice downward into till is a possible way of entraining layers of dirt with higher debris content (Iverson,

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