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better match to the modern ice-surface topogra-
reconstruction of the ice-surface topography (with same
phy than we obtain by using the Greenland rela-
grayscale and shading as in (b)). The divide position and
elevations are a good match to (B) except that the divides are tionship, which yields a maximum elevation of
too peaked. (d) Grayscale map of depth-averaged velocities just under 4 km.
computed using the balance-velocity algorithm of Budd and
These reconstructions illustrate that the sand-
Warner (1996) assuming uniform accumulation. The grayscale
pile model is capable of accurately predicting
map is scaled logarithmically.
ice-sheet geometries if the threshold basal shear
stress is well constrained. However, the principal
either case are small. The ¬‚ow map for Green- objective of glaciological work within the con-
land is indicated in Figure 6.13d and is similar text of geomorphology is to reconstruct former
to the map provided by Bamber et al. (2000b). The ice sheets and glaciers.
grayscale map is scaled logarithmically in this Reconstructions of the Laurentide Ice Sheet
and all subsequent ¬‚ow maps to illustrate the (LIS) have had a long history that includes many
internal ¬‚ow distribution. A linear map, in con- classic contributions. However, despite decades of
trast, would show signi¬cant velocities only at effort there is still vigorous debate over the mean
the outlet areas. thickness of the LIS and the locations of its major
Input data for the East Antarctic Ice Sheet re- domes. These questions bear on many important
construction (Figure 6.14) includes the DEM of issues, including the magnitude of Late Quater-
bed topography provided by Lythe et al. (2001). nary sea-level change. Numerical reconstructions
Lythe et al.™s ice-thickness data were also used to of the Laurentide Ice Sheet at the Last Glacial
derive a mask of ice coverage. Rather than includ- Maximum have generally reproduced a relatively
ing all areas of ¬nite ice thickness in our mask, thick (≥ 2 km) ice sheet with a central east-west
however, we used a threshold ice thickness of 150 divide (e.g. Denton and Hughes, 1981). In con-
meters instead. This elevation was used to elim- trast, Peltier™s (1994) ICE4G relative sea-level (RSL)
6.4 MODELING OF THRESHOLD-SLIDING ICE SHEETS AND GLACIERS OVER COMPLEX 3D TOPOGRAPHY 139



(a) (d)




depth-averaged
bed topography velocity

(b) (c)




4 km
4 km
3 km
3 km




2 km
ice topography
ice topography 2 km
(modeled)
(measured)

Fig 6.14 Reconstruction of the modern Greenland Ice
to account for bed de¬‚ection (e.g. Tomkin and
Sheet from Eq. (6.46) with „ = 18S 0.55 . (a)“(d) are identical
Braun, 2002). The ratio of 0.27 comes from the
to the corresponding images for Greenland in Figure 6.13.
relative density of ice and mantle. This approach
is straightforward to incorporate in the sandpile
model by replacing h with (1 + ρi /ρm )h in the
inversion reproduces a thinner ice sheet with a
more complex ice-surface topography. threshold condition. It should be noted, however,
An inherent uncertainty in many ice-sheet that this approach is valid only for ice sheets
reconstructions is the amount of isostatic re- larger than the ¬‚exural wavelength. Smaller ice
bound experienced by the bed between the sheets and glaciers may be partially or fully ¬‚ex-
time of glaciation and the present. This uncer- urally compensated.
tainty is not signi¬cant for areas that have been Including the effects of isostasy is a much
unglaciated for several tens of thousands of years. greater challenge for LGM ice sheet reconstruc-
In these cases, we can be sure that the modern to- tions because postglacial rebound has not yet run
pography includes no dynamic component: post- its course. It is dif¬cult to estimate the amount
glacial rebound has run its course. It is common of postglacial rebound that has taken place or
in these cases to ignore the time delay of man- to estimate the amount left to occur because
tle response and add an instantaneous de¬‚ection the dynamics of postglacial rebound depend on
amplitude proportional to the local ice load: for the size and shape of the load through time.
every kilometer of ice thickness, approximately In particular, we cannot extrapolate the present
270 meters of ice are added to the thickness rate of sea-level rise along an exponential curve
140 NON-NEWTONIAN FLOW EQUATIONS




Fig 6.15 Reconstruction of the Laurentide Ice Sheet at Last between 0 and 27%. If we believe that postglacial
Glacial Maximum (18 ka) assuming (a) „ = 18S 0.55 (referred
rebound beneath the LIS is nearly complete, we
to as standard basal shear) and (b) „ = 9S 0.55 (one-half the
should choose a value close to 0.27. In contrast, if
standard basal shear). Bed and ice-surface topography are
little postglacial rebound has occurred, the value
illustrated with shaded-relief (100— vertical exaggeration)
should be closer to zero. For our Laurentide re-
grayscale maps. (c) Grayscale map of depth-averaged
constructions, we will assume a ratio of 0.2. This
velocities for the standard basal shear model assuming
value assumes that the majority of postglacial re-
uniform accumulation.
bound has taken place. The uncertainty of this
value, which could plausibly vary between 15 and
25%, introduces a 10% uncertainty into determi-
because the Maxwell-relaxation time of the man-
nation of the LIS thickness.
tle depends on the wavelength of the load (e.g.
Two reconstructions of the LIS and LGM are
Turcotte and Schubert, 2002). Therefore, post-
presented in Figure 6.15. Figures 6.15a and 6.15c
glacial rebound is not a simple function of the
illustrate the ice-surface topography and ¬‚ow us-
local ice thickness but is also controlled by the
ing the basal shear stress observed in East Antarc-
large-scale morphology of the ice sheet, particu-
tica („ = 18S 0.55 ). We refer to this shear--stress re-
larly at long time scales.
lationship as the ˜˜standard™™ basal shear model,
Nevertheless, we can take a similar approach
because it characterizes the observed trends in
to including isostatic de¬‚ection that is often
the Greenland and Antarctic Ice Sheets (Figure
taken in cases with no dynamic topography.
6.11). For comparison, we have also provided a
Rather than adding 27% of the ice thickness to
second reconstruction which uses a shear stress
account for de¬‚ection, we can add some fraction
6.4 MODELING OF THRESHOLD-SLIDING ICE SHEETS AND GLACIERS OVER COMPLEX 3D TOPOGRAPHY 141


equal to one half of the standard basal shear, or and south between Figures 6.15a and 6.15b, but
„ = 9S 0.55 . otherwise the ¬‚ow pattern is unchanged.
The DEM of bed topography was constructed Figures 6.16 and 6.17 illustrate the ice-surface
by projecting the ETOPO5 DEM (Loughridge, topography and ¬‚ow of the Laurentide Ice Sheet
1986) for North America to a Lambert Equal- at 14, 13, 12, 11, 9, and 8 ka. These reconstruc-
Area projection. The ETOPO5 data include tions do not include ice coverage between north-
both bathymetry and topography data. The ern Canada and Greenland (Dyke et al., 2002), and
bathymetry is important in this case because are therefore only approximate in their northern-
the LIS covered many areas that are now sub- most regions.
merged. The ice margins were digitized from The reconstructions of Figures 6.16a--6.16f are
Dyke and Prest (1987) by Eric Grimm of the characterized by a thick central region that
Illinois State Museum (personal communication, changes little as it shrinks in size. By 11 ka the
2002) and recti¬ed to the DEM by the author. The ice sheet has shrunk to less than half its orig-
bed and ice-surface topography are illustrated inal area (including coverage in the Cordillera)
with shaded-relief (100— vertical exaggeration) but still maintains a signi¬cant central region
grayscale maps in Figure 6.15. The bathymetry in with elevations above 3 km. Only by 9 ka has the
these images has been removed so that the coast- overall shape of the ice sheet and the altitude of
lines could be indicated instead. Use of the stan- its central region changed signi¬cantly. At this
dard basal shear model yields an ice sheet with point, the central region of the ice sheet has col-
a central divide striking east-west. The average el- lapsed into three distinct domes. This collapse is
evation and thickness of the ice sheet in Figure followed by the rapid deglaciation of Hudson Bay
6.15a (the standard basal shear case) are 2211 m between 9 and 8 ka.
and 2011 m. Decreasing the shear stress preserves An important caveat should be given regard-
the overall shape of the ice sheet but results in a ing the reconstruction of the ice in the Cana-
migration of the divide to the southwest where dian Cordillera. In alpine terrain it is important
the ice is propped up by the topographic front to use DEM data that enable the model to es-
of the Canadian Rockies. The average elevation timate bed slopes accurately. It is unlikely that
and thickness of Figure 6.15b are 1482 m and the ETOPO5 dataset, with a resolution of 5 km,
1134 m, respectively. Although the average eleva- is an adequate representation of the bed topog-
tion and thickness of the standard reconstruc- raphy in the Cordillera. Coarse DEMs such as
tion are a relatively modest 2.2 and 2.0 km, re- ETOPO5 may underestimate bed slopes, especially
spectively, much of the central core of the ice in high-relief terrain where closely spaced ridges
sheet is quite high, with average elevations above and valleys may not be resolved. This problem
3 km and greater than 3.5 km in thickness. As a can be minimized by using DEMs of the highest-
caveat, it should be noted that we did not re- available resolution. Nevertheless, some size re-
move the post-glacial sediment from Hudson Bay duction may be necessary to maintain reason-
or the Great Lakes prior to this reconstruction. able grid sizes and computing times. When size-
This will not affect the large-scale geometry of reduction must be done, it should be done by
the ice sheet signi¬cantly. Certain areas, however, subsampling the DEM rather than by averaging
including the Great Lakes, will have ice overes- neighboring pixels. Subsampling will tend to pre-
timated thicknesses due to the postglacial sedi- serve the small-scale topographic relief in the
ment ¬ll in the DEM. A more precise reconstruc- DEM better than averaging. In the Laurentide re-
tion would require that any post-glacial sediment construction of Figure 6.15, the thickness of the
be removed from these areas to re¬‚ect the topog- Cordilleran Ice Sheet may be overestimated be-
raphy that the ice sheet experienced. cause of the coarse scale of the DEM (5 km). Appli-
The results of Figure 6.15 indicate that even cations focused on extensive areas of high-relief
though the thickness of the LIS is sensitive to the terrain should be reproduced on multiple scales
basal shear stress, the overall shape is not sensi- to verify that the results are independent of the
tive. The principal divide migrates to the west DEM scale.
142 NON-NEWTONIAN FLOW EQUATIONS



(a) (b)


2 km
2 km
3 km
3 km




13 ka
14 ka

(c) (d)

2 km

3 km
3 km




11 ka
12 ka

(e) (f)


2 km


2 km
3 km




8 ka
9 ka
Fig 6.16 Reconstruction of the ice-surface topography of the Laurentide Ice Sheet from 14“8 ka assuming the standard basal
shear model and using the margin positions of Dyke and Prest (1987). Bed and ice-surface topography are illustrated with
shaded-relief (100— vertical exaggeration) grayscale maps. Ice coverage between Greenland and northern Canada has not been
included in this reconstruction.
6.4 MODELING OF THRESHOLD-SLIDING ICE SHEETS AND GLACIERS OVER COMPLEX 3D TOPOGRAPHY 143



(a) (b)




13 ka
14 ka

(c) (d)




11 ka
12 ka

(e) (f)




8 ka
9 ka

Fig 6.17 Depth-averaged ice velocities corresponding to
the reconstructions of Figures 6.16a“6.16f.

ment in which only a portion of the ice sheet
can be modeled at one time. In addition, the Fin-
Next we illustrate the model application to
ger Lakes are an example where both the bed
a high-resolution reconstruction of the Lauren-
topography and the distance from the ice mar-
tide Ice Sheet in the Finger Lakes Region of New
gin have an important in¬‚uence on ice-surface
York State. This example will illustrate the appli-
topography and ¬‚ow. In contrast, large ice sheets
cation of the model to an ice-marginal environ-
144 NON-NEWTONIAN FLOW EQUATIONS


and alpine glaciers are predominantly in¬‚uenced topography. We will not model the full dynam-
by only one of these variables. For example, the ics of Finger Lakes formation in this chapter but
local topography and ¬‚ow in large ice sheets will rather determine what the ice-surface topog-
is primarily a function of distance from the raphy and ¬‚ow looked like after the Finger Lakes
margin, while the thickness and ¬‚ow in alpine had formed.
glaciers are almost entirely controlled by bed The reconstruction of this region requires the
topography. following inputs: bed topography (Figure 6.18a),
The Finger Lakes Region has been the focus a mask grid incorporating the position of the
of many glacial-geologic studies, including glacio- Valley Heads Moraine, and the threshold basal
logical reconstructions (Ridky and Bindschadler, shear stress. The topography of the region was
1990), stratigraphic studies of meltwater produc- obtained by joining several 90-m resolution USGS
tion and lake in¬lling (Mullins and Hinchey, DEMs. The position of the Valley Heads Moraine
1989), and geomorphic studies of the Finger Lakes was used to create the mask grid by overlay-
themselves (von Engeln, 1956). Geomorphically, ing the sur¬cial geologic map of the area (New
the Finger Lakes Region is dominated by subpar- York State Geological Survey, 1999) onto the bed
allel, glacially-scoured troughs with their south- topography to delineate the ice margin. At the
ernmost extents in Seneca and Cayuga Lakes, the spatial scale of the Finger Lakes, the relation-
two largest of the Finger Lakes. The ¬ve largest ship between basal shear stress and ice-surface
troughs of the region comprise the Finger Lakes slope observed in Greenland and East Antarc-
proper, but there are numerous other troughs cut tica is not useful because ice-surface slopes this
into the Allegheny Plateau of smaller size that close to the margin are uniformly steep. In ad-
are not deep enough to be enclosed depressions. dition, the average basal shear stress observed
Figure 6.18a is a shaded-relief image of the topog- for modern ice-sheet margins (3 bars) may also
raphy of the region. The troughs vary in spacing be inapplicable because shear stresses near ice-
from 10 to 30 km along strike, with the greatest sheet margins are spatially variable and may be
spacing between Seneca and Cayuga Lakes. The much smaller than 3 bars locally. In lieu of a bet-
southern tips of the Finger Lakes coincide with ter constraint on basal shear stresses in the re-
the Valley Heads Moraine (14 ka). As such, the gion, we have used the traditional end-member
scouring of the Finger Lakes Region most likely values of 0.5 and 1.5 bars in two alternative re-
took place when the ice margin was coincident constructions. In addition, we have modi¬ed the
with this moraine, although several phases of ice-¬‚ow portion of the reconstruction so that all
glaciation may have contributed to their forma- of the ¬‚ow originates at the top boundary of the
tion. grid. This modi¬cation represents the incoming
Little is known about the processes and dy- ice ¬‚ow from the Laurentide Ice Sheet, which we
namics of the scouring of glacial troughs in gen- will assume dominates over any locally-derived
eral or the Finger Lakes in particular. The regular accumulation.
spacing of the Finger Lakes suggests a positive- The results for the ice-surface topography and
feedback or instability mechanism in which in- depth-averaged ¬‚ow are given in Figures 6.18b--
cipient depressions in the bed topography focus 6.18c and 6.18d--6.18e for 1.5 and 0.5 bars, respec-
ice ¬‚ow, resulting in enhanced deepening and tively. These values bracket an important tran-
focusing of ice ¬‚ow. In order for this model to sition in ice-sheet behavior. If the basal shear
be valid, the ice-surface topography in the region stress is 1.5 bars or larger, the topography and
must re¬‚ect variations in bed topography in or- ¬‚ow are only weakly dependent on bed topogra-
der for ice ¬‚ow to be focused into troughs. The phy. Figure 6.18b illustrates the ice-surface topog-
¬rst step towards testing this hypothesis is to re- raphy in this case. The reconstructed ice sheet
construct the ice-surface topography and ¬‚ow in in this ¬gure is dominated by a forked central
the region to determine whether the ice-¬‚ow pat- divide and distributary ice ¬‚ow. In contrast, for
terns were likely to have been focused by the bed basal shear stresses of 0.5 bars or smaller, the
6.4 MODELING OF THRESHOLD-SLIDING ICE SHEETS AND GLACIERS OVER COMPLEX 3D TOPOGRAPHY 145



(b) 3 km (c)



2 km




1 km

(a)



depth-averaged
velocity
ice topography
1.5 bars


(d) (e)



1 km




bed topography
30 km




depth-averaged
velocity
0.5 bars ice topography

Fig 6.18 Reconstruction of the ice-surface topography and
depth-averaged velocities in the Finger Lakes Region. Two
Finger Lakes be removed to more accurately re-
reconstructions are presented with uniform basal shear
¬‚ect the bed topography experienced by the ice
stresses of (a) and (c) 1.5 bars and (d) and (e) 0.5 bars. (a)
Shaded-relief image of bed topography. (b) Shaded-relief image sheet. Also, drumlin orientations can be used to
of ice-surface topography and (c) depth-average velocity provide a constraint on the ¬‚ow pattern and,
assuming 1.5 bars. For this case, representing a relatively rigid in conjunction with reconstructions of different
base, ice topography and ¬‚ow are only weakly controlled by
shear stresses, provide an indirect constraint on
subglacial topography. (d) Shaded-relief image of ice-surface
the appropriate value of shear stress. Ridky and
topography and (e) depth-average velocity assuming 0.5 bars.
Bindschadler (1990), for example, used drumlin
orientations to guide their 2D ¬‚ow line recon-
ice-surface is strongly controlled by the subglacial structions in this area.
topography. Ice ¬‚ow is focused into the troughs One additional point should be made regard-
of the Finger Lakes in this case. If the Finger Lakes ing this example. Although the solution of par-
were formed by focused ice ¬‚ow, as the positive- tial differential equations such as Eq. (6.46) typi-
feedback model requires, these reconstructions cally require the application of boundary condi-
suggest that the basal shear stress was probably tions at the edges of the grid, the sandpile algo-
closer to 0.5 bars than to 1.5 bars. Future work rithm does not require boundary conditions at
will require that the postglacial sediment in the grid boundaries in upslope directions. The reason
146 NON-NEWTONIAN FLOW EQUATIONS



Fig 6.19 Reconstruction of the ice
(a) 30 km thickness for the Wisconsin Brooks
Range glaciation assuming uniform
basal shear stresses of 1 bar. (a) Bed
topography and ice margins mapped
by the Alaska PaleoGlacier Atlas
Group (data available at
http://instaar.colorado.edu/QGISL/
data intro.html). (b) Grayscale map
of ice thickness, with brightness
scale at lower left.
bed topography and ice margin


(b)

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