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cal parameters. Importantly, basal sliding in this
model is dependent only on the local ice depth, 50 km in size with a pixel resolution of 100 m.
ice slope, and bed slope. In real glaciers, however, To construct this surface, we allowed all pixels
sliding velocity is controlled by the discharge of in the grid to accumulate ice to simulate an ice
ice, increasing with distance from the divide and sheet with no margin. The slopes in the y direc-
reaching a maximum beneath the ELA, even if tion, S y , were restricted to point down the grid,
the glacier and bed morphology remains con- rather than in either the up or down direction,
stant along the glacier pro¬le. to simulate an ice-sheet ¬‚owing from top to bot-
Here we illustrate the behavior of our glacial- tom. The slope in the x direction, S x , was uncon-
landform evolution model with several numer- strained. The sandpile algorithm for this example
ical experiments. We distinguish three types of was initiated with a uniform ice-surface topog-
depressions: (1) lakes in ice sheet interiors that raphy of 2 km. From this initial state, the sand-
are ¬‚exurally compensated, (2) large lakes that pile method was used to solve for the threshold-
are uncompensated (˜˜great™™ lakes), and (3) elon- sliding ice-surface topography. The resulting ice-
gate ice-marginal depressions (˜˜¬nger™™ lakes). In surface topography dips gently from the top of
each case, topographic analyses are introduced the grid to the bottom (Figure 6.24a). In addition,
for comparison with the model predictions. the ice surface has small-scale variations that re-
¬‚ect the white-noise variations in bed topogra-
6.6.1 Length scales < ¬‚exural wavelength phy. Equation (6.46) requires that the ice-surface
First we consider bedrock erosion beneath a slope and ice thickness be inversely proportional.
gently-sloping ice sheet at scales less than the In the model reconstruction, therefore, pixels
¬‚exural wavelength. In this example we assume with slightly lower bed elevations have lower ice-
that the lithosphere is perfectly rigid. The ini- surface slopes. In this way, variations in the bed
tial bed topography is ¬‚at with random micro- topography are re¬‚ected in the ice-surface topog-
topography characterized by a Gaussian white raphy.
noise with a standard deviation of 10 m. This The pattern of basal-sliding velocities in Fig-
initial topography is certainly an idealization; ure 6.24b was produced assuming uniform ac-
no initial condition would look like this. How- cumulation within the model domain. Although
ever, all topography has some small-scale rough- the ice-surface topography has only minor small-
ness, and it is essential to include these varia- scale variations, the basal sliding is strongly lo-
tions in the initial bed topography because the calized into bedrock depressions in a braided pat-
glacial-erosion instability ampli¬es initial topog- tern. This behavior results from an additive effect
raphy over time. White noise variations in ini- of the ice-surface topography on the basal sliding.
tial conditions are commonly used to initiate The input of ice to the system as accumulation is
¬‚uvial-geomorphic instabilities such as drainage- uniform throughout the domain. At each pixel,
network evolution (e.g. Willgoose et al., 1991) and however, slightly more ¬‚ow is diverted to steeper
hillslope rilling (Hairsine and Rose, 1992), for ex- downslope pixels than to pixels with a gentler
ample. A key question in this section will be slope. As ice ¬‚ows from pixel to pixel downice,
whether the glacial-erosion instability ampli¬es small variations in surface slope act cumulatively
initial topography at all wavelengths or whether to divert ice into bedrock depressions. The re-
certain wavelengths are preferred. As an alterna- sulting areas of intense sliding velocities result
tive to assuming variations in initial topography, in enhanced scouring of the bed. This, in turn,
6.6 GLACIAL EROSION BENEATH ICE SHEETS 153



2020 m iteration 1 iteration 1 iteration 1
(b) (c)
(a)
elev.


flow




2000 m
ice-surface topography basal sliding velocity bed topography
(d) (e) (f)




iteration 10 iteration 10 iteration 10

Fig 6.24 Erosion and basal sliding beneath ice-sheet they depress the ice-surface topography over a
interiors. (a) Shaded-relief image of ice-surface topography,
larger area, focusing more ¬‚ow into the depres-
(b) grayscale image of basal-sliding velocities, and (c) shaded
sion. The dynamics of the model, therefore, can
relief image of bed topography in the ¬rst iteration of the
be characterized as depressions of different size
model. (d) Ice-surface topography, (e) basal-sliding velocities,
competing for ice drainage. The larger a depres-
and (f) bed topography after ten model iterations. Bedrock
sion is, the faster it expands to ˜˜capture™™ neigh-
depressions deepen and expand over time, focusing basal
boring lakes. As a result, Figure 6.24f contains
sliding and erosion in a positive feedback.
lakes or pits of many sizes, while Figure 6.24c
contains only small pits. The braided patterns of
Figures 6.24b and 6.24e meander and shift as the
results in localized bedrock erosion in depres-
bed is eroded and depressions expand.
sions. This is the fundamental glacial-erosion in-
In Figure 6.25, the lakes of Figure 6.24f are
stability. In the case of a gently-sloping surface
compared with the lakes of glaciated topography
of an ice sheet with no margin, the instability
southwest of Hudson Bay (Figures 6.25d--6.25e). In
results in lakes with little or no elongation and
Figure 6.25a, a shaded-relief image of the model
a wide range of sizes. Near the ice margin, in con-
topography from Figure 6.24f is given, with all
trast, lakes are elongated and regularly spaced.
closed depressions ¬lled. The binary image of Fig-
The ice-surface topography, basal sliding ve-
ure 6.25b includes all of the ¬lled areas, or lakes,
locities, and bed topography of this model af-
of Figure 6.25a.
ter 10 iterations are given in Figures 6.24d--6.24f.
The frequency-size distribution was intro-
These ¬gures illustrate that ¬‚ow becomes more
duced in Chapter 1 as a tool for characteriz-
focused into bedrock depressions as the relief of
ing the population of lakes in a given area. The
the bed topography increases through time. Ini-
frequency-size distribution is the number of lakes
tially, all depressions are limited to a few pixels in
as a function of their size. This curve can be used
width. Over time, large depressions deepen and
to compare spatial domains (e.g. lakes, landslides,
expand more quickly than smaller depressions
vegetation clusters) modeled on a computer with
nearby. Large depressions expand faster because
154 NON-NEWTONIAN FLOW EQUATIONS



(b)
(a)




104
(c)
103
N = cA




N (>A)
102

(d) (e) 101

100
101
100 102
10 10
A (km 2)




Fig 6.25 Comparison of modeled and observed topography
resulting from glacial erosion beneath ice-sheet interiors. The
Great Lakes. In this section we refer to these
pixel size in the model is assumed to be 100 m. (a) Shaded
lakes collectively as the ˜˜great™™ lakes of Canada.
relief of bedrock topography with ¬lled depressions. (b) A
binary image of ¬lled depressions (i.e. lakes) from (a). (c) The These lakes are roughly uniformly-spaced in a
cumulative frequency-size distribution of lakes in (b), well ¬t ring around Hudson Bay, with distances between
by the same relationship observed for Canadian lakes less lake centers ranging from 370 km to 1000 km.
than 104 km2 in area: N(> A ) ∝ A ’1 . (d) Shaded relief of
The map of Figure 6.26a was created by extract-
topography in central Canada for qualitative comparison with
ing all the lakes from a 1-km resolution DEM
(a). (e) Binary image of lakes in (d).
of Canada and the northern US. The shading
of each lake is proportional to elevation, with
black indicating lakes near sea level and light
those of populations observed in nature. The cu- gray shades representing lakes as high as several
mulative frequency-size distribution of lakes is hundred meters above sea level. The shading val-
the number of lakes larger than a given area. ues indicate, for example, that the great lakes of
Figure 6.25c illustrates the cumulative frequency- western Canada are separated from Hudson Bay
size distribution for the lakes of Figure 6.25e. The by a broad ridge of topography several hundred
data ¬t a power-law relationship N (> A) ∝ A ’1 . meters above sea level and parallel to the Hudson
This compares well with the frequency-size dis- Bay shoreline.
tribution of small and medium-sized lakes in To investigate the role that ¬‚exure may have
Canada shown in Figure 1.29. in modifying the basic glacial-erosion instability,
we performed a numerical experiment of glacial
6.6.2 Length scales > ¬‚exural wavelength erosion beneath an initially semi-circular ice
The largest lakes in Canada and the US are il- sheet. The 2D ¬‚exure of the lithosphere beneath
lustrated in Figure 6.26a, including (from north- the ice sheet was included in this model ex-
periment. The grid for this model is 3000 km —
west to southeast) Great Bear Lake, Great Slave
Lake, Lake Athabaska, Lake Winnipeg, and the 3000 km with a pixel resolution of 10 km. The
6.6 GLACIAL EROSION BENEATH ICE SHEETS 155


elastic thickness is 50 km. As in Section 6.6.1,
(a) Great Bear
we assumed a ¬‚at initial surface with white
noise variations as the initial bedrock surface.
600 km
Great Slave
The ice-surface topography, ¬‚ow, and bedrock
de¬‚ection in the ¬rst model iteration are given
Athabasca
370
in Figures 6.27a--6.27c. The accumulation pattern
Reindeer
was assumed to be uniform for elevations greater
than 1 km.
1000
The ice-surface topography, ¬‚ow, bedrock de-
Winnipeg

¬‚ection, and bedrock topography of the model
after ten iterations are given in Figures 6.27d--
Superior
950
6.27g. The bedrock topography does not include
500
400 the de¬‚ection under the ice load. Therefore, it
represents what the landscape would look like
400
if the ice load was removed and only the effect
of glacial erosion remained. In this experiment,
(b)
the largest lakes are uniformly spaced in a ring
around the center of the ice sheet, directly un-
derneath the equilibrium line. In the center of
100
the ice sheet the original rough topography re-
80
mains because erosion is minimal there. Near the
60
120 km
equilibrium line the glacial instability is ampli-
¬ed by the de¬‚ection of the lithosphere at wave-
lengths equal to ≈ 500 km. Bedrock depressions
that form at wavelengths of 500 km are ampli¬ed
120 km
faster than smaller wavelengths because their ice
100
80
in¬ll de¬‚ects the lithosphere, further depressing
60
the ice surface and focusing ice drainage into the
depressions.
A grayscale image and contour map of the
elastic thickness of central Canada is given in Fig-
ure 6.26b. The elastic-thickness estimates were ob-
Fig 6.26 The “great” lakes of Canada and their relationship
tained by Tony Watts (personal communication,
to the ¬‚exure of the lithosphere. (a) Topographic depressions
2002) and are based on the coherence method be-
of Canada (all non-white areas) with grayscale shading
mapped to elevation (lakes in black are at sea level, lighter tween gravity and topography. The greatest elas-
tones have water surfaces at higher elevations. The distances tic thickness in Canada exists just southwest of
between the largest lakes are indicated. (b) Grayscale and
Hudson Bay with a maximum value of nearly
contour map of the elastic thickness of the lithosphere from
140 km. To the northwest of this maximum value,
Tony Watts (personal communication, 2002), recti¬ed to (a).
following the lines adjoining the great lakes
The elastic thicknesses are greatest in the areas between
from Winnipeg to Athabasca to Great Slave and
Athabasca Lake, Lake Winnipeg, and Lake Superior, where
Great Bear, elastic thicknesses decrease gradu-
the distances between lakes are also the greatest. The
ally to about 100 km at Athabasca, then more
distances separating lakes are lower in the northwest and
rapidly to about 60 km near Great Slave. Moving
southeast portion of Canada where the elastic thickness
values fall to approximately one half their values in central in the southeast direction from Winnipeg, elas-
Canada. If Reindeer Lake is considered to be a great lake, tic thicknesses also decrease, to 100 km at Supe-
however, the relationship between lake spacing and elastic
rior and 60 km at Ontario. This pattern suggests
thickness is less robust.
that the distances between the great lakes may
correlate with the local elastic thickness, with a
156 NON-NEWTONIAN FLOW EQUATIONS



(d)
(a)




ice-surface topography
iteration10
iteration 1 ice-surface topography

(b) (e)




basal sliding velocity
basal sliding velocity iteration 10
iteration 1

(f)
(c)
ice margin




forebulge
lighting direction
lithospheric deflection
lithospheric deflection iteration 10
iteration 1

(g)


500 km




iteration10 bed topography


Fig 6.27 Numerical experiment illustrating the role of
¬‚exure in lake formation. (a) Shaded-relief image of
Is the relationship between great-lake spac-
ice-surface topography, (b) grayscale image of basal-sliding
ing and elastic thickness consistent with ¬‚ex-
velocities, and (c) shaded-relief image of lithospheric
de¬‚ection in the ¬rst iteration of the model. Basal sliding ure? The elastic thickness is related to the three-
velocities and bedrock erosion are diffuse in this early stage dimensional ¬‚exural parameter by the relation-
of the model. (d) Ice-surface topography, (e) basal-sliding
ship:
velocities, (f) bedrock de¬‚ection, and (g) bedrock topography
1
(no de¬‚ection component) after 10 iterations of the model
D 4
β= (6.53)
run. Domain is 3000 km in length and elastic thickness is
ρg
50 km.
where D is the ¬‚exural rigidity given by
E T e3
D=
proportionality factor of 10. For example, in ar- (6.54)
12(1 ’ ν 2 )
eas where the elastic thickness is 100 km, the dis-
tance between lakes is ≈ 1000 km. It should be E = 70 GPa, ν = 0.25, ρ=
Assuming
2300 kg/m , and g = 10 m/s , and an elastic
3 2
noted that if Reindeer Lake is considered as a
thickness T e = 50 km, the ¬‚exural parameter β
great lake in this analysis, the relationship be-
tween great-lake spacing and elastic thickness is is 77 km. The wavelength of de¬‚ection is related
to the ¬‚exural parameter by » = 2πβ = 480 km,
not as robust.
6.6 GLACIAL EROSION BENEATH ICE SHEETS 157


or about ten times the elastic thickness. This (a) profile 1
analysis indicates that the observed relationship
between great-lake spacing and elastic thickness
profile 2
is consistent with ¬‚exural control.
Although the relationship between lake spac-
ing and elastic thickness suggests that ¬‚exure
is the key process controlling great-lake forma-
tion, the ice margin may be home to many
feedback processes that lead to instabilities. In (b)
addition to the focusing of ¬‚ow by the bed topog-
raphy (through its in¬‚uence on the ice-surface
topography), feedbacks between ice temperature,
¬‚ow, meltwater content, and other variables may
10 km
also be at work. For example, the EISMINT model
comparison project noted that an ice-marginal
basal velocity
0.7 bars
instability driven by a feedback between ice tem-
perature, ¬‚ow, and ice-sheet geometry was re- (c)
produced in all of the models tested (Payne et
al., 2000). Since ice temperature is not a vari-
able in our model, the instability found in the
EISMINT experiments is independent from the
¬‚exural mechanism of our model. The wave-
5“7 km
length of the thermomechanical instability ob-
basal velocity
1.5 bars
served in the EISMINT experiments varied be-
1.5 km 960 m
tween model experiments but was generally profile 2
profile 1

several hundred kilometers. As such, a ther-
momechanical instability could also generate
˜˜great™™ lakes.
950 m
0
150 km
50 km 0
0
6.6.3 Near ice margins
Finger lakes are elongate glacially carved lakes Fig 6.28 Ice-surface topography and ¬‚ow above a rough
surface with „ = 0.7 bars. (a) Shaded-relief image of an ice
formed on the margins of ice sheets. The type ex-
sheet 150 km wide and 50 km from divide (top of image) to
ample is the Finger Lakes Region of New York
margin. Two pro¬les of the ice-surface topography are given:
State. In this chapter we use the term more
pro¬le 1 is along-dip from divide to margin; pro¬le 2 is
broadly to include elongate glacial troughs near
along-strike from left to right. (b) Grayscale image of basal
the ice margin whether or not they impound velocity corresponding to (a) with an ELA of 1 km. Ice-¬‚ow
water. For example, the Finger Lakes Region in- channels with a characteristic spacing of 10 km are present in
cludes ¬ve major lakes, several smaller lakes, and the ¬‚ow even though there is no characteristic scale to the
bed topography. (c) A thicker ice sheet („ = 1.5 bars) results
dozens of glacial valleys of similar shape. By ˜˜¬n-

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