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tion grow to a maximum wavelength equal to viscosity of the rock (or ice, in the case of drum-
about ten times the layer thickness. As expand- lins) can control the time for the instability to
ing regions increase in porosity, their buoyancy occur.
increases with respect to the surrounding sedi- In the second model experiment, a 3D ge-
ment. This buoyancy imparts an upward velocity ometry was considered under uniform pressure
to both the liquid and matrix in these regions, de- (Figures 7.15a and 7.15b, ¬nal model topography
forming the ice--sediment interface above them. shown in color shaded relief). This experiment
Conversely, compacting regions become heavier results in ˜˜egg-carton™™ topography without a pre-
than surrounding regions and sink. This response ferred orientation. This topography is not simi-
can be thought of in terms of an isostatic bal- lar to drumlins, but it does closely resemble the
ance. Mid-ocean ridges sit higher, topographi- hummocky moraine found in large parts of the
cally, than the surrounding ocean ¬‚oor because Canadian prairie (Boone and Eyles, 2001). The for-
of the lower density associated with focused melt mation of hummocky-moraine topography is not
beneath spreading centers. Subglacial bedforms well understood, but this experiment suggests
7.6 HOW ARE DRUMLINS FORMED? 179



Fig 7.15 3D model evolution, map
(a) (c)
30 d view. Shaded-relief and color maps
of the model topography following
compaction. (a) Flat ice sheet, thin
sediment; (b) ¬‚at ice sheet, thick
sediment; (c) sloping ice sheet, thin
sediment; (d) sloping ice sheet, thick
sediment. For a ¬‚at ice sheet,
hummocky moraine is formed, with
the size of the hummocks controlled
by the sediment thickness. For
sloping ice sheets, drumlins are
sloping ice sheet, thin till: l = 0.2
flat ice sheet, thin till: l = 0.2
formed, with the width of drumlins
controlled by the initial
(d)
(b)
sediment-layer thickness. A sloping
ice sheet with thin sediment
reproduces bedforms most similar
to drumlins. With thicker sediment,
wide bedforms are created, some of
which join laterally to form
barchanoid drumlins and Rogen
moraine.




flat ice sheet, thick till: l = 0.6 sloping ice sheet, thick till: l = 0.6


that it may be formed by porewater expulsion un- drumlins can become wide enough to join lat-
der the nearly uniform pressure conditions found erally to form barchanoid drumlins and Ro-
beneath the interiors of great ice sheets. gen moraine (bedforms oriented perpendicular to
The 3D model runs were varied by changing the ice-¬‚ow direction). This suggests that Rogen
the initial till thickness to determine the effect moraine can be formed, in part, by the same in-
on bedform morphology (Figures 7.15a and 7.15b). stability mechanism as drumlins and hummocky
Varying the thickness results in a proportionate moraine.
change in the width of hummocky moraine. In The instability of this model occurs when the
the third model experiment (Figures 7.15c and effective pressure exceeds the yield stress, initiat-
7.15d), an ice sheet with a uniform slope of 0.01 ing viscous ¬‚ow of the matrix. The effective pres-
was assumed. To model a sloping ice sheet, the sure near the base of the sediment layer is given
vector k was replaced with (k + S j) in Eqs. (7.42) by
and (7.43), where S is the ice-surface slope and j is
pe = (1 ’ φ) ρgl (7.48)
the unit vector in the down-ice direction. Sloping
or ≈ 105 Pa for a till layer with a porosity be-
ice results in elongated, tapered bedforms very
similar to drumlins (Figure 7.15b). In this model, tween 0.2 and 0.5 that is at least 20 m thick. Typ-
the blunt up-ice end of the drumlin is associated ical yield stress values for subglacial sediments
are on the order of 104 --105 Pa, with higher values
with the early phase of sediment upwelling. As
the upwelling plume migrates down ice and pore- for well-drained sediments. These values indicate
water is expelled from the matrix, the plume that saturated tills should commonly exceed the
loses buoyancy and narrows. The streamlined threshold stress required for viscous ¬‚ow, as as-
drumlin ˜˜tail™™ represents the last vestige of sumed in the model.
porewater expulsion. The model result obtained The fundamental model prediction is a
with thick till (Figure 7.15d) also suggests that linear relationship between drumlin width and
180 INSTABILITIES



(a) (b)
l


1 w
2




threshold-filtered
shaded relief

(c)



1 w
1 w
2
2

Fig 7.16 Drumlin mapping algorithm. (a) Step 1: Construct shadow that just barely covered the shaded side
a grayscale shaded-relief image from a DEM with the
of drumlins completely. Several illumination di-
illumination direction chosen to be perpendicular to the
rections were used in different portions of each
predominant drumlin axis. (b) Step 2: Apply a threshold ¬lter
drumlin ¬eld to properly illuminate each drum-
to the shaded-relief image to make all shadows black and all
lin from a perpendicular direction. Second, the
other areas white. Step 3: Analyze the binary image for all
shaded-relief image was converted to a black and
connected clusters of black pixels.
white image using a threshold ¬lter. In this step,
shadows remain black while shades of gray are
initial sediment-layer thickness. This prediction converted to white. Third, the black-and-white
is testable using morphological analyses of drum- image was input into a custom software pro-
lins and groundwater well data that include gram that identi¬ed all connected clusters of
depths to bedrock. To do this, drumlin widths black pixels and computed their centroid posi-
were ¬rst mapped using a semi-automated al- tions, areas, lengths, and widths using standard
gorithm (Figure 7.16). This algorithm uses the de¬nitions (Chorley, 1959; Smalley and Unwin,
perpendicular shadow cast by each drumlin in 1968). Drumlin widths were taken to be twice
a shaded-relief image as a proxy for its shape. the cluster width because each cluster repre-
First, a shaded-relief image was constructed using sents only one side of the drumlin in shadow.
a high-resolution Digital Elevation Model (DEM). This algorithm leaves many small clusters re-
For the north-central New York area, 10-m reso- maining (e.g. Figure 7.16b), so the dataset was
lution US Geological Survey DEMs were used. In ¬ltered to remove clusters that are too small
(i.e. less than 0.01 km2 ) to be drumlins. This
Wisconsin, 30-m resolution DEMs were the best
available. The illumination direction was chosen procedure yielded a dataset of approximately
to be perpendicular to the predominant drum- 5800 drumlins in north-central New York, and
lin axis, and the azimuth was chosen to cast a 2900 drumlins in the Wisconsin study area.
7.6 HOW ARE DRUMLINS FORMED? 181



(a) (c)
43.4° N
43.6° N



43.2
43.4


43.0
43.2


42.8
drumlin width
89.2° W
76.0 88.8
76.4
76.8
77.2° W
0 500 m
0 1000 m
(b) (d)




Genessee
depth-to-bedrock
River Valley
60 m
0
0 60 m
areas of thick till and
= wide drumlins

100 m to 500 m in the Wisconsin study area
Fig 7.17 Grayscale maps for average drumlin width and
(Figure 7.17c).
bedrock depth in (a) and (b) north-central New York and (c)
The shadow-mapping algorithm does not pro-
and (d) Wisconsin, east of Madison. (a) and (c) Maps of
duce an ideal representation of the drumlin
drumlin width constructed by averaging widths within a 2-km
form, but it is a robust method. The principal
square moving window. (b) and (d) Maps of bedrock depth
also constructed with a 2-km square moving window and limitation of the shadow-based algorithm is that
using USGS groundwater well data. Curves are drawn to shadows created by other landforms may be in-
highlight areas in which thick sediment and wide drumlins advertently included in the analysis. This was not
(including Rogen moraine) coincide. For color version, see
a signi¬cant problem for most of the study area,
plate section.
however. Two problematic areas were found in
the southwest and southeast corners of the New
York study area. In these areas near the Finger
Finally, drumlin widths were spatially-averaged
using a 2 km — 2 km moving window (Figures Lakes, bedrock slopes are comparable to drum-
lin slopes, resulting in some bedrock ridges and
7.17a and 7.17c). The resulting values for aver-
valleys being included as drumlins. These areas
age drumlin width (including both drumlins and
(shown as shaded in Figure 7.17a) were excluded
Rogen moraine) vary from 100 m to 1000 m in
from the analysis.
the New York study area (Figure 7.17a) and from
182 INSTABILITIES


Till thicknesses were mapped using depth-to- topographic surface
(a)
bedrock (DTB) data from all publicly available
groundwater well records. Data were obtained
from the USGS New York of¬ce and the Wiscon-
sin State Geological and Natural History Survey.
In the New York study area, 2786 wells were avail-
conformable
able for six counties. In Wisconsin, 1349 wells
were available for six counties. DTB data were av- bedding surfaces
eraged using a 2 km — 2 km moving window (Fig- (b)
ures 7.17b and 7.17d). This averaging was done to
minimize the small-scale spatial variability asso-
ciated with well placement and to make the DTB
maps directly analogous to the maps of drumlin
width. DTB data in both study areas vary from 0
to 60 m. DTB data are a good proxy for till thick- unconformable
ness except in areas with signi¬cant ¬‚uvial depo-
sition. In the Genesee River Valley, for example,
(c)
¬‚uvial sediment thicknesses are comparable to
or greater than glacial sediment thicknesses in
the area, making any correlation with drumlin
widths unreliable.
Maps of average drumlin width and sediment unconformable
thickness show a strong correlation (except the
Fig 7.18 Drumlin stratigraphic end members (shown in
Genesee River Valley where ¬‚uvial sediments are
cross section, perpendicular to the drumlin axis). Three types
present). White curves in Figure 7.17 are used
are possible: (a) subsurface bedding broadly parallels the
to highlight several regions in each study area
topography (i.e. “concentric” drumlins), (b) subsurface
where wide drumlins and thick till coincide. In
bedding is undeformed, and (c) subsurface is deformed, but
the New York study area (Figures 7.17a and 7.17b) the deformation is poorly correlated with the topography.
wide drumlins (and, in places, Rogen moraine) oc- Only (a) is consistent with the proposed model.
cur in the northeast corner of the study area and
along a swath trending southwest-to-northeast
near the Lake Ontario shoreline. In the Wiscon- (1971) proposed that till-fabric data are consis-
sin study area (Figures 7.17c and 7.17d), the widest tent with a low-pressure zone along the drum-
drumlins and thickest till occur in the northwest, lin axis. Without a bedrock core to initiate the
south, and east portions of the study area. These low-pressure zone, however, it is unclear how
results prove that the geometry of the till layer low pressure develops in Evensen™s model. The
plays a dominant role in controlling the drum- buoyancy generated by concentrating porewater,
lin geometry and they lend string support to the however, provides a natural explanation for the
proposed model. alternating pressure zones and converging ma-
Sedimentological and stratigraphic observa- trix ¬‚ow inferred by Evensen and others using
tions provide additional constraints on drumlin- till-fabric analyses.
forming mechanisms. Analyses of till fabric (the Figure 7.18 summarizes three possible end
predominant orientation of elongated pebbles in members for drumlin stratigraphy (shown in
the till) provide evidence for convergent ¬‚ow of cross section, perpendicular to the drumlin axis).
sediment upward and inward toward the drum- Figure 7.18a illustrates the case in which the sub-
lin axis (Evensen, 1971; Stanford and Mickelson, surface bedding broadly parallels the topogra-
1985). Based on observations in the Wisconsin phy. Drumlins with this stratigraphy are referred
study area of Figure 7.12b, for example, Evensen to as ˜˜concentric™™ (Hart, 1997). In Figure 7.18b,
7.7 SPIRAL TROUGHS ON THE MARTIAN POLAR ICE CAPS 183


the subsurface bedding bears no indication of to form a self-sustaining poleward-migrating to-
deformation. In Figure 7.18c, sediment deforma- pographic wave. The relationship between this
tion is pervasive, but the deformation is uncorre- model and the spiral morphology of troughs has
lated with topography. Of these three end mem- not been fully established, but Fisher (1993) in-
bers, only Figure 7.18a is consistent with the pro- troduced an asymmetric ice-velocity ¬eld into
posed model, although reworking of drumlins Howard™s model and obtained spiral forms. Re-
by subsequent ice-sheet advances often leads to cent observations from Mars Global Surveyor,
erosional truncations and a more complex however, suggest that ice ¬‚ow near the troughs is
stratigraphy than the simpli¬ed case shown in not signi¬cant (Howard, 2000; Kolb and Tanaka,
Figure 7.18a. Nevertheless, concentric drumlins 2001) and it remains unclear precisely how the
are commonly observed in many drumlin ¬elds spirals form and what controls their spacing, ori-
(e.g. Newman and Mickelson, 1994; Zelcs and entation, and curvature.
Driemanis, 1997). The New York State drumlin In this section we explore a simple mathemat-
¬eld, in particular, is the classic location for con- ical model designed to capture the essential feed-
centric drumlins (Hart, 1997). back processes that couple the topography, the
As a composite of porewater embedded within distribution of solar heating on the surface, and
a matrix of sediment, subglacial sediments be- the resulting accumulation and ablation of ice.
have as deformable porous media. Porosity and The model is based on the processes in Howard™s
buoyancy play a signi¬cant role in the evolu- migrating scarp model and includes the simplest
tion of these systems. First, spatial variations mathematical descriptions of these processes in
in porosity variations can be enhanced during order to determine the necessary conditions for
compaction by a feedback between porosity, realistic spiral troughs. Lateral heat conduction is
permeability, and matrix expansion. Bouyancy as- also included in the model, and this element is
sociated with the resulting regions of high poros- crucial for obtaining realistic troughs. The model
ity provide the driving force for converging ¬‚ow does not include wind erosion or ice ¬‚ow. The
of the sediment into regularly spaced bedforms. equations are similar to those from the ¬eld of
Near the margins of former ice sheets, where excitable media, where solitary and spiral waves
the ice surface is sloping, sediment and porewa- resulting from the dynamics of two interacting
ter are advected down the ice-surface-slope direc- variables have been studied in detail.
tion, creating bedforms very similar to the classic The model equations consist of two coupled,
drumlins of New York and Wisconsin. scaled equations for the deviation of local ice-
surface temperature, T , from its equilibrium
value (the temperature at which no accumula-
tion or ablation takes place) and the deviation of
7.7 Spiral troughs on the Martian
local ice-surface topography, h, below its equilib-
polar ice caps rium value (i.e. troughs have positive h):
‚T
= κ∇ 2 T + f (T , h) (7.49)
The spiral troughs on the Martian polar ice caps
‚t
are one of the most fascinating landforms in the ‚h 1
= T (7.50)
solar system. Spiral troughs form with an insta- ‚t „f
bility in which the ice-surface temperatures of 1
f (T , h) = (T (T ’ T 0 )(1 ’ T ) ’ h) if ∇h · r > 0
steep, equator-facing slopes exceed 0 —¦ C during „i
the summer, sublimating the ice locally to form (7.51)
steeper, lower-albedo scarps (through exposure of 1
= h if ∇h · r < 0
ˆ (7.52)
subsurface dust-rich layers) in a self-enhancing „i
feedback (Howard, 1978). In this model, some of
the water vapor released from the equator-facing The deviations from equilibrium temperature
scarp may accumulate on the pole-facing slope do not include diurnal or annual cycles; these
184 INSTABILITIES



trough deepening
initial scarp formation
(a) Fig 7.19 Diagram of trough
ti < t < tf
0 < t < ti evolution and solution to Eq. (7.52)
high albedo
Sun Sun in 2D. (a) Trough initiation involves a
rapid increase in ice-surface
ice surface ice surface
temperature to its maximum value
as the scarp is steepened and dust
low albedo
warmer
(dust exposure) layers are exposed. During trough
onsun-facing
deepening, the value of h steadily
cooler
ti
slope:
T: T 0 1, h: 0 tf increases to its maximum value over
on sun-facing (shielded)
a time scale „ f = 5 Myr, while the
ti
slope:
T: 1 h: 1
tf temperature decreases in the
deeper portions of the trough due
(b) propagation to shielding. (b) Solution to (1) with
1 t1
Sun
T Runga“Kutta integration and
initial condition: x = 0.2 km, κ = 175 km2 ,
T = small pulse, h = 0
T 0.5 1/2
„i = 0.05, „ f = 1, and T0 = 0.2 at
l = (ktf )
and t0 t1 = 8. The initial condition is a
small pulse with T > T0 at the left
accumulation
0
side of the domain. Modi¬ed from
Pelletier (2004b).
ablation
30 40
20
10
0
x (km)


are assumed to be included in the equilibrium temperatures, bounded by a maximum value of 1.
temperature. Deviations from the equilibrium In the second step, the temperature remains near
temperature initiate ablation and accumulation its maximum value of 1 as the trough deepens
over a longer time scale, „ f . Trough deepening
at any time, t, in the model, although accu-
mulation and ablation of water ice is under- does not continue inde¬nitely, however, because
stood to be restricted to the summer months on deep portions of equator-facing scarps cool due to
Mars. partial shielding from the pole-facing slope. The
heat-absorption function f (T , h) includes an h
Equation (7.52) includes three basic processes:
lateral heat conduction (the diffusion equa- term to represent this negative feedback. As h in-
tion), accumulation and ablation (‚h/‚t = T /„ f ), creases in value, the temperature eventually falls
and enhanced heat absorption on equator-facing below the sublimation temperature and accumu-
scarps (‚ T /‚t = f (T , h)). The initiation and deep- lation begins. The heat-absorption term only ap-
plies to equator-facing slopes (i.e. ∇h · r , where r
ening of troughs is assumed to take place in two ˆ
steps illustrated in Figure 7.19a. In the ¬rst step, is the radial unit vector from the pole).
Equation (7.52) includes four parameters: κ, „i ,
slope steepening and dust exposure are initiated
„ f , and T 0 . The thermal diffusivity of ice is κ =
from an undissected ice surface through a pos-
35 m2 /yr. To estimate the vertical ablation rate
itive feedback between ice-surface temperature,
slope gradient, and albedo. In this step, the ice- on sun-facing slopes, Howard (1978) used Viking
surface temperature is assumed to increase from observations of summertime atmospheric water-
vapor concentration to obtain 10’4 m/yr, imply-
its threshold value for sublimation, T 0 , to a max-
imum value of 1 over a time scale „i . This in- ing „ f = 5 Myr for an average trough of 500 m
stability is represented mathematically in f (T , h) depth. The time of scarp initiation is uncertain,
by a cubic polynomial that generates a nega- but may be small compared with trough deep-
ening. Here we assume „i = 0.05, „ f = 0.25 Myr,
tive feedback if T is below the sublimation tem-
which implies that ≈ 25 m of ablation must

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