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in narrower ¬‚ow channels (assuming the ELA is the same as
ger lakes™™ we will be referring to these features
in (b)).
collectively. In this section we present model re-
sults for the ice-sheet geometry, basal velocities,
and bedrock erosion near linear and curved ice
margins. We show that uniformly-spaced ice-¬‚ow 6.28a and 6.28b. The basal shear stress in this
channels result from ¬‚ow over a rough surface case is assumed to be 0.7 bars and the ELA is
near the ice margin. 1 km. Also given in Figure 6.28a are topographic
The ice-surface topography and basal-sliding pro¬les parallel and perpendicular to the mar-
velocities for an ice sheet sliding over a rough gin. Pro¬le 1 shows the parabolic pro¬le charac-
surface near a linear margin are given in Figures teristic of ice sheets on ¬‚at beds (e.g. Nye, 1951).
158 NON-NEWTONIAN FLOW EQUATIONS



(a) Fig 6.29 Numerical experiment
illustrating the model behavior of
focused erosion near a curved ice
margin. (a) Shaded-relief image of
ice-surface topography, (b) grayscale
image of basal-sliding velocities at
iteration 1 ice-surface topography
the beginning of the model run. (c)
(b) Ice-surface topography, (d)
basal-sliding velocities, and (e) bed
topography of the model after ten
iterations. Finger lakes have formed
with orientations parallel to the
iteration 1 basal sliding velocity ice-¬‚ow directions and lake spacing
is low in zones of converging ice and
(c)
high in zones of diverging ice.




iteration 10 ice-surface topography

(d)




iteration 10 basal sliding velocity

(e)




iteration 10 bed topography



is to create strongly localized ¬‚ow near the ice
Pro¬le 2 shows the microtopography of the ice
margin even though the ice-surface topography
surface along a direction parallel to the margin.
is nearly uniform along the margin. The width
One of the most important ¬ndings of the chap-
of ice-¬‚ow channels depends on the slope of the
ter is illustrated in Figure 6.28: even though the
ice sheet at the ELA. A thicker ice sheet with the
bed topography has no characteristic scale (i.e. it
same ELA (1 km), for example, results in more
is white noise), the resulting ice-surface topogra-
closely-spaced channels, To illustrate this, Figure
phy has a characteristic scale of about 10 km. This
6.28c is a basal-velocity map for an ice-sheet re-
characteristic scale is even more apparent in the
construction with „ = 1.5 bars. Ice-¬‚ow channels
map of basal-sliding velocities (Figure 6.28b). Fo-
in this case are spaced at ≈ 5--7 km instead of
cusing of the ice ¬‚ow occurs as a result of the ad-
10 km.
ditive de¬‚ection of ice ¬‚ow along ¬‚ow paths from
How does the focusing effect differ for a
the divide to the margin. This focusing involves
curved ice margin? Figures 6.29a and 6.29b illus-
several steps. First, the rough bed is re¬‚ected
trate the ice-surface topography and basal-sliding
in the ice surface as microtopographic channels
velocities for a curved margin. The shape of this
(Figure 6.28a). Second, these channels act to focus
margin is identical to the 14 ka ice margin of the
the ¬‚ow a little bit as the ice is routed from pixel
Finger Lakes Region. The bed topography in this
to pixel. The cumulative effect of this focusing
6.6 GLACIAL EROSION BENEATH ICE SHEETS 159



(a) Fig 6.30 Spacing of ¬nger lakes
and troughs in upstate New York.
(a) Shaded-relief DEM image of
Finger Lakes area. White curve
indicates approximate position of
the Valley Heads Moraine (based on
the Finger Lakes and Niagara sheets
of the sur¬cial geologic map of New
York (New York Geological Survey,
1999)). Black lines indicate the
location of the topographic transect
(b)
5 km 10 5 5
20 given in (b). The transect shows
600 m
alternating zones of narrow (≈ 5 km)
and wide (up to 20 km) valley
elevation spacing. Zones of narrow spacing
correspond to regions of converging
¬‚ow in ice embayments. Conversely,
200 m
distance along profile wider spacings correspond with
270 km
0
areas of diverging ice ¬‚ow.

gin was coincident with this moraine, although
example was also chosen to be a ¬‚at surface with
several phases of glaciation may have contributed
white noise variations. The curved margin results
to their formation.
in ice-¬‚ow channels that are alternately closely
Figure 6.30b is the topographic pro¬le of the
spaced and widely spaced. Channels are more
region along the black line of Figure 6.30a. The
closely spaced in margin embayments where ice
pro¬le of Figure 6.30b shows the same pattern as
¬‚ow is convergent. Widely spaced channels occur
the model results of Figures 6.29c--6.29e: closely
beneath ice lobes where ¬‚ow is divergent. Fig-
spaced (≈ 5 km) troughs in margin embayments
ures 6.29c--6.29e illustrate the ice-surface topog-
and widely spaced (10--20 km), deeper troughs be-
raphy, basal velocities, and bedrock topography
neath ice-sheet lobes.
after 10 model iterations. The ice ¬‚ow channels
If ¬nger lake formation is an intrinsic feature
have carved distinct glacial troughs, depressing
of ice margins, why don™t ¬nger lakes occur more
the ice-surface topography and focusing ¬‚ow and
commonly in formerly glaciated terrain? One pos-
erosion into the troughs.
sible reason is that intense glacial erosion and a
Geomorphically, the Finger Lakes Region
stable ice margin are both required for ¬nger-
is dominated by subparallel, glacially scoured
lake formation. If the equilibrium line is rapidly
troughs with southernmost extents in Seneca
migrating during ice-sheet advance or retreat, for
and Cayuga Lakes, the two largest of the Fin-
example, there may not be suf¬cient time for the
ger Lakes. The ¬ve largest troughs of the region
instability between depressions, ice ¬‚ow, and ero-
comprise the Finger Lakes proper, but there are
sion to develop deep troughs. It may be that the
numerous other troughs cut into the Allegheny
Finger Lakes were formed because 14 ka was a
Plateau of smaller size that are not deep enough
period of suf¬cient margin stability for erosion
to impound water. Figure 6.30a is a shaded-
to carve the topography we see today. Alterna-
relief image of the topography of the region
tively, the margin may not have been unusually
with the location of the 14 ka ice margin (the
stable, but erosion may have been particularly in-
Valley Heads Moraine) shown in white. The
tense during this period. For example, meltwater
troughs vary in spacing from 5 to 20 km along
pulses may have driven rapid erosion just prior
strike with the greatest spacing between Seneca
to 14 ka followed by rapid ¬lling of the Finger
and Cayuga Lakes. The southern tips of the Fin-
Lakes between 14 and 13 ka. This hypothesis is
ger Lakes coincide with the Valley Heads Moraine
consistent with the lake stratigraphy of the re-
(14 ka). As such, the scouring of the Finger Lakes
gion (Mullins and Hinchey, 1989).
Region most likely took place when the ice mar-
160 NON-NEWTONIAN FLOW EQUATIONS


and compute pro¬les for each position by numeri-
cally integrating from the terminus up the valley.
6.3 Using the code in Appendix 5 as a guide,
4m model the evolution of a 1D gravity ¬‚ow with
temperature-dependent viscosity ¬‚owing down an
inclined plane of slope S.
6.4 Download a DEM of an area formerly covered by
alpine glaciers. Using the sandpile algorithm code
10 m given in Appendix 5 as a guide, map the thick-
ness of glacier ice assuming a basal shear stress
Fig 6.31 Schematic diagram of lava-¬‚ow margin in Exercise
of 1 bar. Choose an ELA value and assume that ice
6.1.
covers topography only above the ELA.
6.5 The Boussinesq equation
‚h kρg ‚ ‚h
= h (E6.1)
‚t μφ ‚ x ‚x
Exercises
6.1 Lava-¬‚ow margins in the vicinity of Lathrop Wells is used to quantify the evolution of a water table
cone, Amargosa Valley, Nevada, are approximately of height h in an uncon¬ned aquifer of perme-
ability k and porosity φ, where ρ is the density of
10 m wide and 4 m thick in areas of ¬‚at terrain
water, g is gravity, and μ is the dynamic viscos-
(Figure 6.31). Estimate the yield stress for these
¬‚ows based on a 1D perfectly plastic model. ity of water (Turcotte and Schubert, 2002). Plot
6.2 Use a topographic map or DEM to extract a lon- the shape of the steady-state water table in a 1D
gitudinal pro¬le of a steep ¬‚uvial valley. Import aquifer of length 10 km above an impervious stra-
tum with a dip of 1%. Assume k =10’12 m2 and
the pro¬le in two-column format x, h into Ex-
μ =10’3 Pa s. (Hint: map this problem onto the
cel. Using Excel, model the pro¬le of a slow-
moving debris ¬‚ow with yield stress 0.1 bars ¬‚ow- perfectly plastic model for alpine glaciers. See Sec-
ing down the valley bottom using the perfectly tion 9.5 of Turcotte and Schubert for more details
plastic model. Choose several terminus positions on the Boussinesq equation.)
Chapter 7




Instabilities


to spatial and temporal oscillations in arid al-
7.1 Introduction luvial channels. The study of instabilities does
not involve any fundamentally new types of
equations. Rather, most equations involving ge-
Many geomorphic processes involve positive feed-
omorphic instabilities combine diffusion- and
backs. The incision of a river valley into an
advection-type equations similar to those we
undissected landscape is an example of a posi-
have already considered. In most cases, how-
tive feedback process in which an incipient chan-
ever, two or more variables are coupled in such
nel erodes its bed and expands its drainage area
a way as to enhance each other in a positive
in a positive feedback. In some cases, positive-
feedback.
feedback processes give rise to landforms with
This chapter does include one fundamentally
no characteristic scale. Many drainage networks,
new tool: the linear-stability analysis. In this anal-
for example, are branching networks with no
ysis, the fundamental equations that describe
characteristic scale above the scale that de¬nes
the evolution of a particular system are ¬rst lin-
the hillslope-channel transition. In other cases,
earized (i.e. nonlinear terms are neglected) and
periodic landforms are created (Hallet, 1990).
then solved for the case of an initially small
Sand dunes, for example, form by a positive
perturbation of wavelength » (where » is a free
feedback between the height of the dune, the
parameter). Conceptually, a linear stability ap-
de¬‚ection of air ¬‚ow above it, and the result-
proach considers the evolution of the system
ing pattern of erosion and deposition (i.e. en-
with small, incipient landforms with a wide
hanced windward erosion and leeward deposi-
range of sizes (i.e. wavelengths) and determines
tion). Unstable geomorphic systems all include
whether and how fast each perturbation will
some type of positive feedback. In many arid al-
grow to become full-sized landforms. For those
luvial channels, for example, sediment becomes
wavelengths that do grow, the wavelength that
stored preferentially at certain points along the
grows the fastest provides an estimate of the char-
channel pro¬le, thereby causing the channel to
acteristic scale of the landform as a function of
aggrade and the channel bed to widen. Chan-
the system parameters. This fastest-growing wave-
nel widening further enhances sediment stor-
length often, but not always, provides a good es-
age and bed aggradation in a positive feedback.
timate of the spatial scale of full-sized landforms.
This instability continues until a suf¬ciently
As such, linear stability analyses often provide a
steep slope develops downstream of the aggrad-
useful starting point for understanding the emer-
ing zone, triggering channel entrenchment and
gence of characteristic scales in complex geomor-
narrowing. The positive feedback between chan-
phic systems.
nel width and localized erosion/deposition leads
162 INSTABILITIES



y = ’b

r1, m

w x
y=0

r2, m

y=b
y
Fig 7.2 Geometry of the Rayleigh“Taylor instability of a
dense ¬‚uid overlying a lighter ¬‚uid. After Turcotte and
Schubert (2002). Reproduced with permission of Cambridge
University Press.
Fig 7.1 Surface expression of salt domes in the Gulf of
Mexico, from an oblique perspective image of high-resolution
bathymetry. The topography has a hummocky shape that
layers is initially centered on y = 0 but has a si-
re¬‚ects the underlying characteristic scale and spacing of salt
domes. Image reproduced with permission of Lincoln Pratson. nusoidal variation given by

2π x
w(x) = A 1 cos (7.1)
»

7.2 An introductory example: the The goal of the linear stability analysis is to
Rayleigh“Taylor instability predict how w varies with time and to deter-
mine the wavelength » associated with the fastest
growth.
As an introduction to the study of instabilities
The ¬rst step of the Rayleigh--Taylor analysis
in geomorphology, we consider the Rayleigh--
is to solve for the ¬‚ow ¬eld in the two-layer sys-
Taylor instability of a dense ¬‚uid overlying a
tem given a growing interface given by Eq. (7.1).
lighter ¬‚uid. The Rayleigh--Taylor instability de-
The ¬‚ow ¬elds are characterized by four velocity
scribes the formation of salt domes in sedimen-
functions: u1 , v 1 , u2 , v 2 . These functions are the
tary basins (Figure 7.1) and it provides a simple
horizontal (u) and vertical (v ) components of the
model for the initiation of mantle plumes. When
velocity in the upper (1) and lower (2) layers. Tur-
a dense layer overlies a lighter layer, the system
cotte and Schubert (2002) used the stream func-
is gravitationally unstable, i.e. the lighter ¬‚uid
tion method to obtain expressions for these ¬‚ow
will rise and penetrate through the upper layer.
velocities. In this method, the velocity functions
Here we will quantify how this instability works
u and v are obtained by differentiating a single
following closely the approach of Turcotte and
˜˜stream function™™ ψ:
Schubert (2002).
The geometry of the two-layer system is shown
‚ψ
u=’
in Figure 7.2. A ¬‚uid with a thickness b and den- (7.2)
‚y
sity ρ1 overlies a lighter ¬‚uid of thickness b and
‚ψ
density ρ2 . Here we assume that both ¬‚uid layers
v= (7.3)
have the same viscosity μ. We also assume that ‚x
both the top and bottom of the system have rigid
surfaces. The interface between the two ¬‚uid Turcotte and Schubert™s (2002) analysis for this
7.2 AN INTRODUCTORY EXAMPLE: THE RAYLEIGH“TAYLOR INSTABILITY 163


problem gives

» »
2π x 2π y 2π x y 2π b 2π y y 2π y 2π y
ψ1 = A 1 sin + A 1 sin + ’
cosh tanh sinh cosh sinh
» » » » » » »
b 2π b b 2π b
’1
2
» »
1 1 2π b
— + ’ tanh (7.4)
»
sinh 2π b cosh 2π b sinh 2π b cosh 2πb
2πb 2π b
» » » »


The expression for ψ2 is obtained from Eq. (7.4)
Substituting Eq. (7.4) into Eqs. (7.3), (7.3), and (7.8)
by replacing y with ’y.
gives
Equation (7.4) can be used to solve for the evo-
2μA 1 2π x 2π
lution of the interface. The rate of change of the (P 1 ) y=0 = cos
» »
amplitude of the interface, ‚w/‚t must equal the b
»
vertical component of the ¬‚uid velocity at just 1
— +
above the interface. If not, a void would open up sinh » cosh 2π b
2π b
2π b »
in the system. This condition can be written as 2
»
1 2π b
— ’ tanh
‚w »
2πb 2π b
2π b
sinh » cosh »
= (v 1 ) y=0 (7.5)
‚t (7.9)
As an approximation, the velocity in Eq. (7.5) Substituting Eq. (7.9) into Eq. (7.7) gives

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