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the underlying bed topography (e.g. ≈ 100 m for a should be noted, however, that U-shaped valleys
glacier spanning ≈ 1 km in bed elevation). Alpine and hanging valleys are not unique to glacial sys-
glaciers are strongly controlled by bed topogra- tems. Fluvial channels that incise very rapidly
phy, with ice thickness greatest in areas of low- can oversteepen adjacent hillslopes to the point
est bed slope (i.e. valley bottoms). The tendency where water and sediment draining through side
of ice thickness and ¬‚ow velocity to be greatest tributaries will separate from the channel bed
in valley bottoms suggests that glaciers may in- (Wobus et al., 2006). As such, ¬‚uvial systems in tec-
crease subglacial landscape relief more readily tonically very active areas or in areas with strong
than ice sheets. Alpine glaciers are also strongly structural control can also form hanging valleys.
controlled by elevation. The relationship between Figure 1.32 illustrates the large-scale geomor-
the ELA (Equilibrium Line Altitude) and glacial phology of alpine glacial terrain using the Wind
erosion is much tighter in alpine glaciers than River Range as a type example. The Wind River
in ice sheets. Range has three distinct topographic levels. The
Figure 1.31 illustrates examples of two clas- high peaks of the Wind River Range (≈ 4 km a.s.l.)
stand ≈ 1 km above the surrounding low-relief
sic alpine glacial landforms: U-shaped valleys
and hanging valleys. Fluvial valleys are classi- plateau surface. This large-scale pattern of nar-
cally V-shaped, re¬‚ecting the contrasting ero- row high peaks surrounded by a broad plateau
sional power of ¬‚uvial hillslopes process regimes. is common in the glaciated mountain ranges of
In a V-shaped valley, hillslope erosion requires the western US. The formation of these plateau
steep slopes in order to ˜˜keep up™™ with the con- surfaces is not well understood. Did glaciers em-
centrated power of rivers to incise narrowly into anating from the high peaks ˜˜plane™™ the topog-
the valley bottom. In alpine glacial systems, in raphy to a relatively smooth plateau, or was the
contrast, ice ¬lls the valley ¬‚oor, resulting in a plateau exhumed by isostatic rebound respond-
˜˜plug™™ of ice that erodes the valley ¬‚oor and side ing to erosion concentrated in the high peaks?
slopes more uniformly than in the ¬‚uvial case. The most intense glacial erosion occurs near
A hanging valley forms when glacial erosion of the equilibrium line in most alpine glacial

systems. Figure 1.33a illustrates the topography of
(a) a portion of the Uinta Mountains using a shaded-
relief image of a US Geological Survey Digital Ele-
vation Model (DEM). The topography of the Uinta
Mountains is characterized by relatively ¬‚at di-
vide surfaces, steep cirque walls, and overdeep-
ened cirque ¬‚oors. The divides are relatively ¬‚at
because of the limited erosion that takes place
at elevations far above the equilibrium line. Near
the equilibrium line, glacial erosion is concen-
trated spatially, decreasing both up and down-
slope. This local maximum in erosion causes lo-
calized glacial scour. In river systems, bed scour
decreases the local slope, which acts as a negative
2 km
feedback to limit scour. In glacial systems, how-
ever, scour continues as long as the ice surface
(b) cirques
points downslope. Alpine glaciers can maintain
sloping ice surfaces even when the bed beneath
the ice has been scoured to form a closed de-
pression. When the ice retreats, a cirque lake is
formed. Figure 1.33b illustrates the cirque form
at the margin of the Beartooth Mountains in
Fig 1.33 Alpine glacial landforms near the equilibrium line:
(a) Shaded relief image of glacial topography in the Uinta Montana.
Mountains, Utah, illustrating steep cirque walls and The formation of overdeepenings beneath
overdeepened cirque lakes. (b) Cirques of the Beartooth alpine glaciers is illustrated schematically in Fig-
Mountains, Montana.
ure 1.34. Figure 1.34 shows a longitudinal pro¬le


evolves to



of an alpine valley glacier. The basal sliding ve-
Fig 1.34 Conceptual model of ¬‚ow in an alpine valley
locity beneath the glacier is a maximum beneath
glacier. Basal sliding velocity and glacial erosion are
the ELA. Over time, glacial erosion concentrated
concentrated beneath the equilibrium line. Over time,
beneath the ELA will cause the formation of an
concentrated erosion leads to the formation of an
overdeepened bed that forms a cirque lake when the ice overdeepened bed. Numerical models exist that
retreats. form realistic cirques (MacGregor et al., 2000), but

P-T Granite

30 km

Huayna Potosi

P-T Granite


(a) (b)
modern ELA

exhumation begins at 10 Ma,
accelerates towards present
(Benjamin, 1986)

Fig 1.35 Topography, ice cover, and structural geology of
may actually increase the elevations of the high
the Cordillera Real, central Andes. (a) View of the glaciated
peaks of the range. In this ˜˜relief-production™™ hy-
peak of Huayna Potosi, one of the high peaks of the
pothesis, glacial erosion causes isostatic imbal-
Cordillera Real, with the Altiplano in the foreground. (b)
ance. Because the erosion is concentrated at val-
Structural geology of a portion of the Cordillera Real,
ley bottoms with minimal erosion of the high
showing the intrusion of Permo-Triassic granite of the high
peaks, however, the presence of isostatic rebound
peaks into Paleozoic metasediments of the Eastern
together with the absence of signi¬cant erosion
Cordillera. Apatite ¬ssion track (AFT) data indicate that the
high peaks of the Cordillera Real were exhumed rapidly causes the peaks to grow even higher in eleva-
beginning at 10 Ma, accelerating towards the present day. tion. The difference between the glacial-buzzsaw
and relief-production hypotheses is partly an is-
sue of scale. At scales of 100 km or larger, glacial
many questions remain. For example, do most
erosion most likely does limit the relief of moun-
cirques primarily re¬‚ect stable ELAs during LGM
tain belts. At scales of a single valley and peak,
conditions, or are their morphologies controlled
however, the relief-production model appears to
by ¬‚uctuations in ELAs through time?
be most consistent with the results of glacial
The tight coupling between glacial ice cover,
landform evolution models. Feedbacks between
glacial erosion, and elevation suggests that
glaciers and topography may also involve climate
some interesting feedback relationships may ex-
through enhanced orographic precipitation in ar-
ist in glaciated mountain belts. For example, the
eas of higher/steeper topography.
˜˜glacial-buzzsaw™™ hypothesis states that glaciers
The Cordillera Real of the central Andes is one
limit the elevation of mountain belts because
potential example of the role of glacial erosion
any uplift that raises the height of mountain
in enhancing topographic relief. The Cordillera
belts above the ELA will quickly be met with en-
Real comprises the boundary between the Alti-
hanced ice cover and glacial erosion that will
plano to the west and the Eastern Cordillera to
lower elevations back below the ELA (Whipple
the east (Figure 1.35). This region is topograph-
et al., 2004). This model predicts a tight corre-
ically and structurally distinct from both of its
lation between the peak elevation of mountain
surrounding regions. Topographically, the range
ranges and the local ELA. On the other hand, the
is a very narrow (i.e. ≈ 15--20 km) series of peaks
fact that glaciers erode most vigorously in valley
up to ≈ 2 km above the surrounding topography.
bottoms and cirque ¬‚oors suggests that glaciers

Fig 1.36 Schematic model of
ELA isostatic uplift and erosion feedbacks
evolves to
(5 km)
in alpine glacial systems.

isostatic rock uplift

evolves to
(5 km)

broad uplift zone

Structurally, the Cordillera Real is comprised of This rebound would have uplifted the ¬‚anks of
granite exhumed between the Paleozoic sedimen- the Cordillera Real (in addition to its glaciated
tary rocks that make up the Eastern Cordillera. high peaks), increasing the area of the Cordillera
If the Cordillera Real displayed the usual signs Real that stands above the ELA. This increase
of active uplift (modern seismicity, active Qua- in glaciated area would further increase glacial
ternary faults), then its steep, high topography erosion and isostatic uplift in a positive feed-
could simply be attributed to active tectonics. back, causing localized exhumation of igneous
However, active tectonic uplift in the central An- and metamorphic rocks from the subsurface.
des occurs only in the Subandean Zone today,
which is the lowest part of the Andes (i.e. ≈
1.4 Conclusions
1--2 km a.s.l.). Exhumation rates in the Cordillera
Real measured thermochronologically (e.g. Ben-
The remainder of the book is organized accord-
jamin et al., 1987; Gillis et al., 2006), however,
ing to mathematical themes. Starting with Chap-
show high rates from 10 Ma to the present.
ter 2, each chapter focuses on a particular type
Figure 1.36 illustrates a schematic model for
of equation or algorithm and includes example
the exhumation and peak uplift of the Cordillera
applications that span Earth surface processes.
Real. According to this model, the Cordillera Real
This structure serves to emphasize the common
was a broad anticline with peak elevations near
5 km before 10 Ma. Global cooling of ≈ 5—¦ at mathematical language that underlies many of
the disparate process zones of Earth™s surface.
10 Ma lowered the ELA from above 6 km to ap-
Chapter 2, for example, focuses on techniques
proximately 5 km. This ELA lowering would have
used to solve the classic diffusion equation and
initiated glacial erosion in the highest portions
the general class of equations that are similar
of the Cordillera Real. Because glacial erosion in-
to it. Since the diffusion equation can be used
creases as a function of distance from the di-
to describe hillslope evolution, channel-bed evo-
vide, the high peaks of the Cordillera Real would
lution, delta progradation, hydrodynamic disper-
not have been subject to signi¬cant erosion,
sion in groundwater aquifers, turbulent disper-
however, while lower areas within the glaciated
sion in the atmosphere, and heat conduction
zone would have been more intensely eroded.
in soils and the Earth™s crust, Chapter 2 will
As such, an initially broad anticline would have
be relevant to many aspects of Earth surface
been sculpted to a narrow peak. Glacial ero-
sion would also have triggered isostatic rebound.
Chapter 2

The diffusion equation

where ρ is the bulk density of sediment on the
2.1 Introduction hillslope, h is the elevation, κ is the diffusivity
in units of L 2 /T (where L is length and T is
time) and x is the distance from the divide. In
The diffusion equation is perhaps the most
Eq. (2.1), q is a mass ¬‚ux with units of M/L /T
widely used differential equation in science.
(where M is mass, L is length, and T is time).
It is the equation that describes heat trans-
Mass must also be conserved. In this context, con-
port in solids and in many turbulent ¬‚uids.
servation of mass means that any change in sed-
In geomorphology, the diffusion equation quan-
iment ¬‚ux along a hillslope results in either an
ti¬es how landforms, especially hillslopes, are
increase or decrease in the elevation. One way
smoothed over time. Diffusional processes act to
to think about conservation of mass is to con-
smooth elevation in the same way that conduc-
sider a small segment of a hillslope pro¬le (e.g.
tion smooths temperature in solids. The diffusion
the section between x3 and x4 in Figure 2.1).
equation has played a particularly important role
If more sediment enters the segment from up-
in the study of well-dated landforms, such as plu-
slope than leaves the segment downslope, that
vial shoreline scarps and volcanic cones, in which
hillslope segment must store the difference, re-
the initial topography can be relatively well con-
sulting in an increase in the average elevation.
strained at a known time in the past. Certain
Conversely, if more sediment leaves the segment
conditions must be met for the diffusion equa-
downslope than enters the segment upslope (as
tion to apply to hillslope evolution, however. The
in the section between x1 and x2 ), there is a
diffusion equation is applicable to gently slop-
net loss of sediment and the elevation must de-
ing hillslopes comprised of relatively uniform,
crease. The diffusion equation applies this logic
unconsolidated alluvium or regolith subject to
to in¬nitesimally-small hillslope segments in the
creep, rainsplash, and bioturbation. The accuracy
same way that the ¬rst derivative of calculus cal-
of the model breaks down as hillslope gradients
culates the average slope h/ x of a function
increase and mass-movement processes become
h(x) in the limit as x goes to zero. Mathemati-
cally, conservation of mass requires that the in-
The applicability of the diffusion equation has
crease or decrease in the elevation be equal to the
two requirements: slope-proportional transport
change in ¬‚ux per unit length, divided by the bulk
and conservation of mass. First, the ¬‚ux of sedi-
density ρ:
ment per unit length, q, must be proportional to
the hillslope gradient:
‚h 1 ‚q
q = ’ρκ (2.2)
‚t ρ ‚x

where t is time. Substituting Eq. (2.1) into Eq. (2.2) time, however, the rate of erosion and deposition
gives the classic diffusion equation: decreases and the widths of the top and bottom
of the scarps where erosion and deposition oc-
‚h ‚ 2h
=κ 2 curs increases (Figure 2.1f).
‚t ‚x
Many physical processes can be classi¬ed as
If, alternatively, q is expressed as a volumet- either diffusive or advective. Therefore, one rea-
ric ¬‚ux per unit length, the ρ terms are re- son for studying the simple diffusion and advec-
moved from both Eqs. (2.1) and (2.2). We will tion equations in detail is that they provide us
use both the mass and volumetric ¬‚uxes in this with tools for studying more complex processes
chapter. that can be broadly classi¬ed as one of these two
Equation (2.3) is applicable to hillslopes with a process types. Conceptually, diffusive processes
constant cross-sectional topographic pro¬le along result in the smoothing of some quantity (tem-
strike (i.e. a fault scarp). More generally, hillslopes perature, elevation, concentration of some chem-
evolve according to the 2D diffusion equation, ical species, etc.) over time. For instance, if one
given by injects a colored dye into a clear ¬‚uid being
vigorously mixed, the average dye concentration
‚h ‚ 2h ‚ 2h
=κ +2 = κ∇ 2 h (2.4) smooths out over time (rapidly at ¬rst when con-
‚t ‚ x2 ‚y
centration gradients are high, then more slowly)
Figure 2.1a illustrates a hypothetical fault according to the diffusion equation. Similarly, if
scarp 2 m high and 10 m wide after some ero- faulting offsets an alluvial fan, the scarp eleva-
sion has taken place. Figures 2.1b and 2.1c illus- tion smooths out over time as sediment is eroded
trate the gradient and curvature of the scarp, from the top of the scarp and deposited at the
respectively. The diffusion equation states that base.
the sediment ¬‚ux q at any point along a hill- Advection, in contrast, involves the lateral
slope is proportional to the hillslope gradient translation of some quantity without spreading.
(i.e. Figure 2.1b). The magnitude of the ¬‚ux is If a colored dye is injected into a laminar ¬‚uid
illustrated in Figure 2.1a at several points along moving down a pipe, for example, the dye will
the pro¬le using arrows of different lengths. At be transported down the pipe with the same
the top of the scarp, the ¬‚ux increases from left velocity as the ¬‚uid but with little spreading.
to right, indicating that more material is mov- Similarly, if a fault offsets a bedrock pediment
ing out of the section than is being transported instead of an alluvial fan, the fault will evolve
into it from upslope. This results in erosion along primarily by weathering-limited slope retreat in-
the top of the scarp where the change in gradient stead of smoothing. In such cases, slow bedrock
along the pro¬le (i.e. the curvature) is negative. weathering acts to maintain a steep fault scarp
Conversely, ¬‚ux decreases from left to right at over time because transport of the weathered
the bottom of the scarp, indicating that more material is rapid enough that sediment does
material is moving into that segment than out not accumulate at the scarp base and no depo-
of it. The result is an increase in surface eleva- sition occurs. This type of slope evolution can
tion (i.e. deposition) along the base of the scarp be modeled by the advection equation, where
where curvature is positive. The rate of erosion the erosion rate is proportional to the hillslope
or deposition varies with time (Figures 2.1d--2.1f) gradient:
according to the magnitude of curvature. Imme-
‚h ‚h
diately after faulting, for example, the change in (2.5)
‚t ‚x
¬‚ux per unit length (i.e. the curvature in Figure
2.1c) is very large and concentrated right near The variable c is the advection velocity in Eq.
the top and bottom of the scarp. The result is very (2.5), and it equals the rate of horizontal re-
rapid, concentrated erosion and deposition at the treat of the scarp. In general, transport-limited
top and bottom of the scarp, respectively. Over slopes and channels evolve diffusively while

(a) (d)
1 1

h h
(m) erosion
x1 x2


x3 x4
0 0

0.0 0.0
(b) (e)

kt = 25 m2

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