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wide bottom displaces a relatively large volume
tral Andes today. When subduction of the Far-
of high-density mantle. The resulting buoyancy
allon plate ceased beneath the western US, this
requires that the block stand higher above Earth™s
region became a predominantly strike-slip plate
geoid in order to be in isostatic equilibrium. Con-
boundary and the horizontal compressive force
versely, a crustal block with a relatively narrow
that supported the high topography and over-
bottom will displace a smaller volume of mantle
thickened crust of the region could no longer
and will hence stand lower relative to the geoid.
withstand the weight of the overlying topogra-
This kind of adjustment is the fundamental
phy. The result was regional extension (Sonder

Fig 1.1 Hanaupah Canyon and its alluvial fan, Death Valley,

Fig 1.3 Oblique perspective image of the high peaks of
Hanaupah Canyon. Reproduced with permission of
‚h = 0

divide h

Fig 1.2 Schematic diagram of a simple model for Basin and
h regolith
Range extension and isostatic adjustment. In (a), horizontal
extension and vertical compression cause the creation of
trapezoidal crustal blocks bounded by a series of normal
faults. Crustal blocks with wide bottoms (ranges) rise to
maintain isostatic balance, while blocks with narrow bottoms
(basins) fall relative to Earth™s geoid.

reason why extension causes the semi-periodic to-
pography of the Basin and Range Province. channel head
Fig 1.4 Schematic diagram of a hillslope pro¬le from divide
1.1.2 The hillslope system to channel head.
Hillslopes in the high elevations of Hanaupah
Canyon are characterized by steep, planar topo-
graphic pro¬les and abrupt, knife-edge divides because of this change in drainage direction. At
(Figure 1.3). West of the divide, water and sedi- the divide itself, the water and sediment ¬‚ux is
ment drains towards Panamint Valley, while on zero, since this is the transition point from west-
the east side they drain towards Death Valley. If ern drainage to eastern drainage. Sediment ¬‚ux
we quantify the ¬‚ux of water and sediment along on hillslopes is directly related to the topographic
a pro¬le that includes the drainage divide, the gradient. Therefore, since the sediment ¬‚ux is
zero, the topographic gradient, ‚h/‚ x, must also
¬‚ux of water and sediment must be negative on
one side of the divide and positive on the other be zero (Figure 1.4).

Hillslopes in bedrock-dominated landscapes knife-edge drainage divides of Hanaupah Canyon
are composed of a system of two interacting sur- are a signature of landslide-dominated, nonlin-
faces: the topographic surface, with elevations ear transport on hillslopes (Roering et al., 1999).
given by h(x, y), and the underlying weathering Equation (1.2) states that bedrock lowering
front, given by b(x, y) (Figure 1.4). The difference is a maximum for bare bedrock slopes and de-
between these two surfaces is the regolith depth, creases exponentially with increasing regolith
·(x, y). The topographic and weathering-front sur- thickness. This relationship has been inferred
faces are strongly coupled because the shape from cosmogenic isotope analyses on hillslopes
of the topography controls erosion and deposi- (Heimsath et al., 1997). Conceptually, the in-
tion, which, in turn, changes the values of ·(x, y) verse relationship between weathering/regolith-
(Furbish and Fagherazzi, 2001). The values of production rate and regolith thickness is a con-
·(x, y), in turn, control weathering rates. The sim- sequence of the fact that regolith acts as a
plest system of equations that describes this feed- buffer for the underlying bedrock, protecting it
back relationship is given by: from diurnal temperature changes and in¬ltrat-
ing runoff that act as drivers for physical and
‚· ρb ‚b ‚ 2h
=’ ’κ 2 (1.1) chemical weathering in the subsurface. In Figure
‚t ρs ‚t ‚x
1.4, the weathering front is shown as an abrupt
and transition from bedrock to regolith, but in nature
‚b this boundary is usually gradual.
= ’P 0 e’·/·0 (1.2)
‚t Equations (1.1) and (1.2) can be solved for the
where t is time, ρb is the bedrock density, ρs steady-state case in which regolith thickness is
is the sediment density, κ is the hillslope diffu- independent of time at all points along the hill-
sivity, x is the distance along the hillslope pro- slope pro¬le:
¬le, P 0 is the regolith-production rate for bare
ρb P 0 1
bedrock, and ·0 is a characteristic regolith depth. · = ·0 ln (1.3)
ρs κ ’ ‚ 2 h
Equation (1.1) states that the rate of change ‚ x2
of regolith thickness with time is the differ- Equation (1.3) implies that, in steady state, re-
ence between a ˜˜source™™ term equal to the rate golith thickness decreases as the negative (down-
of bedrock lowering multiplied by the ratio of ward) curvature increases, becoming zero (i.e.
bedrock to sediment density, and a ˜˜sink™™ term bare bedrock) where the curvature reaches a crit-
equal to the curvature of the topographic pro- ical value of
¬le. This curvature-based erosion model is the
‚ 2h ρb P 0
classic diffusion model of hillslopes, ¬rst pro- (1.4)
‚ x2 ρs κ
bar e
posed by Culling (1960). The diffusion model of
hillslope evolution, discussed in Chapter 2, is a Equation (1.2) is not universally applicable.
consequence of conservation of mass along hill- As regolith thickness decreases below a critical
slope pro¬les, and the fact that sediment ¬‚ux is value in arid regions, for example, the land-
proportional to topographic gradient if certain scape is unable to store enough water to sup-
conditions are met. The diffusion model of hill- port signi¬cant plant life. Plants act as weath-
slope evolution does not apply to the steep hill- ering agents (e.g. root growth can fracture rock,
slopes of Hanaupah Canyon, however. In steep canopy cover can decrease evaporation, etc.).
landscapes, sediment ¬‚ux increases nonlinearly Therefore, in some arid environments weather-
with the topographic gradient as the angle of sta- ing rates actually increase as regolith thickness
bility is approached and landsliding becomes the increases, rather than decreasing with thickness
predominant mode of sediment transport. The ef- as the exponential term in Eq. (1.2) predicts
fects of mass movements on hillslope evolution (Figure 1.5). Using a vegetation-limited model of
can be captured in a nonlinear hillslope diffu- weathering in landscape evolution models re-
sion model that will also be discussed in Chap- sults in landscapes with a bimodal distribution of
ter 2. The steep, planar hillslopes and abrupt, slopes (i.e. cliffs and talus slopes) similar to many

channel bed and channel bank, and the channel
P0 bed and shoreline, just to name a few.
P0e’h/h0 The transition between the hillslope and ¬‚u-
vial channel system occurs where the shear stress
‚h of overland ¬‚ow is suf¬cient to entrain hillslope
‚t material. In addition, the rate of sediment exca-
vation from the channel head by overland ¬‚ow
must be greater than the rate of sediment in¬ll-
ing by creep and other hillslope processes. Empir-
ically, this transition occurs where the product
h of the topographic gradient and the square root
of contributing area is greater than a threshold
Fig 1.5 Models for the relationship between regolith
value (Montgomery and Dietrich, 1999):
production and regolith thickness, illustrating the exponential
model of Heimsath et al. (1997) and the alternative 1
S A 2 = X ’1 (1.5)
vegetation-limited model of Anderson and Humphrey (1990).
where S is the topographic gradient or slope, A is
The density contrast between the bedrock and sediment is
assumed to be zero for simplicity. the contributing area, and X is the drainage den-
sity (i.e. equal to the ratio of the total length of all
the channels in the basin to the basin area). The
value of X depends on the texture and perme-
arid-region hillslopes (Anderson and Humphrey,
ability of the regolith, hillslope vegetation type
1990; Strudley et al., 2006). It should also be
noted that the parameters P 0 and κ may be ap- and density, and on the relative importance of
different hillslope processes.
proximately uniform along some hillslope pro-
¬les, but in general the hillslope and regolith-
1.1.3 Bedrock channels
thickness pro¬les will in¬‚uence the hillslope hy-
drology (e.g. the ratio of surface to subsurface Channels are divided into alluvial and bedrock
¬‚ow), which will, in turn, modify the values of P 0 channel types depending on whether or not al-
and κ through time. Thinner regolith, for exam- luvium is stored on the channel bed. In bedrock
ple, can be expected to increase surface runoff, channels, the transport capacity of the channel
thereby increasing κ and decreasing P 0 in a pos- is greater than the upstream sediment ¬‚ux. Sed-
itive feedback. iment storage is rare or nonexistent in these
Equations (1.1) and (1.2) suggest that even cases. In alluvial channels, upstream sediment
in well-studied geomorphic systems such as hill- ¬‚ux is greater than the transport capacity of the
slopes, quantifying the behavior of even the most channel and alluvium ¬lls the channel bed as a
basic elements is a signi¬cant challenge. This result. The distinction between bedrock and al-
challenge provides an opportunity for mathemat- luvial channels is important for understanding
ical modeling to play an essential role in ge- how they evolve. In order for bedrock channels
omorphic research, however, because most ge- to erode their beds they must pluck or abrade
omorphic systems are too complex to be fully rock from the bed. Once rock material is eroded
understood and interpreted with ¬eld observa- from a bedrock river it is usually transported
tions, measurements, and geochronologic tech- far from the site of erosion because the ¬‚ow ve-
niques alone. The coupled hillslope evolution locity needed to transport material is typically
model of Eqs. (1.1) and (1.2) also illustrates a much lower than the velocity required for pluck-
larger point: many geomorphic systems of great- ing or abrasion. Bedrock channels erode their
est interest involve the coupling of different pro- beds by a combination of plucking, abrasion, and
cess domains (in this case the hillslope weather- cavitation. Plucking occurs when the pressure
ing and sediment transport regimes). Other key of fast-¬‚owing water over the top of a jointed
process-domain linkages occur at the juncture be- rock causes suf¬cient Bernoulli lift to dislodge
tween the hillslope and the channel head, the rock from the bed. Abrasion occurs as saltating

erosion is usually linear with slope (i.e. n ≈ 1 and
bedload impacts the bed, chipping away small
m ≈ 1/2 (Kirby and Whipple, 2001)).
pieces of rock. Cavitation occurs when water boils
under conditions of very high pressure and shear In steady state, uplift and erosion are in bal-
ance and ‚h/‚t = 0. Equation (1.6) predicts that
stress. Imploding bubbles in the water create
channel slope, S = |‚h/‚ x|, is inversely propor-
pressures suf¬cient to pulverize the rock. Con-
ditions conducive to cavitation are most likely tional to a power-law function of drainage area,
to occur in rare large ¬‚oods. The relative impor- A, for drainage basins in topographic steady state:
tance of plucking and abrasion depend primar- U ’m/n
S= A (1.7)
ily on lithology. Abrasion is generally considered
to be the dominant process in massive bedrock
Equation (1.7) predicts that, as drainage area in-
lithologies such as granite (Whipple et al., 2000).
creases downstream, the channel slope must de-
In sedimentary rocks, abrasion and plucking are
crease in order to maintain uniform erosion rates
both likely to be important.
across the landscape. This inverse relationship be-
Several mathematical models exist for mod-
tween slope and area is responsible for the con-
eling bedrock channel evolution. In the stream-
cave form of most bedrock channels.
power model, ¬rst proposed by Howard and Kerby
Equation (1.7) can be used to derive an ana-
(1983), the rate of increase or decrease in chan-
lytic expression for the steady-state bedrock chan-
nel bed elevation is equal to the difference be-
nel longitudinal pro¬le. To do this, we must re-
tween the uplift rate and the erosion rate. The
late the drainage area, A, to the distance along
erosion rate is a power function of drainage area
the channel from the channel head, x. The re-
and channel-bed slope:
lationship between area and distance depends
‚h ‚h on the planform (i.e. map-view) shape of the
= U ’ K w Am (1.6)
‚t ‚x basin. For a semi-circular basin, the square root
of drainage area is proportional to the distance
where h is the local elevation, t is time, U is
from the channel head. Adopting that assump-
uplift rate, K w is a constant that depends on
tion together with the empirical observation
bedrock erodibility and climate, A is the drainage
m/n = 0.5, Eq. (1.7) becomes
area, m and n are empirical constants, and x
is the distance along the channel. The general ‚h C
=’ (1.8)
form of the stream-power model also includes ‚x x
an additional constant term that represents a
where ‚h/‚ x has been substituted for S and C is a
threshold shear stress for erosion (Whipple and
constant that combines the effects of uplift rate,
Tucker, 1999). The stream-power model is empiri-
bedrock erodibility, climate, and basin shape. In-
cally based: Howard and Kerby (1983) found that
tegrating Eq. (1.8) gives
Eq. (1.6) successfully reproduced observed erosion
rates at a site in Perth Amboy, New Jersey, mea- h ’ h 0 = ’C ln (1.9)
sured by repeat survey. Their analysis cannot,
however, rule out whether other variables corre- where h 0 is the elevation of the base of the chan-
nel at x = L . Figure 1.6 shows the longitudinal
lated with drainage area and slope are the funda-
mental controlling variables of bedrock erosion. pro¬le of the main channel of Hanaupah Canyon
Nevertheless, the stream power is based on the together with a plot of Eq. (1.9). The model ¬ts the
observed pro¬le quite well using C = 0.9 km.
physically-reasonable assumption that the ero-
sive power of ¬‚oods increases as a function of The evolution of a hypothetical bedrock chan-
drainage area (which controls the volume of wa- nel governed by the stream-power model is illus-
ter routed through the channel) and channel-bed trated qualitatively in Figure 1.7a. A small reach
slope (which controls the velocity of that water of the channel™s longitudinal pro¬le is shown at
for otherwise similar drainage areas and precip- times t1 , t2 , and t3 . If the section is relatively
itation intensities). Calibrations of the stream- small and has no major incoming tributaries,
power model in natural systems suggest that the drainage area can be considered uniform

ing in the propagation of knickpoints upstream
(a) as topographic waves. The rate of knickpoint
propagation is equal to K w A m , i.e. knickpoints
propagate faster in larger channels, wetter cli-
mates, and areas of more easily-eroded bedrock.
The stream-power model will serve as the type
example of the advection/wave equation studied
in Chapter 4.
The Kern River (Figures 1.8 and 1.9) pro-
vides a nice example of knickpoint propaga-
3 tion in action. Two distinct topographic surfaces
have long been recognized in the landscape of
the southern Sierra Nevada (Webb, 1946) (Figure
1.8). The Boreal surface is a high-elevation, low-
h (km)

longitudinal profile
relief plateau that dips to the west at 1—¦ (Fig-
model fit
ure 1.8b). The Chagoopa Plateau is an interme-
1 diate topographic ˜˜bench™™ that is restricted to
the major river canyons and inset into the Bo-
real Plateau (Webb, 1946; Jones, 1987). Figure 1.8b
bedrock alluvial
0 maps the maximum extents of the Chagoopa
0 20
5 10 15
and Boreal Plateaux based on elevation ranges
x (km)
of 1750--2250 m (Chagoopa) and 2250--3500 m a.s.l.
Fig 1.6 (a) Shaded-relief image of the Hanaupah Canyon (Boreal). Associated with each surface are promi-
drainage network and alluvial fan. Location of longitudinal nent knickpoints along major rivers. Knickpoints
pro¬le shown as white curve. (b) Longitudinal pro¬le of main along the North Fork Kern River, for example
Hanaupah Canyon channel, together with best-¬t to Eq. (1.8).
(Figure 1.9b), occur at elevations of 1600--2100 m
and 2500--3300 m a.s.l. The stepped nature of the
Sierra Nevada topography is generally considered
to be the result of two pulses of Cenozoic and/or
bedrock late Cretaceous uplift (Clark et al., 2005; Pelletier,
channel 2007c). According to this model, two major knick-
t3 t2 t1 points were created during uplift, each initiating
a wave of incision that is still propagating head-
(b) ward towards the range crest.
Recent work has highlighted the importance
t1 t
2 t3 of abrasion in controlling bedrock channel evo-
lution. In the abrasion process it is sediment, not
water, that acts as the primary erosional agent.
In the stream-power model, the erosive power is
Fig 1.7 Schematic diagrams of the evolution of (a) bedrock
assumed to be a power function of drainage area.
and (b) alluvial channels through time, illustrating the
advective behavior of bedrock channels and the diffusive Although sediment ¬‚ux increases with drainage
behavior of alluvial channels. area, upstream relief also plays an important role
in controlling sediment ¬‚ux. As such, the stream-
power model does not adequately represent the
throughout the reach. If drainage area is uni- abrasion process. Sklar and Dietrich (2001, 2004)
form and n = 1 is assumed, the erosion rate developed a saltation-abrasion model to quan-
is proportional to the channel slope according tify this process of bedrock channel erosion. In-
to the stream-power model. Accordingly, steeper sights into their model can be gained by re-
portions of the bed will erode faster, result- placing drainage area with sediment ¬‚ux in the

118 W
119 W along-dip profile
(b) 1o
profile in (b)
0 50
x (km)
San Joaquin R.
profile in (b)
37 N

inset in (c)
Kings R.

Kaweah R.

along-strike profile
36 N
Kern R.

100 150
0 50
x (km)
3.0 Boreal
profile in (f) (km)
2.0 Chagoopa
Kern R.
x (km)
0 30
Chagoopa (f)
3.0 knickpoints

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