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

2 INTRODUCTION

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

California.

(a)

Fig 1.3 Oblique perspective image of the high peaks of

Hanaupah Canyon. Reproduced with permission of

DigitalGlobe.

crust

mantle

‚h = 0

‚x

(b)

divide h

crust

mantle

b

Fig 1.2 Schematic diagram of a simple model for Basin and

h regolith

Range extension and isostatic adjustment. In (a), horizontal

bedrock

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).

1.1 A TOUR OF THE FLUVIAL SYSTEM 3

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

4 INTRODUCTION

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-

vegetation-

ing by creep and other hillslope processes. Empir-

limited?

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

1.1 A TOUR OF THE FLUVIAL SYSTEM 5

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

Kw

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

n

‚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

x

rates at a site in Perth Amboy, New Jersey, mea- h ’ h 0 = ’C ln (1.9)

L

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

6 INTRODUCTION

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

(b)

have long been recognized in the landscape of

the southern Sierra Nevada (Webb, 1946) (Figure

2

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

(a)

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.

alluvial

Recent work has highlighted the importance

t1 t

channel

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

1.1 A TOUR OF THE FLUVIAL SYSTEM 7

118 W

119 W along-dip profile

3.5

(b) 1o

(a)

h

(km)

along-strike

2.5

profile in (b)

0 50

x (km)

San Joaquin R.

along-dip

profile in (b)

37 N

inset in (c)

Kings R.

Kaweah R.

Chagoopa

Boreal

along-strike profile

36 N

Boreal

3.0

Kern R.

h

(km)

Chagoopa

1.0

100 150

0 50

x (km)

3.0 Boreal

(e)

(d)

(c)

longitudinal

h2.5

profile in (f) (km)

2.0 Chagoopa

Boreal

Kern R.

1.5

x (km)

15

0 30

Chagoopa (f)

3.0 knickpoints

h

(km)

2.0