Environmental Engineering Reference
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remains incompletely understood but has been incorpo-
rated into recent formulations for river incision, either
by adjusting the empirical scaling laws (Finnegan
et al
.,
2005) or by explicitly modelling both vertical and lat-
eral erosion of channels (Stark, 2006; Wobus
et al
., 2006;
Lague, 2010).
Sediment supply and transport by the river can influ-
ence bedrock incision in two different ways: at low
sediment fluxes, sediments impact the bedrock, provid-
ing efficient 'tools' for erosion and increasing the incision
capacity of the stream; large amounts of sediments, in
contrast, partially cover and protect the bed from erosion
(Sklar and Dietrich, 1998, 2001). Beaumont
et al
. (1992)
and Kooi and Beaumont (1994) proposed an incision
algorithm that conceptualized the sediment-cover effect:
∂
where, again,
K
s
is a dimensional constant [L
(3
−
2
m
s
)
T
−
1
],
and
n
s
and
m
s
are dimensionless exponents. This formu-
lation assumes that the river is always at carrying capacity;
incision or deposition is controlled by spatial variations
in the carrying capacity and calculated by combining
Equation 19.8 with the continuity equation (19.1):
∂
h
1
W
∇
t
=−
Q
eq
(19.9)
∂
where
W
is the channel width [L]. Many landscape-
evolution models combine detachment-limited and
transport-limited behaviour to predict fluvial incision,
with
one
or
the
other
constituting
the
rate-limiting
process (Table 19.1).
The above models ignore several potentially important
controls on fluvial incision, including incision thresholds
and stochastic distributions of discharge (Tucker, 2004;
Lague
et al
., 2005), dynamic adaptation of channel geom-
etry (Finnegan
et al
., 2005; Turowski
et al
., 2006; Wobus
et al
., 2006; Attal
et al
., 2008) and the interaction between
sediment and bedrock (Sklar and Dietrich, 1998, 2001).
Inclusion of an incision threshold is based on the well
established observation that a minimum shear stress is
required for incipient motion of sediment on the bed.
It is implemented by modifying the detachment-limited
stream-power equation (19.7):
∂
h
1
WL
f
t
=
(
Q
eq
−
Q
s
)
(19.11)
∂
where
Q
eq
is the carrying capacity as defined in
Equation 19.8,
Q
s
is the actual sediment flux and
L
f
is
a characteristic length for incision. Although this under-
capacity model is conceptual rather than physics based,
it has become very popular and has been implemented
in different landscape-evolution models, including the
widely used model
Cascade
(Braun and Sambridge,
1997; Garcia-Castellanos, 2002; Petit
et al
., 2009; cf.
Table 19.1), because it allows both transport-limited-like
and detachment-limited-like behaviour to be captured
in a single algorithm (Figure 19.3). A similar expression,
in which the characteristic length is applied to deposition
rather than incision and is interpreted as a characteristic
transport length, was derived by Crave and Davy (2001)
and Davy and Lague (2009), and implemented in the
¤
=
k
e
k
t
Q
W
m
t
−
τ
c
a
(19.10)
h
t
=
k
e
(
τ
−
τ
c
)
a
S
n
t
∂
where
τ
c
is critical shear stress,
k
e
and
k
t
are dimen-
sional constants and
m
t
,
n
t
and
a
are nondimensional
exponents. Inclusion of such a threshold has been shown
to change incision dynamics significantly if a stochastic
distribution of discharge is taken into account (Baldwin
et al
., 2003; Snyder
et al
., 2003a), even though discussion
continues concerning the relative size and importance
of the critical shear stress (Molnar, 2001; Tucker, 2004;
Lague
et al
., 2005).
Most of the above formulations implicitly assume
hydraulic scaling laws that have been well established
for alluvial rivers (Hack, 1957; Leopold
et al
., 1964), in
particular scaling of channel width with the square-root
of discharge. However, numerous studies have shown
that this scaling breaks down for rapidly incising bedrock
rivers (e.g., Harbor, 1998; Lave and Avouac, 2001; Duvall
et al
., 2004; Turowski
et al
., 2006; Amos and Burbank,
2007; Whittaker
et al
., 2007). In these cases, rivers appear
to adjust their widths dynamically in order to maximize
shear stress on the bed. Such dynamic width adjustment
ros landscape-evolution model.
Using a different approach, Sklar and Dietrich (1998,
2004) derived, from physical principles, an incision model
that takes both the tools and cover effects into account.
In their model, incision rate is controlled by abrasion
of bedrock by bedload and depends on (1) the rate of
particle impacts per unit time and area, (2) the average
volume of rock detached per particle impact, and (3) the
fraction of river bed made up of exposed bedrock. In its
simplest form, assuming both (1) and (2) constant, the
model can be written as:
1
∂
h
Q
s
Q
eq
=
kQ
s
−
.
(19.12)
∂
t
A more general formulation, allowing for impact rate
and volume that varies with flow velocity and sediment
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