Environmental Engineering Reference
In-Depth Information
on sediment along the drainage network (Figure 19.1).
Thus, the topology of the drainage network must be
established in order to define flow routing.
Two approaches have been developed to track precip-
itation and flow routing in landscape-evolution models:
most models use temporally constant precipitation input,
implicitly supposed to represent the geomorphically effec-
tive events. Alternatively, individual precipitation events
or 'precipitons' can be modelled and tracked down the
grid using random-walker type models (Chase, 1992;
Crave and Davy, 2001), which have the advantage that
precipitation events of different magnitude can be easily
included and that the topology of the drainage network
does not need to be kept in memory. However, trans-
lating 'model' time to 'real' time becomes a nontrivial
issue in these models. Tucker and Bras (2000) and Tucker
et al
. (2001) have proposed an intermediate solution in
which stochastic precipitation events are integrated over
the model time steps.
In most landscape-evolution models, both short- and
long-range processes operate on each grid cell. The ratio-
nale for this approach is that the resolution of the model
grid is usually of the same order of, or coarser than,
the threshold area for channel initiation of 10
4
-10
5
m
2
(Montgomery and Dietrich, 1992) so that all grid cells
contain both hillslopes and channels. However, it has
been argued that for drainage areas
constant' with dimension [L
−
1
]. Note that, although
Equation 19.2 predicts weathering rates to be highest
when bare bedrock is at the surface (
H
s
=
0), there is
evidence that soil production rates in fact peak below a
thin (a few tens of cm) layer of soil or regolith (Ahnert,
1976; Heimsath
et al
., 1997; Small
et al
., 1999). In the
model of Densmore
et al
. (1998), this effect is taken into
account by applying Equation 19.2 below a surficial layer
of linearly increasing soil production rates.
Recent work (Hales and Roering, 2007, 2009) has out-
lined the importance of periglacial regolith production by
frost cracking in high mountain environments. This mode
of production of mobile material is not well described
by Equation 19.2 but has not yet been incorporated in
numerical landscape-evolution models. Most models, in
any case, do not include an explicit regolith production
function at all and thus implicitly suppose that hillslopes
are transport-limited (Carson and Kirkby, 1972).
19.3.2 Short-rangehillslope transport
Models invariably contain a short-range hillslope trans-
port term, represented by a diffusion equation. This
equation is obtained by combining a transport law in
which sediment flux is linearly dependent on slope gradi-
ent with the continuity Equation 19.1:
∂
10
6
m
2
, chan-
nelized sediment transport is controlled by debris flows
rather than purely fluvial processes (Lague and Davy,
2003; Stock and Dietrich, 2003), which should thus be
(but are not generally) explicitly included in the models.
<
∼
h
2
h
t
=−
κ
∇
(19.3)
∂
in which
κ
is the landscape diffusivity [L
2
T
−
1
]. Hillslope
diffusion has been used for several decades to describe
slow, continuous slope-transport processes such as creep
(Culling, 1960; Carson and Kirkby, 1972). The linear
relation between sediment flux and slope gradient has
been verified by studies that used cosmogenic nuclides
to constrain transport rates (McKean
et al
., 1993; Small
et al
., 1999).
In steady state (i.e., if
19.3 Geomorphic processes and model
algorithms
19.3.1 Regolith/soil production
Given the fact that mobile material (soil or regolith)
needs to be produced before it can be transported, it is
perhaps surprising that few landscape-evolution models
explicitly include a soil-production term, exceptions
being the models developed by Tucker and Slingerland
(1994) and Densmore
et al
. (1998; see Table 19.1), and
the earlier work of Ahnert (1964, 1976). In these models,
soil production is modelled as exponentially decreasing
with soil depth
H
s
:
t
is spatially and temporally
constant), the linear diffusion Equation 19.3 predicts
parabolic hillslopes for which both relief and maximum
slope gradient scale linearly with erosion rate, providing
strong potential tests of its applicability (Anderson,
1994). Such tests have shown that the equation is strictly
only suitable to describe the evolution of low-gradient,
soil-mantled hillslopes; at higher slope angles (
>
∼
∂
h
/∂
20
◦
)
hillslopes become linear rather than parabolic and the
linear relationship between erosion rate and hillslope
relief or gradient breaks down (Anderson, 1994; Burbank
et al
.,
∂
H
s
∂
=
ε
0
e
−
α
H
s
(19.2)
t
where
ε
0
is the rate of soil production (or weathering)
at the surface [L T
−
1
]and
1996;
Roering
et al
.,
1999;
Montgomery
and
α
is a 'weathering decay
Brandon,
2002).
Roering
et al
.
(1999)
hypothesized
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