Modelling Landscape Evolution - Environmental Modelling: Finding Simplicity in Complexity - page 302

that this breakdown was due to the onset of shallow
landsliding (i.e. affecting only the regolith layer); Roering
et al . (2001) supported this hypothesis in an experimental
study. These authors suggested an empirical nonlinear
diffusion law that better fits the observed variability in
hillslope morphology:

partly based on the physics of the process, it does predict
realistic magnitude-frequency distributions of landslides,
as well as the expected behaviour of steep hill slopes in
rapidly uplifting terrain (Densmore et al ., 1998; Champel
et al ., 2002).

19.3.4 Fluvial incisionandtransport

2 h

∂ h
∂ t =−

κ ∇

(19.4)

∇ h S c 2

Fluvial incision and transport are the most important
relief-shaping processes in nonglaciated drainage basins,
as rivers both set the lower boundary conditions for
hillslope processes and export material from the system
(e.g. Whipple and Tucker, 1999; Whipple, 2004). The
most generally used fluvial-incision models are based on
the hypothesis that incision rate should be proportional
to either total stream power, unit stream power, or basal
shear stress (Howard et al ., 1994; Whipple and Tucker,
1999). Using drainage area as a proxy for discharge
and empirical relationships between discharge, hydraulic
geometry and flow velocity (Hack, 1957; Talling and
Sowter, 1998; Whipple, 2004), these three formulations
can all be written as a power-law function of drainage area
( A ) and stream gradient ( S ), leading to the well known
'stream-power law' for fluvial incision:
∂

1 −

where S c is a threshold hillslope gradient. It is easy
to see from Equation 19.4 that, if the slope gradient
∇ h S c , the equation simplifies to the linear diffusion
Equation 19.3 whereas, if

S c , erosion rates become
infinite and independent of slope gradient.

∇ h →

19.3.3 Bedrock landsliding

Bedrock landsliding appears as the main process affecting
hillslopes in rapidly eroding, tectonically active mountain
belts (Schmidt and Montgomery, 1995; Burbank et al .,
1996; Hovius et al ., 1997). Although the above empirical
model, or the application of a simple threshold hillslope
gradient above which all material is removed (e.g., van
der Beek and Braun, 1999), may satisfactorily describe
slope transport by landsliding on large spatial and tem-
poral scales, smaller scale studies require capturing the
stochastic nature of landsliding and the temporal persis-
tence of super-critical slopes (e.g. Densmore and Hovius,
2000). Densmore et al . (1998) developed an algorithm to
include bedrock landsliding in landscape-evolution mod-
els, based on the classical Cullman theory of slope stability
(cf. Schmidt and Montgomery, 1995):

h

KA m S n

t =

(19.7)

∂

where K is a dimensional constant reflecting the resis-
tance of the substrate to incision [L (1 − 2 m ) T − 1 ]and n
and m are dimensionless exponents supposedly reflecting
the physics behind the models (Whipple et al ., 2000),
but possibly dependent on other factors such as dis-
charge variability (Lague et al ., 2005). In this algorithm,
bedrock-incision rates are directly coupled to stream
power; hence, it is referred to as a detachment-limited
model (see Tucker and Whipple, 2002 for discussion).
Although such detachment-limited, stream-power mod-
els have been widely used to infer erosion and uplift rates
from landscape form (cf. Wobus et al ., 2006 for a review),
their utility in landscape-evolution modelling on large
spatial scales is limited, as it is obvious from Equation 19.7
that they will predict river channels to incise everywhere,
except in the trivial cases where A

sin β cos φ
1 − cos( β − φ )

4 c
ρ g

h c = (19.5)

where h c is stable slope height [L], C is effective cohesion
[M L − 1 T − 2 ],

is density [M L − 3 ], g is gravitational accel- ρ

eration [L T − 2 ], is

the critical friction angle. In order to model the stochas-
tic nature of the process, a landsliding probability p l is
defined that depends both on the mechanical stability and
the time since the last landsliding event

β is the topographic slope angle and φ

0.
An alternative formulation for fluvial incision argues
that bedrock incision rates are limited by the capacity
of a river to transport eroded materials (Willgoose et al .,
1991; Tucker and Whipple, 2002). Such transport-limited
models are based on the transport capacity Q eq of the river,
which is taken as a function of stream power:

= 0or S =

t slide (crudely
modelling static fatigue and weathering effects):

h
h c +

k 0 t slide
dt char

p l = (19.6)

in which k 0 and dt char are constants. The actual size of the
landslide and the site of deposition are determined by an
empirical rule set. Although the above model is thus only

K s A m s S n s Q eq = (19.8)