Geology Reference
In-Depth Information
substitution allows the reconstruction of long-term
changes in hillslopes under special circumstances (p. 25).
Mathematical models offer another means of probing
long-term changes in hillslope form.
Michael J. Kirkby is a leading figure in the field of
hillslope modelling. He used the
continuity equation
of debris moving on hillslopes and in rivers as a basis for
hillslope models (Kirkby 1971). In one dimension, the
equation of debris on a hillside is:
from the watershed (roughly the distance of overland
flow) and (
dh
/
dx
)
n
represents processes in which sedi-
ment transport is proportional to slope gradient. Empir-
ical work suggests that
f
(
x
)
m
x
m
, where
m
varies
according to the sediment-moving processes in opera-
tion, representative values being 0 for soil creep and
rainsplash and 1.3-1.7 for soil wash. The exponent
n
is typically 1.0 for soil creep, 1.0-2.0 for rainsplash,
and 1.3-2.0 for soil wash (Kirkby 1971). For a hill-
slope catena, the solution of the equation takes the
general form:
=
δ
h
dS
dx
=−
δ
t
=
h
f
(
x
,
t
)
where
h
is the height of the land surface and
S
is the
sediment transport rate, which needs defining by a trans-
port (process) equation for the process or processes being
modelled. A general
sediment transport equation
is:
This equation describes the development of a hillslope
profile for specified slope processes, an assumed initial
state (the original hillslope profile), and boundary con-
ditions (what happens to material at the slope base, for
example). Some of Kirkby's later models demonstrate
the process, and some of the drawbacks, of long-term
hillslope modelling (Box 7.1).
Hillslope models have become highly sophisti-
cated. They still use the continuity equation for mass
f
(
x
)
m
dh
dx
n
=
S
where
f
(
x
)
m
is a function representing hillslope processes
in which sediment transport is proportional to distance
Box 7.1
HILLSLOPE MODELS
Michael J. Kirkby's (1985) attempts to model the effect
of rock type on hillslope development, with rock type
acting through the regolith and soil, nicely demon-
strates the process of hillslope modelling. Figure 7.6
shows the components and linkages in the model,
which are more precisely defined than in traditional
models of hillslope development. Rock type influences
rates of denudation by solution, the geotechnical prop-
erties of soil, and the rates of percolation through the
rock mass and its network of voids to groundwater.
Climate acts through its control of slope hydrology,
which in turn determines the partitioning of overland
and subsurface flow. With suitable process equations
fitted, the model simulates the development of hill-
slopes and soils for a fixed base level. Figure 7.7 is the
outcome of a simulation that started with a gently slop-
ing plateau ending in a steep bluff and a band of hard
rock dipping at 10
◦
into the slope. The hard rock is
less soluble, and has a lower rate of landslide retreat,
than the soft band but has the same threshold gra-
dient for landsliding. Threshold gradients, or angles
close to them, develop rapidly on the soft strata. The
hard rock is undercut, forming a free face within a few
hundred years. After some 20,000 years, a summit con-
vexity begins to replace the threshold slope above the
hard band, the process of replacement being complete
by 200,000 years when the hard band has little or no
topographic expression. The lower slope after 200,000
years stands at an almost constant gradient of 12.4
◦
,
just below the landslide threshold. Soil development
