Non-Linear Dynamics, Self-Organization and Cellular Automata Models - Environmental Modelling: Finding Simplicity in Complexity - page 55

This pattern rarely occurs in reality: each rill has its
own unique characteristics, although it probably shares
a 'family resemblance' with others on the same hillslope.
This assumed uniformity means that the 'patchiness' of
hillslope erosion is not captured very well in models.
Spatial variability is underestimated; in a classic study
using multiply replicated plots, Wendt et al . (1986) found
a variability of both hydrology and erosion that was far
greater that would be estimated by running a conventional
erosion model with data for each plot. The second impor-
tant deficiency takes the form of a logical inconsistency:
on an uneroded surface (such as a freshly tilled field), rills
do not yet exist. Since there are no rills, there are no chan-
nels, and the laws of hydraulic flow in channels cannot
be applied. Thus it is necessary to assume that, in some
sense, rills 'pre-exist' when conventional erosion models
are used. WEPP (Nearing et al , 1989) assumes a 1-m rill
spacing, whereas the user needs to specify initial rill spac-
ing and dimensions for EUROSEM (Morgan et al ., 1998).
Thus we have a 'what comes first: the chicken or the egg?'
problem. To tackle these and other problems, we need
to go beyond an engineering-type approach and focus on
the initiation and temporal development of rill erosion.
Favis-Mortlock (1996, 1998b) constructed the Rill-
Grow 1 model in order to test the hypothesis that the
initiation and development of hillslope rills may be mod-
elled using a self-organizing systems approach, i.e. driven
by simple rules governing systemic interactions on a much
smaller scale than that of the rills. The central idea was
that some rills are more 'successful' than others, since they
preferentially grow in size and sustain flow throughout
a rainfall event. Thus they compete for runoff. From an
initial population of many microrills, only a subset sub-
sequently develops into larger rills as part of a connected
network (Table 4.1).
Erosive modification of the soil's microtopography
produces a positive feedback loop, with the most
'successful' rills (i.e. those conveying the most runoff)

modifying the local microtopography to the greatest
extent, and so most effectively amplifying their chances
of capturing and conveying future runoff. There is a
limit to this growth, however (i.e. an associated negative
feedback); each 'successful' rill's catchment cannot grow
forever, because eventually the whole surface of the
hillslope will be partitioned between the catchments of
the 'successful' rills. The dynamics of this competitive
process give rise to connected rill networks. Thus the
hillslope erosional system is a dissipative system, with
rainfall providing the essential input of matter and energy
to the system, and runoff and sediment being the outputs.
The very simple RillGrow 1 CA model produced appar-
ently realistic results (see, for example, Favis-Mortlock
et al ., 1998) in terms of reproducing the observed char-
acteristics of rill networks on plot-sized areas. However,
it possessed some serious conceptual limitations which
prevented the approach from being more rigorously val-
idated. First, the algorithm used (broadly similar to the
precipiton approach of Chase, 1992, described above)
meant that the model did not operate within a true time
domain, and so validation of relationships with a temporal
aspect (e.g. the effects of rainfall intensity, or time-varying
discharge) was impossible. Secondly, the model assumed
an infinite transport capacity, with erosion being entirely
detachment-limited. Thus the model could only hope to
correctly reproduce situations where deposition is mini-
mal. Finally, the model possesses a rather weak physical
basis.
In many respects, RillGrow 1 was a typical 'first-
generation' geomorphological CA model. In order to
move beyond these limitations, RillGrow 2 was devel-
oped, the aim being to improve the process descriptions
of the first version of the model, while (as far as possible)
still retaining its simplicity. An early version of RillGrow
2 was described in Favis-Mortlock et al . (2000); a more
recent version is summarized here.

Table 4.1 'Successful' and 'unsuccessful' rills.

Category of rill Rate of growth Effectiveness during rainfall event

Successful Higher Becomes major carrier for runoff and eroded soil for part of

hillslope; may 'capture' weaker rills

Unsuccessful Lower Becomes less and less important as a carrier for runoff and

sediment; may eventually be 'captured' or become completely
inactive

Source: From Favis-Mortlock (1996)