The Complex Characteristic Form Model and its Applications

For different hillslope processes, the sediment transport equation can be simply expressed as:

C = K * xm * Sn


C is the (capacity) rate of debris transport,
K is process constant,
x is the distance from the divide,
S is the gradient,
m and n are the power of x and S respectively.

Based upon the characteristic-form slope profiles proposed by Kirkby (1971), the complex characteristic form model (CCFM) is presented here. It is not new, since Dalrymple et. al. (1968) has suggested a nine-unit land surface model for the idea hillslope profile. Also Thornes (1990) has shown a model dealing with the competition among erosional and vegetational domains within a single hilllsope profile. By combining the concept of characteristic form with the concept of domain, however, CCFM allows different processes occupying identical loci within any given hillslope to maintain the relatively stable hillslope form and still keeps a simple mathematical form. Comparing with the models proposed by Carson and Kirkby (1972: 225) and Martin and Church (1997) among many others, CCFM can express explicitly the spatial interaction between domains.

Being independent from the parameters of process laws, two deductive parameters, a and b, are introduced, and set the dimensionless characteristic-form as

y/H = 1 - [ (1/a) * (x/L-b)/(1-b) ]D


D = (1-m+n)/n,
y is the elevation,
H is the total height of the given hillslope,
x is the horizontal distance from the divide,
L is the total horizontal length of the given hillslope.

Both a and b are set by the spatial interaction among different processes on the way of hillslope development.

CCFM can be applied to the hillslopes upon that two or more separated process domains can be identified. This model offers a tool to reveal the threshold angle, the long-term effect of different land covers, and the change of the environment through time and space within complex hilly landscapes. By CCFM, each characteristic form can be treated as an attractor within a stream of attractors defined by environmental parameters and parameters of the process laws. For the creep-wash complex hillslope, the stream of attractors can simply expressed in the figure below, where M is the standardized position (from the divide) of the boundary between creep and wash domains on the given hillslope and G is the ratio between the local gradient at M and the average gradient of the given hillslope. Out of the stream of attractors there are unstable situations (morphology) that hillslopes form have never stayed for long.

Hillslopes will always approach one of these attractors under the constant environmental conditions for one or more working processes. CCFM can show under what conditions a given hillslope will approach or leave from an attractor.
By surveying hillslope profiles that created by CCFM, this paper also identifies the scale factors, EQ and ES, in the sediment transport equation that has been used to calculate the capacity of debris transport for hillsloeps, written as:

C = K * EQ * ES * qm * Sn.

Since K has a structure of k * EQ * ES, it is suggested that the significant variation for the process constant from different studies partly comes from the framework of time and space chosen by the researchers. The effect of time-space framework should be separated from the effect of heterogeneity and variability of the environment.