Hung Fei Lei
Department of Geography, King's College London, Strand, London WC2R 2LS, United Kingdom.
For different hillslope processes, the sediment transport equation can be simply expressed as:
C = K * xm * Sn
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
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:
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.
Keywords: characteristic form, domain, sediment transport equation, process constant, scale factor
Following Kirkby (1971; see also Carson and Kirkby, 1972: 107-109), the capacity of debris transport can be simply expressed as
For transport-limited hillslopes, the capacity of debris transport is approximate to the actual rate of debris transport (Qs).
Generally, (0, 1) is assigned to (m, n) for soil creep, and K is the coefficient of diffusion. For wash processes, (2.0~1.5, 2.0~1.25) is suggested for fine-grained surfaces without the vegetation, and (1.0, 2.0) is suggested for (m, n) to show the influence of low vegetation on morphology; K is often called the erodibility of the soil here. Rainsplash (particles creeping on the surface) working with wash at any point on a given hillslope profile is suggested as:
(see Carson and Kirkby, 1972: 225). Based on the continuity equation and a sediment transport relation, Martin and Church (1997) assigned coefficients for slow continuous mass movements and rapid episodic mass movements separately within one process model in three-dimensional form:
Following Kirkby (1971), if the volume of mineral soil produced from the weathering of unit volume of bedrock is equal to 1, then a reasonable approximation to the characteristic forms is suggested as:
Although some models, e.g. EQ.2 & 3 above, have been used to accommodate different process laws in one equation, EQ.4 alone can not reveal the spatial interactions between those identical processes. The interactions between different processes through space could not be expressed explicitly within the model of characteristic form, since the responses from different processes have been combined together and, then, the information of individual processes is at large.
The model above is called Simple Characteristic Form Model (SCFM) in this paper. A phase-space diagram for defining the domains for different processes through m and n is shown in Figure 1. Within this diagram, values of D are represented by contour lines. Processes are divided into two groups. Group one is creep (or diffusional processes) that can be divided furthermore into different creep processes by the power of gradient. Group two is wash that can be divided furthermore by the power of distance from divide.
SCFM is designed to represent the characteristic form separately for different hillslope processes. Based upon SCFM, Complex Characteristic Form Model (CCFM) that can represent two or more processes in maintaining one complex characteristic form is proposed. It combines the concept of characteristic form with the concept of domain. By using the concept of domain, it allows different processes to occupy identical loci within a given hillslope profile.
CCFM is not new, since Dalrymple et. al. (1968) has suggested a nine-unit landsurface model within one idea profile. Also Thornes (1990) has shown a model dealing with the competition among erosional and vegetational domains within a single hilllsope profile. However, CCFM has a simple mathematical form being capable of showing the characteristic forms maintained by the interaction of different geomorphic processes within a single profile. Although different processes work interweaving into each other at every point on a single hillslope (as being suggested by EQ.2 & 3), the domain where only one process is predominantly responsible for the local morphology does exist on hillslopes, and gives the place for a simple and approximate description.
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. Between each pair of attractors there are unstable situations (morphology) that hillslopes 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 be used to show under what conditions a given hillslope will approach or leave from an attractor (equilibrium).
In this paper, the basic characters will be explored firstly for the case of creep-wash domain. Some implications of CCFM will follow later for more complex erosional landscape. In the end, the structure of the sediment transport equation will be modified by scale factors surveyed upon profiles created by CCFM. Being different from lab tests and experiments in the field, CCFM offers a simple tool for decoding the complex hillslope environment by just surveying the morphology, although it is based upon what we have already know about the mechanisms and processes in the nature. It is suggested that the threshold slope angle, the long-term effect of different land covers, and the change of the environmental conditions through time and space in complex hillsope environments can be identified through this model.
SSFM is a representation of equilibrium family for the response of hillslope processes. If the function form of SSFM can be precisely modified further to allow different processes to play simultaneously at different loci within one hillslope profile, then it will be more useful in studying some hillslopes, where more than one kind of hillslope processes operate at different time and place. In first step here, only slow processes (soil creep and wash without gully) are considered for keeping the profile differentiable. Failures with threshold angles will be considered later.
Two parameters, A and B, are introduced. They are for defining the natural magnitude of creep-dominated and wash-dominated transport-limited hillslopes respectively. This means that creep/wash will exhaust the total given relief by the slope length A/B, if all the relief is controlled by creep/wash domain and the slope angle is still much lower than the threshold of failures at the bottom/top slope. In the field, three situations are common in blanketing out A/B. Firstly, far before A/B can be achieved, either the relief has been totally consumed or the threshold of failure is approached at the bottom/top slope. Secondly, other processes appear to occupy a part of the given hillslope, not allowing creep/wash complete its work alone. Finally, the assumption of transport-limited condition keeps no longer because of the deposition or the total loss of the soil mantle on the given hillslope.
For a hillslope with a creep domain on the top and a wash on the bottom, EQ.4 can be re-written as:
where A and B have been standardized by the length of a given hillslope into a and b. If a = 1 and b = 0, i.e. the given process (with the given D) occupies the whole hillslope, then the equation converges back to EQ.4. For a creep-wash complex hillslope, (b, D) = (0, DC) and a < 1 is set for the creep domain; (a, D) = (1, DW) and b>0 is set for the wash domain.
By setting a new reference point on x-y coordinate, it is easy to prove that parameter a and b are redundant when only one process operates on one hillslope profile. Both SCFM and CCFM are analytical and dimensionless models. If we change the scale, we will not change any character of these models. In numerical simulation, however, the parameters of the sediment transport equation, that is the base of this model, are functions of scale. There should be place for scale factors linking between the analytical and numerical approaches.
Parameter a and b are basically deductive concepts that exist only when more than one process law apply upon the same transport-limited hillslope. They are set up by the interaction of the given process law and its environmental conditions where the given hillslope has been evolving.
Based on the theoretical framework above, parameter D, a, and b should be determined for each domain on the given hillslope by the given process law and environmental conditions. The boundary, M, of the domain can be expressed as
If (D, b, a) = (2, 0, 0.42) for the creep domain upslope and (D, b, a) = (2/3, 0.2, 0) for the wash domain downslope, then M will be 0.3. For a continuous hillslope profile that is occupied by processes defined above, EQ.6 is deduced from the following condition: the coordination and the gradient at the boundary of two domains must be the same by approaching from wash domain and creep domain respectively. For any given time, M can be determined by EQ.6, a function of D, bW, and aC. In reality, M will shift by the interaction of processes and set values of aC and bW on the way of developing a hilllsope. Thus, we have:
Given the value g (=3 for the case above), parameters a and b change only with M (shown in Figure 2).
Parameter b will approach '0', when the wash domain is extended upward with gullying or instability and/or enlarged downward with accumulating debris on the base slope, meanwhile M will move upward the standardized profile. Both of the situations will reduce the energy available for sediment transport, and gradually slow down the shifting speed of b to '0'. Whatever direction they shift, the shifting speed of b will be slower than the shifting speed of M.
Parameter a will approach '1', when the creep domain is extended downward without pushing the slope angle over the threshold of instability. By this way, creep offers more and more regolith over the capacity of wash. The closer to '1', the more difficult for a to approach '1'; as the drifting force from the gradient for one unit hillslope length becomes less and less. This situation will make the creep domain be defeated easier by the wash domain, especially the wash with gully or channel heads. This condition (a => 1) may offer a chance of studying the sensitivity of landscape response.
Both wash and creep are regulated by the negative feedback. This complex characteristic form can be maintained only when the removal regime at the foot slope and the earth-climatic regime over the whole hillslope keeps in step with the equilibrium transport rate and also when there is no threshold of instability setting on the way of slope development.
If M=0.3 and the slope angle at position M keeps a threshold gradient (tanY=tan30), then, with the given hillslope length (L=100 metre), the total relief should be around 35 metre, calculated from:
This is illustrated in Figure 3. Based upon EQ.8, if the processes laws are given, a morphological and landslide survey on a series of hillslopes, upon given earth surface materials, will reveal tanY in situ. The implications of EQ.8 will be discussed later.
Parameter bW can be a measured variable only when a hillslope has been capped by strong surface materials that has allowed the top hillslope to avoid significant particulate or mass creep, if a series similar hillslopes is found. Parameter aC can be a measured variable only when a hillslope has been covered well by the soil mantle and vegetation which allows the whole hillslope to avoid wash. With EQ.7a and 7b, measured aC and bW can be used to check the accuracy of our process laws, under a view of long-term evolution. Finding a right place to measure a and b in the field is difficult, if not impossible. Thus, it is easier to treat a and b as linkages between M (a measurement of morphology) and D (a measurement of process).
Wihtin creep domain, the slope angle along the hillslope profile for creep domain can be written as:
where H/L is mean slope angle for the whole given hillslope. The slope angle within the creep domain is a decreasing function of M and L, i.e., if the given hillslope is shortened (L decreases) or by undergoing considerable wash from the base upward (M decreases), for a corresponding change of base level or climate, the slope angle within the creep domain will increase. Through this way, the creep domain offers more sediment moving downhill to prevent the extension of the wash domain. If the threshold angle for instability has been approached before the new equilibrium is achieved, then a new slope segment, with new threshold angle, will be created.
Under a given H/L, M can approach down the hillslope as long as the slope angle remains less than the threshold of instability for the soil mantle. Keeping tana under the threshold for instability, M increases with H/L and decreases when L increases. Slope angle at position M will move ahead when the mean slope angle increases. Incision at this position will cause more risk of triggering a series of shallow landslides through the shifting of main scarp.
CCFM also offers a chance to understand the situation of whole hillslope by just analyzing the local position M. Setting x/L = M, EQ.9 can be re-written as:
where D = tana/[DC*(H/L)]. The solutions of M in EQ.10 can be analyzed easily by setting a curve shown in Figure 4. Therefore, the solutions of EQ.10 will be the intersections of f(M) and a family of horizontal lines: y = (D-3/4). Since reasonable solution for M only exists within the interval of [0,1], (D-3/4) must satisfy the condition: 0.5 >= (D-3/4) >= 0. Thus, tana >= 0.79 * DC * (H/L).
Setting tana = G * (H/L), if DC=2, DW=2/3, and tana is assigned to be the gradient at M, then the relationship between M and G can be shown in Figure 5. G and S will be minimum when M = 0.5.
Whatever domain holds the superior position, tana will increase while M moves toward each end. It shows the trajectory of attractors on G-M phrase-space, and each attractor marks a steady state of the interaction between two domains. This location is a function of both the environment and the parameters of processes. The whole given hillslope will be conquered by creep, if 0.79*Dc*(H/L) >= tana and if tana is still far smaller than the threshold for developing failures or channel heads. By setting tana as the threshold tanY, a creep-domain hillslope will exist only when the mean slope angle is less than tanY/(0.79*DC). If Y is 15 degrees then the mean slope angle must be less than 9.5 degrees. For soils with strong cohesion Y can be as high as 30 degrees, then the mean slope angle must be less than 20 degrees. Under the proposed process laws, if we want to assess the long-term effects of creep or diffusive processes on hillsope morphology through surveying a catchment or even larger area, the condition of slope angle above should be first satisfied.
Increasing tana or L will decrease D-3/4 from 0.5 downward and assign M (the division of domains) two identical equilibrium positions. The choice of the system depends on the history and the interaction between these two domains. There exists a bifurcation for the evolution of the hillslope. If the capacity of wash increases relatively slowly under the given regimes during the change (which can be true for those gentle and well-vegetated hillslopes or for hillslopes well drained through soil matrix), then the creep domain and M will increase. When the mean slope angle is less than half of the threshold, creep will dominate the whole hillslope. Otherwise, M decreases and creep domain can occupy only the very upper part of the given hillslope, and will totally loose its territory if the slope angle is too high.
Moving toward wash domain, the gradient within this domain should, thus, be expressed as:
where DW is 2/3. Following the analysis strategy above, only two local conditions (x/L=1 and x/L=M) will be surveyed. When x/L=1, tana presents the gradient of the given hillslope on foot and is equal to (2/3)*(H/L)/[1-(2/3)*M], which is always greater than `0`. Since M keeps a value only within [0,1], tana must satisfy the condition below
That is a statement for defining the transport-limited wash domain on a hillslope. If M=1, which means creep must occupy the whole hillslope, then tana must be 2(H/L) and less than the threshold, tanR. Therefore, (H/L) will remain less than a half of tanR. This is the same conclusion as found above. If M=0, wash will dominate the whole hillslope, and tana is (2/3)*(H/L) at the foot slope. Thus, for the equilibrium hillslope that is totally dominated by wash domain long enough to achieve the characteristic form, transport-limited wash process stop at the place where tana = (2/3)*(H/L). This can be verified in the field.
Where to stop the profile measuring on the hillslope bottom can not be determined by the theory or profile form as has been suggested by Young (1972), but may be determined by the sediment analysis, or by process study. By practicing this method within the field with long stable environmental conditions, we are able to determine how well the surveyed S can be fitted to the estimated S from equation EQ.11. This may be used as another validation strategy for the wash process law.
Comparing with the studies in the lab, CCFM offers a chance to evaluate the effect of some parameters that only show their powers significantly at a certain spatial scale in the field. By the consistency character of CCFM through scale, the human-specific scale can be set up for any purpose.
In Figure 5, it shows that G is relatively sensitive while M shifts at low value domain. M is basically a function of a, b, and parameters of the process laws. If geological, climatic, or land use regimes change, then it should be reflected on the changing of M. For surveying the environmental change, those hillslopes with small M will be better choice. An environment of wash-dominated hillslopes with gullies should be this case and has been recognised by many researchers for long. Parameter G is actually a ratio between the local slope angle at M and the average angle for the whole hillslope. It can be surveyed easily through a hillslope profile or a catchment by DEM. By these results, the minimum G can be determined with the loci of M.
Two iterative approaches are proposed here to reveal the structure of macro landscape. Firstly, for a given uniform physical site, Gmin is simply a linear function of DC. If the coupling M (=MG) also keep constant, then DW and DC can be determined by equations as following:
The parameters of process laws can be determined simply through this survey, while a and b are fixed. Secondly, if the coupling M change through space, i.e., we do not have a single, precise G-M curve from the surveyed area, then parameter a and b mark their variation through space with the constant parameters of process laws. Surveying through space by these two method iteratively will reveal the structure of erosional landscape.
By using the strategy above, the long-term effect of different land covers upon erosional processes can be determined by showing the value variation of parameters in process laws, if a and b have been specified.
Turning toward a more complex model of hillslope characteristic form adding with the threshold concept, CCFM also offers a complementary quantitative method for finding tanY other than by the soil test in the lab, meanwhile different parameters for describing different geomorphic processes can be linked through the model. Surveying on a series of hillslopes with the relief changing gradually around the critical value, as mentioned before, we can not only measure tanY in situ, but also find out M value for the right tanY. By EQ.8, therefore, we will have a chance to check out from only macro morphology the parameters of process laws for slow hillslope processes.
When slope angle achieves or goes beyond the threshold angle that regolith can support, the segment will appear on hillslopes. Although the later might still maintain continuous transport-limited processes (creep & wash), failure will dominate the development of the segment in most cases. If slope angle increases because of the increasing intensity of processes, or the significant dropping of local thresholds, then the segment can appear at any locus on the hillslope. If the segment is originated from downcutting, undercutting, or dropping of base level at foot slope, then it will be initiated at the foot slope. If wash domain downcuts and couples with the removal at a particular speed, then it is possible to have a segment starting to develop at the boundary of creep and wash domains (Figure 6).
Suppose that the wash domain has approached upward to the threshold tanYw before it can keep balance with creep domain at M where the slope tangent is tana (>tanYw). Above the wash domain, a new segment will be created with a threshold slope angle tanYw. If the creep domain is far from its threshold tanYc, then the new segment will develop a threshold slope of tanYw first. Further upward development of wash will push the creep domain back to the hill top, increase the slope angle (as the argument shown before), and, later, a complex segment with another threshold slope of tanYc appears on the top. This segment will be even more complex if the erosion power within wash domain is still unsatisfied by the sediment supply and is going to wipe out the whole soil mantle. Thus, a coupling slope section of cliff-scree will appear in the middle of the complex segment.
The situation will be different if the creep domain approaches its threshold faster than wash domain (tanYw > tanYc). In this case, the new segment will develop a threshold slope of tanYc at the top and a maximum slope downslope next to it. For tanYw > tanYc is an unstable condition on the junction point and only a segment with strong cohesive material can stabilize this condition. It is similar to the case: putting dry sand on top of a steep slope of loess that always keeps a strong effective cohesion under the dry condition. More progressive wash process will go further to create a cliff. Down to this step, there will be no significant difference between this case and the case above.
For calculating the (capacity) rate of debris transport on the erosional landscape, one of the widely used empirical equations is expressed as EQ.1, but with x being replaced by q for short term study. By the parameters K, m, and n determined from measurements in the field or scale-down tests in the lab, this equation can be set up as a statement for a given specialized landscape process. It also has been used as an equation in the algorithm of discrete simulation by many researchers. However, the effect of chosen units for space and time is ignored in most studies.
There are two objectives of this section. Firstly, under the different conditions, investigate of how the gradient term behaves, if only the space unit for measuring the gradient changes from the whole hillslope size down to 1/128 of the hillslope. The value n of 4/3, 5/3, and 6/3 will be surveyed. This investigation is practised on the ideal hillslope profiles that have been produced by CCFM. The average gradient of the whole hillslope changes from 0.1 to 0.7. The hillslope form is allowed to change from the gentle convex-concave form to the deeper one-main-segment form, and later the form with one escarpment and one segment in the middle of the hillslope. Secondly, investigate of how the runoff term changes, if the time unit for measuring the hydrograph changes from 1/2 of the duration of a given event to 1/64 of that. The value m of 1.5, 2, 2.5, and 3 will be surveyed. The magnitude of the event is set from 22.5mm to 96 mm in total within 1200 seconds. The pattern of the hydrograph is allowed to change from the one with a climax in the beginning to that in the middle, and later in the end of the event.
Two strategies are common in studying sediment yield. The first strategy is that researchers use small-scale tests (in the lab or field) to outline process laws, and later apply them to the whole hillslope as a single unit in the algorithm using the same process constant. However, the effect of slope angle on the capacity of erosion will be underestimated by this approach. The second strategy is that researchers use large-scale measurements to figure out the parameters in process laws. Then, apply them into each cell of the simulating model using the same process constant. In this situation the size of the cell in modelling is always smaller than the area where the single-point measurement at the outlet is taken from. Therefore, the effect of slope angle on the capacity of erosion will be overestimated. These statements are implied in the expression written as:
In the first strategy above, we use the right side of EQ.14 to define n from measurements and later use it on the left side of EQ.14 in the algorithm in modelling with identical K value. The result of simulation must be smaller than the measured one. Keeping in bay this simple function form for describing landscape processes, if the result of modelling is expected to maintain the consistency with the ideal field measurements, then K in modelling must be bigger than K defined by measurements. In the second strategy above, an opposite argument can be set up.
As shown in Figure 7, the simulation error is stable for every case after scale factor N is greater than 10. That means the percentage of the overestimation or the underestimation will be stable if the size of whole hillslope is more than ten times of the size of one unit or cell. Simulation error can be approximated by
where ES is the simulation error in percentage form the gradient factor and S is the average gradient for the whole given hillslope.
Hillslope profile form has little influence and can be ignored, with the error less than 5%. If K, n, and m in process laws are determined from plots in the field or lab tests and will be applied to estimate the capacity of sediment yield of the whole hillslope as a unit in modelling, the first modification of the equation should be written as:
If the parameters in process laws are determined from the data at a given outlet of the whole hillslope and will be applied to simulate the process-response system through each segment or small hillslope section, then the modification term is (1/ES).
'Time' is an abstract concept and should owns no place in the operational world. It is process or event itself that gives a stage for the word 'time'. 'Time' always means one process is completed after another one has been run certain times. Time, basically, is just the representation of the relationship among different events in the operational world. In calculating sediment yield by the sediment transport equation, the statements above can be simply summarised as:
i.e., after the running water has applied its power many time units (the right side of EQ.17), an event is set up (at the left side of EQ.17; that indeed the symbol of '=' means). Given EQ = k1/k2, we can have:
where a and b are parameters defined by the hydrograph pattern. a is almost a constant (=0.91) and has a variation level less than 1%. b is 0.57 when the peak runoff takes place in the middle period of a given event, and 0.68 for the climax happening at the beginning or the end of the event. For convenience, if 0.6 is assigned to b, then the error will be still about 17% for the worst situation. The magnitude of the event is not relevant at all.
If k is determined by the short-time-scale test, for example a simulation of rainfall with uniform intensity on a plot, and is applied to a long-term simulation with the data event by event. The modified equation should be, then, written as:
The modification term will be 1/EQ, if the approach is scale-down one. Since the term of slope angle and that of runoff are totally independent to each other, EQ.16 and 19 can be combined easily, expressed as:
while k is determined form the measurements on small plot by the short time unit. Move two modified terms ahead, it shows clearly that K parameter in EQ.1 actually holds on a structure of k*EQ*ES. The significant variation for parameter K from different studies partly comes from the framework of time and space chosen by the researchers. This part should be separated from the effect of heterogeneity and variability of the environment.
In this survey, only hillslopes and small catchments are considered. They are represented by two-dimension profiles based upon CCFM. A further survey should be done on 3D space, e.g. DEM. This survey also leaves the pattern of a series event and the scale effect of vegetation unchecked, although the last factor is important in semiarid and agricultural areas and the former one is important for the long-term predictions.
In the place where complex processes will often be the case, one question should be answered: where and under what conditions can the study of long-term effects of a given process be undertaken safely? CCFM is, therefore, presented as a theoretical base that can be used to answer this question. It can be applied to those hillslopes within which two or more process domains can be separated clearly.
This model also offers a chance to identify the threshold angle, the long-term effect of different land covers, and the change of the environmental conditions through time and space within the complex hilly landscape.
By surveying hillslope profiles that created by CCFM, this paper also identifies the scale factors, EQ and ES, in the sediment transport equation for calculating the capacity of debris transport.
Scale-down and scale-up approaches always cause serious problem in comparison among studies with different chosen scales. It is suggested that the process laws should keep scale factors for space and time in modelling. The simulation results from different scale will, thus, have a chance for comparison. That is the way to keep the simple process laws stable in landscaping geomorphology and to keep the uncertainty in charged.
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