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Lab. for Experimental Geomorphology, Catholic Univ. of Leuven, Redingenstraat 16, 3000. Leuven, Belgium
* Corresponding author (rafael.gimenez{at}geo.kuleuven.ac.be)
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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Soil detachment has mainly been studied using relatively small soil samples with fixed geometry and a smooth surface (e.g., Poesen et al., 1999; Nearing et al., 1991, 1997; Shainberg et al., 1994; Glass and Smerdon, 1967; Ghebreiyessus et al., 1994; Van Klaveren and McCool, 1998; and Ciampalini and Torri, 1998). Under such circumstances, flow detachment rates can be successfully related to hydraulic parameters such as shear stress (e.g., Ghebreiyessus et al., 1994). However, real rills have a rough irregular bed with headcuts and knickpoints. Whereas on smooth beds, the total shear stress is equal to the grain shear stress. A considerable amount of form shear stress is generated on such irregular rill beds, thereby reducing the flow velocity and increasing flow depth.
It may therefore be questioned whether information on soil detachment derived from flat bed experiments can directly be applied to actively eroding rills with a rough, irregular bed geometry. There are only a few studies on rill detachment under realistic conditions. Brown and Norton (1994) carried out studies in field plots to evaluate the effect of crop residues on interrill and rill (ridge) erosion. Similar studies were done in the field by Van Liew and Saxton (1983) to assess both the slope steepness and incorporated residue effects on rill erosion rates. Nearing et al. (1999) evaluated soil detachment rate as a function of shear stress, stream power, and hydraulic friction in artificial rills created on the field in a stony soil. Field experiments were conducted on a loam and a silt loam soil by Franti et al. (1999) to determine the effect of tillage on soil detachment in concentrated flow channels and to define a shear stress-based detachment model including a soil strength index.
In these papers no attempt was made to relate the results for natural rills with flow detachment information obtained on small soil samples with a smooth surface. Nearing et al. (1997) attempted to combine data obtained from natural rills with data obtained from small soil samples with a smooth surface. However, the data obtained were not directly comparable as the soil types used and the slopes and discharges applied in both types of experiments were different. Lei et al. (1998) took a different approach and developed a dynamic finite-element model for rill development model based on the relationships of Nearing et al. (1997) for sediment transport and detachment derived from smooth bed experiments. Experiments showed that the model was capable of simulating observed patterns of rill erosion. However, the application of such a model on a routine basis is not possible because of the required input data and computer processing power.
A comparison of flow detachment data obtained from smooth bed experiments with those obtained from natural rills is useful for several purposes. First, it is worthwhile to investigate if a flow detachment parameter exists that can be used to predict detachment on surfaces with varying bed geometry. On rough surfaces a large amount of the flow's energy is dissipated on form roughness, so that form shear stress is often more important than grain shear stress. Govers and Rauws (1986) and Govers (1992a) showed that the sediment transporting capacity of overland flow on irregular beds is not determined by the total shear stress, as an important part of the shear stress is dissipated on macroroughness elements and therefore not effective for sediment transport. For fine sediments, transporting capacity on irregular beds was therefore much better related to grain shear stress and unit stream power. Whether this also holds for sediment detachment remains unclear. Second, if a relationship between flow detachment and flow hydraulics can be established that is valid both on smooth and irregular beds, it becomes much easier to transfer the results from experiments with small soil samples to real rills. This is important, as experiments with small soil samples are much easier to carry out and to replicate than simulations of natural rills.
Govers (1992b), Takken et al. (1998), and Nearing et al. (1997) showed that flow velocities in rills eroding loose materials are well related to the total rill discharge. The power relationships they proposed are similar to the ones used in studies of river geometry (e.g., Langbein, 1964) and recognize that flow velocity is determined by the geometry of the whole cross-section. The finding that flow velocity in rills can be well predicted from total discharge may also have some implications for the prediction of flow detachment in rills. Usually, sediment detachment is related to parameters that are calculated on a unit width or unit surface basis (e.g., flow shear stress, stream power). This requires that rill width and, in some cases, also average flow velocity have to be predicted first, before rill detachment can be calculated. This may result in a loss of accuracy. We may therefore be more successful in predicting rill detachment using hydraulic parameters based on total discharge or on a unit length basis. An example of a parameter based on total discharge is the total stream power (
T) that can be defined as:
 | [1] |
Where, 
equals density of the fluid, g equals gravitational acceleration, Q equals total discharge, and S equals slope.
An example of a hydraulic parameter calculated on a unit length basis is the unit length shear force
 | [2] |
Where, Wp represents wetted perimeter, R equals hydraulic radius, and A equals wetted cross section.
A series of laboratory experiments was set up to address these issues. Our objectives are therefore (i) to evaluate to what extent the relationship between flow detachment and flow hydraulics is affected by bed geometry and to investigate whether there exists a hydraulic parameter that may be used to predict flow detachment independent of bed geometry, and (ii) to evaluate the potential of flow hydraulic parameters calculated on a total discharge or unit length basis to predict sediment detachment in rills.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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Natural Rill Experiments
The natural rill experiments were carried out in a 4.30-m long, 0.4-m wide, and 0.45-m deep flume using a set-up similar to the one described by Giménez and Govers (2001). The upstream part of the flume was filled with soil and then covered with a 1.5-m long plastic sheet over which the water was led to the entrance of the 2.80-m long test section without causing any erosion. The bottom 0.2 m of the test section was filled with a silt loam soil which was manually compacted to simulate a subsoil. This soil was left in place for all experiments. Before each experiment, the top 0.25 m of the test section was filled with soil that was air-dried and sieved at 0.02 m to simulate fine seedbed conditions. The surface was smoothed with a rake, creating a flat-bottomed longitudinal depression of around 0.15 m wide and around 0.05 m deep to avoid water flowing down along the flume wall.
Before each experiment, the soil's topography was determined with a 1-mm resolution in the transverse direction and a 2-mm resolution in the longitudinal direction using a laser microreliefmeter driven by computer controlled stepping motors, similar to the system described by Huang et al. (1988). The accuracy of the height measurements is around 0.1 mm. A detailed topographic map with a horizontal resolution of 1 mm was constructed from the raw data by kriging using the Surfer software (Golden Software, 1999). Soil moisture, bulk density, and vane shear strength were also measured (Table 1).
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Experiments were continued as long as necessary to reach equilibrium flow conditions, i.e., flow velocities no longer changed significantly over time. Such conditions were achieved much quicker when the flow was more erosive. Experiments therefore lasted anywhere between 5 and 60 min.
After the experiments, soil moisture, bulk density, and vane shear strength were again determined (Table 1). The flume was then replaced in a horizontal position and a new laser scan of the surface was made.
Experiments were conducted at four slopes (3, 5, 8, and 12°) and five inflow rates (0.2, 0.4, 1, 2.2, and 3.6 m3 h-1). Experiments were carried out with two different topsoils (Table 2). In total, 20 experiments were carried out with a silt loam loess-derived topsoil, while 17 experiments were carried out with a loamy sand topsoil.
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Next, estimates of flow depth, wetted perimeter, and hydraulic radius were obtained for each experiment using a series of 10 to 12 rill cross-sections obtained with the laser scanner and spaced 0.1 m apart. From the measured values of discharge and flow velocity the water height in a cross-section was estimated by matching the calculated discharge (using the assumed water height and the measured flow velocity) with the measured discharge using an iterative procedure. The values of flow depth, hydraulic radius, and cross-sectional area were calculated for each cross-section and then averaged. Such an estimation procedure neglects local variations in flow velocity as well as the fact that the water surface in a cross-section is not horizontal. Therefore, the estimated values should be treated with caution. For example, it was noticed that estimates hydraulic mean depth showed a coefficient of variation of 10 to 35% for different cross-sections obtained after one experiment, so that the coefficient of variation of the average value for 10 to 12 cross-sections varied between 3 and 10%. The true uncertainty is probably even greater, as flow velocity is not constant along the rill.
Small Sample Experiments
The flume used was similar to that described previously by Ciampalini and Torri (1998) and Poesen et al. (1999). It is made of Plexiglass and measures 2-m long by 0.098-m wide. It contains a plot box (0.39-m long, 0.098-m wide, and 0.09-m deep) in its bottom.
The soil box was filled with air-dried soil sieved at 0.02 m. Before the experiment the soil sample was saturated by capillarity and then drained to field capacity. The soil box was then inserted in the flume, making sure that the soil surface was at the same level as the bottom of the flume. The flume was then set at the desired slope and a preset discharge was applied. Flow velocities were measured by recording the travel time of the dye cloud over a distance of 0.4 m. As the flow velocities in these experiments were rather high, the movement of the dye cloud was recorded on a video tape that was analyzed later. The same benchmark within the dye cloud and velocity correction factor as those in natural rill experiments were used. Again, runoff samples were taken at regular time intervals to determine sediment load and detachment rates. Because of the small size of the soil sample, it was not possible to determine soil moisture and bulk density on the samples used for the experiments. Therefore, two to three sample boxes were prepared with each soil using exactly the same procedure as the one applied to the sample boxes used in the experiments. From these boxes, two to three subsamples were taken to determine soil moisture and bulk density.
Experiments were run at three slopes (5, 8, and 12°) and four inflow rates (0.4, 1, 2.2, and 3.6 m3 h-1) for both soils. Each experiment lasted 5 min. Experiments were carried out with the same two soils used in the natural rill experiments (Table 2). In total, 10 experiments were carried out with each soil.
Flow depth was estimated by dividing the total discharge by the product of measured flow velocity and flume width (0.098 m) while the hydraulic radius was calculated by the estimated flow cross-section by the estimated wetted perimeter (= w + 2d, where w = the flume width and d = the flow depth).
Flow Detachment
For each experiment in both flumes, sediment concentration measurements over the first 5 min were averaged and the net detachment was calculated as the amount of sediment leaving the flume per unit time and per unit of rill length, DL, or per unit of rill bed area, DA. The rill bed surface area was calculated as the product of rill length and flow width.
 | [3] |
 | [4] |
Where Qs is solid discharge (kg s-1) and L (m) and Ax (m2), are the rill length and the rill surface area, respectively.
Hydraulic Parameters
Using the measured or estimated data of flow depth and velocity, the following flow hydraulic parameters were calculated:
Hydraulic shear stress (Nearing, 1997)
 | [5] |
 | [2] |
Stream power (Bagnold, 1966)
 | [6] |
Unit stream power (Yang, 1972; Moore and Burch, 1986)
 | [7] |
Effective stream power (Bagnold, 1980; Govers, 1992a)
 | [8] |
 | [9] |
Where, equals the water density (kg m-3), g corresponds to the gravitational acceleration (m s-2), Wp is the wetted perimeter (m), A represents the wetted cross-section area (m2), R equals the hydraulic radius (m) = A/Wp, S corresponds to the slope (sin), v is the average flow velocity (m s-1), d equals the flow depth (m), and fg represents the grain roughness friction factor (-).
The variable 
g was estimated following the approach of Govers and Rauws (1986): fg is estimated from the unit discharge, the slope, the grain roughness (D90), and the water temperature using the algorithm of Savat (1980). Next, the grain shear stress can be calculated using Eq. [9].
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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Reynolds's numbers in the small sample experiments were similar to those in the natural rill experiments, that is between 1000 and 8800. However, in the small flume experiments much higher flow velocities were obtained and therefore much lower values of flow depth and hydraulic radius. Consequently, Froude numbers were much higher and ranged between 2.4 and 5.7. Flow width was constant at 0.098 m while flow depth ranged from 0.002 to 0.01 m.
Soil Conditions
The results of the measurements of characteristics of the two soils before and after the natural rill experiments are presented in Table 1. Some variation in initial conditions occurred between experiments, despite the efforts to keep soil conditions as constant as possible. Initial soil moisture content was rather high, so that the soil was relatively resistant to runoff erosion (Govers et al., 1990; Bennett et al., 2000). This was considered to be an advantage as rill evolution was relatively slow, so that changes could well be monitored even on steep slopes. Bulk density and vane shear strength of the silt loam soil did not show a systematic evolution during the experiment, while they increased slightly in the loamy sand soil. The initial average bulk density for the silt loam and loamy sand soils in the natural rill experiments were 1.39 ± 0.04 and 1.48 ± 0.03 Mg m-3, respectively.
The soil moisture before the small sample experiments (0.26 g g-1) was somewhat higher (p < 0.01) than that before the natural rill experiments (0.23 g g-1). Although the erosion resistance of soils is known to vary with the moisture content prior to the erosion event (Govers et al., 1990; Bennett et al., 2000; Bryan, 2000), the relative dependency of the runoff erosion resistance is considered to be rather small for initial moisture contents exceeding 20% for the soil types used in our experiments (Govers, 1991).
Initial bulk densities in the small sample experiments were similar to those measured in the natural rill experiments. The average bulk density for the silt loam and loamy sand soil were 1.40 ± 0.03 and 1.50 ± 0.03 Mg m-3, respectively.
Evolution of Sediment Concentration Over Time
In the natural rill experiments, sediment concentration did not show a clear, systematic evolution over time. However, in most experiments an initial peak in sediment load was measured just after the start of the experiment. This is thought to be because of the washing out of readily detachable soil material from the top of the soil surface. After this initial peak has passed, the variation in sediment yield remains considerable. This is probably due to local sediment production events, such as the generation of a headcut or a scour hole (Poesen et al., 1999; Bennett et al., 2000), as well as the slumping of the rill walls.
The fact that the variation of sediment load over time is to some extent determined by random events implies that there is some experimental uncertainty associated with the 5 min average values that were used throughout the analysis. Therefore, a perfect correlation between flow detachment and average hydraulic parameter values cannot be expected.
Effect of Sediment Load on Detachment
A problem that may arise in evaluating flow detachment is the effect that sediment load may have on flow detachment: Foster and Meyer (1972) proposed that the flow detachment capacity is a linear function of the difference between the sediment load (Qs) and the transporting capacity (Tc):
 | [10] |
Where 
equals rate control constant.
If the presence of sediment affected flow detachment significantly in our experiments one would expect that the final rill cross sections would decrease with increasing distance downslope. To investigate this, the eroded volumes of the rills were plotted against the downslope distance in the rill (Fig. 1) . The rill cross-sections show a considerable variation but there is no systematic decrease of the rill's cross-section with increasing distance downslope. This suggests that, at least for our experiments, the effect of sediment load on flow detachment is not important. This is in agreement with the findings of Huang et al. (1996); they found that in most of their experiments sediment export from natural rills increased linearly with rill length. Sediment detachment in the rills appeared to be independent of actual sediment load and they concluded from their experiments that rill detachment and transport should not be considered as coupled processes. On the other hand, Merten et al. (2001) working with small soil boxes connected to each other to form a narrow long canal, found that, in general, detachment decreased with increasing sediment load but in a way not fully explained by current detachment-transport coupling theory. The fact that our results are different from those of Merten et al. may be because of several factors, one of which is the difference in experimental set-up (a concatenation of soil boxes versus a continuous rill). It is also important to realize that sediment loads in our experiments were always well below the flow's transporting capacity (see below), so that any effect of sediment load on detachment rates may have been quite small. However, the absence of a sediment load effect on rill development in our experiments cannot be considered as an artifact because of the baselevel in our experiments was periodically lowered; the level of the the downslope wall of the flume was always kept slightly above the rill bed level, so that no regressive erosion could occur.
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, as the independent variable and detachment per unit length, DL, as the dependent one (Table 3). The performance of the shear stress, which is the most frequently used predictor of flow detachment is also good. The relationships found for the unit length shear force and shear stress appear to be valid both for rough and smooth beds (Table 3, Fig. 2b)
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This finding is somewhat surprising: for a given discharge and slope flow detachment rates are relatively high on a rough rill bed where flow velocities are low. On a smooth bed where flow velocities are high flow detachment rates are relatively low. The absolute magnitude of the effect of different bed geometries on flow detachment is accounted for when the unit length shear force or the total shear stress are used as predictor variables. This result is different from what was found with respect to the transporting capacity of overland flow. The transporting capacity of overland flow for fine-grained sediment can be predicted using the unit stream power or the grain shear stress, that is, the part of the shear stress absorbed by individual grains. Total shear stress is a poor predictor of transporting capacity on irregular beds (Govers and Rauws, 1986; Govers, 1992b). On the other hand, grain shear stress appears to be a poor predictor of sediment detachment in natural rills (Table 3). This may be qualitatively explained as follows. A large part of the form shear stress in natural rills is dissipated on knickpoints, headcuts, and other bed irregularities; these are indeed the locations where most of the sediment detachment occurs. However, the presence of these bed irregularities slows down the water flow and reduces locally the slope gradient, thereby reducing the flow's transporting capacity. Thus, for a given slope and discharge, the transporting capacity of flow over a rough, natural rill bed will be lower than the transporting capacity over a smooth bed while its detachment capacity will be higher.
As long as the whole rill bed consists of erodible soil, the form shear stress contributes effectively to sediment detachment while it does not contribute to sediment transport. The implication of this is that a simple coupled modelling of sediment detachment and transport, as proposed by Foster and Meyer (1972) and implemented in various erosion models, is not possible because sediment detachment and transport are controlled by different hydraulic parameters. The relationship between the flow's detachment capacity and the flow's transporting capacity is therefore dependent on bed roughness.
This can be illustrated by comparing values of (see Eq. [10]) for both types of experiments. Govers (1992a) proposed the following equation for the calculation of the transporting capacity of overland flow on smooth and rough beds:
 | [11] |
Where, D equals the grain size (m) and Tc represents the transporting capacity (kg s-1). Assuming that sediment load has no effect on sediment detachment and considering flow detachment per unit length rather than per unit surface, Eq. [10] can be rewritten as:
 | [12] |
can then be calculated as:
 | [13] |
Table 4 shows that the average values of for the natural rill experiments are almost 10 times higher than those for the small sample experiments. The use of the -value derived from small sample experiments to predict detachment in rills would therefore lead to a considerable underestimation of flow detachment in rills.
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is dependent on bed geometry and do not imply that sediment load does not affect flow detachment. The presence of a sediment load may indeed reduce the flow's ability to detach sediment. However, a different modelling concept than that proposed by Foster and Meyer is necessary to describe this interaction; the relationship between transporting capacity and detachment capacity cannot be described using a simple rate control constant (). A possible approach is to model detachment rate (Dr) as a function of both detachment capacity (Dc) and transporting capacity, using different hydraulic parameters to predict detachment capacity and transporting capacity:
 | [14] |
 | [15] |
a equals constant, Ev represents explanatory variable such as or and Tc may be calculated using Eq. [11].
Our data also indicate that predicting flow detachment rate on a per unit length basis using unit length shear force as a predictor gives somewhat better results than prediction on a unit surface area basis using shear stress. This is interesting as former research has shown that flow velocity, and therefore also the wetted cross-section in rills can be quite well predicted from total rill discharge, using equations of the form:
 | [16] |
 | [17] |
Using over 400 datapoints, Govers (1992b) obtained values of 3.52 and 0.294 for a and b and 0.34 and 0.732 for c and d, respectively for rills eroding loose homogeneous materials. Further research has shown that, for rills developing in a homogeneous substrate, the effect of soil type or soil conditions on rill flow velocities is rather limited (Takken et al., 1998). However, flow velocities appear to be lower when rock fragments are present (Takken et al., 1998; Nearing, 1999; Govers et al., 2000) and can be very strongly reduced by the presence of vegetation cover (Takken et al., 1998).
Thus, for rills developing on bare soils without rock fragments, the unit length shear force can be written as:
 | [2] |
 | [18] |
Therefore, flow detachment can be predicted directly from the knowledge of total discharge and slope. Figure 3 indeed shows that the unit length shear force as calculated by Eq. [18] is a good predictor of the soil detachment per unit length in a natural rill. It is not necessary to predict flow width or the exact rill geometry. It should be kept in mind though, that Eq. [16] and [17] are valid for rill flow only.
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONCONCLUSIONSREFERENCES |
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The fact that different hydraulic variables control sediment detachment and sediment transport has implications for the way the interaction between these two processes are modelled. Different equations are needed to predict sediment transport and sediment detachment: both processes cannot be linked through a simple rate constant.
Our results also show that prediction of sediment detachment on a unit length basis may have some advantages compared with prediction on a unit area basis because rill flow velocity and wet cross-section can easily be predicted from total discharge. Using this approach may lead to simpler, yet more performant models of rill erosion.
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ACKNOWLEDGMENTS |
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Received for publication September 3, 2001.
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eff equals effective stream power; 
