Experimental and modeling approach to the study of the critical slope for the initiation of rill flow erosion

WATER RESOURCES RESEARCH, VOL. 41,, doi:10.1029/2005wr003991, 2005 Experimental and modeling approach to the study of the critical slope for the initiation of rill flow erosion Xuejun Shao, Hong Wang, and Huiwu Hu Department of Hydraulic Engineering, Tsinghua University, Beijing, China Received 26 January 2005; revised 29 August 2005; accepted 22 September 2005; published 1 December 2005. [1] Critical conditions for rill erosion are determined through laboratory experiments and one-dimensional numerical simulations. Relationships between rill flow characteristics and slope angle q or plot size are studied both experimentally and numerically for realistic rainfall intensities. The experimental results show that maximum rill flow velocities occur within a specific range of slope angles, q =30 40, which is independent of plot size and degree of overland flow concentration. For a given plot size and rainfall intensity the threshold slope angle for rill erosion depends on a number of factors, including soil erodibility, the degree of overland flow concentration, and the effect of gravity on soil erosion. The effect of plot size on the stable rill width generated by rill erosion is also discussed along with its implications for soil loss predictions. Citation: Shao, X., H. Wang, and H. Hu (2005), Experimental and modeling approach to the study of the critical slope for the initiation of rill flow erosion, Water Resour. Res., 41,, doi:10.1029/2005wr003991. 1. Introduction [2] Concentrated flows on hillslopes lead to the initiation and development of rills when flow shear exceeds the critical threshold, and are responsible for the transportation of eroded soil particles through a system of rills and gullies from runoff contributing areas where soil particles are detached by raindrop splash and Hortonian overland flow. [3] Rill erosion is an important component of processbased soil erosion models. Previous research has focused on the critical conditions for rill initiation and development and on their dependence on slope, soil erodibility [Bennett, 1999; Bryan, 2000] and plot length [Watson and Laflen, 1986; Fox and Bryan, 1999]. The hydrodynamic characteristics of rill flow have been studied experimentally during the last few decades [e.g., Yoon and Wenzel, 1971; Shen and Li, 1973; Abrahams et al., 1986; Abrahams and Parsons, 1991; Nearing et al., 1997; Bryan and Rockwell, 1998]. [4] The critical shear stress, t c, required for particle entrainment, has been widely used as a key indicator of soil erodibility and of soil shear strength. Studies on loessderived soils have shown that there is a minimum shear stress required for gully development [Ciampalini and Torri, 1998; Nachtergaele et al., 2002]. Further investigations are needed to identify and define relationships between the critical shear stress of overland flow erosion and key factors related to plot configurations, such as its length, width and slope angle, for realistic plot configurations and storm events. [5] Rills in the laboratory exhibit alternating regions of deposition and detachment for the steady state, uniform initial slope case that were affected by small-scale random variations in the initial conditions of the rill [Lei et al., 1997]. A deterministic model that is not able to capture such natural variability tends to overpredict soil erosion for small measured values, and under predict soil erosion for large Copyright 2005 by the American Geophysical Union. 0043-1397/05/2005WR003991 measured values [Nearing, 1998, 2000; Nearing et al., 1999; Gomez et al., 2001]. [6] The transition through various erosion forms represents a continuum as scales increase [Poesen et al., 2003]. A number of approaches to problems of upscaling have been developed in the context of climate and soil erosion [Kirkby et al., 1996; Kirkby, 1999], and both empirical and physically based models have been applied at quite different scales [Amore et al., 2004]. One important issue is to assess to what extent process identification and parameterization carried at one scale can be extrapolated to a different scale [Cerdan et al., 2004]. It has been shown that the homogeneous sediment transport equations reflect only responses at low or medium rainfall intensities, as experimental evidence showed significant departure from the general trend in cases of steep slopes and high rainfall intensities [Mathier and Roy, 1996]. [7] This paper presents the results of laboratory experiments and numerical simulations of rill flows, and investigates relationships between hydraulic factors (flow velocities or critical shear stresses) and plot configurations (e.g., plot length, slope angle, rill width). 2. Experimental Results [8] Both flumes and soil plots are generally used in experimental studies on rill erosion [e.g., Bennett, 1999; Bryan, 2000]. Typical plot layouts are shown in Figures 1a and 1b. Plots with constant horizontally projected area (Figure 1a) always receive the same amount of total rainfall regardless of slope gradient, resulting in constant total runoff from the plots for the same rainfall intensity, if their surfaces are impermeable. In contrast, on plots with constant slope length L (Figure 1b), the total amount of rainfall received can vary significantly as slope angle changes (Figure 2). [9] In this experimental study, a ridged plot (Figure 1c) was used, with an impermeable rigid surface made of Plexiglas. Cloth patches were used to adjust the roughness 1of11

SHAO ET AL.: CRITICAL SLOPE FOR RILL FLOW EROSION Figure 1. Ridged plot configuration and two types of rainfall simulation plot layout. (a) Constant horizontally projected area, (b) constant slope length, and (c) ridged plot. of the plot surface. This plot is a laboratory version of Wooding 0 s open-book catchment schematization [Wooding, 1965], which is frequently used in numerical simulations. The plot has a fixed slope length of L = 1.5 m, plot width B = 1.0 m and two rill widths b = 0.05 and 0.1 m. During the test runs, the slope angle q varied from 15 to 65 to generate different total runoff. The extreme steep slope angles are required for the completeness of this experimental study. For this plot configuration the overland flow characteristics were found to be quite sensitive to variation of slope angle, and the measured rill flow velocities can be used as an ideal database to verify the numerical model and to test its capability for predicting the effects of slope angle on rill flow hydrodynamics. [10] All the experimental runs were conducted in a recirculating system, which had a rainfall module 1.8m above ground level, with a total of 1341 capillary openings at the bottom of an open tank to produce rainfall with steady, uniform intensity (Figure 3). Raindrops from these capillary openings had diameters of approximately 3 mm. Under the rainfall module a ridged plot shown in Figure 1c was installed, and its slope angle was adjustable by an interval of 2. A reservoir underneath the plot collected runoff and raindrops falling outside the plot. A pump supplied water from the reservoir to the rainfall module. [11] The spacings of capillary openings on the rainfall module bottom were 0.1 m in the transverse direction and 0.01 m in the longitudinal direction (i.e., parallel to rill flow). The amount of rainfall collected on the plot would change following adjustments of slope gradient by a step of 2. Water levels in the rainfall module were kept constant by regulating the weir height at the free falls, to achieve different intensities of steady rainfall. Rill flow velocities were measured using a video camera installed at the toe of the plot, which recorded tracer (dye) movement at the end of the rill from a fixed viewpoint, with a speed of 24 frames per second. The velocity was measured from the leading edge of the dye cloud. [12] The experimental setup is a simplification of the field situation because changes in bed roughness associated with sediment transport [e.g., Gimenez and Govers, 2001] are neglected. In this study, the plot surface had a fixed 2of11 roughness, and all the variations of roughness coefficient for the sheet and rill flows were caused mainly by the transition from laminar to turbulent flow conditions. In each test run, the rainfall simulator was adjusted to generate a steady rainfall intensity to produce steady rill flow on the model plot. The steady runoff discharge, Q, at the plot outlet was then measured by collecting the total runoff within a given time interval. The net rainfall intensity was then calculated on the basis of the measured values of runoff and the slope-dependent projection of the plot area on a horizontal plane. [13] As shown in Figure 4 and Table 1, in this study, the maximum velocities, U max, were found to occur within a specific range of slope angles, i.e., q =35 40. This is because slope angles within this range are large enough to create sufficient gravitational acceleration for water flow, and at the same time small enough for the plot to have relatively large projection areas that collect sufficient net rainfall. Figure 2. Variation of total runoff (Q) with the slope angle (q) on a test plot with constant slope length (simulated plot size is 6 m 1.5 m; net rainfall intensity is 1.5 mm/min).

SHAO ET AL.: CRITICAL SLOPE FOR RILL FLOW EROSION Figure 3. Sketch of the setup for rainfall and rill flow simulation on a 1.5 m 1 m plot. 1, ridged plot; 2, rainfall module; 3, pump; 4, reservoir; 5, video camera. p[14] ffiffiffiffiffiffiffiffiffiin this p ffiffiffiffiffiffiffiffi study, the rill flow shear velocities, u * = t 0 =r = ghj, were found out based on estimated flow depth, h = Q/b/U. The hydraulic slope is defined as J =sinq. Experimental data shown in Figures 4 and 5 were used to test the numerical model calculations. 3. One-Dimensional Mathematical Model for Rill Flow [15] In process-based models of overland flow and soil erosion, hillslope runoff processes are often represented either as broad sheet flows or as rill flows with rectangular cross sections. The hydraulic characteristics of overland flow are usually calculated using the kinematic wave simplification [Kirkby 1990; National Soil Erosion Research Laboratory, 1995] of the dynamic wave model. Baird et al. [1992] considered rilled or gullied topography by using a flow strip concept in which the complex hillslope surface was represented by a series of profile depths of unit width which operate independently of their neighbors. Tayfur and Kavvas [1994] employed an openbook-type schematization of rill topography on a hillslope, and studied both 1-D rill flow and 2-D interrill area overland flows by using the kinematic wave approximation. In this paper the full dynamic wave model is used for rill flow velocity and depth calculations, because it accurately represents the unsteady, nonuniform nature of overland flows and can provide more detailed and reliable results of the hydrodynamic characteristics required in the estimation of potential for overland flow erosion. 3.1. Dynamic Wave Model [16] We assume that all rills are straight and parallel to each other with relatively small spacing, and that the interrill drainage areas, represented by regular shaped strips, are tilted from the divide toward the rill, as shown Figure 4. Observed relationship between normalized rill flow velocity (U/U max ) and slope angle, q, for different rainfall rates, r, and b/b ratios. U max is the maximum rill flow velocity for the same values of r, b/b, and slope angle. 3of11

SHAO ET AL.: CRITICAL SLOPE FOR RILL FLOW EROSION Table 1. Observed Rill Flow Velocity for Various Rainfall Intensity and Slope Angles Net Rainfall Intensity r, mm/min Relative Rill Width b/b Rill Flow Velocity U for Various Slope Angles q, m/s 10 15 20 25 30 35 40 45 50 55 60 65 2.0 0.1 0.332 0.371 0.389 0.577 0.557 0.61 0.594 0.502 0.45 0.423 0.324 0.313 2.0 0.1 0.392 0.419 0.514 0.574 0.604 0.601 0.503 0.446 0.407 0.362 0.337 0.25 2.1 0.1 0.424 0.445 0.485 0.53 0.564 0.551 0.66 0.528 0.551 0.472 0.472 0.465 2.1 0.1 0.401 0.412 0.429 0.508 0.54 0.561 0.522 0.451 0.426 0.397 0.378 0.368 2.5 0.05 0.408 0.484 0.541 0.666 0.75 0.702 0.857 0.782 0.711 0.687 0.65 0.598 2.7 0.05 0.624 0.693 0.702 0.724 0.784 0.831 0.871 0.939 0.869 0.827 0.668 0.686 4.5 0.05 0.722 0.773 0.795 1.081 1.135 1.118 1.241 1.201 1.124 1.124 1.084 0.866 4.7 0.05 0.772 0.874 0.897 0.925 1.018 1.064 1.06 0.997 0.975 0.94 0.823 0.789 5.1 0.05 0.662 0.767 0.924 0.959 1.039 1.301 1.121 1.17 1.097 1.045 0.964 0.823 5.3 0.05 0.723 0.768 0.809 0.927 1.129 1.125 1.209 0.99 0.975 0.919 0.836 0.795 in Figure 1c. The lateral flow discharge into a rill is considered as a linear source of uniform inflow (Figure 6), perpendicular to the rill flow direction. The numerical model is one-dimensional. The hydrodynamic characteristics are calculated for only one rill. [17] The continuity equation of the dynamic wave model is @A dt þ @Q dx ¼ r cos q N and the momentum equation is @Q @t þ @ Q 2 ¼ ga @Z @x A @x þ Q2 C 2 A 2 R in which x is distance from the top of the plot, r is net rainfall intensity (i.e., total rainfall intensity less infiltration rate), A is area of the wetted cross section, N is number of rills per unit width of the plot, Z is the surface elevation of rill flow as shown in Figure 6, and C is Chezy coefficient. The uniform inflow into the rill due to rainfall on interrill area is treated as a linear source term involving r in the derivation of equation (2), which can be cancelled out with other terms if the continuity equation (equation (1)) is ð1þ ð2þ introduced in the derivation, resulting in the conventional form of the Saint-Venant equation. In this system of equations the net rainfall intensity is the driving variable, and the effects of the infiltration process and the impact of raindrop are not included in the momentum equation of rill flow. This assumption is acceptable if overland flow becomes concentrated and rill flow momentum becomes dominant. [18] Figure 7 shows an overall acceptable agreement between model simulations and the experimental results (Table 1) for the cases of rainfall intensity, r, of 4.5 mm/min and 5.1 mm/min, and b/b = 0.05 (see Figure 1 for a definition of b and B). Such hypothetical net rainfall intensities were used in this study to achieve better accuracy in experimental measurements and numerical calculations. 3.2. Stable Rill Width [19] Assuming that rill initiation is simultaneous to particle entrainment, the threshold for rill initiation can be expressed as the critical average flow velocity, U c,for soil detachment using for example Hjulstrom s curve [Hjulstrom, 1935; Vanoni, 1975, Figure 2.46]. In this study stable rill widths are determined through a trialand-error method calculating the widths associated with overland flow velocities matching the critical velocity for Figure 5. Observed relationship between shear velocity, u *, and slope angle, q, at the toe of a 1.5 m 1 m ridged plot. 4of11

SHAO ET AL.: CRITICAL SLOPE FOR RILL FLOW EROSION Figure 6. A schematic representation of the onedimensional rill flow. particle entrainment. This procedure was repeated for a number of combinations of net rainfall intensity, slope angle, critical flow velocity of soil detachment and fixed slope length. Such stable widths can be regarded as the maximum dimension at the final stage of rill development during a storm event of steady rainfall intensity, since flows in rills with widths larger than the stable one will have an average flow velocity below the critical value, resulting in siltation of rills or gullies and reduction in their cross-sectional area, most probably through the decrease of rill width. Therefore this is a transport-limited erosion process. [20] Relationships between stable rill width and q are established for various U c values and plot sizes (Figures 8 and 9), by assuming that the values of U c are independent of slope angle and the system of concentrated overland flows on the hillslope plot will finally integrate into one dominant rill or gully at the toe of the slope, through the processes of cross grading and micropiracy. It should be noted that the largest stable widths are always achieved within a range of q =30 40 regardless of the plot size and the value of U c, which is consistent with the rill flow velocity calculations. Calculated stable widths of rills are very sensitive to the magnitude of U c, which is affected by a number of factors including soil types, farming practices or environmental conditions. A nonlinear increase of stable width b in response to the reduction of U c is implied by the b-q relationships shown in Figures 8 and 9. [21] Stable rill widths become almost independent of slope angle q after U c exceeds a certain magnitude, and such widths are obviously related to the size of the plot. In addition, on a plot with fixed length L, larger slope angles result in smaller stable rill widths if U c remains a constant value, due to the decrease of runoff as q increases. [22] The variation of the plot size can also result in nonlinear changes in stable rill widths. For instance, a maximum width of b = 0.13 m is formed on a 6 m 1.5 m plot for U c = 0.4 m/s (Figure 8), but on an 80 m 20 m plot it becomes b = 18 m for the same U c (Figure 9), i.e., theoretically, the rill at the toe will be about 140 times wider on a plot about 13 times longer, for the same type of soil. Since the lateral dimensions may be viewed as an indication of the amount of soil erosion occurring due to flow shear within rills or gullies with fixed length, the above result implies that rill erosion in the Figure 7. Observed and calculated rill flow velocities, U, in the 1.5 m 1.0 m plot as a function of the slope angle, q. 5of11

SHAO ET AL.: CRITICAL SLOPE FOR RILL FLOW EROSION Figure 8. Calculated stable rill width, b, as a function of the slope angle, q, for various values of entrainment velocity, U c (simulated plot size is 6 m 1.5 m; net rainfall intensity is 1.5 mm/min). form of a rill-gully continuum could generate a total soil loss much larger than the summation of those from many shorter and separate rills with the same total length. 3.3. Threshold Slope Angle for Rill Initiation: Constant u * c [23] Basic detachment processes in rill erosion include channel scour, headcutting, slaking, and sidewall sloughing. Process-based erosion models often use critical flow shear, t c, as the primary parameter. The detachment rate is assumed as zero below this value. Even though it is still unclear how its value should be estimated [e.g., Zhu et al., 2001], the critical shear stress is widely used as a key indicator of soil erodibility. Studies on loess-derived soils have shown that the minimum shear stress required for gully development is about t c =5Pa[Ciampalini and Torri, 1998; Nachtergaele et al., 2002], which corresponds to a critical shear velocity of u * c = 0.07 m/s. In this study two extreme cases are considered for the critical shear velocity u * c, i.e., a constant u * c value independent of slope angles (cohesive soil or negligible gravitational effects), and a nonconstant u * c whose value varies with slope angle q (noncohesive soil or strong gravitational effects). [24] Rill flow shear velocities were calculated with b/b = 0.05, r = 2.0 mm/min, and various slope angles. Figure 10 shows the distance required for flow shear to achieve the critical value (u * c = 0.07 m/s) for various slope angles. It is obvious that to achieve the same shear velocity of rill flow, a larger value of x is required if slope angle q is smaller, and vice versa, as identified by Horton [1945]. Shear velocity becomes less dependant on q for very large values of q, i.e., the same increment of Dq will causes a much fast increase in u * for smaller q values (gentler slope) than for larger ones. The same is true for the u * x relationships. Figure 9. Calculated stable rill width, b, as a function of the slope angle, q, for various values of entrainment velocity, U c (simulated plot size is 80 m 20 m; net rainfall intensity is 1.5 mm/min). 6of11

SHAO ET AL.: CRITICAL SLOPE FOR RILL FLOW EROSION and 13), which show that for higher degrees of flow concentration, smaller slope angles are needed for the flow shear stresses to exceed the critical value. [27] For overland flows with more than one rill on the plot the results of Figures 12 and 13 may still be used to indicate an average magnitude of u * in the rills for any value of (1 b/b), except that b should be replaced by Sb. A turning point at (1 b/b) = 0.85 can be identified in the u * (1 b/b) relationships of Figure 13, after which u * increases much faster with (1 b/b). Figure 10. Relationship between the shear velocity, u *, and the slope angle, q, ona6m 3 m plot for b/b = 0.05 and r = 2.0 mm/min; x is distance from top of the plot. [25] Figure 11 shows shear velocities at the toe of the 6m 3 m ridged plot calculated for r = 2 mm/min and various values of b/b, which have a similar trend as measured data (Figure 5). Assuming a constant critical shear velocity of u * c = 0.07 m/s (the thick solid line in Figure 11), the critical slope for rill erosion in existing rills (that is, given values of b/b) can be determined in Figure 11 as intersection points with the line u * c = 0.07. For instance, rill erosion is found to begin at the toe of the 6 m 3m plot when the slope angle is 3 or above, for the relative rill width of b/b = 0.03. However, for b/b = 0.17, such erosion will not start until the slope angle becomes greater than 10. [26] In this study, the value of (1 b/b) is used to represent the degree of sheet flow concentration on a plot. The condition of b = B indicates no concentrated flow, and b B indicates a very high degree of flow concentration. Therefore the value of (1 b/b) = 0.17 for a 6 m 3m plot indicates a low degree of overland flow concentration, with a rill width of b = 2.5 m and the overland flow spreading over 83% of the plot width. This can be viewed as an ideal sheet flow. In such a way the calculated results are interpreted as u * (1 b/b) relationships (Figures 12 3.4. Threshold Slope Angle for Rill Initiation: Nonconstant u * c [28] In addition to rill flow shear stresses, the component of particle gravity down the slope is another major cause of the incipient motion of soil particles, if the soil type is noncohesive or the gravitational effects are nonnegligible. The factor of gravitational action can be taken into account by modifying the expression for t c to let it depend on q [Brooks, 1963]. Starting from a more general case [Vanoni, 1975, equation 2.126], the following relationship can be obtained if the flow direction is parallel to the weight component down the slope, such as in the case of rill flows, t c ¼ cos q sin q t c0 tan f where t c0 is the critical shear stress for particles on a horizontal surface, f is the repose angle. In this study, equation (3) is used to calculate the critical shear velocity u * c for various slope angles, with the value of f being 31. The repose angle, f, may also be viewed as a general parameter of soil erodibility accounting for the effect of gravity on soil particle entrainment (e.g., due to slope failure or to soil detachment caused by the collapse of the walls of a rill or a gully). The value of f represents a threshold of slope angle above which other factors such as gravitational effects dominate soil detachment and turn rill erosion into a transport limited process. The rill will be widened to its stable width faster than in the case of a constant u * c since gravitational effects will accelerate the process. ð3þ Figure 11. Relationship between the shear velocity, u *, and the slope angle, q,atthetoeofa6m3m ridged plot for r = 2.0 mm/min and various values of b/b. 7of11

SHAO ET AL.: CRITICAL SLOPE FOR RILL FLOW EROSION Figure 12. Relationship between the shear velocity, u *, and the slope angle, q,atthetoeofa6m 3m ridged plot for r = 2.0 mm/min and various values of (1 b/b). [29] The critical slope angles, q c, for overland flow erosion initiation at the toe of the plot, are determined by the intersection points of the calculated u * q curves with the u * c q relationship given by equation (3) (Figure 14). The critical slope angles found in this way vary greatly depending on the value of b/b. Compared with Figure 12, in which no other effects are considered except the rill flow shear stresses, Figure 14 shows a significant reduction in q c. Because equation (3) accounts for more realistic factors such as noncohesive soils and the effect of gravity on erosion, it predicts a higher erosion potential. According to equation (3), for a sheet flow with (1 b/b) = 0.03 (i.e., overland flow covers 97% of the plot width) the critical value of shear velocity at the toe of the plot is achieved at q c = 15. If the overland flows are more concentrated, the critical slope will be much smaller, for instance q c =3 for (1 b/b) = 0.97, which corresponds to a rill width of 0.1 m on this plot. [30] The results are summarized in Figure 15: curves A and B represent values of q c estimated with constant and nonconstant values of u * c, respectively. All soil types exhibit a nonerosion zone even when gravitational effects are taken into account. In the transition zone erosion occurs only for high soil erodibility or strong gravitational effects. The third zone predicts rill flow erosion for cohesive soils even without considering the gravitational effects. Curve A provides a more realistic prediction, in that it accounts for factors commonly encountered in field situations. On the basis of predictions of curve A, for a 6 m 3 m plot with ideal sheet flow, erosion will not occur at the toe of the slope for slope angles up to 15 if the net rainfall intensity remains a steady 2.0 mm/min. It should be noted that the Figure 13. Relationship between the shear velocity, u *, and the degree of flow concentration, (1 b/b), at the toe of a 6 m 3 m ridged plot. 8of11

SHAO ET AL.: CRITICAL SLOPE FOR RILL FLOW EROSION Figure 14. ridged plot. Relationship between the shear velocity, u *, and slope angle, q, atthetoeofa6m 3m theoretical predictions suggest that the initiation of rill erosion depends strongly on the degree of flow concentration on the plot in question, and the differences in soil types or erosion agents are not important after rills or gullies are formed and overland flows become concentrated. For instance, it can be seen in Figure 15 that for a highly concentrated flow, i.e., (1 b/b) > 0.8, the value of q c is almost identical regardless of soil conditions or gravitational effects, but for (1 b/b) < 0.1 the respective value of q c differs from each other by a factor of up to 200%. Such results may help to find the reason why small-scale random variations in the initial conditions of the rill can lead to variations in erosion that a deterministic model is not able to capture [Nearing, 1998, 2000; Nearing et al., 1999]. 4. Concluding Remarks [31] Threshold conditions for rill erosion on ridged plots with fixed length are studied on the basis of both rainfall simulation experiments and numerical calculations Figure 15. Relationship between the degree of flow concentration, (1 b/b), and critical slope angle, q c,atthetoeofa6m 3 m plot for a net rainfall intensity of 2.0 mm/min. 9of11

SHAO ET AL.: CRITICAL SLOPE FOR RILL FLOW EROSION with a one-dimensional mathematical rill flow model. Experimental observations and calculated results are summarized as follows. [32] 1. By using a Hjulstrom-type critical velocity for particle entrainment, U c, stable rill widths are determined through a trial-and-error method calculating the widths associated with overland flow velocities matching the critical velocity for particle entrainment. The maximum widths occur within a range of q = 30 40, consistent with that for the maximum values of U. [33] 2. Calculations show that the effects of plot sizes are significant, e.g., the calculated rill width is about 140 times wider on a plot about 13 times longer. Such results suggest that, the total amount of rill or gully erosion from a series of shorter and nonconnected rills can be much smaller than that from a rill-gully continuum with the same total length as that of the shorter rills put together, provided that the lateral dimension of rills and gullies with fixed lengths can be regarded as an indication of the total amount of rill flow erosion occurred. [34] 3. Factors affecting the erosion processes, particularly the gravitational effects, can be accounted for in this numerical model by using a nonconstant critical shear stress for particle entrainment depending both on repose angle, f, and on the slope angle, q. The model indicates that the threshold of rill erosion depends closely on the degree of flow concentration, while differences between values of u *c are not important after rills or gullies are formed and overland flows become concentrated. Notation A area of the wetted cross section, m 2. B plot width, m. b rill widths, m. C Chezy coefficient, m 1/2 /s. g gravitational acceleration, m/s 2. h rill flow depth, m. J hydraulic slope of rill flow, J =sinq. L slope length, m. N number of rills per unit width of the plot. Q rill flow discharge, m 3 /s. r net rainfall intensity, mm/min. t time, s. U rill flow velocity, m/s. U c critical rill flow velocity, m/s. U max maximum rill flow velocity, m/s. u * shear velocity of rill flows, m/s. u * c critical shear velocity of rill flows, m/s. x distance from the top of the plot, m. Z water surface elevation of rill flows, m. q slope angle, degrees. q c critical slope angles, degrees. r density of water, kg/m 3. t 0 boundary shear stress of rill flows, N/m 2. t c critical shear stress for rill erosion, N/m 2. t c0 critical shear stress for particles on a horizontal surface, N/m 2. f repose angle, degrees. [35] Acknowledgments. Partial financial support of the work presented in this paper from the National Natural Science Foundation of China (grant 50179015), the NSFC Science Fund for Creative Research Groups (grant 50221903), and China s National Key Basic Research and Development Program for its support for this project (grant 2003CB415206) are gratefully acknowledged. The authors are greatly indebted to the three anonymous reviewers for their comments and discussions, which made this paper publishable. References Abrahams, A. D., and A. J. Parsons (1991), Resistance to overland flow on desert pavement and its implications for sediment transport modeling, Water Resour. Res., 27(8), 1827 1836. Abrahams, A. D., A. J. Parsons, and S.-H. Luk (1986), Resistance to overland flow on desert hillslopes, J. Hydrol., 88, 343 363. Amore, E., C. Modica, M. A. Nearing, and V. C. Santoro (2004), Scale effects in USLE and WEPP application for soil erosion computation from three Sicilian basins, J. 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