The impact of slope length on the discharge of sediment by rain impact induced saltation and suspension

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1 EARTH SURFACE PROCESSES AND LANDFORMS Earth Surf. Process. Landforms 34, (2009) Copyright 2009 John Wiley & Sons, Ltd. Published online 16 June 2009 in Wiley InterScience ( The impact of slope length on the discharge of Chichester, EARTH Earth 9999 ESP1828 Research Copyright John 2006Journal Wiley Science Surf. Surface SURFACE Articles Process. & UK of 2006 Sons, Processes the PROCESSES John British Ltd. Landforms Wiley Geomorphological Landforms AND & Sons, LANDFORMS Ltd. Research Group sediment by rain impact induced saltation and suspension P. I. A. Kinnell* Institute of Applied Ecology, Faculty of Science, University of Canberra, Canberra, Australia Received 3 February 2008; Revised 11 March 2009; Accepted 23 March 2009 * Correspondence to: P. I. A. Kinnell, Institute of Applied Ecology, Faculty of Science, University of Canberra, Canberra, Australia. peter.kinnell@canberra.edu.au ABSTRACT: Simulations using a mechanistic model of raindrop driven erosion in rain-impacted flow were performed with particles travelling by suspension, raindrop induced saltation and flow driven saltation. Results generated by both a high intensity storm, and a less intense one, indicate that, because of the effect of flow depth on the delivery of raindrop energy to the bed, there is a decline in sediment concentration, and hence soil loss per unit area, with slope length when particles are transported by raindrop induced saltation. However, that decline is reversed when the critical velocities that lead to flow driven saltation are episodically exceeded during an event. The simulations were performed on smooth surfaces and a single drop size but the general relationships are likely to apply for rain made up of a wide range of drop size. Although runoff is not always produced uniformly, as a general rule, flow velocities increase with slope length so that, typically, the distance particles travel before being discharged during an event increase with slope length. The effect of slope length on soil loss per unit area is often considered to vary with slope length to a power greater than zero and less that 1 0. The simulations show that effect of slope length on sediment discharge is highly dependent on the variations in runoff response resulting from variations in rainfall duration-intensityinfiltration conditions rather than plot length per se. Consequently, predicting soil loss per unit area using slope length with positive powers close to zero when sheet erosion occurs may not be as effective as commonly expected. Erosion by rain-impacted flow is a complex process and that complexity needs to be considered when analysing the results of experiments associated with rain-impacted flow under both natural and artificial conditions. Copyright 2009 John Wiley & Sons, Ltd. KEYWORDS: rainfall erosion; slope length; rain-impacted flow Introduction The loss of soil material on hillslopes has both onsite and offsite consequences. The effect of slope length on sediment discharged in runoff from an eroding area is important to the prediction of soil loss from natural and disturbed areas within hillslopes and the impact of those areas on water quality. In theory, the longer the slope, the more runoff will accumulate and gather energy and cause more erosion. Many experiments have been conducted where the effect of slope length on the amount of soil lost from an area has been investigated. Generally, the results of field studies indicate that a power relationship exists between slope length and sediment yield (mass per unit area); Y L α l m (1) where Y L is the sediment yield for an area with a slope length of l. Zingg (1940) found that soil loss per unit area varied with m = 0 6. Later, Wischmeier et al. (1958) observed that the value of m was strongly influenced by changes in soil, plant cover and crop management. In the Universal Soil Loss Equation (USLE) (Wischmeier and Smith, 1978), m varies with slope gradient. In the revised version of the Universal Soil Loss Equation (RUSLE; Renard et al., 1997), m varies with the ratio of rill to interrill erosion. As a general rule, the amount of material contained in a volume of water is given by the product of that volume of water and the amount of that material per unit volume of water. This rule applies irrespective of whether the material is dissolved or remains as a solid in the water. Consequently, the sediment discharged (Q s, mass per unit width of flow) from an area can be expressed as the product of the volume of water discharged (Q w, volume per unit width of flow) and sediment concentration (c s, mass of sediment per unit volume of water discharged); Q s = Q w c s (2) If runoff is generated uniformly over an area then there is a direct relationship between sediment concentration and sediment yield. Consequently, it follows from Equation 1 that

2 1394 EARTH SURFACE PROCESSES AND LANDFORMS when runoff is generated uniformly over an area, sediment concentrations will vary directly with slope length to the power m, c s α l m (3) While experimental data supports the use of values of m close to 0 5 when areas show active rilling, data such as that collected on low slopes under artificial rainfall by Mutchler and Greer (1980) indicate that positive values of m less than 0 2 are appropriate for slopes of 0 5% where sheet erosion controls soil loss and the ponding of water on the surface influences the amount of raindrop energy causing detachment of soil particles. Low values of m indicate the sediment concentrations from areas experiencing sheet erosion are largely insensitive to variations in slope length. Sheet and interrill erosion are recognized as being forms of erosion where detachment and transport processes are highly dependent of the energy dissipated when raindrops impact the soil surface. In this paper, a mechanistic model of the detachment and transport processes associated with raindrops impacting surface water flows is used to examine the effect of slope length on sediment transported by rain-impacted flows on areas varying in length from 2 30 m when runoff is generated uniformly or non-uniformly as slope length varies. The Simulation Model When natural rain falls on the ground, raindrops impact on the ground surface randomly in time and space. On sloping surfaces covered with flowing water, particles detached from the cohesive soil matrix by raindrop impact move downslope at any given time (a) by raindrop splash, (b) by raindrop induced rolling, (c) by raindrop induced saltation, (d) by flow driven rolling, (e) by flow driven saltation or (f) as a result of remaining in suspension long enough not to return to the bed before they are discharged. The mode of transport adopted depends of the flow conditions (Figure 1A) and the size and density characteristics of the particles involved. Very fine particles that remain suspended in the flow move down stream without any further stimulus by raindrop impact but particles that fall back to the bed, or roll, move a limited distance (X p,d ) after a drop impact. That distance depends on the size (p) and density of the particle, the size (d) of the impacting raindrop, and the flow conditions. In the simulations reported here, the effects of the various modes of transport associated with randomly spaced drop impacts are modelled when rainfall intensity varies in time and flow depths and velocities vary in time and space. Raindrop induced saltation is largely responsible for the transport of silt, sand and silt and sand sized aggregates in the rain-impacted flows that dominate sheet and interrill erosion. With raindrop induced saltation, particles lifted from the soil surface into the flow produced a cloud of particles that travels downstream as the particles fall back to the bed under the force of gravity. The distance travelled is given by the product of the velocity of the flow (u) the time the particles are suspended in the flow (t p,d ). In the simulations reported here, the rain is assumed to be made up of 2 7 mm drops (i.e. d = 2 7 mm), the area of the bed impacted by each drop impact assumed to be 5 mm by 5 mm square, and the cloud 11 mm Figure 1. Schematic of (A) how raindrop energy and flow shear stress influence the detachment and transport mechanisms associated with sand, silt and fine particles in rain-impacted flows and (B) the boundaries controlling the discharge of material transported by raindrop induced saltation. E c, critical raindrop energy to cause detachment. Raindrop induced erosion occurs when drop energy exceeds E c. A, line for E c when raindrops are detaching soil particles from the soil surface prior to flow developing. The slope on this line is used to indicate increasing resistance to detachment caused by, for example, crust development. B, line for E c when raindrops are detaching soil particles from the soil surface when flow has developed. The slope on this line is used to indicate increasing utilization of raindrop energy in penetrating the flow when flow depth increases as flow power increases. τ c(loose), critical flow shear stress required to transport loose (pre-detached) soil particles. τ c (bound), critical shear stress required to detach particles bound within the soil surface (held by cohesion and inter-particle friction). RD-ST, raindrop detachment and splash transport. RD-RIS, raindrop detachment and raindrop induced saltation. RD-FS, raindrop detachment and flow suspension. FD-FDS, flow detachment and flow driven saltation.

3 THE IMPACT OF SLOPE LENGTH ON THE DISCHARGE OF SEDIMENT BY RAIN IMPACT INDUCED SALTATION AND SUSPENSION 1395 Figure 2. flow. Schematic of the uplift and travel of 0 46 mm particles of sand and coal in the simulation when 2 7 mm drops impact a 7 mm deep by 11 mm square. The value of t p,d varies not only with raindrop size and particle characteristics but also with flow depth (h). Figure 2 provides a schematic of the way particle uplift and travel associated with 2 7 mm drops impacting 7 mm deep flows over a mixture of 0 46 mm sand and coal is simulated by the model used in this paper. Data obtained in laboratory experiments by Kinnell (2001) indicate that 0 46 mm particles of sand and coal remain suspended 0 20 s and 0 55 s respectively when these particles are lifted into the flow by 2 7 mm drops impacting 7 mm deep flows. On cohesive soil surfaces, particles moving by raindrop induced saltation lie on top of the cohesive layer and provide a degree of protection (H R ) against the detachment of particles from that underlying layer. Consequently, if M p,d.m is the mass of p sized material lifted into the flow when no previously detached particles are present (H R =0) and M p,d.pdl is the mass of p sized material lifted into the flow when a layer of previously detached p sized particles completely protects the underlying soil surface from detachment (H R = 1), the value of M p,d at any time is given by (Kinnell, 1993a): M p,d = M p,d.m (1 H R )+H R M p,d.pdl (4) This equation gives M p,d.p.m when H R =0 and M p,d.pdl when H R = 1 and a linear transition between these two values as H R varies between zero and one. Particles moving across the surface by raindrop induced saltation result in H R varying in time and space. Keeping account of the masses of the particles sitting on the surface provides a mechanism for calculating H R for each drop impact during the simulations. When flow velocities exceed the critical flow velocities associated with the onset of flow driven saltation, particles lifted into the flow by drop impacts are moved by flow driven saltation rather than by raindrop induced saltation. Consequently, the model deals dynamically with the downstream movement of particles lifted into the flow by drop impacts depending on the specific characteristics of particles and of the flow at the point of drop impact. The rainfall-runoff conditions used in the simulations were produced by using a runoff simulation model in conjunction with rainfall intensities observed during a convective rainfall event that occurred near Canberra, Australia so that flow depths and velocities varied in time and space. The runoff simulation model produces runoff by solving the exact forms of the kinematic overland flow equations as described by Moore and Kinnell (1987). Figure 3(A) shows how the model performed in comparison to the actual runoff rates measured when the storm used in the simulations actually occurred on a 40 m long bare fallow plot near Canberra, Australia. Two storms were used in the simulations: (a) the high intensity event shown in Figure 1(A) and (b) a more moderate event using one third of these intensities. These two storms were applied to two runoff producing situations: (i) The situation where runoff was produced uniformly over the surface of a plane where the infiltration rate controlling runoff was spatially uniform and set at 2 mm h 1. This case mimicked the situation for the bare fallow plot at Canberra. (ii) The situation where runoff was produced on the surface of a plane where the infiltration rate controlling runoff varied spatially depending on the depth of flow. That infiltration rate was set at 2 mm h 1 when flow depth was 0 mm and increased linearly with flow depth at a rate that gave an infiltration rate of 150 mm h 1 when the flow depth was 10 mm. This results in the runoff coefficient decreasing with slope length (Figure 3B). Reductions in the volume of runoff produced per unit area with increases in slope length have been observed in some rainfall erosion experiments producing sheet erosion under natural rainfall (e.g. Yu and Rosewell, 2001; Rejman and Usowich, 2002). The cause of such reductions in runoff may be interpreted as resulting from the surface not being fully submerged so that water entry from flow occurs over a greater area as rainfall intensity increases. However, in the simulations, deeper flows are perceived to result in greater hydrostatic head which enhances infiltration as sheet flow accumulates downstream. Runoff was simulated for two slope gradients (5% and 0 5%). In all the simulations, Manning s n was set at The simulations were performed using a mixture of six different materials: 0 1 mm sand, 0 2 mm sand, 0 46 mm sand, 0 9 mm sand and 0 46 mm coal were used as examples of materials that move as a result of raindrop induced saltation when flow shear stresses are below the critical value necessary to cause flow driven saltation once particles are detached from the cohesive soil matrix. Coal has a density close to that of soil aggregates. Very small particles that remain suspended in the flow and do not return to the bed before being discharged

4 1396 EARTH SURFACE PROCESSES AND LANDFORMS Figure 3. (A) Rainfall, observed runoff and modelled runoff rates for the natural rainfall event on a 40 m long bare plot with a slope gradient of 4% (Kinnell, 1985) and Manning s n = 0 03 and (B) runoff ratios for the rainfall and infiltration conditions used in the simulations. from the eroding area were used to simulate suspended load. Laboratory experiments with raindrops impacting flows over non-cohesive beds of uniform sized particles show that the mass of particles lifted in the cloud (M p,d ) is influenced by particle characteristics, drop size and flow depth. The characteristics the particles used in the simulation of raindrop induced saltation were chosen because laboratory experiments with these materials provide data that facilitate operation of the model when d = 2 7 mm. Kinnell (1990) observed that when particles are lifted into the flow and fall back to the bed, the discharge of sediment across any arbitrary boundary is controlled by the impacts of drops within the distance X p,d of that boundary (Figure 1B). If M p,d is the mass of p sized material lifted into the flow by a drop of size d, then q s (p,d) =M p,d (F d X p,d ) (5) where q s (p,d) is the mass of sediment of size p discharged per unit width of flow in unit time (in g m 1 s 1 ) and F d is the spatially averaged frequency of the impacts of drops of size d (number of drops in m 2 s 1 ) in the zone that extends the distance X p,d (in metres) upslope of the boundary. Given that X p,d = t p,d u (6) where, as noted earlier, t p,d is the time particles of size p remain suspended in the flow following disturbance by the impact of drops of size d, and u is the depth averaged flow velocity, q s (p,d) =M p,d (F d t p,d u) (7) Experimental data exist that enable values of M p,d and t p,d to be obtained when flows are impacted by 2 7 mm drops. Kinnell (2001) observed that the effective average distance 0 46 mm particles of coal travelled in experiments with 2 7 mm drops impacting 7 mm deep flows was 22 mm when the flow velocity was 40 mm s 1. Consequently, t p,d for 0 46 mm coal when 2 7 mm drops impact a 7 mm deep flow is 0 55 s. The data obtained for 0 46 mm sand gives a value for t p,d of 0 2 s for that material under the same rain and flow conditions. No data for travel distances of 0 1 mm, 0 2 mm and 0 9 mm sand were obtained by Kinnell (2001). However, assuming that t p,d is directly related to the settling velocity of the particles in water (v p ), the data for the 0 46 mm sand and coal materials indicate that t p,d =1.886h v p 1 h <9mm (8) provides a reasonable estimate of the effects of flow depth (h) and particle size and density on t p,d for flows impacted by 2 7 mm drops when h is in millimetres, v p is in mm s 1, and t p,d is in seconds. The limit to h results from the fact as h increases beyond about 3d, detachment and uplift result from the collapse of the Raleigh jet rather than during the

5 THE IMPACT OF SLOPE LENGTH ON THE DISCHARGE OF SEDIMENT BY RAIN IMPACT INDUCED SALTATION AND SUSPENSION 1397 development and collapse of the cavity that forms when drops impact the water layer (Moss and Green, 1983). This change influences the height to which material is lifted into the flow and hence t p,d. Particles lifted into the flow within the distance X p,d of the boundary pass across the boundary without returning to the bed. Particles lifted into the flow at a distance greater than X p,d of the boundary return to the bed before the boundary and provide a degree of protection (H R ) to the underlying cohesive soil surface so that Equation 4 applies. From the experiments reported by Kinnell (2001), M p,d.pdl for 0 46 mm sand is 10 mg, and for 0 46 mm coal, 17 mg when 2 7 mm drops impact 7 mm deep flows. These data, together with data on sediment discharges produced by 2 7 mm drops impacting beds of 0 11 mm, 0 2 mm and 0 9 mm sand (Kinnell, 2001), indicate that M p,2 7.PDL = 135 v p (9) provides a reasonable estimate of the values of M p,d.pdl in 7 mm deep flows impacted by 2 7 mm drops when v p is expressed in mm s 1 and the loose layer is made up of particles that all have the same size and density. However, when, as in the simulations, the loose layer contains a number of particle sizes and particles with different densities, if it is assumed that the movement of material from this layer into the cloud is considered to be not affected by particle characteristics, Equation 9 can be used to generate a value for the mass of material in the area impacted that gives H R = 1 0 taking into account the relative proportions of the various materials in the layer. If the mass of material in the layer of loose particles is less than this value, then all the particles sitting on the bed are lifted into the cloud. However, when the mass of material in the layer of loose particles is greater than the H R =1 associated value, the total mass of material lifted into the cloud is limited to that H R = 1 0 associated value, and the value of M p,d.pdl for each of material is determined as a proportion of that mass. Some or all of the energy possessed by raindrops is absorbed in the water layer that occurs above the bed. As a result, both M p,d.m and M p,d.pdl are influenced by flow depth. The sediment concentration (c s ) of a material in the outflow is given by dividing sediment discharge by water discharge. Since, the water discharge is given by the product of flow depth and velocity, Equation 7 leads to c s (p,d) =q s (p,d) h 1 u 1 = M p,d F d t p,d h 1 (10) Consequently, when, as indicated by Equation 8, t p,d aries directly with flow depth, the effect of flow depth on M p,d is directly related to the effect of flow depth on sediment concentration when F d is constant. Kinnell (1993a) observed that for beds of sand, sediment concentration varied with flow depth according to the equation c s (p,d) =k p I d f[h,d] (11) where k p is a coefficient that varies with particle size and density, I d is rainfall intensity of the rain produced by the d sized drops. The function f[h,d] accounts for the effects of flow depth and velocity on sediment concentration when F d is constant. Kinnell observed from laboratory experiments with three sizes of sand (0 1 mm, 0 2 mm and 0 9 mm) and five sizes of raindrop (0 8 mm, 1 3 mm, 2 7 mm, 3 7 mm and 5 1 mm) that that function was expressed by f[h,d] = exp( h) h h c (12a) Figure 4. The relationship between sediment concentration and the effect of flow depth on sediment concentration predicted by Equation 11 for rainfall with 2 7 mm drops impacting flows over 0 2 mm sand in experiments used by Kinnell (1993a). f[h,d] = exp( h b d (h h c )) h > h c (12b) when drop size (d) and flow depth (h) are expressed in millimetres, and h c = ln (d) (13) b d = exp( d) (14) While this broken-stick exponential model is consistent with the observation that, as flow depths decrease, sediment concentration tends towards a value that is independent of drop size, because f[h,d] is directly related to c s (p,d) I d 1, Figure 4 shows that, for 2 7 mm drops impacting 0 2 mm sand and the flow depths used in the simulations (less than 9 mm), Equation 12 can be replaced by a polynomial equation; f[h,2 7mm] = h h h <9mm (15) Although experiments with non-cohesive surfaces under rain impacted flows provide data on the effect of flow depth on sediment discharge when H R = 1, experiments with soil (Kinnell and Wood, 1992) have shown that the form of the sediment discharge to flow depth relationship observed for beds of sand also applies to cohesive soil surfaces where H R < 1. Given values for M p,d.m and M p,d.pdl when flow depth = 7 mm, Equation 15 results in M p,2 7.M = M* p,2 7.M ( h h ) h <9mm (16) where M* p,2 7.M is the value of M p,2 7.M when h = 7 mm, and M p,2 7.PDL = v p (14 36 h h ) h < 9 mm (17) when Equation 9 applies. Thus, M* p,2 7.M for the fine material was arbitrarily set at 0 01 mg per drop. For all the other materials, M* p,2 7.M was arbitrarily set 0 25 mg per drop. Consequently, when, in the simulations, a 2 7 mm drop impacted a 7 mm deep flow, it lifted 1 26 mg of material into the flow when H R =0. For all materials transported by raindrop induced saltation, the product of M p,d and t p,d accounts for the effect of flow depth on sediment discharge (Equation 7). That product varies directly with the result of multiplying Equation 15 by flow depth (h),

6 1398 EARTH SURFACE PROCESSES AND LANDFORMS and Meyer, 1972; Kirkby, 1980; Iverson, 1980; Wright and Webster, 1991) but higher shear stresses are required in overland flow (Govers, 1989). Larionov et al. (2006), in experiments with sands varying in size from 0 11 to 2 0 mm developed relationships for the threshold flow velocity (u c ) associated with sediment transported in various modes in shallow overland flow. For flow driven saltation, and the observation by Larionov et al. (2006) that threshold flow velocity for 0 11 mm sand is 0 19 m s 1 leads to u c (p) =4 525 [g (ρ p ρ w ) p ρ w 1 0 ] 0 5 (19) Figure 5. The effect of flow depth on M p,d t p,d for 0 2 mm sand and 2 7 mm raindrops. M p,d t p,d α h h h h < 9 mm(18) and produces a peak at h = 3 3 mm (Figure 5). This depth was exceeded during the simulations, particularly when the high intensity event occurred on 0 5% slopes (Figure 6). As slope lengths and gradients increase, some particles moving across the surface by raindrop induced saltation may, at some point along the plane, subsequently move by flow driven saltation once the value of the flow shear stress exceeds a critical value (τ c(loose), Figure 1A). The classical Shields-curve is often used to determine the critical conditions for the initiation of flow driven saltation in overland flow (e.g. Foster where g is gravity (in m s 1 ), ρ p is the density of the particle (in kg m 3 ), ρ w is the density of the water (in kg m 3 ) and the particle size (p) is expressed in metres. Apart from u c =0 19ms 1 for 0 11 mm sand, Equation 19 gives u c = m s 1 for 0 46 mm coal and greater or equal to m s 1 for sand greater or equal to 0 2 mm. The threshold velocities obtained using Equation 19 were used to determine the position of the transition from raindrop induced saltation to flow driven saltation on the slopes during the simulations. As a result, flow driven saltation of 0 46 mm coal and 0 11 mm sand occurred on plots longer than 15 m in the simulations associated with the high intensity event when runoff is produced uniformly on the planes inclined at 5% (Figure 7). Once flow driven saltation occurs, particles move downstream at average velocities that are linearly related to the shear velocity of the flow and, at any given shear velocity, larger particles travel faster than smaller ones (Govers, 1989). Figure 6. Peak brink depths for plots where the controlling infiltration rate is (A) 2 mm h 1 and (B) varies from 2 mm h 1 when the flow depth is 0 mm to 150 mm h 1 when the flow depth is 10 mm.

7 THE IMPACT OF SLOPE LENGTH ON THE DISCHARGE OF SEDIMENT BY RAIN IMPACT INDUCED SALTATION AND SUSPENSION 1399 Figure 7. Peak brink flow velocities for plots where the controlling infiltration rate is (A) 2 mm h 1 and (B) varies from 2 mm h 1 when the flow depth is 0 mm to 150 mm h 1 when the flow depth is 10 mm. Also, the average velocities may exceed that of the depth average velocity of flow (u) when sediment concentrations are low but particles travel at velocities less than u at higher sediment concentrations (Abrahams and Atkinson, 1993). As a first approximation, it is assumed that once flow driven saltation occurs, 0 46 mm particles of coal move downstream at the velocities equal to u while 0 11 mm sand particles move at 80% of that velocity. As noted earlier, in the simulations, each drop impact disturbed a 5 mm by 5 mm (25 mm 2 ) area of the 500 mm wide bed. A square rather than round disturbed area was used for simplicity. The size of the disturbed area was based on the fact that a drop impact produces a crater whose maximum depth is 3d (Engel. 1966). Consequently, an area of about 5 mm in diameter will be disturbed when 2 7 mm drops impact flows that are 7 mm deep. Also, as noted earlier, the position of the drop impacts was random to the extent that it was chosen using a computer based random number generator. Each material had a two-dimensional array for keeping account of the spatial distribution of that material in the loose layer or in suspension within a grid of 1 mm cells during each event. At the start of all simulations, H R was set to zero. Each drop impact lifted a mixture of all six materials into a cloud that had an 11 mm by 11 mm (121 mm 2 ) horizontally projected area. The size of the cloud was arbitrary and compares with the 18 mm by 18 mm (324 mm 2 ) projected area used by Kinnell (1994) for a 5 mm drop impacting a 5 mm deep flow. The detached material was distributed uniformly through the cloud. The fine particles that remain suspended when lifted into the flow were added to those already in the flow and moved horizontally at the depth average velocity of the flow (u). The actual size of the fine particles was immaterial to the simulation. Detachment of material from the cohesive soil mass was not selective in terms of the size and density of the particles. Also, the uplift of loose particles sitting on the surface was not determined in any way by particle characteristics. Consequently, the relative proportions of the materials lifted into the cloud were directly related to the relative proportions in the cohesive soil mass and the layer of loose particles sitting on the surface. At any given time, H R was calculated summing the ratios of respective masses of loose material sitting on the surface to the values of M p,d.pdl with the limit to H R being 1 0. As a result, M p,d varies in time and space during the simulations through the combination of Equations 4, 16 and 17. Because the fine material that travels as suspended load does not return to the bed, the value of M p,d.pdl for the fine material is always zero, and this causes the discharge of the suspended load to decline as the mass in the layer of previously detached material increases (Kinnell, 2006) and protects the surface of the cohesive soil mass. Results Amounts discharged and sediment yields The moderate intensity event produced the lowest peak brink flow velocities recorded when the infiltration rates increased with flow depth on 0 5% slopes (Figure 7B). It also produced flow depths that were well below the depth that produces

8 1400 EARTH SURFACE PROCESSES AND LANDFORMS Figure 8. (A) Amounts discharged and (B) sediment yields produced during the moderate event on 0 5% slopes when the infiltration rate varied with flow depth. the peak in M p,d t p,d (Figure 6B). In addition, on plots 15 m and longer in length, the time of concentration exceeded the time to the first peak in rainfall intensity so that the peak brink depths (Figure 6B) and flow velocities (Figures 7B) did not increase with slope length on plots longer that 15 m. Figure 8 shows the amounts of sand, coal and fine material discharged during this event (mass/width/event) together with the sediment yields (mass/area/event). There is an almost linear increase in the discharge of the fine material as plot length increases but the increase in the discharge of the 0 1 mm sand is less linear (Figure 8A). There is an increase in the discharge of all the coarser materials between the 2 m and the 30 m lengths but the increases with plot length are not often regular. However, in all cases, sediment yield decreases non-linearly with plot length from a peak at the 2 m length (Figure 8B). Figure 9(A) shows the amounts of sediment discharged during the high intensity event on 5% slopes when the runoff controlling infiltration rate is constant at 2 mm h 1. In this case, flow velocities exceed the critical velocities for flow driven saltation on 0 46 mm coal (Figure 7A) and 0 1 mm sand when slope length increases beyond the 20 m length. As a result, the discharges of these two materials increase rapidly with plot length beyond the 20 m length in the case of 0 46 mm coal, and the 25 m length in the case of 0 1 mm sand (Figure 9A). This change in transport mechanism on the lower part of these plots reversed the decline in sediment yield with plot length that occurred for these two materials on the shorter plots (Figure 10A). Figure 9(B) shows the amounts of sediment discharged during the high intensity event on 0 5% slopes when the runoff controlling infiltration rate is constant at 2 mm h 1. In this case, the peak brink flow velocities did not exceed critical flow velocities for flow driven saltation on the slope lengths being considered. However, peak brink depths exceeded the depth for the peak in M p,d t p,d when plot lengths were more than 5 m (Figure 5A). Although this had little impact on the non-linarity of the effect of plot length on the discharge of the fine material, it did cause the discharges of the material travelling by raindrop induced saltation to peak at the 15 m length. Even so, as with the moderate intensity event on 0 5% slopes, sediment yield decreases non-linearly with plot length from a peak at the 2 m length (Figure 10B). Without exception, sediment yields associated with material travelling as a result of raindrop induced saltation produced during the simulations undertaken here decreased non-linearly with plot length from a peak at the 2 m length. Sediment concentrations When runoff is generated uniformly over the plots, event sediment concentrations and sediment yields (Y s ) are directly related to each other (Figure 10) because, in

9 THE IMPACT OF SLOPE LENGTH ON THE DISCHARGE OF SEDIMENT BY RAIN IMPACT INDUCED SALTATION AND SUSPENSION 1401 Figure 9. Amounts of sediment discharged during the high intensity event from (A) 5% slopes with uniform infiltration characteristics and (B) 0 5% slopes with uniform infiltration characteristics. Y s = q w c s L 1 (20) where q w varies directly with plot length (L). However, this is not the case when infiltration rates vary spatially. Figure 11(A) shows the sediment concentrations produced when the moderate intensity event occurred on 0 5% slopes where infiltration rates varied with flow depth. When compared with the sediment yields shown in Figure 8(B), the decline in sediment concentration for the materials moving by raindrop induced saltation between the 2 m and the 30 m lengths is less severe that the decline in the sediment yields between those lengths. Also, the sediment concentration for the fine material increases with plot length where as the sediment yields for this material decline albeit slightly as plot length increases (Figure 8B). Figure 11(B) shows the sediment concentrations for the high intensity event on 5% slopes where infiltration rates varied with flow depth. Composition of sediment discharge Figure 12 shows that the composition of the sediment discharge became finer as slope length increased when the infiltration rate varied with flow depth on 0 5% slopes. However, the compositions produced by the high intensity event (Figure 12B) were coarser than those produced by the moderate intensity event (Figure 12A). Also, Figure 12 shows that the compositions were finer than the composition of the original soil source. The same trends occurred for the moderate event on 5% slopes and where the runoff was produced uniformly. Flow driven saltation enhanced the fining of the sediment discharge on the plots when it occurred on the 5% slopes (Figure 13B). Discussion The simulations provide qualitative data on the discharge of particles detached from a cohesive soil surface by raindrop impact and subsequently transported to the outfall in suspension, by raindrop induced saltation and episodically by flow driven saltation. While the simulation model is based on the relationships described by Equations 4 19, Figure 1, Equations 4 7 and 10, and Figures 4 7 are central to understanding the results. For simplicity, consider just one material travelling by raindrop induced saltation. Equation 5 is based on the understanding that if a particle travelling by raindrop induced saltation is to be discharged directly across the downstream boundary, the drop impact that induces the saltation must occur within a limited distance from the boundary (Figure 1B). That limit is determined by the distance X p,d that a particle travels after it has been induced to saltate. Laboratory experiments with non-cohesive beds (Kinnell, 1991) where H R is maintained constant at 1 0 have shown conclusively that when flow depth is held constant, sediment discharged as the result of raindrop induced salutation in flowing water varies directly with the depth averaged velocity of the flow (u). The implication of this result is that particles transported by raindrop impact induced saltation move downstream at rates that vary directly with the depth averaged flow velocity. The simulation model is based on this general relationship between downstream particle travel rate and flow velocity, and the fact that also X p,d depends on the time the particles remain in the flow during a saltation jump (t p,d ). Consequently, X p,d varied directly with the product of u and t p,d (Equation 6) so that the mass of a particular material discharged as a result of raindrop induced

10 1402 EARTH SURFACE PROCESSES AND LANDFORMS Figure 10. Sediment yields produced by the high intensity event on (A) 5% slopes with uniform infiltration characteristics and (B) 0 5% slopes with uniform infiltration characteristics. saltation depends directly on the product of u and t p,d (Equation 7). Flow depth also influences particle travel rate because flow depth influences the height particles are lifted into the flow and hence t p,d. At low flow depths, the height of the water surface restricts the height particles can be lifted from the bed where as in deeper flow, the absorption of raindrop energy in the water layer limits the height particles can be lifted from the bed. In the simulations, flow depth and velocity increase with slope length (Figures 6 and 7) in a manner that causes X p,d to increase as slope length increases. However, variations in flow depth influence M p,d, and the effect of flow depth on M p,d counters the effect of the increases in X p,d on the rates of discharge. The influences of flow depth on M p,d and t p,d are responsible for the amounts of the materials discharged by raindrop induced saltation peaking at 15 m when the high intensity event occurred on the 0 5% slopes that had spatially uniform infiltration characteristics (Figure 9B). When flow driven saltation occurs, X p,d increases markedly as flow length increases, and this causes sediment concentrations to increase with slope length (Figure 10A). With fine material moving in suspension, the effective value of X p,d is the length of the plot. Consequently, the discharge of fine material is not affected by flow depths near the downstream boundary to the same extent as the discharge of material moving by raindrop induced saltation. Although the simulations were restricted to rain containing only 2 7 mm drops, the general forms of the relationships for the effects of flow depth on sediment concentration (i.e. sediment concentration declines with flow depth) and sediment discharge when flow velocity is held constant (i.e. sediment discharge peaks at some flow depth before declining as flow depth increases) apply to rain made up of a wide range of sizes (Kinnell, 1993a). Consequently, the relative amounts of the materials discharged during the simulations should be reasonably indicative of the relative amounts that would be discharged under more natural rain and flow conditions even though the rain used in the simulations was made up of only 2 7 mm sized drops. There have been suggestions that the maximum distances particles move during a rainstorm do not increase as slope length increases beyond a certain length (Rejman and Usowich, 2002; Parsons et al., 2006). As a result, sediment yields increase with slope length before peaking and then declining as slope length increases (Parsons et al., 2006). However, such an effect was not observed during the simulations undertaken here. In all cases for the moderate event, sediment yields declined non-linearly with slope length irrespective of whether runoff was generated uniformly or not (Figure 8B), and for the high intensity event on 0 5% slopes. Figure 10(B) shows that this decline even occurred when peaks in the amounts discharged did occur as slope length increased on 0 5% slopes (Figure 9B) because flow depths exceeded depth that produced a peak in the product of M p,d t p,d (Figures 5 and 6). Given that the slope gradients used in the experiments analysed by Rejman and Usowich (2002) and Parsons et al. (2006) were much greater than the 5% gradient used in the simulations, flow depths in their experiments were probably not great enough to have exceeded those which cause M p,d t p,d values to peak for the drops that contribute most to raindrop driven erosion under natural rainfalls. Also, as shown in Figures 9(A), some particles travelling by raindrop impact induced saltation may, when the flow velocity exceeds a critical value at some point

11 THE IMPACT OF SLOPE LENGTH ON THE DISCHARGE OF SEDIMENT BY RAIN IMPACT INDUCED SALTATION AND SUSPENSION 1403 Figure 11. Event sediment concentrations for (A) the moderate intensity event on 0 5% slopes and (B) the high intensity event on 5% slopes when infiltration rates varied with flow depth. along the flow line, subsequently move by flow driven saltation. When this happens, the rate at which these particles move downstream increases because they do not need to wait for drop impacts before they move with the result that the sediment yield of these particles increases with slope length (Figure 10A). In effect, the onset of flow driven saltation markedly increases X p,d and, as a result, the rates particles of size p are discharged increase. Arguably, the critical flow velocities for the change between raindrop induced and flow driven saltation used in the simulations may be too high and consequently, the effect of flow driven saltation is underestimated. However, the effect of flow driven saltation on the transport of material detached from the cohesive soil surface is likely to become more important as slope gradients increase towards those used by Rejman and Usowich (2002) and Parsons et al. (2006). The simulations illustrated the fact that the composition of the sediment discharged may differ substantially from the original composition of the eroding cohesive surface, and that the high intensity rainfall event produced coarser sediment discharges than the moderate intensity event (Figure 12). The reason for this is that, for any given flow depth-velocity condition, a time of concentration exists for each particle size density situation, and the higher intensity event generates shorter times of concentration. Also, the proportions of the various materials in the sediment discharge will not be the same as that in the original cohesive surface until after the times of concentration for all the moving particles have been satisfied. Material in transit that does not reach the outfall during a storm is available for transport during the next storm. Subsidiary simulations tracing the movement of particles in a 7 mm deep 150 mm s 1 velocity flow over a 2 m long segment produced times of concentration of 18 minutes for 0 46 mm coal and 49 minutes for 0 46 mm sand when 2 7 mm drops produced rain at 60 mm h 1. Consequently, many rainstorms are needed before steady state conditions occur under natural rainfall, particularly as slope length increases. There are approaches other than the one used here that are designed to model the downslope movement of material lifted into the flow by raindrop impact. In the Hairsine and Rose (1991) approach, material moving by flow driven saltation is considered to contribute to the protection of the underlying material from raindrop impact but this is not considered in the approach used here. Consequently, although the protection provided by the material sitting on the bed in the simulations declines when flow driven saltation occurs, because that protection is largely provided by coarse sand, the reduction may not have had a substantial impact on the results. Also, in comparison, the Hairsine and Rose approach does not deal with the dispersive effects that are associated with the temporal and spatial random nature of raindrop impact which are relevant to the movement of soil material from source areas (Kinnell,

12 1404 EARTH SURFACE PROCESSES AND LANDFORMS Figure 12. Cumulative proportions of discharged sediment for (A) the moderate intensity event and (B) the high intensity event of 0 5% slopes when infiltration rate varied with flow depth. 2008) or deal with the effect of the episodic change between raindrop induced saltation and flow driven saltation that can occur during an event. In addition, the time varying solutions for the equations for raindrop detachment, the plucking of soil particles from the cohesive soil mass, followed by raindrop induced saltation associated with the Hairsine and Rose approach tend to be both complex and computationally demanding (Hairsine et al., 1999). In common with the Hairsine and Rose (1991) approach, the model used in the simulations does not deal with the downstream movement of material that rolls after it is disturbed by raindrop impact. That transport process is perceived to enhance the discharge of coarse material beyond expectations that are derived from considering the effect of settling velocity on raindrop induced saltation. Certainly, given appropriate data on raindrop induced rolling, the model could be extended to include the movement of material larger than 1 mm in size. Such a development is a matter for the future. Also, the model used in the simulations does not deal with the transport of material by aerial splash. Splash erosion occurs for considerable periods of time on bare surfaces in many locations because considerable amounts of rain occur at intensities that produce little or no runoff. Because splash transport is a highly inefficient transport system, rain can generate considerable amounts of loose pre-detached material which is available for transport when surface water flows subsequently occur. The model used here takes no account of this, detachment only occurs when there is flowing water on the surface. Models such as PSEM_2D (Nord and Esteves, 2005), consider the build up of loose material on the surface when flow driven saltation does not occur and so include an effect on the material available for transport in runoff resulting from splash erosion but fail to deal with the effect of the downstream transport of detached material by raindrop induced saltation. While the simulations reported here may generate total amounts for the materials discharged during an event that are below those that would be produced when detachment by raindrop impact in all conditions is considered, the relative effects of slope length on the results produced here are unlikely to change substantially. The conclusion that the amount of sediment discharged during an event by raindrop induced saltation and episodically by flow driven saltation tends to increase with slope length because particle travel rates are controlled by flow velocities that tend to increase with slope length is unlikely to change. Also, the gross effect of flow depth on M p,d t p,d is independent of the amount of loose material sitting on the bed, so that the peak in the amounts discharged for the high intensity event on the 0 5% slope when infiltration is uniform will occur irrespective of whether the effect of splash erosion is considered in the model or not. MAHLERAN is a soil erosion model that considers the transport rates of particles moving in rain-impacted flow (Wainwright et al., 2008). For transport by unconcentrated flow and rain with a more natural drop size distribution than used here, the virtual velocity for a particle of size ϕ is given by v p.u.ϕ = 0 525R 2 35 ω M ϕ 1 (20)

13 THE IMPACT OF SLOPE LENGTH ON THE DISCHARGE OF SEDIMENT BY RAIN IMPACT INDUCED SALTATION AND SUSPENSION 1405 Figure 13. Compositions of the total sediments discharged for the high intensity event on (A) 0 5% slopes and (B) on 5% slopes when the runoff controlling infiltration rate was spatially constant. where R is rainfall intensity, ω is the stream power of the flow, and M ϕ is the mass of the particle. For concentrated flow, the virtual velocity is given by v p.c.ϕ = (ω ω *ϕ ) (21) where ω *ϕ is the threshold stream power for the initiation of movement of particles of size ϕ by the flow. In the context of the modelling approach presented here, it is perceived that Equation 20 applies to raindrop induced saltation and rolling (given that the data upon which it was derived only relate to relatively coarse particles) and Equation 21 applies to flow driven saltation and rolling. In MAHLERAN, the effect of flow depth (h) on detachment by raindrops is given by where β ϕ is an empirical parameter and ε u. ϕ = ε r. ϕ e β ϕ h (22) ε r. ϕ = a KE b ε r. ϕ = a KE b S c S =0 (23a) S > 0 (23b) where KE is the kinetic energy per unit quantity of rain, and a, b, c are empirical parameters that are a function of particle size and density. Equation 22 is used despite the fact that detachment, the plucking of particles from the cohesive soil matrix, is considered not to be particle size selective in most physical models, and the fact the exponential model of Torri et al. (1987) was not based on measurement of the actual amount of material plucked from the cohesive soil matrix under the flow. It was based on measurement of material transported by drop splash, and the relationship between the amounts transported by drop splash and detached under water varies with flow depth and drop size because, for example, flow depth influences, among other things, splash trajectories (Kinnell, 2005). Also, some raindrops may pass into the water layer and disturb the underlying surface without causing any splash (Moss and Green, 1983). Conclusion On sloping surfaces covered with flowing water, particles detached from the cohesive soil matrix by raindrop impact move downslope at any given time (a) by raindrop splash, (b) by raindrop induced rolling, (c) by raindrop induced saltation, (d) by flow driven rolling, (e) by flow driven saltation or (f) as a result of remaining in suspension long enough not to return to the bed before they are discharged. The mode of transport adopted depends of the flow conditions and the size and density characteristics of the particles involved. Temporal and spatial variations in flow conditions may result in a particle switching modes one or more times during its travel between the point of detachment and the point where it is discharged. The rate at which a particle travels at any given time will vary depending on which mode of transport is adopted and the size and density of the particles.