Water Resources Research

Response of a soil-mantled experimental landscape to exogenic forcing

Authors

  • Lee M. Gordon,

    1. Department of Geography, State University of New York at Buffalo,Buffalo, New York,USA
    2. New York State Energy Research and Development Authority,West Valley, New York,USA
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  • Sean J. Bennett,

    Corresponding author
    1. Department of Geography, State University of New York at Buffalo,Buffalo, New York,USA
      Corresponding author: S. J. Bennett, Department of Geography, State University of New York at Buffalo, Buffalo, NY 14261, USA. (seanb@buffalo.edu)
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  • Robert R. Wells

    1. USDA-ARS National Sedimentation Laboratory,Oxford, Mississippi,USA
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Corresponding author: S. J. Bennett, Department of Geography, State University of New York at Buffalo, Buffalo, NY 14261, USA. (seanb@buffalo.edu)

Abstract

[1] Soil erosion can severely degrade landscapes, and concentrated flows such as rills and gullies can be the dominant contributor to the soil losses. This paper examines the growth, development, and spatiotemporal evolution of rills and rill networks using a soil-mantled experimental landscape subjected to simulated rain and downstream base level lowering. Rill incision and network development and extension occurred due to actively migrating headcuts formed at the flume outlet by base level lowering. The communication of this wave of degradation due to this exogenic forcing occurred very quickly in space, and resulted in nearly the same amount of bed incision throughout the network. Rill incision, channel development, and peaks in sediment efflux occurred episodically, yet these were in direct response to the downstream base level adjustments. Although flows were supply limited, most of the sediment efflux was genetically linked to headcut development and migration. The geometry of the eroded rills and the rates of headcut migration were well correlated to overland flow rate. These findings have important implications for the prediction of soil loss, rill network development, and landscape evolution where headcut erosion can occur.

1. Introduction

[2] Soil erosion is a critical concern worldwide for the sustainable management of agricultural regions. Soil erosion rates from agricultural areas are consistently much higher than both the rates of soil production and losses on nonagricultural hillslopes [Montgomery, 2007], soil degradation can devastate crop yields and cause wide-spread famine and starvation [Lal, 2009], and the sequestration and export of organic carbon from agricultural soils now are being critically assessed [Van Oost et al., 2007; Lal and Pimentel, 2008]. Moreover, the economic costs of soil erosion in the U.S. have been estimated to be tens of billions of dollars annually [Pimentel et al., 1995; Uri and Lewis, 1999; Lal, 2001].

[3] While soil erosion can be the primary cause of soil degradation, erosion in concentrated flows can be the dominant contributor to this sediment efflux. Rill development on landscapes can drastically increase rates of sediment discharge, accounting for a significant proportion of the total soil losses observed [Govers and Poesen, 1988; Römkens et al., 1997; Yang et al., 2006]. Similarly, gully development also may be the dominant sediment source within some watersheds [Watson et al., 1986; Poesen et al., 1996, 2003; Bennett et al., 2000b; Casalí et al., 2000; Valentin et al., 2005]. Taken together, rill and gully erosion represent discrete erosional processes that can be the principal sources of total soil erosion and the primary drivers of landscape degradation. The focus of the current paper is on rill erosion processes.

[4] Five mechanisms have been suggested to explain the formation of rills on soil-mantled landscapes. These include (1) a critical slope length for soil bed incision, (2) a threshold shear stress, stream power, or slope for soil bed incision, (3) the existence of seepage erosion and exfiltration, (4) overland flow hydraulics, and (5) headcut development and migration [e.g., Savat and DePloey, 1982; Torri et al., 1987; Slattery and Bryan, 1992; Huang et al., 2001; Yao et al., 2008]. These mechanisms need not be mutually exclusive, and all can be considered internal to the system (endogenically forced rill development). In contrast, experimental studies using relatively large physical models typically rely upon a mechanism for rill formation external to the system (exogenically forced), specifically base level lowering [Parker, 1977; Gardner, 1983; Pelletier, 2003; Douglass and Schmeeckle, 2007]. This base level adjustment produces a localized headcut that is communicated upstream, bifurcating as it does so to create a rill network, or incising and enhancing a preexisting drainage pattern.

[5] The significance of headcut erosion in upland concentrated flows such as rills, crop furrows, and ephemeral gullies has long been recognized. Headcut development and migration have been linked to rill and gully initiation and drainage network evolution [Parker, 1977; Bryan and Poesen, 1989; Bryan, 1990; Slattery and Bryan, 1992; Smith, 1993; Casalí et al., 1999; Foster, 2005] and significant increases in soil loss and sediment yield [Mosely, 1974; Römkens et al., 1997; Brunton and Bryan, 2000]. Because headcuts are critically important geomorphic phenomena in upland areas, their morphodynamic and hydrodynamic characteristics have been examined systematically using specialized experimental facilities [Bennett, 1999; Bennett et al., 2000a; Bennett and Alonso, 2005, 2006; Wells et al., 2009a]. This discrete scour process, however, has not been extensively studied in an experimental landscape where channel width is unconfined and where rill networks can develop freely. Moreover, the links between network development, sediment efflux, and the rill erosion processes over time and space remain poorly defined, primarily because the requisite data have not yet been collected.

[6] This research sought to define the processes of rill formation and network development within a soil-mantled landscape subjected to an exogenically forced perturbation (base level lowering). The objectives of the current study were (1) to develop a comprehensive data set of rill network evolution in an experimental landscape subjected to simulated rain and episodic base level lowering with a very high temporal and spatial resolution; (2) to define the relationship between rill network development and sediment efflux using direct and indirect sampling methods; and (3) to quantitatively define the response of a soil-mantled landscape over time and space to this exogenic forcing and its effect on the evolving rill network system. These results should facilitate the further development of soil erosion prediction technology to mitigate and manage the degradation of agricultural landscapes. Preliminary results of this work have been presented elsewhere [Gordon et al., 2010, 2011], and the current paper extends both the data analysis and discussion of these results.

2. Experimental Equipment and Procedure

2.1. Flume and Rainfall Simulator

[7] Two experiments were conducted in a 7.0 m long by 2.4 m wide flume (see Gordon et al. [2011] and Figure 1). These experiments, runs 1 and 2, were similar in design and execution, but they were not replicates. The flume was set at 5% slope, had 0.3-m high sidewalls, and had diversions at the downstream end to direct runoff through a 0.16 m wide weir. This slope was selected to maximize the application of the results to the study of landscape evolution and the degradation of agricultural areas. As such, slope was selected on the basis of two criteria: (1) it was hypothesized that a steeper slope was required to promote rill erosion endogenically, more characteristic of hillslope environments; but (2) the slope should not be too steep so as to negate the potential for agricultural activity. The weir was fully adjustable so that base level with respect to the soil bed could be modified during an experiment. The bottom of the flume was perforated and covered with layers of pea gravel and landscape fabric to facilitate free drainage of the soil bed. Simulated rainfall was delivered by four cone-jet nozzles (Lechler 460.848.30BK) at a mean fall height of 3.5 m. Municipal water was pumped to each of the nozzles at approximately 10 psi. A similar rainfall simulator produced about 75% of the kinetic energy equivalent of natural rainfall [Brunton and Bryan, 2000]. Rainfall intensities were measured at 250 rain gauges (mean gauge spacing of 0.25 m) arranged in a grid pattern within the flume prior to each experimental run and before the soil bed was prepared. Rainfall intensities averaged 87 mm h−1 with a standard deviation of 31 mm h−1 (Figure 2a) and total runoff rate collected at the downstream weir during rainfall calibration was 22.1 l min−1.

Figure 1.

Experimental landscape facility looking upstream showing flume, eroded soil surface, and sample collection weir. Flume length is 7.0 m and flume width is 2.4 m.

Figure 2.

Contour maps of (a) rainfall rate (contour interval of 10 mm h−1), and initial surface topography for (b) run 1 and (c) run 2 (contour interval of 0.01 m). Flume is oriented from upstream (top) to downstream (bottom).

2.2. Soil Material and Landscape Preparation

[8] The same soil was used in both experiments. The Ap horizon of the Cheektowaga fine sandy loam from Erie County, New York was chosen for its propensity to exhibit surface sealing characteristics conducive to headcut development and rill erosion. Soil was air dried to a moisture content of about 5%, mechanically crushed, and passed through a 2 mm sieve to remove gravel and large organic material. The prepared bulk soil material consisted of 20% clay, 22% silt, and 58% sand. Soil was added to the flume in layers up to 0.04 m thick and shaped incrementally to create a broad central depression representative of a natural drainage basin in the downstream half of the flume (Figures 2b and 2c). The prepared soil had a surface roughness on the order of millimeters, although this was not quantified. Final soil depths were about 0.22 m at the flume periphery and about 0.14 m along the central depression. Initial bulk density was 1380 and 1360 kg m−3 for runs 1 and 2, respectively.

2.3. Rainfall Application and Sediment Sampling

[9] For each experiment, rainstorms of the same intensity but different durations were applied to the prepared soil bed. For run 1, 15 rainstorms were applied with the following durations: 87, 14, 27, 24, 30, 30, 30, 45, 60, 80, 60, 80, 60, 75, and 85 min. For run 2, 20 rainstorms were applied with the following durations: 85, 15, 20, 20, 20, 20, 20, 30, 50, 90, 120, 180, 180, 20, 20, 20, 20, 30, 50, and 50 min. Each experimental run lasted up to 3 days, and the time between storms varied greatly (from several minutes to 10 h). The soil bed was covered with plastic overnight to prevent it from drying. No effects related to starting and stopping of the experiment were observable in the data. The first rainstorms were applied to dry soil beds and lasted about 85 min. After this initial wetting storm, the base level at the weir in both experiments was dropped by 0.03 m, creating a preformed step in bed elevation at the flume outlet. A second base level adjustment of 0.03 m was applied to run 2 after 850 min of rainfall application. Initial attempts to force rill development to occur endogenically using just rainfall application within this experimental setup were unsuccessful. An external (exogenic) forcing to the system (i.e., base level adjustment) was required to produce significant rill development using the imposed boundary conditions. For similarly sized experimental landscapes and rainfall application rates, much higher bed slopes than that used here (5%) were required to form incised rill channels endogenically (20% [Gómez et al., 2003]; 9% [Raff et al., 2004]; 10% [Yao et al., 2008]). Following this base level change, storms with relatively short durations were employed to capture digitally these stages of network evolution. As the experimental landscapes reached steady state conditions (see below), storm durations were increased. Rates of runoff and sediment discharge were measured during each storm at the flume outlet at an interval equal to approximately one third of a storm duration. Sediment samples captured in 1.0 L containers were weighed, decanted, and oven-dried, while runoff rates were measured by weighing a 30 s sample.

2.4. Digital Photogrammetry

[10] Bed surface topography was quantified using photogrammetry. A 10 Mpixel digital camera (Canon EOS 20D) was mounted to a movable carriage on a rail system suspended about 3.4 m above the soil bed. Before each storm, a series of nine photographs with 67% overlap were acquired along the longitudinal axis of the flume. Fifty photogrammetric control targets with a spacing of approximately 0.3 m were secured to the top of the flume sidewall. These targets were surveyed with an electronic total station (Pentax PCS-1s) from two fixed locations near the downstream end of the flume. Bearings (horizontal and vertical angles) to the targets from the two stations were converted to three-dimensional coordinates using intersection and triangulation methods as described by Chandler et al. [2001]. These control coordinates, interior and exterior camera orientation information, and lens distortion model parameters were linked to eight image pairs using commercially available software (Leica Photogrammetry Suite v. 9.2). An average of six control points were manually identified within each image pair and 150 radiometrically and/or geometrically similar tie points were identified automatically. These points then were used to obtain a triangulated solution for each image pair. A single digital elevation model (DEM) was generated for each image pair at a cell size of 3 mm, and these were combined to produce a DEM of the entire flume. These DEMs, therefore, became the primary source of all morphodynamic data. Uncertainties from all sources (surveying, control point digitizing, triangulation, and DEM interpolation) produce combined errors on the order of 1 to 3 mm in the horizontal and vertical dimensions, respectively [see Chandler et al., 2001; Heng et al., 2010].

3. Results

[11] As noted above, preliminary results from these experiments were presented by Gordon et al. [2010, 2011]. These results are briefly summarized below, expanded to include additional analysis and discussion.

3.1. Rainfall Application and Initial Landscape Response

[12] After landscape preparation, with the terminal weir spillway set at the height of the soil bed, 85 min of rainfall was applied. Water ponding was observed after about 5 min as the soil surface became saturated. Initial runoff occurred at about 10 min and averaged 7.0, 11.0, and 14.8 l min−1 at 20, 45, and 75 min, respectively (Figure 3). Time, as used herein, refers to the cumulative duration of successive rainfall applications and it is synonymous with experimental runtime. A wetting front observable through the flume sidewalls progressed quickly through the soil bed, reaching the bottom of the flume during the first storm. As the wetting front propagated downward, the soil bed and bed foundation (pea gravel and fabric) settled on the order of 0.02 m. Raindrop impact and sheet flow caused the formation of a thin, pliable soil seal as aggregates were dispersed and fines were washed into voids. In the broad central thalweg, approximately 1 m upstream of the weir, a very shallow headcut formed, a few millimeters in height, which migrated and expanded upstream to create a continuous, wide shallow scarp.

Figure 3.

Variation in applied and cumulative rainfall and overland flow rate (Q upper), total sediment discharge rate (Qs middle), and rill network drainage density (DD lower) as a function of time for (left) run 1 and (right) run 2. Timing of base level adjustments also are shown.

[13] Exogenic forcing was accomplished by lowering the terminal weir by 0.03 m at the conclusion of the initial wetting storm. This created a preformed step at the flume outlet measuring 0.16 m wide and 0.03 m high. Once rainfall recommenced, runoff occurred almost immediately, passing over the preformed step at the outlet, eroding its face and causing it to migrate quickly upstream. The step, now termed the primary headcut, took the form of an arcuate-shaped headcut with a vertical face as it migrated upstream along the path of concentrated flow. As the primary headcut passed the center portion of the flume, it encountered the topographic transition between the broad central depression (thalweg) and the upland hillslope (see Figure 2). The primary headcut then bifurcated into lower order tributary headcuts at this transition. Concurrently in the downstream half of the flume, additional headcuts (termed secondary headcuts) were initiated within the now incised primary channel. These secondary headcuts propagated through the network and into preexisting tributaries in semiregular waves of degradation. While the primary headcut originated directly from the preexisting step at the base level drop, the secondary headcuts tended to form at channel confluences and at small undulations in the channel bed where locally high shear stresses might be expected to occur [Giménez et al., 2009]. The most upstream areas of the flume in both experiments remained uneroded (clearly demarcated in Figures 4a and 5c below). These observations are in general agreement with the experiments summarized by Schumm et al. [1987].

[14] Both experimental landscapes were subjected to significant amounts of rainfall. In total, 1141 and 1532 mm of rainfall was applied to the soil in runs 1 and 2, respectively. Runoff rates during each experiment, however, quickly attained nearly constant values after brief periods of rainfall application. In both experiments, a constant runoff rate of 0.00027 m3 s−1 (16.2 l min−1) was achieved after approximately 80 min of applied rainfall, and this runoff rate did not change as a result of base level adjustment and variable rates of soil erosion and network development within the landscape (Figure 3). Given that the rainfall rate applied to the entire soil surface was 22.1 l min−1, as measured during rainfall calibration, approximately 27% of the applied rain infiltrated into the soil bed, while 73% ran off the soil surface. Pore water pressure conditions are important in rill erosion. Erosion rates and rill incision processes may be enhanced in the presence of positive pore pressures and exfiltration [Bryan and Rockwell, 1998; Gabbard et al., 1998] and suppressed in the presence of negative pore pressures [Römkens et al., 2002; Wells et al., 2009a], and Brunton and Bryan [2000] observed both infiltration and exfiltration using a similar facility. While no measurements of pore water pressure were taken, it is assumed here that (1) all pore water pressures were negative because of the free-drainage of the soil landscape, and (2) the infiltration rate was constant in space.

3.2. Sediment Signatures and Steady State Erosion

[15] Peaks in soil erosion occurred in direct response to the development and upstream migration of the primary headcut formed due to base level adjustment. Figure 3 shows that the peak in sediment discharge Qs following base level lowering was 55.3 g s−1 for run 1, and 40.4 and 32.5 g s−1 for the two base level adjustments in run 2. This peak in Qs represents an increase of approximately 10 to 20 times that observed just prior to base level adjustment. Sediment discharge decreased quickly with time following base level adjustment, achieving asymptotic rates of 5 g s−1 in run 1 and 4 g s−1 in run 2. This nearly constant soil erosion rate was maintained until the conclusion of the experiment (run 1) or until base level was again adjusted (run 2). Erosion rates after the second base level drop in run 2 were comparable to those following the initial base level drop, but the experiment appears to have been terminated prior to the attainment of an asymptotic rate of sediment discharge. The attainment of nearly constant rates of sediment discharge with time following base level adjustment is consistent with previous observations, wherein steady state rates of soil erosion typically are observed in experimental plots [Parker, 1977] and postulated conceptually [Schumm et al., 1987].

[16] The base level adjustments and their peaks in sediment discharge caused much of the subsequent soil loss from the landscape. The contribution of sediment discharge associated with base level adjustment and total sediment discharge can be determined simply by integrating these values over time t:

display math

[17] Where inline image is the time of base level adjustment and inline image is the time when the peak sediment discharge has abated (assumed here to be the time when Qs approaches asymptotically to within 20% of that rate observed at steady state). Using (1), the asymptotic time of steady state for Qs following the (first) base level adjustment is 440 min for run 1 and 433 min for run 2, and the ratio of the base level sediment flux to the total sediment flux is 0.69 for run 1 and 0.71 for run 2. In both cases, approximately 70% of the sediment discharge from the evolving soil landscape occurred because of the imposed 0.03 m base level adjustment.

[18] Cumulative sediment efflux also can be calculated using the geospatial data. The incremental soil loss measured at the flume outlet can be compared directly to the volume differences calculated from successive DEMs. Experimental observations indicated that two adjustments were necessary to derive sediment effluxes using the DEMs. First, soil loss during the initial wetting storm, as calculated by subtracting the postwetting storm DEM from the initial dry-soil DEM, was higher than sediment discharge measured at the flume's exit (which was negligible). Observations indicated that this volumetric loss was due to a flume-wide settling (∼17 mm) of the soil and bed foundation (as noted above). Second, soil bulk density was observed to vary with time, which affected the calculation of sediment efflux. Preliminary experiments and observations demonstrated that soil bulk density increased slightly with time during rainfall application. Soil bulk density was approximately 1380 and 1360 kg m−3 at the start of runs 1 and 2, respectively, and these values increased to 1520 and 1560 kg m−3 by the end of the experiments (an increase of about 15%). As such, a linear time variation in bulk density on the basis of these end-member values was applied to convert the volumetric effluxes determined from the DEMs to mass fluxes at the flume outlet. Figure 3 shows the very close agreement between cumulative sediment discharges measured at the flume outlet and those determined from the DEMs employing these adjustments.

[19] All soil eroded during base level adjustment and through rill incision was removed from the landscape. Figure 4 shows time-successive cross sections for run 2 obtained from the DEMs. Very little deposition occurred within the landscape, except for the fan-like accumulation in the extreme downstream end of the flume as conditioned by the terminal weir (similar results were observed in run 1). The sequential DEMs showed no sediment storage on rill interfluves or within the rill channels themselves as the landscape evolved. This observation suggests strongly that both experiments were dominated by supply-limited flow conditions, which will be further discussed below.

Figure 4.

Morphometric analysis of run 2 showing (left) an orthophotograph of the final landscape for run 2 (1060 min), select cross-section locations, and centerline rill (white arrows), and (right) successive cross sections with time at the selected locations following two base level adjustments. Flume length is 7.0 m and flume width is 2.4 m.

3.3. Network Development

[20] As the landscape evolved subject to base level adjustment and headcut erosion, a rill network system was created. The DEMs were used to define objectively this drainage network in the rapidly evolving landscape using the following procedure, implemented in Geographic Information System (GIS) software. First, DEM cells with a contributing area of 2000 cells (0.018 m2) were delineated as the flow network. Second, any segment of the flow network that was located in an area of the soil bed that had not incised at least 0.02 m was eliminated from the defined rill network. Due to the nature of the headcut-driven erosion in these experiments and the expansion of the initial outer scarp, the resulting rill network agreed well with the erosional features observed in the orthophotographs (Figure 5).

Figure 5.

Orthophotographs of run 2 with rill networks defined at (a) 120 min, (b) 228 min, and (c) 1051 min. Flume length is 7.0 m and flume width is 2.4 m.

[21] The drainage density of the rill network can be determined using the geospatial data. Figure 3 shows the time variation of drainage density DD, defined here as the total rill channel length divided by total drainage area. Drainage density rapidly increased with time following the first base level adjustment, reaching an asymptotic value of 3.3 m−1 in run 1 and 3.9 m−1 in run 2. In both cases, DD tended to parallel the time variation in cumulative sediment discharge from the landscape, initially increasing rapidly, but attaining a quasi-steady state after approximately 400 min. These times to achieve steady state DD values are similar to the asymptotic times of steady state for Qs noted above. After the second base level drop at 850 min for run 2, sediment efflux continued to increase while drainage density remained relatively constant as the network has effectively filled the available space. Drainage density of evolving rill networks have been observed to increase with time [Parker, 1977; Rieke-Zapp and Nearing, 2005], as well as decrease slightly from a maximum value with time [Parker, 1977; Gómez et al., 2003], and this latter observation was due to the amalgamation of channels during landscape evolution.

3.4. Headcut Erosion and Longitudinal Profiles

[22] The development and upstream migration of headcuts was the primary driver of sediment detachment and drainage network development. The broad shallow headcut scarp (see Figure 5) that formed early in the experiments (before a base level drop was applied) continued to expand for the duration of the experiments. This headcut scarp appeared to be driven largely by rain splash and sheet wash erosion of the soil surface seal. While this initial region of erosion disturbed part of the soil surface, it did not cause significant incision of the landscape or the development of the rill network.

[23] Rill incision on the landscape was the result of headcut development and migration in direct response to exogenic forcing. The base level drop at the terminal weir constituted a preformed step in the elevation of the soil bed. As this step migrated upstream (Figure 5), it evacuated a nearly rectangular rill channel initially (Figure 4). This type of headcut, associated directly with defining the rill channel and its headward growth and bifurcation of the rill network, can be identified as a type A headcut using Bryan and Poesen's [1989] nomenclature. Type A headcuts resulted from exogenic forcing (base level adjustment) and migrated through the relatively undisturbed landscape areas that exhibited a surface seal of sufficient strength to support vertical rill banks. As such, type A headcuts were found at every upstream rill tributary head for the entire duration of each experiment (Figure 6). Headcuts or knickpoints also were observed to form in rills previously incised by type A headcuts, and these are termed type B headcuts [Bryan and Poesen, 1989]. Type B headcuts typically formed, and were most easily observable, at tributary confluences, possibly due to locally high shear stresses, and these features then migrated upstream (Figure 6). In addition, several type B headcuts would usually form in close spatial proximity, and they could be found in series with regular spacing within a single rill.

Figure 6.

Representative photographs of headcut forms observed: (a) two type A headcuts; (b) type A headcut with a well-defined scour pool; (c) type A headcut bifurcating along lines of concentrated flow; (d) type A headcuts approaching outer headcut scarp and a type B headcut at a rill channel confluence; (e) type B headcuts at channel confluences; and (f) type B headcuts with semiregular spacing. Photographs are decimeter scale in width, black arrows show overland flow direction, and white arrows show headcuts.

[24] Headcut migration rate was not strongly correlated to easily identifiable parameters. Migration rates were determined by tracking individual headcuts, over time and space, on successive DEMs. Gordon et al. [2010, 2011] showed that headcut migration rates were not significantly correlated to headcut (scour) depth, distance from the flume outlet, or time following base level adjustment. The initial headcut steps formed by base level adjustment (type A) clearly migrated much faster than those headcuts developed at tributaries or within preexisting channels. In general, type A headcuts migrated upstream at a rate of approximately 0.9 mm s−1, whereas the average migration rate of type B headcuts ranged from 0.5 mm s−1 for those originating in the downstream reaches of the network to 0.2 mm s−1 for those originating in the upstream reaches. In rill erosion studies, observed headcut migration rates can vary from 0.01 to 2.5 mm s−1 depending upon headcut height, overland flow rate, and soil erodibility [Bryan and Poesen, 1989; Bryan, 1990; Slattery and Bryan, 1992; Bennett et al., 2000a; Wells et al., 2009b].

[25] While headcut migration rate varied within the evolving landscape, headcut depth of scour was a function of base level adjustment. Gordon et al. [2010, 2011] showed that headcut incision depths within individual rill channels clustered around the magnitude of base level drop. The average (±standard deviation) incision values were 0.031 ± 0.006 and 0.033 ± 0.007 m for the first base level adjustments in runs 1 and 2, respectively, and 0.062 ± 0.006 m for the second base level adjustment in run 2. That is, following a base level drop regardless of how much time had passed or where in the network a particular headcut was located, the scour depths tended to be distributed about this forced perturbation.

[26] The slopes of rills remained nearly the same as the network incised, developed, and extended upstream. Figure 7 shows longitudinal profiles along the centerline rill of run 2 at various times (see Figure 4), and both type A and B headcuts can be observed. Although the flume slope was 5% in run 2, the initial bed slope along the centerline was 5.85% owing to the imposed bed topography (Figure 7). Starting at the terminal weir, type A headcuts would gradually decrease in depth during their migration from 0.03 to approximately 0.01 m at their most upstream progression (a reduction of about 0.003 m m−1). Additional bed incision along this centerline rill would be accomplished by the transient, ephemeral type B headcuts. By 370 min of runtime following base level adjustment, the entire rill incised approximately 0.031 m on average and now had a longitudinal slope of 6.18%. This represents only a slight increase in overall bed slope (about 6%) as compared to the initial bed slope. The second base level adjustment imposed in run 2 caused a similar longitudinal response. By 1060 min of runtime, an additional 0.038 m of bed incision occurred on average along the rill, with higher incision rates observed in the lower 1 m of the landscape. The rill now had a longitudinal slope of 6.29% (which is only 2% higher than the bed slope at 310 min and only 8% higher than the original soil surface; Figure 7). While the type A headcuts were not entirely effective in communicating the base level adjustment along the entire rill length (they decreased slightly in size with distance), this incision process produced secondary waves of bed incision (type B headcuts), which further reduced the rill bed elevation to the new base level elevation and retained nearly the same longitudinal bed slope. This observation of parallel downwasting along rills following base level adjustment and headcut migration was also noted by Parker [1977].

Figure 7.

Soil elevations along the primary (centerline) rill of run 2 (see Figure 4) as a function of linear distance upstream of the terminal weir following base level adjustments after (a) 85 and (b) 850 min.

3.5. Sediment Budgets

[27] The DEMs now can be used to construct sediment budgets for each important erosion phenomena of the evolving landscape. For each storm, digital orthophotographs were used to manually categorize the evolving landscape into four dominated erosion regimes: (1) erosion of the outer scarp by rain splash and sheet wash; (2) extension of the rill network by type A headcut erosion; (3) incision of the existing rill network by type B headcut erosion; and (4) erosion of vertical rill banks by rain splash and sheet wash. This was accomplished using the following procedure. For each DEM, discrete areas were manually digitized into one of four erosion regimes. The areal progression of the outer scarp, type A headcuts, and rill banks were easily classified, and the remaining portion of the rill network subject to vertical incision was classified as type B headcut erosion. Volumetric variations in each of the four erosion regimes as a function of time were determined for successive DEM pairs using GIS, and then converted to mass. This procedure allowed for the construction of sediment budgets over time and space. Figure 8 presents the time-dependent sediment budget for run 2 using the four classified erosion regimes (a sediment budget constructed for run 1 was nearly identical).

Figure 8.

(a) Time series and (b) total relative contributions of four erosion classifications for run 2.

[28] The contribution of each erosion regime can be deduced for the entire landscape. The erosion regime and its contribution to the total sediment budget are as follows (averaged over the entire experiment after base level adjustment): (1) 22% for erosion of the outer scarp by rain splash and sheet wash; (2) 49% for extension of the rill network by type A headcut erosion; (3) 9% for incision of the existing rill network by type B headcut erosion; and (4) 20% for erosion of vertical rill banks by rain splash and sheet wash (Figure 8). Erosion of the outer headcut scarp began before base level lowering initiated, when it was the only source of sediment. Erosion by type A headcuts dominated the sediment budget following base level adjustment (70% to 80%), which then decreased to about 40%. Type A headcuts created nearly vertical rill banks, and these were slowly widened and their sidewall angles reduced due to rain splash and sheet flow. Given that the average contribution of erosion of the outer scarp by rain splash and sheet wash following base level adjustment was about 22%, approximately 78% of all soil erosion that occurred on the landscape in run 2 could be genetically linked to headcut development and migration.

4. Discussion

[29] A number of observations were made as the soil-mantled experimental landscape responded to rainfall and exogenic forcing. The discussion of these observations now will focus on those concepts of interest in the evolution of landscapes and the prediction of soil losses. The boundary conditions employed here (e.g., bed slope, soil erodibility, rainfall intensities, flume size) may limit the application of these results elsewhere.

4.1. Exogenic Forcing and Rill Erosion

[30] Rill incision and network development as described herein was genetically linked to headcut erosion. While four erosion regimes were identified, the dominant landscape erosion process was headcut erosion, which was exogenically forced through base level lowering.

[31] Initial attempts to create rill networks by rainfall alone, or through endogenic forcing, were not successful. Preliminary experiments not reported here attempted to create rill networks and significant landscape evolution via rainfall application alone resulted in very low erosion rates and the development of a shallow headcut scarp similar to that observed in Figure 5a. Only by imposing a base level adjustment, or forcing the system exogenically, could significant soil loss and network development occur. These results indicate that for this slope and topographic configuration, the soil material and preparation procedures resulted in a very stable soil surface. Once the landscape was forced externally, a primary headcut proceeded upstream and significant rill incision occurred whose width and depth decreased in the upstream direction, and whose bifurcations and tributary channels formed concomitantly. Thus the network developed in a similar fashion as described by Howard [1971] and Hancock and Willgoose [2001a], wherein channels grow headward and bifurcate, filling the available space.

[32] Studies examining the critical conditions for rill initiation have focused almost exclusively on endogenically forced processes. For similar experimental landscapes and rainfall applications, previous work has shown that much higher bed slopes (see references above) or a more erodible soil [Brunton and Bryan, 2000], or some combination thereof, are required to produce significant soil erosion and rill network development endogenically. In the experiments of Yao et al. [2008], the upstream movement of the rill headcut was not significant when compared with the development of the rills downstream of this point of initiation, which is the exact opposite of the process of drainage network development observed here. Moreover, both Gómez et al. [2003] and Raff et al. [2004] observed rill networks that evolved by a general downward erosion, or superimposition, of a drainage network into the landscape. This suggests that the mode of rill initiation, i.e., endogenic versus exogenic forcing, has important implications for the subsequent evolution of the landscape and rill drainage network.

4.2. Episodic Adjustment, Steady-State Erosion, and Supply-Limited Conditions

[33] Rill incision, channel development, and sediment efflux occurred episodically in time, related directly to the imposed base level adjustments. While these rapidly migrating headcuts caused rill incision and spikes in sediment efflux, the secondary waves of degradation driven by the minor headcuts were not expressed in the sediment discharge time series. Moreover, sediment efflux and drainage density approached nearly constant or asymptotic values with time during most of the experiment, despite the continuous application of rainfall. Thus sediment efflux and rill network development achieved steady state erosion conditions similar to the declining equilibrium landscape of Hancock and Willgoose [2001b] when the system was not perturbed by the downstream base level control.

[34] No significant storage of sediment was observed at any point in time or space within the landscape. This observation suggests that sediment transport within these eroding rills was limited by the supply of sediment. To quantify this, a simple sediment transport capacity rate can be used [e.g., Prosser and Rustomji, 2000]:

display math

where qs is sediment transport capacity per unit width, q is discharge per unit width, S is local slope gradient, and β and γ are constants. Considerable experimental and theoretical work has been undertaken on the sediment transport capacity of overland flow, which has yielded a range of values for these coefficients (Table 1). Using direct measurements of sediment discharge, runoff, and bed slope approximately 1 m upstream of the terminal weir, estimates of sediment transport capacity for runs 1 and 2 can be determined (Table 1). For all coefficients, the calculated sediment transport capacity for the flow was greater than the steady state sediment discharge observed (Figure 3), indicating that supply-limited conditions persisted in the lower portions of the flume where data were readily available. While some calculated sediment transport capacity values were less than the observed sediment discharge rates immediately following base level adjustment, all detached sediment was subsequently removed.

Table 1. Examples of Erosion Model Parameterizations of β and γ in Equation (2) and Calculated Values of Sediment Transport Capacity for Runs 1 and 2
ReferenceβγQs (g s−1) Run 1Qs (g s−1) Run 2
Kirkby [1993]2.001.00161.3230.6
de Roo et al. [1995]1.200.7853.058.0
Flanagan and Nearing [1995]0.901.0512.112.0
Yu et al. [1997]1.401.3017.420.3
Morgan et al. [1998]0.780.7324.323.1
Prosser and Rustomji [2000]1.401.4012.915.1

4.3. Hydraulic Geometry Relationships for Incised Rills

[35] Leopold and Maddock [1953] demonstrated that the bankfull width w and depth d of river channels are dependent on bankfull flow rate for a wide range of systems. Here, channel dimension becomes a function of discharge defined as

display math
display math

where a and c are coefficients of the power functions and b and f describe the rates of change of the river channel width and depth with increasing flow rate, respectively. These hydraulic geometry relationships recently were derived for rills and ephemeral gullies. Nachtergaele et al. [2002] compiled available data from field and laboratory settings and found that inline image for rills and inline image for ephemeral gullies, and that significant departures in these exponent values can occur due to variations of soil erodibility in time and space. Torri et al. [2006] extended this database and noted that inline image for rills and inline image for ephemeral gullies. Moreover, Salvador Sanchis et al. [2009] found that inline image for both rills and gullies, but b can vary from 0.35 for relatively small-width rills (∼0.05 m) to 0.6 for relatively large-width gullies (∼5 m).

[36] Hydraulic geometry relationships for rill width and depth can be derived for each experimental landscape. Rill dimensions for depth and width were measured using DEMs and orthophotographs at various points in the network where the channels were well-defined and easily demarcated, and at three times following base level adjustment: (1) at 787 min for run 1; (2) at 580 for run 2; and (3) at 1060 min for run 2. Given that the applied rainfall, infiltration, and runoff rates all were invariant with time, GIS techniques were used to determine flow rate at-a-point Q* at each rill dimension measurement. This was determined by defining the upstream contributing area at each rill measurement location, and by calculating the total overland flow rate at that location as the summed difference in the applied rainfall rate (spatially varied) minus the observed rate of infiltration over the upstream area. In this case, the calculated Q* is used in (3) and (4). Moreover, given the implicit error in Q*, both ordinary least squares (OLS) and reduced major axis (RMA) regressions (A. J. Bohonak and K. van der Linde, RMA: Software for reduced major axis regression for Java, 2004, available at http://www.bio.sdsu.edu/pub/andy/RMA.html) were employed.

[37] Plots of rill hydraulic geometry are shown in Figure 9. In general, inline image, inline image (0.612 to 0.693 using all data), and inline image (0.360 to 0.480 using all data), and these results broadly agree with previous observations [Leopold and Maddock, 1953; Nachtergaele et al., 2002; Torri et al., 2006; Salvador Sanchis et al., 2009]. Interestingly, both b and f increased in run 2 after the landscape was subjected to another episode of base level adjustment, and this could be attributed to increased drainage area and overland flow rates due to the further areal expansion of the network. These observations also can be extended to consider mean flow velocity u using

display math
display math

where k is a coefficient (and inline image) and m is the rate of change of flow velocity with discharge. In this case, inline image (−0.173 to 0.028 using all data) and inline image. These formulations suggest that rill channels get significantly wider and deeper to accommodate the increase in flow discharge in the downstream direction, but flow velocity can increase or decrease slightly, or remain unchanged. This is in contrast to observations by Govers [1992] [Govers et al., 2007] who suggested that inline image for bare soils. In the current experiment, both the hydraulic radius R and width-to-depth ratio inline image increase significantly in the downstream direction (by factors of 2 and 3, respectively). Given that bed slope S is relatively constant along the rills (Figure 7), this large increase in R and modest change in u in the downstream direction must be accompanied by a significant increase in flow resistance, as defined using the Darcy-Weisbach friction factor ff,

display math

where g is acceleration due to gravity. The computation of overland flow rate at-a-point using the above technique clearly may not be entirely accurate, and overland flow would most likely not fill the entire width and depth of the rill within the experiments. Moreover, rill depth was markedly deeper in the plunge pool just downstream of the migrating headcuts (see Figure 7), but these increased depths were not included in this analysis. Nonetheless, these relationships aid in characterizing the hydraulic geometry of the observed rill networks.

Figure 9.

Hydraulic geometry relations derived for rill width w (upper) and depth d (lower) as a function of calculated overland flow rate Q* for runs 1 and 2 at discrete times during each experiment and for all data combined. Coefficients for the derived power function relationships and correlation coefficients r2 are provided using ordinary least squares (OSL; regression line shown) and reduced major axis (RMA) regression techniques.

4.4. Predicting Headcut Migration Rates

[38] The rate headcuts or knickpoints migrate upstream is a key determinant in soil erosion fluxes from hillslopes [Bennett et al., 2000a] and landscape response to base level perturbations [Whipple et al., 2000]. As such, several approaches have been used to predict these rates, and three are examined here using the current experimental data set: (1) a diffusion model for the evolution of an incised channel; (2) an empirical relationship between headcut migration rate and flow rate; and (3) an analytical model of headcut erosion and migration in upland concentrated flows. The following analysis is limited to type A headcuts (32 in total) because the outer headcut scarp was driven by rain splash and sheet flow and because type B headcuts were ephemeral and their migration varied over time and space.

[39] The degradation of alluvial channels in response to base level lowering has been described as a mass diffusion process. Using expressions for sediment mass continuity, Begin et al. [1980] derived the time-variation of bed elevation y at-a-point x as

display math

where K is the diffusion coefficient and B is the volume of lateral inflow of sediment per unit flow width and per unit length of channel (assumed here to be zero, which is consistent with the presumed supply limited conditions). Begin [1987] provided the necessary algorithm to solve (8). In the application here, K was used as in input parameter, and was varied from 100 to 2000 cm2 min−1 as per the derived values presented by Schumm et al. [1987] (500 to 1800 cm2 min−1) for the experiments of Parker [1977].

[40] The application of this diffusion model for the evolution of an incised rill channel is shown in Figure 10. Here, bed profile predictions for the primary rill trajectory at the end of run 2 (1060 min) are predicted using (8) with K = 100, 1000, and 2000. These results demonstrate a significant limitation in the prediction of headcut-driven rill incision using a diffusion approach. The diffusion model does not communicate the base level drop into upstream portions of the channel as it was observed herein (compare Figure 10 to Figure 7), as it predicts the rotation and flattening of the initial vertical headcut as it migrates upstream. As such, this model cannot be used to predict rill incision and evolution within this experimental landscape.

Figure 10.

Initial (0 min) and final (1060 min) smoothed bed profiles for run 2 and final (1060 min) profiles predicted using the formulation of Begin et al. [1980] for three diffusivity coefficients.

[41] Headcut or knickpoint migration also can be predicted empirically using flow rate. Several studies have developed simple empirical relationships for headcut or knickpoint migration rate as a function of slope S and drainage area AD [e.g., Hayakawa and Matsukura, 2003; Bishop et al., 2005; Crosby and Whipple, 2006]. Reasoning for this stems from the parameter stream power QS, often used as the driving force for the evolution of drainage networks in landscape models [e.g., Howard, 1994; Whipple and Tucker, 1999]. Implicit in this assumption is that drainage area can be used a proxy for discharge.

[42] A simple empirical expression for headcut migration can be derived from the current experiments. A slope-drainage area formulation was not used here since (1) bed slope was nearly invariant in space (see Figure 7), and (2) drainage area was not a good proxy for discharge because rainfall intensity was spatially varied (see Figure 2). Derived overland flow rate Q* will be employed instead. Migration rate M for type A headcuts averaged over individual storm durations were measured at 12 and 20 locations in runs 1 and 2, respectively, and plotted against the calculated overland flow rate Q*, spatially averaged between each starting and stopping location of each headcut (as bounded by the DEMs). Figure 11 shows that this simple empirical approach predicts well the rate of headcut migration, and it substantiates the observation that type A headcuts migrated at a faster rate in the lower reaches of the experimental landscape in response to higher overland flow rates.

Figure 11.

Headcut migration rate M as a function of overland flow rate Q*. Also shown are the power functions derived using ordinary least squares (OSL) and reduced major axis (RMA) regression techniques.

[43] Finally, analytic formulations also have been proposed to predict headcut erosion in upland concentrated flows. Alonso et al. [2002] derived explicit equations on the basis of jet impingement theory to predict the depth of scour, migration rate, and sediment efflux of headcuts in rills and crop furrows typical of agricultural fields. Headcut migration rate M can be determined using:

display math
display math

where Ve is the jet entry velocity at the headcut brinkpoint, SD is the headcut scour depth, h is the vertical distance from the brink to the tailwater surface, kd is the erodibility coefficient of the soil, and θ is the jet entry angle. To apply these formulations, a number of assumptions were made. Flow depths upstream of the headcuts were observed to vary between 0.01 m at the flume outlet and 0.001 m at the most upstream location, h was assumed to be 10% of the headcut height at the maximum discharge and 50% of the headcut height at the minimum discharge, and flow rates at-a-point were determined using the hydraulic geometry relationships presented above. Jet entry velocity and angle were determined using the relationships presented in Alonso et al. [2002; equations (13), (14), (26), and (28)], and scour depth SD was measured from the DEMs. Finally, the soil erodibility coefficient kd was used as fitting parameter to the model, where its final value (5.6 × 10−7 m3 N−1s−1) was very similar to those previously used [Alonso et al., 2002]. Figure 12 compares the headcut migration rates observed and those predicted using the model of Alonso et al. [2002]. In general, the predicted values fall within one order of magnitude of the observed, which is a positive result given the assumptions above. But the correlation coefficient derived from this comparison is near zero. Much uncertainty exists in the prediction of headcut migration rate, as noted by Alonso et al. [2002], primarily due to the uncertainties in the soil erodibility and jet entry angle values. Moreover, the flow rates observed here are much smaller than those used to derive and test the original equations, and it seems likely that the hydrodynamic characteristics of the reattached wall jet would be modulated greatly by these boundary conditions [Bennett and Alonso, 2005, 2006]. Based on these comparisons, simple empirical relationships using rate of overland flow are an effective predictor for headcut migration rate in this evolving landscape.

Figure 12.

Observed and predicted headcut migration rates using the formulation of Alonso et al. [2002]. Dashed lines represent mean uncertainty range.

5. Conclusions

[44] Soil loss due to rill and gully erosion remains a critical concern in agricultural regions since these processes can lead to significant landscape degradation. Experiments were conducted to examine the processes of soil erosion and rill network development in the presence of simulated rain and changes in the downstream base level control. The following observations were made.

[45] 1. Base level adjustment caused the formation of a primary headcut that migrated upstream and created an incised rill network having a drainage density of about 3 to 4 m−1. Sediment discharge rates peaked during the incision process, accounting for about 70% of the total efflux, but then decreased asymptotically to steady state values.

[46] 2. Erosion rates derived from geospatial techniques agreed well with direct measurements, which showed that nearly all sediment eroded during rill network development was removed from the landscape because flows were deemed supply limited.

[47] 3. Headcut development and migration was the dominant erosion process within the landscape, and these were formed by base level adjustment, at rill tributary junctions, and within rill channels. Moreover, approximately 80% of all erosion that occurred in the landscape could be genetically linked to headcut development and migration.

[48] 4. For the boundary conditions used, exogenic forcing (base level lowering) was required for significant rill incision and network development to occur.

[49] 5. Base level adjustments strongly conditioned the heights of the headcuts and ultimately the depth of rill incision, yet the longitudinal slopes of the primary rill channel did not change appreciably in response to landscape degradation.

[50] 6. Hydraulic geometry relationships derived for the evolving rills corroborated the observations that an increase in discharge in the downstream direction is accommodated by rill widening and deepening, and not necessarily by increasing mean flow velocity.

[51] 7. Prediction of headcut migration rate on the basis of diffusion and jet impingement theory did not perform as well as a simple empirical relationship using rate of overland flow.

[52] Future work will consider the characteristics of the rill network and additional boundary conditions for the expression of soil erosion and landscape evolution.

Acknowledgments

[53] This research was supported by NSF (EAR0640617), the University at Buffalo, and by the USDA-Agricultural Research Service. We thank Toby Gardner and Peter Ashmore for kindly providing technical guidance on photogrammetry, and Rorke Bryan for the flume. This paper was reviewed by three anonymous referees and the two editors, all of whom offered many helpful suggestions.