Analysis and modeling of gully headcut dynamics, North American high plains

Authors


Abstract

Gullies are dynamic fluvial features that can be the primary driver for landscape dissection and sediment production in many settings. This research exploits a well-constrained field area near West Bijou Creek, Colorado, U.S., in order to develop a natural experiment in which we explore gully headcut erosion rates, the controls on gully headcut height, and the morphology of gully longitudinal profiles. Analysis of headcut retreat using aerial photography and airborne lidar imagery indicates that headcut retreat rates correlate with the square root of drainage area approximately. We investigate how a drainage area control on headcut retreat translates into the longitudinal profile morphology over time using a simple numerical model. The model combines fluvial erosion, deposition, and headcut retreat to identify the necessary and sufficient conditions needed to reproduce longitudinal profiles observed in the field. Field profiles are typically concave-upward, predominantly aggradational channel profiles with retreating headcuts whose height varies with catchment position. Systematic variation of environmental parameters in the model showed that the most successful model was achieved when highly resistant vegetation is applied throughout the channel, excluding a bare soil zone downstream of the headcut. This model scenario maintained an abrupt headcut over hundreds of model years and produced a realistic longitudinal profile that aggrades downstream of the headcut over time. The vegetation pattern used in the best model fit is observed at the field site, where easily erodible, sparsely vegetated soil downstream of the headcut grades into a more resistant grassy channel downstream.

1 Introduction

Gullies are common features in dryland areas, and they are major contributors of sediment erosion, supplying 10–94% of the total sediment yield in some watersheds [Poesen et al., 2003]. Within gullies, headcuts play a special role. Headcuts are near-vertical steps that erode the valley network by migrating upstream over time [Bull and Kirkby, 2002] and add mobile sediment to gully channels downstream [Tucker et al., 2006]. The origins of headcuts are debated, with formation variously attributed to locally steep channel reaches [Patton and Schumm, 1975; Bull, 1997; Tucker et al., 2006], areas of weak vegetation [Bull, 1997; Graf, 1979; Tucker et al., 2006; Yetemen et al., 2010], the combination of overland flow and subsurface flow exceeding a critical drainage-area and slope threshold [Montgomery and Dietrich, 1989, 1994; Dietrich and Dunne, 1993; Poesen et al., 2002], or an abrupt base level drop [Berlin and Anderson, 2007].

Considerable experimental work has been performed to understand how headcuts behave once they have been initiated. Flume experiments suggest that headcut propagation depends on the balance between erosion at the headcut lip and undercutting at the headcut base [Stein and Julien, 1993]. Headcut growth and burial are less well understood in field settings at decadal timescales, however. Some field investigations have documented particular failure mechanisms that lead to headcut retreat [Bradford and Piest, 1980; Montgomery, 1999], but little investigation has been done linking headcut retreat processes to watershed characteristics such as drainage area and vegetation type [Istanbulluoglu et al., 2005]. Moreover, the long-term morphological influence of headcut retreat on gully longitudinal profiles remain unclear. What is the geomorphic legacy of headcut retreat? How is the longitudinal profile shape influenced by the vegetation and discharge in nature? What factors determine whether a headcut will grow, shrink, or maintain a constant height as it migrates upstream?

The objective of this study is to answer these questions at a field site in eastern Colorado, U.S., where headcuts are actively incising into a gully network. Digital elevation model (DEM) analysis and historic aerial photography are used to examine the rate of headcut migration and how this rate changes as a function of upstream drainage area. Additionally, we use field observations of headcut failure and morphological characteristics from airborne light detecting and ranging (lidar) data to formulate a numerical model of headcut retreat. The model is used to investigate the relationship between headcut erosion and long-term longitudinal profile development.

2 Background

2.1 Headcuts and Knickpoints

Knickpoints are steep, near-vertical channel segments [Hayakawa and Oguchi, 2006] frequently found in bedrock-floored channels that correspond to waterfalls. Generally, headcuts have been referred to in the literature as near-vertical steps in rills or gullies that are formed in sediment and often located at the initiation point of a rill/gully. Because the processes driving knickpoint retreat parallel those involved in headcut retreat, in the following literature review we reference knickpoint studies as relevant corollaries for understanding headcut retreat.

2.2 Potential Mechanisms for Headcut Migration

Mechanisms thought to be responsible for headward migration of channel steps include plunge pool erosion, mass wasting, and seepage erosion. We briefly review each of these processes.

Many researchers have observed plunge pool erosion as a cause of headcut retreat through undercutting. A jet of water plunging over a headcut can be very erosive if it contacts the base of the plunge pool bed, but if the jet diffuses in the standing water of the plunge pool, erosion is diminished [Stein et al., 1993]. Often plunge pool erosion exploits layered strata that comprise a resistant top layer overlaying weaker layers [Gilbert and Hall, 1907; Gardner, 1983; Wohl et al., 1994; Frankel et al., 2007], but it can also occur in uniform soils [Bennett, 1999]. Plunge pools reach a stable equilibrium depth when erosive shear stress equals resisting strength in the pool [Stein et al., 1993]. Flume experiments performed by Bennett et al. [2000] revealed that once equilibrium depth is achieved, there is relatively little change in headcut/plunge pool geometry as the headcut migrates upstream. Flores-Cervantes et al. [2006] used models and observations developed from plunge pool erosion experiments [Stein et al., 1993; Stein and Julien, 1993, 1994; Bennett, 1999; Alonso et al., 2002] to incorporate plunge pool erosion into the Channel-Hillslope Integrated Landscape Development model [Tucker et al., 2001], a three-dimensional numerical model of landscape evolution.

Mass failure at the headcut has also been identified as a key process in extending a gully network and generating sediment [Bull and Kirkby, 2002; Montgomery, 1999]. Several modes of mass failure have been observed in field settings, including deep-seated failure to bedrock, slab failure, base or pop-out failure, and overhang slab failure [Bradford and Piest, 1980]. Mass-failure increases as weathering processes weaken soil [Robinson et al., 2000]. Istanbulluoglu et al. [2005] and Montgomery [1999] conceptualized a headcut as a fracture-bound 3-D slab of sediment subject to Coulomb failure, such that a fractured slab at the headcut face will remain upright until soil weight plus the hydrostatic pressure force exceed the resisting force of soil cohesion. Lamb and Dietrich [2009] suggested an alternative mode of headcut failure for fractured bedrock, in which fluid flow across the lip of the headcut causes sufficient torque to counteract the partially immersed weight of the fracture-bounded block, leading to rotational failure.

In contrast, several researchers have proposed that seepage erosion is the key to headcut retreat [Abrams et al., 2009; Laity and Malin, 1985; Dunne, 1980]. Howard and McLane [1988] developed an analytical model for headcut migration due to seepage erosion in noncohesive sediment. This model assumes that the differential pressure at the interface between a saturated headcut and the atmosphere will allow for the movement of sediment grains away from the face of the headcut. Numerical modeling further suggests that seepage erosion leads to a parabolic planform at headcuts [Pelletier and Baker, 2011]. However, Lamb et al. [2006] note that there is little definitive evidence of seepage erosion as the primary driver of headcut erosion in consolidated material such as basalt.

2.3 Headcut Migration Rate

In addition to the specific processes that cause headcut retreat, several researchers have explored the relationship between drainage area and the retreat rate of headcuts in bedrock rivers [Begin, 1988; Rosenbloom and Anderson, 1994; Weissel and Seidl, 1998; Stock and Montgomery, 1999; Crosby and Whipple, 2006; Berlin and Anderson, 2007]. Generally, these studies assume that the rate of vertical channel erosion correlates with either shear stress or stream power. Such a correlation can be expressed with an erosion equation of the following form [Howard and Kerby, 1983]:

display math(1)

where inline imageis the erosion rate; z is elevation; t is time; A is the upstream drainage area (a surrogate for water discharge); inline image is the local slope; x is distance downstream; K is a factor that amalgamates information about material properties, hydrology, and other effects [Whipple and Tucker, 1999]; and m and n are exponents that can be manipulated to make this equation equivalent to an excess shear stress law or an excess stream power law [Whipple and Tucker, 1999]. Headcut migration can be modeled as n = 1, which assumes that the headcut will retreat as a linear kinematic wave [Tucker and Whipple, 2002]. With this assumption, equation (1) can be rearranged [Rosenbloom and Anderson, 1994] to get:

display math(2)

Equation (2) has the form of a wave celerity equation [Whipple and Tucker, 2002; Tucker and Whipple, 2002].

Although equation (2) provides a reasonable empirical description of observed headcut retreat rates, it lacks mechanical basis when flow over the headcut is mostly unsteady, nonuniform, and not necessarily in contact with the bed. For this reason, when formulas like equation (2) are used to describe headcut migration, they are best viewed simply as indicating a general correlation between retreat rate, basin area, and the average power or stress.

3 Study Site

The existing studies on headcut retreat illustrate a variety of mechanisms for headcut failure and propagation, but most of the previous studies were done in laboratory flume settings [Gardner, 1983; Stein et al., 1993; Stein and Julien, 1993, 1994; Bennett, 1999; Alonso et al., 2002; Lamb and Dietrich, 2009; Howard and McLane, 1988] with the notable exception of Montgomery [1999]. We explore headcut dynamics at a field site on the eastern plains of Colorado, U.S., in the Great Plains physiographic province (Figure 1). We use the principles of headcut erosion from the previous models and field observations to develop a conceptual model for headcut erosion in section 4 and a numerical model in section 6.

Figure 1.

(a) Site location shown as black star on map of U.S. (b) Shaded relief image generated from a 1m lidar DEM (based on airborne laser-mapping data acquired on 23 April 2007). Highlighted watersheds are used for modeling the relationship between drainage area and headcut migration. White star indicates location of photo. (c) Photo of typical gully headcut.

3.1 Geology and Climate

The site lies along an escarpment that forms the western edge of the West Bijou Creek drainage basin in eastern Colorado, U.S. The underlying rocks are subhorizontal sedimentary deposits of the Denver Basin, which is a foreland basin east of the Front Range containing strata emplaced during the Laramide orogeny [Barclay et al., 2003]. The late Cretaceous (70–80Ma) to Eocene age (35–55Ma) Denver Formation is composed of sandstone, mudstone, and lignite [Barclay et al., 2003]. Most rock on the site is fissile and weakly resistant to erosion. The soils on the site are typically loam or clay-loam, and an X-ray diffraction analysis of six soil samples from headcuts showed that smectite, a shrink-swell clay, composed 20% of the clay minerals identified in the soil.

The West Bijou Creek site has a semiarid climate with an average annual precipitation on the order of 300–450mm, the majority of which (70–80%) falls between April and September [Doesken et al., 2011]. The winter storms are typically low-intensity frontal rainstorms and snowfall. Summer precipitation comes from convective thunderstorms, which can deliver 100mm of rain in a few hours [Doesken et al., 2011].

3.2 Gully Networks

A low-relief surface occupying the western portion of our site is incised by gullies cutting down toward West Bijou Creek (Figure 1b). The gullies show signs of cut/fill cycles resembling the discontinuous ephemeral channels described by Bull [1997]. These signs include abandoned alluvial terraces and headcuts that primarily cut into alluvial fill that was deposited during a previous fill cycle. Headcuts at our site fit the morphologic description of abrupt channel heads [Montgomery and Dietrich, 1989] (Figure 1c), and often, two or three such steps occur in a series along the same valley. Typical gully longitudinal profiles show overall upward concavity but local convexity at the headcut lip and just downstream of the headcuts (Figure 2). The convex reaches downstream of headcuts are similar to the in-channel fan deposits described by Montgomery [1999], where much of the sediment eroded from the headcut is stored. In flume experiments, Bennett [1999] showed that regardless of the slope upstream, sediment deposited downstream of the headcut had a slope of 0.024. At our study site the first 10m downstream of a headcut tend to be much steeper than the average slope from the headcut to the drainage basin outlet (Figure 2).

Figure 2.

(a) Four tributaries of a gully network overlain on a 2006 aerial photo. Square pixels define stream paths to each headcut location (circles). (b) Longitudinal profiles of tributary streams shown in Figure 2a. The headcuts correspond to the same terrace level (black line) which is mapped from the terrace in the main valley. An apparent offset is simply an artifact of projecting the data in 2-D. Three of these headcuts cluster near longitudinal distances of 300m, and the fourth headcut (with a much smaller drainage basin) is at 200m. A headcut from an older generation (with a higher matching terrace level) is at 380m. The upper limit of each profile is the drainage divide (not visible in Figure 2a). Profiles were obtained from a 1m DEM generated from filtered airborne laser swath mapping data flown in 2007.

3.3 Hydrology

Qualitative observations of the hydrologic drivers of headcut erosion were obtained from monthly site visits and three time-lapse cameras, which take photos every hour during daylight hours, annually. Two time-lapse cameras are located 10m from the headcuts, and one is positioned within 2m of a headcut face. Overland flow within valley bottoms was only observed during brief periods following heavy convective rainstorms during the summer and autumn. These flows typically last for one to a few hours. We have never observed seepage flow at the base of headcuts during our monthly visits to the study area.

3.4 Observed Headcut Failure

Centimeter- to decimeter-scale fractures lace the alluvium and bedrock exposed in headcuts and gully side walls. The dominant fracture orientation is subparallel to the walls. These subparallel fractures commonly delineate decimeter-scale slabs of material. Fracture-bounded slabs occasionally slide or topple from the walls, leaving heaps of debris along the base, which is also observed by Istanbulluoglu et al. [2005] and Montgomery [1999]. Time-lapse photography reveals examples of slab failures that are variously correlated with summer flash floods (Figure 3a), winter snowmelt (Figure 3b), and prolonged summer dry periods (Figure 3c).

Figure 3.

Headcut block failure under different soil moisture conditions. (top) Prefailure surface image and (bottom) postfailure image (arrows indicate location of failure). (a) Failure following a thunderstorm with channelized overland flow (headcut height 3m). (b) Failure following snowmelt (headcut height 2m). (c) Failure during dry period of 11 days without precipitation (headcut height 3m).

Although fracturing results in uneven blocks of soil on the headcut face, headcut erosion is not random. Rather, there is a typical erosional sequence observed at our field site and by other researchers [Bradford and Piest, 1980; Montgomery, 1999]. Slab failures occur along the face of the headcut below the root zone, leaving behind an overhanging block retained by grass roots along the headcut rim. Overhang failure follows when the block weight exceeds the grass root strength. Sediment from both types of mass-failure event is stored at the headcut base until overland flow removes the sediment (Figure 4).

Figure 4.

(a) Wall pop-out failure leaving an overhang (indicated by arrow). (b) Overhang failure. (c) Fluvial erosion moves sediment away from the headcut base on 17 June 2011. This headcut is approximately 3m tall.

Based on field observations shortly after flash floods, relatively little erosion occurs immediately upstream of headcuts due to a dense vegetative mat of native grasses, such as blue gramma (Bouteloua gracile), green needle grass (Stipa viridula), and buffalo grass (Buchloa dactyloides). In the first several meters downstream of headcuts, the soil tends to be mostly bare, with sparse Canada thistle (Breea arvense) and sunflower (Helianthus annuus). These species are known to thrive in wet and disturbed soils, indicating that erosion events frequently uproot or bury plants in this zone (L. Gilligan, Colorado Natural Heritage Program, personal communication, 2013). The area of sparse vegetation varies in length from 0 to 10m and grades into more dense vegetation downstream. In the aftermath of flash floods, one commonly observes fresh deposits of sediment eroded from the sparse vegetation zone and deposited on top of fluvially matted grass stems downstream.

4 Conceptual Model

The observations of gully morphology and dynamics at West Bijou Creek suggest the following conceptual model (Figure 5). Gully headcuts and sidewalls retreat primarily by weathering and collapse of exposed soil, sediment, and bedrock. Similar to findings by Montgomery [1999], we observe that drying of these materials between storms leads to contraction as the smectite clays dehydrate, which forms networks of fractures and creates fracture-bounded blocks subparallel to gully heads and walls. These blocks then collapse either when crack propagation proceeds past the point where the cohesion force can support the block's weight or when temporary near-surface saturation adds pore pressure [Istanbulluoglu et al., 2005; Montgomery, 1999] and reduces apparent cohesion. Winter block detachment may also occur through ice-lens growth and subsequent melting. The detached material accumulates along the base of the walls and headcut. Debris delivered to the base of the headcut is remobilized during summer flash floods and redeposited further downstream. These processes contribute to the upstream migration of the headcut as well as widening of the valley downstream of the headcut, as walls gradually retreat from the valley centerline.

Figure 5.

Conceptual model of the processes occurring at the headcut.

In laboratory headcuts with a relatively small ratio of height to flow depth, plunge pool scour plays an important role in undermining the step face. For headcuts with a large ratio of height to flow depth, such as those shown in Figure 4, we hypothesize that the direct influence of plunge pool scour is relatively minor. For example, following a plunge pool erosion event in June 2011, we observed that the center of the plunge pool was located 1.5m from the headcut face, with an approximate diameter of 0.5m and a depth of 0.3m. Because the margins of the pool were separated from the face, the plunging jet had limited contact with the wall face and was fully aerated. Moreover, similar to Montgomery [1999], we saw no evidence of undercutting of the headcut from either a plunging jet or turbulence from a plunge pool. Therefore, we assume that jet impingement erosion is negligible.

One implication of this conceptual model is that headcut height is set by the competition between sediment delivery to the headcut base by wall collapse, and sediment evacuation by flash flood erosion. If indeed headcut retreat is driven primarily by the generation and episodic release of fracture-bounded blocks, then the retreat rate must be set by the rate of fracture propagation (presumably by clay shrinkage [Montgomery, 1999]), by the frequency of headcut saturation events sufficient to destabilize the blocks, or both. Moreover, it is beneficial to understand if saturation is primarily influenced by local processes (e.g., rain and snowmelt) or water discharge arriving from upstream during flash floods. We can assess the relative control of local or upstream discharge on headcut saturation by determining the degree of correlation between retreat rate and drainage-basin area, a surrogate for water discharge.

5 Relation Between Headcut Retreat Rate and Drainage Area

To determine whether headcut retreat rate correlates with upstream drainage area, as in equation (2), we compare the positions of headcuts in four tributaries of the gully network shown in Figure 2. We studied headcuts that have a common point of initiation through field mapping of a paired terrace, which extends downstream from all of the headcuts to a main valley (Figure 6). This terrace is also observed in longitudinal profiles from the airborne lidar (Figure 2b). The common terrace at each headcut shows that as the central headcut moved upstream, it bifurcated at each tributary junction. Headcuts on the larger tributaries appear to have migrated further upstream than those on smaller tributaries, as has been found in studies of bedrock knickpoint propagation [Crosby and Whipple, 2006; Berlin and Anderson, 2007] (but see Weissel and Seidl [1998] for a counterexample). And while the location of some bedrock knickpoints can result from a channel incision threshold, this is clearly not the case for the headcuts in this study, which are actively migrating.

Figure 6.

Series of time steps showing how headcuts migrate upstream into an alluvial filled valley and leave behind terraces.

To analyze headcut migration patterns, we used airborne lidar that was collected by the National Center for Airborne Laser Mapping (NCALM) on 23 April 2007 over our field location, resulting in a 1m resolution digital elevation model (DEM) of the site. We identified headcut locations by calculating the topographic curvature in the profile direction from the 1m DEM. Locations in the DEM with curvature less than −15 cm−1 effectively highlight topographic convexities. From these areas of negative curvature we select DEM cells with a drainage area greater than 5000m2 to exclude any steep cliffs on channel or valley walls (Figure 7). Headcuts identified by these thresholds correspond to headcuts observed in the field, and no headcuts visible in aerial photos were missed by the algorithm.

Figure 7.

Identification of headcut locations based on curvature. Dashed circles outline the location where large drainage area (> 5000m2) crosses with negative curvature (<−15) to identify the headcut pixel.

We selected a set of four headcuts that can be identified in both a 2006 and a 1957 aerial photograph of the site (Figure 2). Using these image pairs, we estimated the total migration distance of each headcut over the 49year period between images. These measurements were then used to calibrate values of K and m in equation (2). To accomplish this, we modeled the upstream migration of each headcut along flow paths delineated from the DEM using a steepest-descent routing algorithm (Figure 2). The headcut transit time across each cell, dt, is given by inverting equation (2):

display math(3)

where dx is the distance between cells (either 1m or inline image m, depending on whether the flow path direction is orthogonal or diagonal). The total predicted transit time (Tp) is then the sum of the individual cell travel times. The best fitting parameters were found by repeating the calculations with different values of K and m and calculating the misfit for each parameter pair as the sum of the position differences between modeled and observed headcuts:

display math(4)

where E is the misfit, Tk is the known time of retreat (49years), and Tp is the predicted time of retreat. The resulting best fit values are m = 0.57 and K = 0.0021 m−0.14 yr−1. These parameters result in an error (E) of ±3years. A 10% change in either the m or K value results in error ranging from ±7 to 42years. The roughly square root dependence on drainage area is consistent with a study of bedrock knickpoint retreat [Berlin and Anderson, 2007]. Additionally, this analysis assumes uniform K values in a watershed, which is not unreasonable considering the similar soils throughout the study area.

We applied the calibrated model in three watersheds by finding headcuts that had moved up tributaries and originated from a single headcut on the main stream of the channel (Figure 1). We ensured that headcuts used in our analysis had a common origin through field mapping of paired terraces and only applied the model on watersheds with limited anthropogenic interference. If equation (3) is accurate, then each headcut in a given watershed should have the same total transit time Tp. As might be expected, Tp varies somewhat among the various headcuts, but in each case it fell within 35% of the mean transit time for headcuts within the same watershed (Table 1). This suggests that equation (2) provides a reasonable description of the relationship between migration rate and drainage area, and from this we can infer that saturation from overland flow may play a role in headcut migration, as speculated in section 4. This analysis implies that it has been 150–400years since the headcuts in each basin diverged from a single headcut in the main channel. We also compare the total travel distance of a headcut to the summation of the drainage area at each cell along a headcut path (Figure 8). Because headcuts originate at larger drainage areas and have faster retreat rates early in time, the summation of drainage area accurately accounts for the full migration path of the headcut. A simple regression between drainage area at the current headcut location and travel distance does not account for changes in retreat rate, and therefore a poor relationship is observed (Figure 8).

Table 1. Time for Headcut Propagation in Three Watersheds
 Predicted TimeaResidual Time 
 Tpabs(inline image)Normalized Time
Watershed(Years)(Years)Residual/inline image
  1. a

    Predicted times are determined from the summation of equation (3), and an average time inline imageis determined for each watershed independently.

WS824620.257.61%
WS827912.754.79%
WS82736.752.54%
WS82670.750.28%
WS634433.512.58%
WS635027.510.33%
WS640729.511.08%
WS640931.511.83%
WS526769.535.18%
WS512869.535.18%
WS22114.32.10%
WS216838.618.71%
WS224134.316.61%
Figure 8.

(a) Total travel distance from a point of headcut initiation in relation to the accumulated drainage area from each pixel within the path of the headcut as it moves upstream. A best fit power law trend line obtains an exponent that approximates the square root of accumulated drainage area. (b) A weak relationship between drainage area and travel distance is obtained when the accumulated drainage area is not taken into account.

6 Numerical Model

We have identified a strong relationship between headcut retreat and drainage area (Figure 8), and we would like to know how the processes involved in headcut retreat influence the overall gully longitudinal profile. We hypothesize that gully morphology arises from a relatively simple interaction between weathering, mass wasting, and water-driven sediment transport. To test this hypothesis, we have formulated and solved numerically a model of gully profile evolution. The model expresses the conceptual view (Figure 5) in mathematical terms, using process equations that are amenable to field and/or laboratory testing. The model is based on water and sediment mass conservation with steady, nonuniform discharge, headcut weathering, and fluvial sediment transport. The hydrology of each year is treated the same, with a single rainfall value that is assumed to be representative of the natural sequence of storms and droughts. Moreover, we model headcut retreat as the integral of annual weathering rather than individual events. These model equations are described next, and specific parameter values are noted in section 7.

6.1 Conservation of Mass: Water

For overland flow, we assume steady state discharge per unit width specified by:

display math(5)

where P is precipitation rate, I is infiltration rate, B is the bottom width of the channel (uniform), and Ais drainage basin contributing area. Basin area is calculated from downstream distance along the valley profile, x, using Hack's law [Hack, 1957]:

display math(6)

with the values f = 0.19and H = 2.3, which were derived from the DEM of the study catchment. The flow depth h is calculated using a modified form of the Manning roughness equation that uses unit discharge and where hydraulic radius is replaced by the flow depth (a valid assumption in this model where the median flow depth is 0.17m),

display math(7)

where nm is the Manning's roughness coefficient (Table 2) and S is the channel bed slope inline image.

Table 2. Table of Parameter Values
ParameterValue
k1.4 × 10−6 sm2/kg
n for grass0.05
n for bare soil0.022
τc for grass140Pa
τc for bare soil6.2Pa
l1m
Rc0.5m/yr
B1m
νs3.47 × 10−3 m/s
φ0.5

6.2 Conservation of Mass: Sediment

To model sediment transport, we use an equation that explicitly represents the entrainment of particles from the bed into the flow (either through suspension or bedload transport) and resettling of particles back onto the bed, with a continuity of mass equation expressed as:

display math(8)

where hT is bed elevation (defined as the immobile interface below the active bed), d is the deposition flux (volume per unit time per unit bed area) of sediment settling from the water column and active bed layer onto an immobile channel boundary, e represents the erosion flux from the immobile channel boundary into the active bed layer and water column, and φ is the porosity of sediment on the bed. When combined with mass conservation for the water column, defined below, this formulation allows for a lag in the flow hydraulics and can be simplified into the Exner equation if the length scale of sediment transport approaches zero [Davy and Lague, 2009].

The erosion flux is modeled using a generic excess-shear stress equation [Howard and Kerby, 1983; Sanford and Maa, 2001],

display math(9)

where k is the erodibility constant, τ is the bed shear stress, and τc is a critical shear stress below which no erosion occurs. Bed shear stress for steady, uniform, wide-channel flow is:

display math(10)

where ρ is water density and g = gravitational acceleration.

The deposition rate depends on the product of settling velocity and sediment concentration. Following Davy and Lague [2009], this can be expressed as:

display math(11)
display math(12)

where d is a dimensionless parameter representing the ratio of near-bed sediment concentration to the vertically averaged concentration, cs is the vertically averaged total sediment concentration in the water column, νs is the settling velocity, and qs is the sediment discharge per unit width derived from the stream bed. We account for the coarse and fine grain size fractions separately by using different d and νs values for different grain size classes. Consequently, the values of d and νs are large for the coarse sediment in the active layer near the bed and small for the small particles occupying the suspended and wash load (see Appendix A for a detailed description).

In addition to fluvial sediment, we also route the sediment derived from headcut wall retreat. The headcut is treated as a local source of sediment with a volume flux per unit valley width:

display math(13)

where Vr is the rate of sediment eroded per unit valley width, R is the retreat rate, and Hc is the height of the headcut. Motivated by the observation that slump deposits tend to accumulate within a few meters of the main headcut face, we describe the depositional pattern of headcut-derived sediment using an exponential pattern that decays downstream from the headcut face:

display math(14)

where dhc is the deposition rate due to wall collapse downstream of the headcut, l is the distance downstream of the headcut, and l∗is a characteristic length scale for wall collapse deposits. This rule mimics the observation that the thickness of slump deposits tend to taper downstream over a distance of one to several meters.

We combine equations (8)(14) to obtain an equation for bed evolution:

display math(15)

Equation (15) necessitates an expression for the sediment discharge, qs. To provide this, we look at the continuity of mass (sediment) undergoing active transport in the water column. Sediment concentration in the water column is a function of any erosion or deposition from the stream bed, the net flux of sediment in transport, and the incoming sediment from lateral sources. In one dimension, this can be expressed as:

display math(16)

where qslat represents the sediment flux (volume per bed area per time delivered to the water column from tributaries and hillslopes (qslat=1 × 10−3 m/yr). We assume a quasi-steady sediment concentration with time such that inline image and solve for qs to obtain:

display math(17)

Equations (17) and (15) are then solved numerically.

6.3 Headcut Erosion

The final ingredient in the model is the upstream migration of the headcut. The observed relation between headcut retreat rate and drainage area could be due to a variety of erosional processes. Here we consider two alternative hypotheses: retreat rate is a function of (1) discharge (which varies directly with drainage area (equation (5)) or (2) headcut height, which may vary autogenically with drainage area. These hypotheses will be formulated in relation to a constant retreat rate, which is derived from aerial photo observations. The headcut is treated as a discrete step in the solution space, and it moves upstream with a prescribed velocity. The constant retreat rate is Rc=0.5 m/yr, based on the average retreat rate (0.48m/yr) for four headcuts measured between aerial photos from 1957 and 2006 (note that this may not reflect the average value of all gullies in the study region).

We define a discharge-dependent retreat rate rule as:

display math(18)

where Rw is the water discharge retreat rate, inline image, q0 is discharge at the gully outlet, and qh is the discharge at the headcut. All discharge at the headcut is normalized by q0, which represents the maximum flow in the watershed. Because discharge is a function of drainage area (equation (5)), equation (18) mimics our lidar observations showing that headcut retreat goes approximately as the square root of drainage area. Additionally, we multiply the fraction inline image by a factor of η = 2 (a model assertion) so that when inline imageis half of inline image, Rw will achieve the average retreat rate observed in the photos. In studies which show that headcut migration is driven by upstream drainage area, the variable of drainage area is typically used as a proxy to describe water discharge [e.g., Burkard and Kostaschuk, 1997; Vandekerckhove et al., 2001]. Consequently, equation (18) allows us to test whether field-observed relationships between headcut migration and water discharge can produce realistic topography in our model. Mechanistically, water discharge influences headcut migration through processes such as headcut saturation mass failure and removal of sediment buttressing the headcut [Montgomery, 1999].

A height-dependent retreat rule is formulated as:

display math(19)

where Rh is the height-dependent retreat rate, inline image, hh is the height of the headcut, hmax is the maximum expected headcut height (based on the maximum field-observed height), and we assign ζ = 2. This rule linearly scales the observed average retreat rate as a function of headcut height relative to a maximum headcut height. Empirical observations show that headcut height is related to retreat rate [Vandekerckhove et al., 2001], and modeling shows that as headcut height increases, stability decreases, leading to headcut failure [Istanbulluoglu et al., 2005]. Because mechanical erosion from tension crack weathering [Montgomery, 1999] or local soil saturation from snowmelt are not influenced by upstream drainage area, this retreat equation provides an alternative hypothesis to discharge-dependent migration. Consequently, this formulation enables us to explore whether headcut retreat would autogenically decrease with upstream drainage area, without explicitly specifying this condition.

6.4 Numerical Model Setup

The numerical model is implemented using an explicit finite-difference solution in which the longitudinal profile is divided into cells of length Δx. The initial model longitudinal profile is an idealized version of an actual field slope, with a 2m drop representing a headcut near the downstream end of the model domain. The total length of the profile is 469m, which is the distance between a drainage divide and a channel outlet at our field site. To implement upstream headcut migration, the headcut is initially positioned at the boundary between two cells. During each model time step, its horizontal position is updated. To accommodate this motion, the length of the downstream cell is increased while that of the upstream cell is shortened. When the length of the downstream cell reaches 1.5Δx, the cell geometry switches such that the headcut becomes the boundary between the next pair of cells upstream. The length of the original downstream cell returns to its original value (Δx), while the length of the new headcut cell pair is adjusted to accommodate their new position. In this manner, the headcut migrates “through” the array of cells that represents the longitudinal profile. In addition, the cell immediately upstream of the headcut is treated as an internal boundary by assigning it a slope value equal to that of its upstream neighbor. The slope for the rest of the profile is determined using inline image. This method prevents the model from treating the headcut as a steep ramp.

7 Model Experiments and Results

7.1 Sensitivity Analysis

To map out the behavior of the model and isolate the role of each of its components, we performed sensitivity experiments that introduce elements of reality one by one. The first model run represents a case with a constant rate of headcut migration (equation (13), Rc=0.5 m/yr) and exponential downstream deposition (equation (14)) but without additional fluvial erosion above or below the headwall (Figure 9a). The basal deposit length scale, l, is assigned a default value of 1m, comparable to the scale of wall collapse deposits in the field. With this scale of deposition and without fluvial erosion, the initial headcut accumulates debris at its base each time it moves back, becoming shorter until a step no longer exists (Figure 9a, inset). This behavior contrasts with the pattern of evolution when l∗=100 m. This case represents a situation in which the transport distance of slump deposits is much greater than the height of the headcut. Although l∗=100 m is physically unrealistic, the fact that it creates a more realistic profile indicates that sustained headcut retreat requires evacuation of sediment from the headcut face.

Figure 9.

Model calculations showing valley longitudinal profile evolution. Rate of headcut retreat is constant in each case. (a) Deposition from headcut retreat using l∗=1 m and l∗=100 m, with no fluvial processes. Inset shows close-up of l∗=1 m. (b) Fluvial erosion and uniform discharge. Slope surface is clay-loam soil. (c) Fluvial erosion and uniform discharge. Slope surface is grass upstream of the headcut, and bare clay-loam soil downstream. (d) Fluvial erosion and downstream-increasing discharge (equations (5) and (6)). Slope surface is grass upstream of the headcut and bare clay-loam soil downstream. (e) Fluvial erosion and downstream-increasing discharge. Slope surface is entirely grass. (f) Fluvial erosion and downstream-increasing discharge. Slope surface is grass excluding a 4m clay-loam soil zone downstream of the headcut.

The model run shown in Figure 9b introduces fluvial erosion and deposition. Here, the discharge has a constant value of 0.1m2/s, which is consistent with discharge estimates for typical flash floods in the study catchment. The substrate is assumed to consist of bare soil; the critical shear stress τc and erodibility coefficient k are based on field experiments on clay-loam soil [Elliot et al., 1989] (Table 2). For completeness, the model simulates erosion upstream and downstream of the headcut. Fluvial erosion downstream of the headcut is sufficient to transport sediment eroded from the headcut, and therefore the headcut grows in height until upstream erosion causes headcut lip lowering. The easily erodible bare soil upstream of the headcut results in a much different profile from model runs where upstream erosion is limited by highly resistant vegetation.

Active gullies in the field area usually have a vegetated valley surface upstream of the headcut and a bare, actively scoured channel downstream. Figure 9c illustrates an analogous case in which the upstream valley floor has a resistant vegetation carpet, represented by a high critical shear stress [Prosser and Dietrich, 1995] and a higher roughness (Table 2). In this case (using a constant discharge), erosion is concentrated downstream of the headcut, so that the headcut height grows through time as it migrates upstream.

In Figure 9d, discharge increases downstream according to Hack's law (equations (5)(6)). The upstream boundary condition for discharge is set by the drainage area of the first cell (0.0075m2), and discharge increases downstream according to equations (5)(6). The precipitation used in this model is 35mm/h, and infiltration is 20mm/h. The corresponding downstream increase in shear stress is sufficient to exceed the high erosion threshold along a portion of the grass-covered valley upstream of the headcut, creating a slope break. This has the additional effect of eroding the headcut lip.

The unrealistically tall headcut heights achieved in Figures 9c and 9d suggest that another process is needed to prevent such deep incision downstream of the headcut; thus, in Figure 9e we show a model where the entire profile has the resistance of grass. This inhibits fluvial erosion of material deposited during headcut weathering. Consequently, the headcut rapidly dwindles as wall collapse deposits accumulate at its base.

In Figure 9f, we introduce a frequently observed vegetation pattern at our field site, which is a bare soil zone that extends four meters downstream from the base of the headcut that grades into dense grass further downstream. This model shows deposition extending downstream from the bare soil-grass zone interface, which produces a local convexity downstream of the headcut. Deposition overpowers erosion downstream of the headcut because (1) the erosive shear stress near the headcut is weak due to the low slope, and (2) downstream vegetation retards erosion.

Figure 9f also shows a dynamic headcut height adjustment. The headcut height starts at 2m. At the downstream end of the model, erosion capacity is higher than deposition from headcut weathering, and so the headcut height increases to more than 3m. As the headcut moves upstream, erosion capacity declines, but headcut weathering remains the same, resulting in a final headcut height of 2m. Among the sensitivity experiments illustrated in Figure 9, it is this scenario (Figure 9f) that most closely resembles gullies at our study site. Key similarities include relatively limited change in step height as the headcut migrates upstream, aggradation within the valley downstream of the headcut, and a subtle convexity in the longitudinal profile downstream of the headcuts, near the transition from erosion to deposition (Figure 2).

Among the sensitivity tests, we also varied the bottom channel width with the relationship B = kwA0.5. However, we found little change in the model results, which are not shown because they are nearly identical to those shown in Figure 9. In this system, where the maximum channel widths are tens of meters across, we found that changing channel width did not significantly alter the longitudinal profile. In this case, doubling the channel width results in approximately halving of the water depth, which cuts the shear stress by a similar factor. By contrast the effective critical shear stress (vegetated versus unvegetated) can vary over 2 orders of magnitude. We suspect, therefore, that in a grassland environment like that of our study area, channel width dynamics are subservient to vegetation dynamics. We acknowledge that undulating narrow and wide channel reaches could affect erosion and deposition patterns; however, we did not add this complexity into the model.

7.2 Headcut Migration Rate

Figure 10 compares the longitudinal profile evolution in cases with (1) constant headcut migration rate, (2) a migration rate that depends on discharge (equation (18)), and (3) a migration rate that depends on headcut height (equation (19)); the remaining parameters are the same as in Figure 9f. The three different migration rates are shown at two model times, 250years and 700years. At all times, the constant retreat rate moves the headcut further upstream than either the discharge or height-dependent retreat rates. This result is specific to the chosen value for constant retreat. After 250years the height-dependent headcut is retreating faster than the discharge-dependent headcut. The change in retreat rates over model time can be compared using a nondimensional number, herein referred to as the retreat ratio, RR = inline image(Figure 11). The retreat ratio expresses the balance of sediment derived from headcut weathering versus sediment settling from fluvial erosion. Finally, the height-dependent rule does show an autogenic decrease in headcut height with upstream drainage area (Figure 12).

Figure 10.

Discharge-dependent and height-dependent weathering rates compared to a constant weathering rate of 0.5m/yr. Model profiles shown for t = 250 years and t = 700 years.

Figure 11.

Comparison of the retreat ratio inline image for both the discharge-dependent and height-dependent retreat models.

Figure 12.

Comparison of a field longitudinal profile (HC1 in Figure 2) with modeled profiles. This model uses a bare soil zone of 4m, variable discharge, and precipitation = 35mm/h. (a) Model result using the height-dependent retreat rate. Total model time shown is 500years. (b) Model result using the discharge-dependent retreat rate. Total model time shown is 400years.

7.3 High Versus Low Rainfall

Comparison of models with low or high precipitation intensity (Figure 13) illustrates the importance of flash flood discharge for gully morphology and evolution. The low rainfall intensity scenario shows a headcut becoming buried as it moves upstream because there is not enough water discharge to move sediment away from the base of the headcut (Figure 13). The higher rainfall intensity scenario can move more sediment away from the base of the headcut, so initially it creates a larger headcut. However, the larger rainfall can also erode through the grass upstream of the headcut and therefore it begins to cut down the headcut lip which ultimately decreases headcut height (Figure 13). Note that these models use a constant headcut migration rate, discharge increases downstream according to Hack's law, and full grass cover excluding a 4m zone downstream of the headcut, as done in the Figure 9f scenario.

Figure 13.

Models showing the change in longitudinal profile with variations in rainfall. Aside from precipitation, all model parameters are the same as in Figure 9f. (a) Rainfall intensity = 22mm/h. (b) Rainfall intensity = 45mm/h.

7.4 Model-Field Comparison

When the variable retreat rate models are run with the combination of processes and parameters shown in Figure 9f, the computed longitudinal profile downstream of the headcut bears a strong resemblance to the longitudinal profile of the study catchment (Figure 12). The models retain a distinct headcut, and the slope downstream from the headcut shows a gently concave-upward profile. The models also produce a short convex-up reach directly downstream of the headcut where sediment is deposited from headcut weathering. This feature is similar to the subtle convex-up in-channel fans 1–10m downstream of the headcuts observed at our field site. The discharge-dependent retreat model slows as it retreats upstream (Figure 12), which better fits the field observations (Figure 8).

8 Discussion

We find that headcut retreat rate is related to drainage area at our field site, which agrees with existing gully research [Burkard and Kostaschuk, 1997; Vandekerckhove et al., 2001; Radoane et al., 1995]. In fact, for equation (2) the best fit exponent of 0.57 is comparable with statistical models of the same form [Burkard and Kostaschuk, 1997; Vandekerckhove et al., 2001], and bedrock knickpoint retreat [Berlin and Anderson, 2007]. Perhaps headcuts with greater drainage area migrate faster because more water discharge is available for headcut saturation, accelerating slab failure [Istanbulluoglu et al., 2005]. Alternatively, it may point toward a cross-correlated variable such as headcut height, which could influence retreat rate by, for example, allowing for deeper tension cracks that destabilize headcuts and increase retreat. At present, our data do not permit us to distinguish among these possibilities. Because equation (2) is not sufficient to explain all of the processes that contribute to headcut migration, we developed a numerical model to isolate the competing factors.

We explored the controls on headcut morphology with the numerical model. We found that a headcut will grow, shrink, or remain relatively constant in height from the competition between headcut weathering and fluvial erosion, largely influenced by the channel critical shear stress. A substrate of bare soil downstream of the headcut (low critical shear stress) is vulnerable to fluvial erosion, which increases headcut height (Figure 9c). Headcut height decreases when deposition outpaces fluvial transport of material from the headcut base (Figure 9e). A nearly constant headcut height is maintained when debris from headcut failure can be eroded by fluvial processes, but the downstream channel is relatively resistant to fluvial erosion (high critical shear stress) (Figure 9f).

The numerical model was, likewise, helpful in identifying the necessary and sufficient conditions required to maintain a near-vertical headcut, a key morphological feature of headcuts observed at our field site. Even in our model in which vertical and migrating headwalls are imposed, we found that in the absence of fluvial erosion sediment will accumulate immediately downstream of the headcut eliminating the abrupt slope discontinuity (Figure 9a), as conceptualized by Montgomery [1999]. Consequently, fluvial erosion is necessary to preserve a near-vertical drop, by moving sediment away from the headcut base. Additionally, the role of fluvial erosion in maintaining a sharp headcut is modified by the substrate erodibility. When sediment downstream of the headcut is bare soil, a stepped headcut can be achieved (Figure 9c), but if the substrate is covered by hard-to-erode grasses, fluvial erosion is insufficient to keep a headcut from becoming a graded slope (Figure 9e).

Gullies observed at our field site have an overall concave profile. The numerical model helps to explain how to achieve this profile while maintaining a migrating headcut. A steep convex-up channel is generated if sediment downstream of the headcut is easily eroded and sediment does not accumulate downstream of the headcut (Figure 9c). Overall channel concavity is achieved with a short bare soil zone downstream of the headcut and grass throughout the rest of the channel. This allowed for continuous aggradation but not so much that it buried the headcut (Figure 9f). The precipitation rate influences channel concavity by affecting the rate of fluvial incision (Figure 13).

Numerical model simulations show that the key to reproducing a realistic channel profile is the pattern of vegetation. A model run with a grass-lined channel, excluding a 4m vegetation-free zone downstream of the headcut, produced the closest match to the longitudinal profile of our field site (Figure 12). This configuration allows fluvial erosion to remove wall collapse debris, which maintains headcut height. Material scoured from the bare soil zone is deposited downstream, aggrading the channel. Consequently, the numerical model reveals that both headcut height and the downstream slope evolution are largely controlled by the vegetation pattern. These data confirm the conceptual model proposed by Montgomery [1999] for abrupt channel headcut evolution.

The controls on vegetation growth at the field site appear to be related to the frequency of disturbance by both fluvial and headcut failure. Site monitoring shows that in the early spring vegetation may grow back near the headcut, but headcut failure often buries this new sprouting vegetation. Moreover, because the zone immediately downstream of the headcut is sparsely vegetated, fluvial erosion is especially active near the base of the headcut. Vegetation reestablishment would thus require a rare scenario; cessation of headcut sedimentation, a hiatus from fluvial erosion, and sufficient soil moisture to nourish seedlings. Consequently, as observed in the numerical model, the bare soil zone creates a feedback that helps to maintain headcuts.

Although both the height-dependent and discharge-dependent retreats can reproduce the observed topography (Figure 12), numerical modeling supports the hypothesis that total discharge influences headcut retreat rate. Model runs with a height-dependent retreat rate show little decrease in migration speed as the headcut moves upstream in the first 500 years (Figure 11). By contrast, calculations with a discharge-dependent retreat rate predict that migration should slow as drainage area shrinks (Figure 11), similar to the patterns observed at the field site. The numerical model and field observations (Figure 8) point toward the importance of total discharge flowing over a headcut as a primary control governing headcut retreat.

This research highlights the long-term geomorphic consequences of headcut migration and fluvial erosion in gullies. Continuous upstream headcut migration without periods of headcut reburial result in paired terraces on either side of the gully channel, which are observed at our field site. The concave-up longitudinal profile results from fluvial erosion that increases in the downstream direction, but is not strongly influenced by headcut deposition. Moreover, the vegetation pattern found in the study channel proves to be important in controlling the headcut height and the downstream slope.

9 Conclusions

This study uses remotely sensed field observations and numerical modeling to explore the controls on headcut retreat, headcut height, and channel slope downstream from a headcut. We show that a simple headcut retreat equation (equation (2)) can explain headcut positions within a watershed, suggesting that headcut retreat diminishes with decreasing drainage area. A numerical model with a physically based rule set is developed to better understand the effects of headcut erosion and deposition on a channel profile. The model results match the morphology observed in natural gullies such as in-channel fans and channel concavity. The model runs most closely resembling the field observations indicated that headcut height and channel slope downstream of the headcut are controlled by the pattern of vegetation within a gully channel.

Appendix A

In this section, we justify the use of a single “effective” value of the adjusted sediment deposition-rate coefficient inline image. We derive an expression for inline image under the assumption that there are N grain size fractions and that the proportional concentration of each fraction in the flow is constant.

Sediment deposition in our numerical model is described by equation (11). In order to account for the deposition of different size fractions, we consider that the total sediment concentration is the sum of the individual concentrations of sediment in each size fraction. Therefore, equation (11) can be represented as:

display math(A1)

where N is the number of grain size categories, csi is the concentration of sediment for a grain size category i, νsi is the settling velocity for a grain size category i, and di is the near-bed sediment enrichment factor d, calculated according to Davy and Lague [2009] for a grain size category i. The fraction of each concentration in a grain size category can be expressed as:

display math(A2)

where fi is the fraction of the sediment concentration within a grain size category, and csTotal is the total sediment concentration. We assume that the volumetric proportion, fi, of each size fraction i is constant. Substituting the fractional concentration into equation (A3), we obtain:

display math(A3)

Given the relationship:

display math(A4)

we simplify equation (A3) to obtain:

display math(A5)

Consequently, this approach takes into account the variety of grain sizes within the fluvial system. Some grain sizes will be prone to wash load, therefore they will experience d values close to unity, and low settling velocities. By contrast, sediment moving as bed load will have high d∗ values and high settling velocities.

To estimate inline image for our study site, we averaged six grain size distributions obtained from sediment samples in the gully channel (Table A1). Although we only report the average values here, the relative proportions are similar among the six samples. We determined the settling velocity for each size fraction according to the following equation [Ferguson and Church, 2004]:

display math(A6)

where R is the submerged specific gravity (assumed to be 1.65, representing quartz in water), D is the representative diameter of the grain, C1 is 18, and C1 is 0.4. Therefore, using the values in Table A1 and equation (A3), we account for the different deposition rates of different grain size categories that are entrained by water. The data in Table A1 result in a value of 5.78 ×10−3 for inline image.

Table A1. Grain Size Distribution
 Representative Grain SizeSettling Velocity  
Grain Size Category(μm)(m/s)dFraction of Grain Size
Very coarse sand15002.4 × 10169680
Coarse sand7501.32 × 1011330.0019
Medium sand3755.76 × 1028.50.016
Fine sand187.51.95 × 1022.10.053
Very fine sand93.755.57 × 1031.20.108
Silt > 34μm48.251.55 × 1031.10.133
Silt < 34μm192.46 × 1041.00.374
Clay41.1 × 1051.00.311

Notation

A

drainage area (m2)

a

ratio used in the discharge-dependent headcut retreat rule

B

channel width (m)

b

ratio used in the height-dependent headcut retreat rule

cs

sediment concentration

d

deposition flux (m/s)

d

parameter representing the ratio of near-bed sediment to suspended sediment

dhc

deposition rate of wall collapse (m2/s)

e

erosion flux (m/s)

E

erosion rate (m/yr)

f

coefficient for Hack's law

g

gravitational acceleration (m2/s)

H

exponent for Hack's law

Hc

headcut height (m)

h

water depth(m)

hh

headcut height (m)

hmax

maximum headcut height (m)

hT

elevation of immobile bed (m)

I

infiltration (mm/h)

K

erosion/hydrology factor

k

erodibility constant

l

length scale for wall collapse (m)

l

distance downstream of the headcut (m)

m

drainage area exponent

n

slope exponent

nm

Manning n

P

precipitation (mm/h)

q

unit discharge (m2/s)

qsLat

annual sediment flux applied to channel bed (m2/yr)

qh

unit discharge at the headcut (m2/s)

q0

unit discharge at the gully outlet (m2/s)

qs

unit sediment discharge (m2/s)

R

retreat rate (m/s)

Rh

height-dependent headcut retreat (m/s)

Rc

constant headcut retreat (m/s)

Rw

discharge-dependent headcut retreat (m/s)

S

slope

t

time (s)

Tk

known retreat time measured from aerial photos

Tp

predicted retreat time from the summation of equation (3) in a headcut path

Vr

volume headcut retreat per time and valley width (m2/s)

x

distance in the upstream direction (m)

z

elevation (m)

η

coefficient in the discharge-dependent headcut retreat rule

ρ

water density (kg/m3)

φ

porosity

τ

shear stress (Pa)

τc

critical shear stress (Pa)

νs

settling velocity (m/s)

ζ

coefficient in the height-dependent headcut retreat rule

Acknowledgments

This study was supported by the National Science Foundation grant EAR-0952247. We are grateful to the Plains Conservation Center for access to the study site and NCALM for the airborne lidar data. We would like to thank David Montgomery, Joel Johnson, Alex Densmore, John Buffington, and an anonymous reviewer for their helpful suggestions.

Ancillary