Runoff during intense rainstorms plays a major role in generating debris flows in many alpine areas and burned steeplands. Yet compared to debris flow initiation from shallow landslides, the mechanics by which runoff generates a debris flow are less understood. To better understand debris flow initiation by surface water runoff, we monitored flow stage and rainfall associated with debris flows in the headwaters of two small catchments: a bedrock-dominated alpine basin in central Colorado (0.06 km2) and a recently burned area in southern California (0.01 km2). We also obtained video footage of debris flow initiation and flow dynamics from three cameras at the Colorado site. Stage observations at both sites display distinct patterns in debris flow surge characteristics relative to rainfall intensity (I). We observe small, quasiperiodic surges at low I; large, quasiperiodic surges at intermediate I; and a single large surge followed by small-amplitude fluctuations about a more steady high flow at high I. Video observations of surge formation lead us to the hypothesis that these flow patterns are controlled by upstream variations in channel slope, in which low-gradient sections act as “sediment capacitors,” temporarily storing incoming bed load transported by water flow and periodically releasing the accumulated sediment as a debris flow surge. To explore this hypothesis, we develop a simple one-dimensional morphodynamic model of a sediment capacitor that consists of a system of coupled equations for water flow, bed load transport, slope stability, and mass flow. This model reproduces the essential patterns in surge magnitude and frequency with rainfall intensity observed at the two field sites and provides a new framework for predicting the runoff threshold for debris flow initiation in a burned or alpine setting.
 Sparsely vegetated alpine areas and recently burned steeplands are susceptible to debris flows following intense rainfall. These areas commonly have abundant loose channel or hillslope sediment situated downslope of lower permeability surfaces, such as bedrock in alpine areas [e.g., Berti et al., 1999; Coe et al., 2008a] or hydrophobic soils in burned areas [e.g., Wells, 1987; Shakesby and Doerr, 2006]. Unlike more humid settings, debris flows in burned and alpine areas often do not initiate from clearly identifiable landslides, which are triggered by increases in pore water pressure at a slip surface [e.g., Iverson, 1997, 2000; Montgomery et al., 2009; Baum et al., 2010]. Instead, debris flow initiation is typically attributed to runoff from low-permeability surfaces during rain storms, which at a critical discharge threshold, mobilizes loose sediment downslope into a debris flow [e.g., Johnson and Rodine, 1984; Cannon et al., 2001; Berti and Simoni, 2005; Larsen et al., 2006; Coe et al., 2008a, 2008b; Gregoretti and Fontana, 2008]. While a variety of conceptual and physically based models have been proposed for this initiation process, there is a lack of consensus regarding the mechanics by which water runoff can transform loose sediment into flowing debris.
 Mechanistic theories for debris flow initiation by runoff can be grouped into two categories: mass failure of the channel sediment by sliding along a discrete failure plane and grain-by-grain bulking by hydrodynamic forces. The former style of initiation requires a sudden large impulse of sediment to be added to and/or entrained within the water flow, such as from the failure of the sediment-filled bed of the channel or failure of the channel banks caused by channel erosion. The most well-developed theory for this style of initiation is by Takahashi [1978, 1981, 2007], who used static 1-D Mohr-Coulomb analysis to predict the depth of failure of a sediment-filled bed. In this model, frictional strength at depth decreases as surface water flow increases pore pressure. In burned settings, similar slope stability concepts have been used to argue that mass failure of small, centimeter-scale soil slips at the heads of rills play an important role in triggering post-fire debris flows [Wells, 1987; Gabet, 2003]. An alternative view of debris flow initiation by runoff has been motivated by sediment transport experiments in steep flumes [e.g., Tognacca et al., 2000; Gregoretti, 2000; Armanini et al., 2005]. In some of these experiments, a critical discharge of water is observed to create a debris flow surge by eroding the sediment by hydrodynamic forces from the top down rather than by sliding at a failure plane several to many grains below the surface. Models for the incipient motion of particles on steep slopes [e.g., Armanini and Gregoretti, 2005; Lamb et al., 2008] provide a framework for predicting the critical discharge needed to destabilize the bed.
 Unfortunately, a lack of direct field observations of debris flow initiation has made it difficult to identify which mechanism(s) are primarily responsible for generating a debris flow for a given set of topographic and hydrologic conditions. Such observations are scarce, because events are usually infrequent and the terrain of the initiation zone is typically difficult and dangerous to access. Consequently, efforts to predict runoff-initiated debris flows usually rely on empirical rainfall intensity duration thresholds as an indirect way of specifying the critical level of discharge for debris flow initiation [e.g., Cannon et al., 2011]. Such thresholds are usually more distinct than thresholds developed for shallow or deep-seated landslides, because they are not controlled strongly by additional hydrologic factors, such as antecedent soil moisture [Berti et al., 2012]. At closely monitored sites, calibrated rainfall-runoff models have been used to get a step closer to the process, by providing estimates of the rainfall needed to produce the critical level of discharge for debris flow initiation [Berti and Simoni, 2005; Coe et al., 2008a; Gregoretti and Fontana, 2008]. While successful, both empirical rainfall thresholds and discharge thresholds derived from rainfall-runoff models are site specific and typically require years of data collection to implement.
 Important clues about the runoff initiation process are available from a diverse but limited set of measurements made in or near debris flow initiation zones. Sites include Acquabona Creek, Italy [Berti et al., 1999, 2000], Mount Yakedake, Japan [Suwa et al., 1993], the Ichinosawa catchment, Japan [Imaizumi et al., 2005], Chalk Cliffs, USA [e.g., McCoy et al., 2010, 2013], and the Arroyo Seco burned watershed in southern California, USA [Kean et al., 2011]. A common feature of these observations is that debris flows near the initiation zone usually consist of many surges. Often, these surges are quasiperiodic and have a frequency (typically 0.5 to 2 surges per minute) that is higher than that of fluctuations in rainfall intensity and higher than the surge frequency observed downstream. The presence of multiple surges is certainly not unique to sites where runoff is believed to be the trigger. For example, numerous qualitative observations and direct measurements of multiple surge fronts have been made downstream of initiation areas where failures from shallow landslides or a mixture of processes including slope failures are believed to initiate debris flows [e.g., Sharp and Nobles, 1953; Pierson, 1980; Costa and Williams, 1984; Pierson, 1986; Davies et al., 1992; Zhang, 1993; Marchi et al., 2002; McArdell et al., 2003]. Yet the fact that high surge frequency is often observed very close to runoff initiation areas suggests that the processes controlling debris flow initiation by runoff and surge formation may be linked.
 While a connection between debris flow initiation and headwater surge frequency has not been established, surge development in debris flows has received considerable study both at the above-mentioned field sites and in a variety of experimental [e.g., Davies, 1990; Iverson, 1997; Coussot, 1994; Iverson et al., 2010] and theoretical investigations [e.g., Takahashi, 1981; Hungr, 2000; Zanuttigh and Lamberti, 2004a, 2004b]. Following Davies , Zanuttigh and Lamberti  distinguished between two types of debris flow surges: those resulting from progressive or regressive instabilities. Progressive instabilities, also called roll waves, have received the most analysis [Zanuttigh and Lamberti, 2007, and references therein] and have features in common with sediment-free water waves that develop spontaneously in flow on steep slopes [e.g., Brock, 1967]. Progressive debris flow instabilities are usually smaller than the first main surge and can translate through stationary slurries of freshly deposited material. In contrast, regressive instabilities exhibit time-dependent release of material created by the formation and failure of temporary dams of large grains. Based on field observations in New Zealand and China, Davies  suggested that regressive surges are characteristic of flow in small channels and gullies close to debris flow initiation zones, whereas progressive instabilities are characteristic of debris flows in larger, lower gradient channels downstream.
 Here we present field data and a model that explores the connection between regressive instabilities (temporary dams) and debris flow initiation by runoff. The data consist of stage and rainfall measurements associated with 22 debris flows recorded in the headwaters of two separate monitoring sites: a bedrock-dominated basin in Colorado (Chalk Cliffs, 0.06 km2) and a recently burned catchment in southern California (Arroyo Seco, 0.01 km2). Data analysis shows that the magnitude and frequency of surges at these two sites follow similar patterns that correlate with rainfall intensity. Based on these data and additional video observations of surge formation at Chalk Cliffs, we hypothesize that longitudinal decreases in sediment transport capacity (such as that caused by a local, low-gradient section of channel) play a key role in debris flow initiation by surface water runoff. Specifically, we hypothesize that these short, low-gradient reaches act as “sediment capacitors,” which temporarily store sediment from ordinary bed load by water flow and/or very small debris flows and periodically release the accumulated sediment as debris flow surges. To explore this hypothesis, we develop a simple one-dimensional morphodynamic model of a sediment capacitor using approaches from both river and debris flow mechanics. In the absence of constraints needed to make a quantitative test of the model (i.e., a controlled experiment), we focus on the qualitative agreement between model predictions and field observations. We then use the model to investigate potential controls on debris flow initiation and surge magnitude and frequency.
2 Study Sites
 The Chalk Cliffs and Arroyo Seco sites (Figure 1 and Table 1) were selected because both sites have (1) debris flows triggered primarily by surface water runoff (no meter-scale landslides observed), (2) monitoring stations located close to the debris flow initiation areas, (3) similar instrumentation and sampling rates, and (4) recorded events over a range of rainfall intensities. Debris flows at both sites are generated rapidly after rain bursts that represent some of the lowest documented rainfall intensity thresholds in the world [Cannon et al., 2008]. Recorded events occur within minutes after 5 min bursts of rainfall exceeding just 9 mm/hr at Chalk Cliffs [McCoy et al., 2010, 2011] and about 20 mm/hr at Arroyo Seco [Kean et al. 2011]. The longitudinal channel profiles at both Chalk Cliffs and Arroyo Seco contain a series of undulating meter-scale steps, which are comparable in height to observed flow depths (Figures 1c and 1d). These topographic features are common in steep channels worldwide and can substantially influence the flow dynamics.
Table 1. Summary of Site Topographic Characteristics
 Chalk Cliffs is located in a band of hydrothermally altered and fractured quartz monzonite in the Sawatch range of central Colorado. Debris flow monitoring in a 0.3 km2 basin along the cliffs began in 2004 with field visits and automated measurements of rainfall and soil moisture [Coe et al., 2008a]. Monitoring expanded in 2008 to include an automated sensor network of three-instrumented cross sections and three video cameras [McCoy et al., 2010, 2011, 2012, 2013]. An average of three debris flows per year occurs in the study basin between May and October. The high frequency of debris flows is due to a large percentage of exposed bedrock (60%) and sparsely vegetated colluvium (40%), which rapidly concentrates runoff during summer rain storms into channels loaded with loose sediment accumulated from ravel and rockfall generated primarily during the winter months [Coe et al., 2008a]. In some years, debris flows earlier in the summer storm season remove nearly all of the sediment from the channels in the upper parts of the basin. This results in a more flood-like response to late season storms. To avoid potential complications regarding sediment supply, we restrict our analysis to 12 debris flows that do not appear to be supply limited.
 We focus here on data recorded at the uppermost stage-monitoring station in the Chalk Cliff network (Upper Station, 0.06 km2 upslope drainage area) and on videos of debris flow initiation (Figure 1). The videos were recorded near the Middle Station (0.16 km2 drainage area) and near the top of the East Channel (0.0028 km2). Our knowledge of the West Channel initiation zone above the Upper Station is somewhat limited by the difficulty of access; however, it is likely that the flow processes in the West Channel initiation zone are similar to those at the top of the more accessible East Channel, which is monitored with a video camera. Additional details of the grain-sized distribution and channel geometry at the Upper Station are given in McCoy et al. .
2.2 Arroyo Seco, California
 The Arroyo Seco site is located in the San Gabriel Mountains of Southern California. The site was burned in 2009 by the Station Fire, resulting in almost complete removal of all vegetation in the 0.01 km2 study area (Figure 1b). The underlying bedrock is Cretaceous granitic rock, which prior to the fire was covered by a thin (<1 m) mantle of colluvium. After the fire, erosion progressively exposed saprolite on the upper slopes of the study area. Erosion by dry ravel in the first months after the fire also contributed greatly to the supply of loose sediment in the channels. A debris flow monitoring station was installed in November 2009, 1 month after the end of the fire. The Arroyo Seco station was one of five post-fire debris flow monitoring sites established in Southern California in 2009 [Kean et al., 2011]. Of these sites, we study Arroyo Seco, because it is the closest to the initiation area and has the largest number of recorded debris flows.
 Unlike the long-term monitoring site at Chalk Cliffs, Arroyo Seco was maintained for only 2 years because rapid regrowth of vegetation at the site substantially reduced the probability of debris flow. During the first winter storm season after the fire, a series of four storms produced 13 separate episodes of debris flow at Arroyo Seco [Kean et al., 2011; Kean and Staley, 2011]. Half way through the end of the third storm of the first winter storm season, there was a distinct change in the measured stage characteristics, from well-defined surges to a more agitated flow surface characterized by high-frequency (~0.5 Hz), small-amplitude fluctuations. Kean et al.  attributed this change in stage characteristics to a decrease in sediment supply. As at Chalk Cliffs, we restrict our analysis to the first 10 debris flows, which were not supply limited. No debris flows were observed in the second year after the fire, despite storms having similar rainfall intensity as the first year. Additional details of the grain-sized distribution and channel geometry at Arroyo Seco are given in Kean et al. .
3 Measurement Methods and Data Analysis
 To identify patterns in surge frequency and magnitude and their association with rainfall intensity, we analyzed measurements of debris flow stage (H) and 5 min rainfall intensity (I5) at the two study sites. Sensor specifications are given in McCoy et al.  (Chalk Cliffs) and Kean et al.  (Arroyo Seco). We measured rainfall at each station using a tipping bucket rain gage located near the channel cross section and sampled it every 2 s. Peak 5 min rainfall intensities were calculated for each event. We measured flow stage at both sites using laser distance meters suspended ~2 m above the channels by small bridges and sampled at 10 Hz. The respective datums for the Chalk Cliffs and Arroyo Seco stage gages were the elevations of a 232 cm2 force plate and pressure transducer, both of which were installed in the bedrock channel floor directly beneath the lasers. Prior to installing lasers at Chalk Cliffs in 2010, stage was measured using ultrasonic distance sensors. Data from the ultrasonic sensors is inadequate for this study because of a low sampling rate (2 s) and frequent missed measurements during high flows. Rather than exclude the five 2009 events recorded by ultrasonic sensor at Chalk Cliffs, we developed surrogate 10 Hz stage time series from 100 Hz data measured by the force plate beneath the laser. McCoy et al.  showed that when the force plate was covered with at least a 5 cm layer of stationary bed sediment, the force time series closely tracked stage fluctuations associated with surges. In addition, McCoy et al.  showed that temperature data from a colocated pressure transducer could be used to identify whether or not the protective layer of sediment above the force plate was present. We developed a surrogate stage time series for the 2009 events by (1) isolating times when the force plate was covered by sediment, (2) linearly scaling these data to match the maximum and minimum measured stage, and (3) resampling the data to 10 Hz. The main uncertainty in the surrogate stage time series arises from changes in flow density during the event, which is estimated to result in a stage accuracy of ± 5 cm.
 We applied Fourier analysis to the stage time series to quantify three measures of surge amplitude/frequency for each event. Total spectral power (Ptot), which is equivalent to the variance of the time series, was used to quantify the amplitude of the fluctuations. The frequency of the peak in the power spectrum (fp) was used to identify the dominant frequency of the fluctuations. Lastly, the ratio of power in the low-frequency band between 0.01 and 0.1 Hz (Plow) to total power was used to summarize the relative distribution of power between low and high frequencies (Rlow = Plow/Ptot). Prior to spectral analysis, the stage data were detrended using a high-pass Butterworth filter (1/100 Hz cutoff frequency). A second low-pass filter (1 Hz cutoff frequency) was used to remove the highest-frequency fluctuations, which were likely associated with splashes and are not resolved in the 2009 stage data derived from the force plate. Spectral results are sensitive to the time window used in the analysis. To avoid bias from a subjective choice of this time window, we calculated spectrograms for each event to isolate the time window having the maximum spectral power and used this window to calculate the surge characteristics Ptot, fp, and Rlow for the event (see Percival and Walden  for a description of spectrogram calculation methods). We used a window spanning the duration of the shortest event recorded at each site (4 min at Chalk Cliffs and 15 min at Arroyo Seco) and moved the window in 10 s increments. To preserve the connection between Ptot and variance, we report results using a rectangular window (no tapering); however, similar trends were obtained using a Hamming window. We excluded time windows that contained large, nonrepeating, initial surge fronts in order to focus on the amplitude and frequency characteristics of repeating fluctuations in the data.
 We obtained additional information on surge characteristics and debris flow initiation from three rain-triggered video cameras at Chalk Cliffs (Figure 1e). Arroyo Seco was not equipped with video cameras.
4 Field Observations
4.1 Debris Flow Initiation
 Videos of debris flow initiation at the top of the East Channel confirm that water runoff and associated sediment transport are instrumental in generating debris flows at Chalk Cliffs. The videos depict an intermediate intensity event on 28 June 2010 (I5 = 52 mm/hr; Figure 2 and video S1 in the supporting information) and a high-intensity event on 30 July 2010 (I5 = 149 mm/hr; video S2). The camera views a steep (35–40°) section of colluvium that is situated below bedrock cliffs defining the top of the drainage. During the initial minutes of both events, bores of concentrated runoff from the bedrock cliffs impacted the colluvium at multiple locations in the field of view (Figure 2). These water bores quickly formed small (~2 cm high), slow-moving (~10 cm/s) debris flows that eventually coalesced and accelerated in the steep channel below. As both storms progressed, water runoff from the cliffs rapidly cut rills into the colluvium and debris flow surges formed farther down the channel. Similar rills were observed on the hillslopes at Arroyo Seco after the first debris flow at the site [Kean et al., 2011; Schmidt et al., 2011]. During the peak of both storms, additional sediment was mobilized from the jet impacts of newly formed waterfalls. These impacts punctuated and enhanced sediment delivery to the channel.
 Video of surge formation below the Middle Station at Chalk Cliffs provides additional insight into the mechanics of debris flow initiation at Chalk Cliffs (video S3). This video shows that debris flow initiation at Chalk Cliffs is not confined to the upper parts of the basin such as the East Channel initiation area but can also occur in lower gradient channels downstream. Key frames of the video are shown in Figure 3. The beginning of the sequence shows bed load and small (< 10 cm high) debris flows entering the field of view from upstream of the Middle Station (Figure 3a). Much of this sediment supply was deposited in a lower gradient section about 20 m downstream of the Middle Station. The deposit formed a permeable dam that blocked incoming sediment from upstream while sieving nearly all the incoming water flow through the dam face (Figure 3b). The combination of a permeable dam and near-steady sediment supply upstream appears to have created a positive feedback that allowed the dam to grow rapidly in size. Had the dam been less permeable, such that incoming water accumulated and then overtopped the dam, the dam would have grown more slowly or not at all due to erosion of the dam face by water. As seen in Figure 3c, the dam grew in height until the frictional forces within the dam could apparently no longer resist the downstream-directed gravitational and pressure forces. At this point, the front began to slowly creep downstream. In the video, the dam face can be seen to move a short distance before stopping. The stationary dam grew quickly again in size and finally accelerated into a moving debris flow surge (Figure 3d). Similar qualitative field observations of dam formation and subsequent debris flow have been made elsewhere [e.g., Costa and Williams, 1984; Davies et al., 1992; Berti et al., 1999].
4.2 Surge Magnitude and Frequency
 At both study sites, debris flow surge characteristics are distinct at low, intermediate, and high rainfall intensity. Representative examples of stage data and power spectra for events at three levels of rainfall intensity are shown in Figures 4 (Chalk Cliffs) and 5 (Arroyo Seco). Corresponding photographs and video of the flow during the three Chalk Cliff events are shown in Figure 6 and in the supporting information (videos S4, S5, and S6). At low rainfall intensity (< 30 mm/hr), the stage time series at both sites consists of a number of small, quasiperiodic surges (fp = 0.04 and 0.02 Hz at Chalk Cliffs and Arroyo Seco, respectively). At intermediate rainfall intensity (30–60 mm/hr), larger-amplitude, regularly repeating surges are observed at a similar frequency. Finally, at high rainfall intensity (> 60 mm/hr), the flow changes character substantially with the appearance of a single, large-surge front followed by small, high-frequency fluctuations about a more steady background flow. The power spectra show that in general, the majority of the power is located in the low-frequency band between 0.01 and 0.1 Hz. The only exception is the Chalk Cliffs high-intensity case, where the power is more evenly distributed between high- and low-frequency bands (Rlow = 0.46). Typically, there is a single pronounced peak in the spectrum. This distinct peak indicates that the processes controlling surge generation are periodic. The power spectra also show the unexpected result that the greatest power occurs at intermediate rainfall intensity, not high-intensity rainfall. This result means that, with the exception of the first surge front, the largest surge amplitudes are generated during intermediate rainfall intensity.
 A summary of the trends in surge magnitude and frequency across all events at the two sites is shown in Figure 7. The top row shows that the total power (Ptot) for both sites is greatest at intermediate rainfall intensity on the order of 50 mm/hr. The middle row shows that while most of the power is concentrated in the low-frequency 0.01 to 0.1 Hz band, there is a systematic decrease in the fraction of power in this band (Rlow) with increasing I5. This trend corresponds to an increase in the amplitude of high-frequency stage fluctuations with rainfall intensity. The lowest row in Figure 7 shows that the frequency of the peak in the power spectrum (fp) always resides in the 0.01 to 0.1 Hz frequency band, despite the decrease in Rlow with increasing rainfall intensity. We will revisit these trends in the context of a model in section 6.
5 Sediment Capacitor Model
 The above observations of debris flow initiation and surge magnitude and frequency lead us to the hypothesis that local decreases in the sediment transport capacity can create sediment capacitors, which transform near-steady supplies of sediment by water flow and/or thin debris flows into distinct and potentially dangerous debris flow surges. Provided a steady, high sediment supply from upstream, the surge cycle should repeat at a regular frequency set by the slope, rainfall intensity (or water discharge), grain size, and length scale of the low transport zone. Here we develop a physically based version of the sediment capacitor conceptual model described in Figure 8. The mathematical model consists of five components and is constructed in one dimension for a short reach shown in Figure 8e. We have limited the number of processes in the model in an effort to keep it as simple as possible. The model reach contains three sections: two steep sections of bed slope θo, bracketing a lower gradient middle section of slope βo and horizontal length λ. This reach is intended to provide a simple representation of the uneven topography of a steep, natural channel. We focus here on the transition of incoming bed load by water flow into debris flow; however, the model can be more generally considered a filter that acts on all incoming sediment, including incoming sediment that is already moving as debris flow.
5.1 Coupled Surface-Subsurface Flow
 The water discharge per unit width of channel, qw, is modeled as the sum of porous media flow through sediment stored in the channel, qp (if any), and free surface water flow above the bed, qf (if any):
 The porous media discharge is described as
where is the Darcy velocity through the porous media and min (hw, hs) is the lesser of either the thickness of the sediment (hs) or the depth from the water surface to the bedrock surface (hw). The bedrock is assumed to be impervious. Flow through coarse-grained sediment in steep channels can be sufficiently high that inertial forces affect fluid motion causing the flow to deviate from Darcy's Law. This deviation becomes significant at particle Reynolds numbers (Rep) around 1–10 [Bear, 1972], where , ρf is the fluid density, μf is the fluid viscosity, and D is the grain size. To accommodate inertial forces, we model using the Ergun equation, which has been found to be in good agreement with laboratory data up to Rep of about 3000 [Ergun, 1952]. The Ergun equation is given by
where x is downstream coordinate direction, n is the porosity of the bed sediment, and ∂p/∂x is the pressure gradient, which we take equal to − ρfg sin θw, where g is the acceleration of gravity, and θw is the slope angle of the water table (or water-free surface if there is surface flow). This slope is not assumed to be equal to the local bedrock slope but rather is calculated iteratively as part of the complete flow solution. At low flow velocities, the second (inertial) term on the right-hand side becomes negligible and equation ((3)) follows Darcy's Law given by
where K is hydraulic conductivity.
 To accommodate the high flow resistance associated with shallow flow on steep, rough slopes, we model the free surface water flow using the variable power flow resistance equation (VPE) of Ferguson . The VPE equation captures the gradual change in the velocity profile between deep and shallow flows that is caused by the increase in relative roughness of the boundary. Rickenmann and Recking  evaluated several flow resistance relations and showed the VPE equation to have the best overall agreement with a large field data set of velocity measurements that included many steep channels. For a uniform grain size, D, the water discharge (qf) associated with the VPE resistance relation can be written as
where hf is the thickness of the free surface water flow above the sediment layer or bedrock, is the depth-averaged free surface water velocity, u* is the shear velocity, and a1 and a2 are empirical coefficients set by Ferguson  to 6.5 and 2.5, respectively. The shear velocity is defined as , where τ is the local boundary shear stress of the surface water flow, which is scaled by the friction slope, Sw, (τ = ρfghfSw). This slope is not assumed to be equal to the bed surface slope but rather is calculated iteratively in the solution to the complete flow equations. For simplicity, we assume that the bedrock surface has the same roughness as the bed sediment surface. We also neglect the effects of high suspended sediment concentrations on the velocity profile and flow dynamics. This assumption is appropriate given our focus on the initial transition of water flow into debris flow but is not suitable if the sediment concentrations are high enough to damp the turbulence or violate the Newtonian fluid mechanics assumptions behind equation ((5)).
 Our solution for the water depth profile captures the essential role of runoff in the initiation process, which is principally to deliver bed load to the low-gradient section. We model this process using a 1-D diffusion wave approximation to the St. Venant equation governing unsteady shallow water flow. The continuity and momentum equations for this approximation are
where t is time, θ is the local slope of the bedrock surface, and Sw is the friction slope. We further assume the flow is quasi steady during each bed evolution time step, eliminating the time derivative in ((6a)). These approximations allow backwaters to form behind sediment dams and have the computational advantage of being much easier to solve than more complete equations that include unsteady conditions, inertial accelerations, and 3-D flow. These omitted effects are clearly present during rain bursts on steep, irregular topography (e.g., videos S1, S2, and S3); however, they are not essential for obtaining the desired physical insight into debris flow initiation processes. Our steady flow approximation is justified because the model reach is short and because prior to mass failure of accumulated bed sediment in the reach (described in sections 5.3 and 5.4), the rate of bed sediment elevation change is slow compared to the rate at which the water-surface elevation can change. We continue to approximate the water flow as quasi steady in the reach after failure and subsequent motion of the bed sediment, because after this time, we are primarily concerned with the motion of bed sediment as debris flow. Solution of the simultaneous evolution of water and bed sediment elevations during dam break failure are better handled using more sophisticated models such as that of Fraccarollo and Capart . By also neglecting inertial accelerations, our solution for the water depth profile does not accommodate water flow instabilities, such as roll waves or hydraulic jumps. Small roll waves would punctuate sediment delivery to the low-gradient section, but they would not fundamentally change the time scale of surge formation from that of steady flow. Larger hydraulic jumps could have a more substantial effect on sediment transport in the reach, but addressing this complication, as well as 3-D effects, is left for future study.
5.2 Bed Load Transport
 Bed load transport per unit width of channel, qs, is modeled using the sediment transport relation of Rickenmann , which is given by
where ρs is the density of the sediment, τ* is the nondimensional shear stress equal to τ/((ρs − ρf )gD), τ*c is the nondimensional critical stress for the initiation of bed load transport, and Fr is the Froude number. Nitsche et al.  found equation (7)) to be in good agreement with bed load transport measurements made in 13 Swiss mountain streams, provided it was used in conjunction with an appropriate flow resistance-partitioning scheme to separate the flow energy dissipated on drag from the flow energy available for sediment transport. One simple scheme, which they found worked well for the entire data set, used the VPE approach of Ferguson  (equation ((5)) to reduce τ* based on the decreased energy slope associated with high flow resistance at shallow depths. We adopt a similar approach here, except that instead of reducing τ* from its total value, we inflate the critical shear stress using the VPE resistance-partitioning scheme described by Ferguson . Nitsche et al.  found that more complicated resistance-partitioning schemes, such as that of Yager et al. , performed slightly better at sites with large immobile boulders; however, we do not use these more complicated schemes because immobile boulders are not present in the channels of our study sites (Figures 1c and 1d). For simplicity, we also do not consider the contribution from suspended load or the effects of nonuniform grain size on sediment transport.
 For a bed of uniform grain size, the nondimensional critical shear stress (τ*c) proposed by Ferguson  reduces to the equations
 In these equations, θs is the slope angle of the sediment bed, a0 is a constant equal to 8.0, hfc is the flow depth at the threshold of motion, and τ ′ * c is the base value of critical shear stress, which is constant and taken equal to 0.026. For a given ratio of grain size to critical flow depth (D/hfc), equation ((8a)) can be substituted into ((8b)) to solve for the slope and corresponding critical nondimensional shear stress. Although derived from bulk flow properties, equation (8) yields a similar increase in the critical shear stress with slope to that predicted from the more mechanistic approaches of Lamb et al.  and Recking .
 At slopes greater than about 15°, the downslope component of grain weight, which aids mobility, is large enough to significantly alter the slope dependency in critical shear stress described by equation (8). We follow Ferguson  and include gravitational effects by correcting the base value of critical shear stress τ′* c using the Parker relation
where τ′* co = 0.026 and ϕ is the sediment friction angle (G. Parker, 1D sediment transport morphodynamics with applications to rivers and turbidity currents, 2004, e-book available at http://hydrolab.illinois.edu/people/parkerg//morphodynamics_e-book.htm). Using this correction, the nondimensional critical shear stress for uniform sediment size increases from the background value of 0.026 at very low slopes to a maximum value of 0.14 at a slope of 23°. Above slopes of 23°, τ* c decreases rapidly to zero at θs = ϕ = 40°. Similar decreases in τ* c at very high slopes are present in the theories of Lamb et al.  and Recking .
5.3 Slope Stability
 The stability of stationary sediment in the bedrock channel can be evaluated using a Mohr-Coulomb static force balance as implemented by Iverson and Denlinger . We adopt their formulation because it not only describes the static state of limiting equilibrium but it also describes the depth-averaged flow of sediment once failure has occurred (next section). Assuming failure occurs at the bedrock/sediment interface, the balance of forces in the x direction at the static limit to motion is given by
where ρsb is the bulk density of the sediment-water mixture, pbed is the pore fluid pressure at the bedrock-sediment interface, ϕbed is the Coulomb frictional angle at the bedrock-sediment interface, gx = g sin θ, gz = g cos θ, and kact/pass is an earth pressure coefficient, which we set equal to 1 for simplicity. The first term on the left-hand side of ((10)) represents the resisting basal shear stress. The second and third terms represent longitudinal normal stresses, and the term on the right-hand side is the driving stress due to the gravitational body force. Under static conditions, pbed is assumed to be equal to hydrostatic pressure, ρf gz hw. Equation ((10)) can be used to calculate the ratio of the resisting forces (left-hand side) and driving force (right-hand side) as a function of distance along the channel. Failure of a column of sediment in the channel occurs when this ratio or factor of safety is less than one.
5.4 Flow of mass failures
 We use a simplified version of the Iverson and Denlinger  x momentum equation to compute the depth-averaged velocity of the bed sediment, , after failure. We neglect the effects of stresses on lateral boundaries and the typically small basal stresses associated with fluid viscosity. Eliminating these terms, the Iverson and Denlinger  conservation of x momentum equation reduces to
where t is time. In this equation, all the terms in equation ((10)) have been grouped on the right-hand side. In addition, the basal stress term (first term on the right-hand side of ((11))) has been modified from its static value in ((10)) to include the direction of force (sgn()) and changes in normal stress caused by centripetal accelerations associated with variations in channel slope.
 Once the bed sediment begins to move, pbed will deviate from hydrostatic pressure due to deformation of the pore spaces. Although George and Iverson  have recently developed advanced techniques to calculate the complex evolution of pbed during motion, we model the change in pbed using the simpler approach originally used by Iverson and Denlinger  and Denlinger and Iverson . Their formulation is based on experimental observations of debris flow initiation experiments [Iverson et al., 1997; Reid et al., 1997]. During the first 1 sec of movement, pbed is specified to increase linearly to 0.9 times the total basal normal stress, ρsb gz hs. After this time, the high pore pressure is modeled to diffuse upwards through the sediment column. We follow Iverson and Denlinger  and model the decrease in pbed back to hydrostatic levels using the approximation
where δ is the ratio of pbed to normal stress and Df is the pore pressure diffusivity. To limit the number of processes that depend on grain size, we do not include a grain size dependence on Df. Instead, we use a constant value of 10−4 m2/s, except in places where ∂hs/∂x < 0, where we follow Denlinger and Iverson  and simulate the effects of low pressure on surge fronts due to size segregation using a high value of Df = 10−2 m2/s.
 Like pbed, the bulk density of the sediment, ρsb, will likely evolve as the sediment pile fails and mixes with water runoff. Given our simplified treatment of the water flow during mass flow, we do not attempt to model this interaction and assume ρsb is constant. The complex evolution of ρsb could be handled using more sophisticated models designed for dam break flows, such as that of Cao et al. .
5.5 Bed Sediment Elevation
 We model the evolution of bed sediment thickness as the sum of spatial changes in the flux of bed load and the flux of bed sediment as debris flow. This sum is given by the equation
where cb is the volumetric concentration of sediment on the bed.
5.6 Boundary Conditions and Numerical Implementation
5.6.1 Water Discharge
 Steady water discharge is supplied at the upstream end of the reach as a boundary condition. The incoming steady water discharge is equivalent to the runoff produced by constant rainfall (at rate I) over an impervious catchment upstream of the reach having an area (A) equal to 10,000 m2. This conversion provides a connection to the field sites where water discharge is unmeasured, but rainfall is known. We limit the present analysis to simple, steady boundary conditions in order to obtain a basic insight into the model behavior; however, the model can accept gradually varying inputs of water and sediment discharge associated with a particular storm. Variable boundary conditions could be defined using a distributed hydrologic model that converts time series of rainfall into water and sediment discharges upstream of the reach. The upstream water depth boundary condition, hw(t, x = 0), is fixed by distributing the water discharge evenly across a flat-bottomed channel of width, W = 1 m.
5.6.2 Sediment Supply
 The incoming steady sediment discharge is specified as the transport capacity at the entrance to the reach given an unlimited sediment supply. We use an unlimited sediment supply because it is similar to the conditions that produce debris flows at the study sites. To focus on the transition of bed load transport to debris flow, we start each simulation with a bare bedrock channel, hs(t = 0, x) = 0. Although we do not consider it here, it is important to note that limiting the sediment supply boundary condition will result in very different model behavior. For example, if the supply of sediment from upstream is less than the transport capacity of the low-gradient reach, no sediment would accumulate in the low-gradient section and no surges would be produced. This condition may be more typical of vegetated steeplands. The threshold value of sediment supply that must be exceeded to produce sediment accumulation (and debris flow surges) can be estimated by evaluating equation ((7)) in the low-gradient section, with τ = ρfghf sin βo.
5.6.3 Numerical Methods
 We use an iterative Newton-Raphson scheme to solve ((1)), ((2)), ((3)), ((5)), ((6a)), and ((6b)) for the water depth profile (hw) at the beginning of each bed evolution time step. The solution begins with a guess of the hw profile and converges once the profile satisfies ((6a)) and ((6b)).
 We solve equation ((11)) for the depth-averaged velocity of bed sediment () using a less sophisticated numerical scheme than the Riemann methods used by Denlinger and Iverson . Our solution method is intended to capture the most important role of debris flow motion in this problem, which is merely to evacuate failed sediment a short distance λ out of the modeling domain. As such, we are not attempting to calculate the precise shape of the surge, which would require special numerical methods. Instead, we solve ((11)) using a simple explicit finite difference scheme, which only permits downstream-directed (positive) velocities. Our numerical solution results in surges that have approximately the same front velocity as the experimental debris flow surges of Iverson et al.  but are more compact in shape and do not contain secondary roll waves in the tail. We consider these two deficiencies acceptable given the present goal of making a qualitative connection between dam formation and debris flow initiation, but we acknowledge that our numerical approach is not appropriate for modeling longer reaches.
6 Model Results
 Complex, yet realistic, flow dynamics emerge from the coupled interaction of basic water flow, bed load transport, and slope stability. Representative examples of modeled stage time series and power spectra at three levels of rainfall intensity (or water discharge, qw = I A/W) are shown in Figure 9. The displayed stage hydrograph is sampled near the downstream end of the model reach (Figure 8e). The power spectra is calculated for a 2 min time window that starts at t = 30 s. Corresponding videos of the evolving water and bed sediment elevations over the entire reach for each case are shown in the supporting information (videos S7, S8, and S9). The parameters in the simulations are θo = 30o, βo = θo/4, λ = 2 m, D = 2 cm, ϕ = ϕbed = 40o, n = 0.4, μf = 1.0 10−3 kg/(m s), ρf = 1000 kg/m3, ρs = 2600 kg/m3, ρsb = 1960 kg/m3, cb = 0.6, and g = 9.8 m/s2.
 The model behavior resembles the dynamics observed at the field sites, in that the model produces small, regularly repeating surges at low rainfall intensity, larger repeating surges at intermediate intensity, and a single large surge followed by smaller-amplitude fluctuations at high rainfall intensity. The primary difference between the model and field observations is the lack of high-frequency fluctuations in the model at high rainfall intensity. High-frequency fluctuations at this level of rainfall intensity are probably associated with roll waves and/or waves generated by the irregular topography at the field sites, neither of which is included in the model. Despite the absence of high-frequency power, the model exhibits a similar shift towards increased surge frequency at high rainfall intensity (Figure 9f) as observed in the field.
 The magnitude of individual surges in Figure 9 depends on the volume of stored sediment that fails (V). Likewise, the frequency of surges (f) is governed by the time it takes for each sediment pile to grow to the limit of stability. Volume and frequency are linked through the expressions
where Hp is the characteristic height of a sediment pile at failure, αV and αf are geometric coefficients related to the shape of the pile upon failure, and dhs/dt is the rate at which the sediment pile grows in height. Prior to failure, the rate of pile growth depends on the difference between the incoming and outgoing bed load flux to the flat section as given by equation (13)) with = 0. While algebraic scaling relations for dhs/dt in terms of basic variables (like I, λ, and θo) can be derived for the sediment boundary conditions used in Figure 8, similar scaling relations for both Hp and the geometric coefficients αV and αf cannot be derived algebraically. This is because the shape and maximum size of a pile include complications from several factors. These factors are the following: (1) piles evolve to different shapes depending on the flux of bed load to the reach; (2) based on limit equilibrium (equation ((10)), the volume of sediment that mobilizes into a surge depends strongly on the shape of the pile (i.e., distribution of longitudinal stress) and the distribution of basal pore pressure; and (3) for each surge cycle, the initial conditions such as hs(x) and pbed(x) are not the same but are instead inherited from the previous surge.
 These three factors help explain the patterns in surge magnitude and frequency shown in Figure 9. Surges are small at low I (Figure 9a) because sediment piles only grow to a small height before failure due to the relatively low rate of sediment supply. The sediment piles are small because deposition progresses in a thin layer from the flat section up the steep section (video S7). As deposition progresses up the steep section, a Takahashi-style failure [Takahashi, 1981] initiates at the upstream end of the sediment pile and subsequently triggers mass movement downslope through the flat section. In contrast, at intermediate and high rainfall intensity, the increased rate of bed load transport creates a different deposit distribution, resulting in much thicker sediment piles than at low rainfall intensity (videos S8 and S9). The first pile to fail at both intermediate and high I generates a large initial surge. These surges are very similar in magnitude to each other (Figure 9c and 9e). The similarity in magnitude is due to the fact that the size and shape of each pile at failure is nearly identical. A key difference between the initial surges at intermediate and high I, however, is the timing with which they occur. The first surge at high I appears several seconds before the first surge at intermediate I. The earlier onset of initial failure under high I results from the greater rate of bed load sediment delivery.
 Although the initial surges at intermediate and high I are very similar, there are substantial differences between the average amplitudes of subsequent surges at the two levels of rainfall intensity. As in measured debris flows, the modeled amplitude of subsequent surges at high I are relatively small (about 40% of the initial surge amplitude), whereas at intermediate I, the later surges are of moderate size (about 60% of the initial surge amplitude). The differences in amplitude are related to the manner in which the sediment piles evolve and fail. Pile evolution and failure is partially controlled by the initial conditions at the beginning of a surge cycle. Failures do not usually evacuate all of the sediment out of the reach, so the next sediment pile starts to develop on a remnant distribution of sediment and pore pressure left over from the previous surge. The distribution of pbed at the beginning of the first and second surge cycles can be quite different. This is because the pile grows under hydrostatic conditions up to failure during the first surge cycle, whereas at the beginning of the second cycle, pbed can be elevated above hydrostatic conditions as a legacy of the previous surge (according to equation (12)). Elevated pore pressures in the second and following surges can cause the sediment piles to fail sooner and at smaller sizes than the first pile. Differences between the amplitude of the first and subsequent surges are less pronounced at low I (Figure 9a), because the excess pore pressures diffuse quickly back to hydrostatic levels through the thin remaining sediment layers.
 Additional differences in pbed and the rates of sediment supply with rainfall intensity account for the remaining differences between surge amplitudes in Figure 9. As mentioned above, sediment piles grow more rapidly at higher rainfall intensities. However, the increased rate of growth of the pile does not necessarily translate into larger surges because the greater water depth (pressure) at high I also helps destabilize the pile. This is the case for the high-intensity simulation in Figure 9e (video S9). After the initial surge, a new pile rapidly develops in the flat section. Soon after, however, high pore pressures generate a small failure at the thinning downstream end of the pile. This failure mobilizes into a small surge, while the rear of the pile remains stored in the reach. As sediment continues to enter the reach from upstream, similar small failures are generated in rapid succession creating the high-frequency surges shown in Figure 9e. In contrast, at intermediate I, the pore pressures at the front of the pile are not as large as at high I (due to the shallower flow). This creates a more stable front that permits the pile to grow to a greater size before failure.
7 Controls on Surge Magnitude and Frequency
 Further insight into the controls on surge magnitude and frequency can be obtained by systematically varying key parameters in the model. We focus here on the variations of total surge power (Ptot) and peak surge frequency (fp) with changes in four variables: rainfall intensity (I), the length scale of the low-gradient section (λ), slope at the top and bottom of the modeled reach (θo), and grain size (D). For each set of simulations (Figure 10), we vary one variable (I, λ, θo, or D), while keeping the other parameters fixed at the values used to create Figure 9.
7.1 Rainfall Intensity
 Rainfall intensity controls the water flow depth and rate of sediment supply to the reach. Variation of I (with λ = 2 m, θo = 30°, βo = θo/4, and D = 2 cm) provides a more complete picture of the trends described in the previous section. As rainfall intensity increases, surge power rapidly grows to a peak at 32 mm/hr and drops to near zero above 81 mm/hr (Figure 10a). The asymmetrical peak in total power is very similar to what was observed at both Chalk Cliffs and Arroyo Seco (Figures 7a and 7b). In addition, as in the field cases, the range in modeled fp is confined to a fairly narrow, low-frequency band. Within this band, peak surge frequency gradually increases with I up to 80 mm/hr (Figure 10e). The slight increase in fp with I is driven by the increased rate of sediment delivery to the reach, which causes piles to grow faster and reach their failure geometry sooner than at lower I. The abrupt drop in power above I = 81 mm/hr is triggered by elevated bed pore pressures created by the first surge. The increased pressure, combined with high background pressure of the water discharge, prevents the next pile from developing and consequently stops surge production.
 The length scale λ controls the amount of sediment that can be stored in the reach. Variation of λ (with I = 30 mm/hr, θo = 30o, βo = θo/4, and D = 2 cm) shows an intuitive trend of increasing surge power with λ (Figure 10b). In other words, larger capacitors generate larger surges. Similarly, as the capacitor grows in size, it takes longer to fully charge. Consequently, there is an overall decrease in the frequency of surges with increasing λ (Figure 10f). The greater scatter in fp for λ > 3 m is due to an increase in the variability of surge height produced by the model. This variability results in a broader peak in the Fourier power spectrum than that shown in Figure 9 and leads to more variability in fp. Figure 10b also indicates that there is a minimum value of λ (1 m) required to accumulate enough sediment to produce a surge.
 Slope θo controls both the flux of bed load into the reach and the stability of the accumulated sediment. Variation of θo (with I = 30 mm/hr, λ = 2 m, βo = θo/4, and D = 2 cm) shows a distinct threshold for debris flows (Figure 10c). Using the values of I, λ, and D given above, no debris flow surges are observed below θo = 17.5°, only steady bed load transport. Without surges, the bed load transport completely fills in the low-gradient section, eventually forming a stable planar slope that is parallel to the incoming bedrock slope. The maximum thickness of this layer of sediment is set by the geometry of the reach at hs = λ sin(θo − βo)/cos βo. The threshold slope is approximately equal to the slope defining a Takahashi-style failure for this thickness of sediment and the water depth, hw, which is set by the incoming water discharge. Interestingly, the greatest surge power (and lowest frequency) is observed just above the threshold slope. This occurs because sediment piles are more stable at lower slopes and can grow to a larger size before failure occurs. Conversely, surge frequency increases with slope (Figure 10g) because sediment piles on steep slopes are less stable and fail more quickly, resulting in smaller and more frequent debris flow surges. At very high slopes above 40°, the model produces a single first surge followed by an almost steady flow with negligible stage fluctuations.
 It is important to note that the debris flow threshold shown in Figure 10c is also specific to the boundary conditions used in the calculation. We assume a steady input of water and sediment at the transport capacity of the upstream end of the reach. Different thresholds and surge characteristics would emerge for supply-limited conditions or episodic boundary conditions.
7.4 Grain Size
 Grain size D controls the boundary roughness, the permeability of sediment piles, and the critical shear stress for bed load transport. Variation of D (with I = 30 mm/hr, λ = 2 m, θo = 30o, and βo = θo/4) shows that surge power decreases as grain size increases (Figure 10d), while surge frequency remains nearly constant (Figure 10h). The modeled decrease in surge power with D is primarily related to differences in the water elevations with grain size. The water depth across most of the sediment pile increases with D because of increased boundary roughness and reduced surface flow velocity. This leads to higher basal pore pressures, which cause sediment piles to fail at smaller volumes than at lower water depths and pore pressures. The tendency for water depths to be greater for large grains than small grains is partially counteracted by higher subsurface flow associated with more permeable large grains; however, the bed sediment does not reach a sufficient thickness in these simulations for this effect to overwhelm the influence of boundary roughness on flow depth. It is important to note that real debris flows have a wide range of grain sizes, which segregate quickly after motion begins [e.g., Iverson, 1997; Johnson et al., 2012]. This process causes large grains to collect at the front and sides of the surge and will likely produce a more complicated trend in surge power with D than that modeled using a uniform grain size and one-dimensional model.
 Field observations of runoff-generated debris flows in the headwaters of two small catchments reveal systematic patterns in surge characteristics that correspond with rainfall intensity. Field evidence also suggests that the observed flow patterns are regulated by variations in channel slope, in which low-gradient sections act as sediment capacitors, temporarily storing incoming sediment and periodically releasing the sediment as debris flow surges. The salient features of this process can be modeled through a system of coupled equations for water flow, bed load transport, slope stability, and mass movement. Unlike other models for debris flow initiation by runoff, the sediment capacitor model describes both the initiation process and the observed variations in surge magnitude and frequency. These patterns emerge from a complex interaction between sediment deposition within the reach and mass failure of stored sediment piles.
 As a filter operating on incoming sediment, the sediment capacitor model accommodates other processes that may contribute to debris flow generation, including soil slips, bank failures, and rapid sediment entrainment by hydrodynamic forces and fire hose impacts. Debris flows initiated by these other processes during a storm are likely to be distributed throughout the headwaters and occur with irregular intervals and magnitudes. However, when filtered by the uneven topography that is common in steep, natural channels, these irregular flows can be organized into a more regular series of surges in a manner analogous to how simple resistor-capacitor electrical circuits can be used to filter a noisy signal into a precise frequency band. For this reason, the sediment capacitor model may be applicable to sites beyond those where runoff is believed to be the primary trigger, that is, sites where debris flows are also mobilized from landslides on soil-mantled hillslopes. Interesting directions for future research are to understand how a series of sediment capacitors might interact and influence flow dynamics downstream, as well as to understand how variable boundary conditions and limits on sediment supply affect the flow dynamics.
 Initial exploration of the model parameter space shows that the sediment capacitor model reproduces two other important features of the observations. First, it defines distinct slope thresholds above which debris flows occur. Second, it identifies a relatively narrow range of intermediate rainfall intensities that produce large surges. The reproduction of these two observations suggests that the sediment capacitor framework has the potential to predict a runoff threshold for debris flow initiation in a given catchment, as well as to estimate the flow conditions that will generate the most destructive surges. Given that the present model is intentionally constructed with basic components and contains a number of simplifying assumptions, considerable testing and model refinement (such as including 3-D effects) will likely be needed before accurate field predictions are possible. Nevertheless, the initial qualitative agreement between the field observations and our model invites quantitative testing through controlled experiments and more detailed field measurements.
water velocity profile constants, dimensionless
catchment area above model reach, L2
concentration of bed sediment, dimensionless
grain size of bed sediment, L
hydraulic diffusivity, L2/T
frequency of surges, 1/T
frequency of peak in power spectrum, 1/T
Froude number, dimensionless
gravitational acceleration, L/T2
downslope component of gravitational acceleration, L/T2
bed-normal component of gravitational acceleration, L/T2
debris flow stage, L
free surface water depth, L
flow depth at threshold of bed sediment motion, L
height of sediment pile at failure, L
bed sediment depth, L
water depth, L
rainfall intensity, L/T
5-minute peak rainfall intensity, L/T
hydraulic conductivity, L/T
earth pressure coefficient, dimensionless
porosity of bed sediment, dimensionless
pore pressure at the bed, M L−1T−2
spectral power in 0.01 to 0.1 Hz band, L2
total spectral power, L2
unit discharge of free surface flow, L2/T
porous media unit discharge, L2/T
bed load volumetric unit discharge, L2/T
total water unit discharge, L2/T
Particle Reynolds number, dimensionless
fraction of total power in 0.01 to 0.1 Hz band, dimensionless
friction slope, dimensionless
depth-averaged free surface water velocity, L/T
porous media flow velocity, L/T
shear velocity, L/T
volume of sediment pile, L3
depth-averaged velocity of moving bed sediment, L/T
width of model reach, L
downstream coordinate direction, L
slope normal coordinate direction, L
geometric shape coefficients of sediment piles, dimensionless
bed slope angle of low-gradient section of reach, dimensionless
ratio of pore pressure at bed to normal stress, dimensionless
horizontal length scale of low transport zone section, L
local bedrock slope angle, dimensionless
local slope angle of water-free surface, dimensionless
bedrock slope angle at top and bottom of modeled reach, dimensionless
bed sediment slope angle, dimensionless
local slope angle of water table or water surface, dimensionless
sediment friction angle, dimensionless
friction angle at the bedrock-sediment interface, dimensionless
fluid density, M/L3
sediment density, M/L3
bulk sediment density sediment-water mixture, M/L3
water flow boundary shear stress, ML/T2
dimensionless shear stress
dimensionless critical shear stress
base value of dimensionless critical shear stress
low slope base value of dimensionless critical shear stress
 This research was supported by the USGS Landslide Hazards Program and NSF grants EAR 0643240 and EAR 0952247. We thank R.M. Iverson, three anonymous reviewers, and the Associate Editor Jon Major for many helpful comments that improved this manuscript.