Steep streams occupy a large fraction of mountainous drainage basins and partially control the sediment supplied to downstream rivers. In these channels, sediment transport equations typically over-predict bedload flux by several orders of magnitude because they do not account for sediment-supply limited conditions. Thus, accurate predictions of bedload flux require an estimate of the sediment available for transport in a given event. We demonstrate through field measurements that boulder step protrusion is a proxy for sediment availability. Protrusion is also a function of the time elapsed since an extreme event and this simple relationship can be used to estimate the relative sediment availability at any given time. In addition, bedload transport predictions in a steep channel were only accurate if they included this variable protrusion. Predictions of sedimentation hazards, water quality, river restoration success, long-term channel network evolution, and channel stability may therefore require estimates of sediment availability for transport.
 An unequal sediment supply and transport capacity (a river's ability to transport sediment) could cause a river to avulse or significantly change its form by eroding or depositing sediment on the bed or banks. Thus, predictions of sediment supply are integral to controlling sedimentation hazards, water quality and the stability of restored river reaches. Sediment supply can also significantly alter channel grain sizes [e.g., Dietrich et al., 1989; Buffington and Montgomery, 1999; Nelson et al., 2009], which impact the available aquatic habitat for a number of threatened and endangered aquatic species [e.g., Buffington et al., 2004]. Long-term calculations of landscape evolution will inaccurately predict channel form unless the sediment supply is known because it influences erosion magnitudes [e.g.,Sklar and Dietrich, 2006]. In addition to the hillslope sediment supply, the amount of readily mobile, in-channel sediment sources are also important [e.g.,Garcia et al., 1999] and we use the term sediment availability to integrate these two effects.
 We currently lack mechanistic models that use simple field measurements to determine the relative sediment availability for transport at any given location or time. This is particularly problematic in steep streams (gradients greater than 3%) in which sediment transport rates and bed morphologies are tightly coupled to the sediment supply from episodic landslides and debris flows [e.g., Bathurst et al., 1986; Benda and Dunne, 1997] and in-channel sediment sources. Steep channels make up a majority of mountain drainage networks and partially control the amount, timing, and grain-size distribution of sediment delivered to downstream rivers. Most sediment transport equations over-predict bedload transport rates in steep streams by several orders of magnitude [e.g.,Bathurst et al., 1987; Rickenmann, 2001; D'Agostino and Lenzi, 1999; Yager et al., 2007; Mueller et al., 2008; Yager et al., 2012] because they fail to account for the influence of large, relatively immobile boulder steps, wide range of grain size mobility, and relatively little sediment availability. Even when they account for the first two variables, sediment transport prediction errors in steep channels are relatively large [e.g., Nitsche et al., 2011; Yager et al., 2012]. This is because sediment transport equations are generally based on the assumption that the sediment availability is equal to the transport capacity, but this almost never occurs during individual events in steep streams.
 Estimates of the relative sediment availability in lower gradient channels have been previously based on the degree of pool filling with fine sediment or streambed coarsening/fining [e.g., Dietrich et al., 1989; Lisle and Hilton, 1999; Buffington and Montgomery, 1999]. Most of these qualitative measures cannot be directly incorporated into sediment transport predictions. Sediment availability estimates previously used to modify transport equations [e.g., Barry et al., 2004] require some a priori knowledge of the sediment supply, are not universally applicable, and do not allow the sediment availability to vary between events within a given channel.
Yager et al. demonstrated that a simple bed parameter, boulder protrusion, varied with the upstream sediment supply but they used experiments of single grain size transport through regular arrays of fixed spheres (to represent boulders). Such experiments lack the complexities in grain size and channel morphology that are definitive features of steep channels. Here, we show that boulder-step protrusion (amount steps protrude above the surrounding bed sediment) is a proxy for the relative sediment availability in a natural steep channel. We hypothesize that extreme events increase sediment availability and decrease protrusion and this sediment is progressively winnowed through time (increasing protrusion) by subsequent smaller events. We also demonstrate that a simple equation for protrusion can estimate the relative sediment availability through time and significantly improve calculations of sediment flux.
 Following Yager et al. , we divide the streambed into finer, more mobile sediment patches and large grain steps with a median grain diameter (D) and an average downstream spacing (λx) and length (λw). The total boundary shear stress (τt = ρghaS) is partitioned between the stress borne by the immobile steps (τI) and that borne by the mobile sediment (τm),
The combination of the continuity (U = q/ha) and stress partitioning equations yields a solution to calculate the flow hydraulics (U, τm, ha). In these equations reach-averaged parameters are used and ha is flow depth, S is slope, ρ is water density, q is discharge per unit width (w), Cm is drag coefficient of the relatively mobile sediment, and U is flow velocity. Yager et al.  then modified the Parker  bedload transport equations to use the following: 1) τm rather than τt, 2) the grain-size distribution of the mobile sediment (excludes boulder steps) rather than the entire bed, and 3) a hiding function that usesτm and the mobile grain sizes. They assumed that protrusion and therefore sediment availability were constant in all of their predictions of sediment flux.
 We conducted a series of measurements in the Erlenbach torrent (9.8% gradient reach, 0.74 km2 drainage area) in Switzerland. Here, the Swiss Federal Institute for Forest, Snow, and Landscape Research (WSL) uses impact sensors to record bedload transport rates [see Rickenmann and McArdell, 2007]. We used bedload transport data from 2006 to 2010 for a total of 64 events, which included a wide range of sediment yields (62 m3 in 2004 to 2675 m3 in 2010), and two extreme events (June 20, 2007 and August 1, 2010) that we define by the transport of boulder steps and increased subsequent bedload fluxes [Turowski et al., 2009].
 We calculated the ratio of the total measured transported sediment to water volumes (Qsw) for all events, which is a simplified version of the balance between the sediment yield and total transport capacity (a function of discharge) and indicates the relative sediment availability. Sediment availability and bedload transport rates can be highly variable between events but are also related to the occurrence of extreme events [Turowski et al., 2009; Lenzi et al., 2004]. To average over this substantial variability, we calculated the mean Qsw over the 6 month period that preceded each of our protrusion measurements. If an extreme event occurred within this time frame, we only used events up to and including this extreme flood.
 In a reach immediately upstream of the bedload sensors, we made detailed bed measurements five times (in 2004, 2007, 2009 and twice in 2010) to represent intervals with a range of sediment yields and timings since an extreme event (1 month to 9 years). We measured the channel slope along the thalweg, the bed topography and the outlines of the relatively mobile sediment patches (gravel and cobbles) and boulder steps using a total station (see Yager et al.  for specific details on measurement techniques). We used this survey and pebble counts on the patches to calculate parameters needed to predict sediment flux (λx, λw, w, S, D, grain sizes of mobile sediment, protrusion). We assumed that boulder steps were composed of large grains that rarely move (see Figure S1 in the auxiliary material). All other patches were assumed to be composed of relatively mobile sediment [Yager et al., 2012]. The step protrusion (p) was the average (for all steps) of the elevation differences between upstream step edges and paired points on the downstream boundaries of neighboring mobile patches (see Table 1 and Figure S2). The percent protrusion (P) was the protrusion divided by the median grain diameter of the steps.
Uncertainties are standard errors. # denotes the number of samples for each protrusion measurement. Time denotes the time elapsed since an extreme event.
0.16 ± 0.02
0.059 ± 0.015
0.12 ± 0.01
0.11 ± 0.02
0.078 ± 0.016
 In the equations of Yager et al. , the step protrusion directly controls the flow resistance and total bed-perpendicular area occupied by boulder steps and indirectly influences sediment fluxes (through U and thereforeτm). Calculations for a given transport event also required values for Cm, drag coefficient of boulder steps (CI), and the measured flow hydrograph (q) [see Yager et al., 2012].
3. Results and Discussion
 Protrusion systematically declined with greater measured relative sediment availability (Qsw) (Figure 1a). This confirms results from simplified flume experiments by Yager et al.  and suggests that step protrusion could be used as a proxy for sediment availability in steep mountain channels. Qsw likely increases with lower protrusion (lower drag) partially because this decreased drag augments the transport efficiency out of the reach [see Yager et al., 2007], but this effect does not fully explain the observed changes in Qsw (see auxiliary material). The fit in Figure 1a probably changes between streams, which would necessitate knowledge of transported sediment volumes in events that preceded the one of interest, and such information is normally not available. Therefore, an equation that does not explicitly use known sediment transport rates is required for streams without instrumentation.
Turowski et al.  and Lenzi et al. demonstrated that sediment transport rates, and therefore sediment availability, increased following extreme events. Such events move boulder steps, have greater sediment delivery from landslides, and disrupt bed surface packing and armoring, which also can increase the in-channel sediment supply. Thus, the measured protrusion and sediment availability should vary with the time elapsed since an extreme event.
 As hypothesized, the measured protrusion increased with greater time elapsed since an extreme event (Figure 1b) and fitting a power law function to the data gives
where time is in units of months. This implies that extreme events supply sediment that buries steps and is slowly winnowed over time through subsequent smaller events (see Figure S3) until the next extreme flood occurs. Our protrusion measurements were taken after two extreme events, which completely re-established the step arrangement and bed grain size distribution, and therefore this trend could be applicable for a number of steep streams. A simple history of extreme events combined with a number of protrusion measurements over time may be used to determine the protrusion for a given event. Such an indirect measure of sediment availability could estimate the relative impact of land-use practices (e.g., logging, grazing, road construction) or hillslope restoration projects on the sediment supplied to a given channel over time.
 We now test if protrusion, as a proxy for sediment availability, influences sediment transport predictions using the modified Parker equation [Yager et al., 2012]. We hold all other parameters constant and estimate boulder protrusion using the following scenarios: 1) constant high protrusion (2004 value), 2) constant low protrusion (2007 value), 3) random protrusion sampled from a normal distribution (mean = 0.14 m, standard deviation =0.05 m), 4) protrusion determined from equation (2), and 5) constant 2004 protrusion combined with a variable grain size distribution through time. We used a random protrusion because landslide sediment delivery to steep channels and in-channel sediment mobility are often highly stochastic and therefore protrusion may be as well. The variable grain size distribution used different measured grain sizes (seeFigure S4) through time, depending on the temporal proximity to an extreme event. The exact timing of a given grain size distribution had little effect on the predicted sediment fluxes and therefore we only show one possible scenario as an example. We also directly calculated the sediment flux, without the modified Parker equation, using the regression in Figure 1c and the known total transported water volume (for a given event) and timing since an extreme event. This is a completely empirical approach that is only applicable to the Erlenbach.
 Use of constant protrusion values caused systematic under- (high protrusion, 2004) or over-prediction (low protrusion, 2007) of the measured sediment volumes (Figure 2 and Table 2). The low 2007 protrusion represents bed conditions (buried steps) with relatively high sediment availability because it was measured immediately following an extreme event, whereas the high 2004 protrusion was measured during a period of low sediment availability (9 years after an extreme event). Thus, sediment availability is not constant through time and assuming such will result in large prediction errors. Use of a random protrusion removed some of this systematic bias in predicted sediment volumes, but significantly reduced the prediction accuracy (Figure 2 and Table 2). Conversely, use of equation (2) caused almost all predicted sediment volumes to be within an order of magnitude of the measured values and had the lowest RMSE of all tested scenarios. When we used a completely empirical approach (regression in Figure 1c), the predicted bedload fluxes were slightly less accurate (Figure 2 and Table 2) than those from the modified Parker equation combined with equation (2). This is likely because the empirical fit in Figure 1conly uses the total water volume, which does not account for the non-linear relationship between transported sediment flux (and grain sizes) and shear stress.
Table 2. Statistics for Predicted Bedload Volumesa
% Over-Predicted by Over an Order of Magnitude
% Under-Predicted by Over an Order of Magnitude
% Within an Order of Magnitude
Percent values are the percent of 64 measured bedload transport events, time since extreme event denotes equation (2)was used, variable grain size distribution means that the grain size distribution was allowed to vary with respect to the time since the last extreme event and was combined with a constant 2004 protrusion. RMSE values were log-transformed values, seeauxiliary materialfor details. Time-Qsw fit denotes that no protrusion values were used and bedload flux was calculated directly from the empirical relationship in Figure 1c.
Time since extreme event
2004 and variable grain size distribution
N/A, Time-Qsw fit
 Protrusion directly incorporates changes in bed roughness and sediment deposition depths, which are functions of the upstream and in-channel sediment supply. It does not account for other variables that may change with supply such as grain interlocking, bed armoring, and grain size. Grain size is often cited as the dominant parameter influenced by sediment supply [Dietrich et al., 1989; Buffington and Montgomery, 1999] and we measured a finer grain size distribution after an extreme event (Figure S4). However, when incorporated into the modified Parker equation these grain size changes had relatively little influence on bedload volume predictions and did not produce accurate results (Table 2 and Figure 2). This may be partially caused by the relatively small changes in the grain size distribution over time. Thus, the influence of roughness and overall sediment availability (as represented by protrusion) could be relatively more important than grain size fluctuations for bedload flux predictions in steep channels.
 Our calculations demonstrate that the effects of sediment availability cannot be neglected and the integrated history of a channel may be important for calculations of event-scale transport. Our bedload transport equation is the first that mechanistically accounts for the effects of sediment availability and allows this parameter to vary through time. Other parameters such as the local flow conditions, step spacing, supplied grain-size distribution, channel width and slope may influence the protrusion and these parameters will vary between streams [Yager et al., 2007]. Further work is needed to better understand the variation of protrusion over time and between sites before such a relationship could be systematically applied. Our calculations may under-estimate sediment flux at long time intervals because we assume there are no sediment sources besides extreme events. We have also simplified a highly spatially and temporary variable process, which is influenced by turbulence fluctuations, grain to grain interactions, and local flow and grain size conditions. No current sediment transport equations explicitly incorporate these local effects and this could explain why although our bedload flux predictions are better than most estimates, they could still be improved.
 Our field measurements demonstrate that step protrusion may be used as a proxy for sediment availability. The use of an empirical relationship between step protrusion and time since an extreme event eliminated systematic over- and under-predictions of sediment volumes caused by assumed static bed conditions. In steep streams, step protrusion may provide a new method to improve sediment transport predictions and calculations of channel conditions and stability throughout drainage networks.
 Funding was provided by an NSF-CAREER grant (0847799) to E.M. Yager. Field assistance was provided by Heidi Schott and Pat Thee. The authors acknowledge thoughtful reviews by Peter Nelson and one anonymous reviewer.
 The Editor thanks Peter Nelson and Ellen Wohl for assisting with the evaluation of this paper.