**Short communication**
31 Aug 2020

**Short communication** | 31 Aug 2020

# Short communication: Field data reveal that the transport probability of clasts in Peruvian and Swiss streams mainly depends on the sorting of the grains

Fritz Schlunegger, Romain Delunel, and Philippos Garefalakis

**Fritz Schlunegger et al.**Fritz Schlunegger, Romain Delunel, and Philippos Garefalakis

- Institute of Geological Sciences, University of Bern, 3012 Bern, Switzerland

- Institute of Geological Sciences, University of Bern, 3012 Bern, Switzerland

**Correspondence**: Fritz Schlunegger (schlunegger@geo.unibe.ch)

**Correspondence**: Fritz Schlunegger (schlunegger@geo.unibe.ch)

Received: 07 Dec 2019 – Discussion started: 02 Jan 2020 – Revised: 23 Jun 2020 – Accepted: 22 Jul 2020 – Published: 31 Aug 2020

We present field observations from coarse-grained streams in the Swiss Alps
and the Peruvian Andes to explore the controls on the probability of
material entrainment. We calculate shear stress that is expected for a mean
annual water discharge and compare these estimates with grain-specific
critical shear stresses that we use as thresholds. We find that the
probability of material transport largely depends on the sorting of the bed
material, expressed by the *D*_{96}∕*D*_{50} ratio, and the reach gradient but
not on mean annual discharge. The results of regression analyses
additionally suggest that among these variables, the sorting exerts the
largest control on the transport probability of grains. Furthermore, because
the sorting is significantly correlated neither to reach gradient nor to
water discharge, we propose that the granulometric composition of the
material represents an independent, yet important control on the motion of
clasts in coarse-grained streams.

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It has been proposed that the transport of coarse-grained material in
mountainous streams occurs when flow strength – or bed shear stress – exceeds
a grain-size-specific critical shear stress (Miller et al., 1977; Tucker and
Slingerland, 1997; Church, 2006). This has been documented based on flume
experiments (e.g., Meyer-Peter and Müller, 1948; Dietrich et al., 1989;
Carling et al., 1992; Ferguson, 2012; Powell et al., 2016) and field
observations (e.g., Paola and Mohring, 1996; Lenzi et al., 2006; Mueller et
al., 2005; Lamb et al., 2008), and related concepts have been employed in
theoretical models (Paola et al., 1992; Tucker and Slingerland, 1997).
Whereas flow strength is mainly a function of discharge, energy gradient and
channel width (e.g., Slingerland et al., 1993; Hancock and Anderson, 2002;
Pfeiffer and Finnegan, 2018; Wickert and Schildgen, 2019), the threshold
shear stress itself has been considered to depend on grain-specific
variables, such as grain size and the arrangement of clasts including hiding and
protrusion effects (Carling, 1983; Parker, 1990; van den Berg and
Schlunegger, 2012; Pfeiffer and Finnegan, 2018), but not on the shape of
individual grains (Carling, 1983) – or at least this variable plays a minor
role only (Komar and Li, 1986). In addition, the threshold shear stress has
also been related to the reach gradient (Lamb et al., 2008; Turowski et al.,
2011; Pfeiffer and Finnegan, 2018). Here, we provide field data from
coarse-grained single-thread streams in the Swiss Alps and braided rivers in
the Peruvian Andes to illustrate that amongst the various variables, the
sorting of the grains exerts the largest control on the transport
probability. The field sites are located close to water gauging stations so
that we have good constraints on the streams' discharge in our analyses. We
determined the grain size distribution of gravel bars at these locations and
calculated, within a probabilistic framework using Monte Carlo simulations,
the likelihood of sediment transport for a mean annual water discharge
*Q*_{mean} and for discharge percentiles. We explored whether the related
flows are strong enough to shift the *D*_{84} grain size, which is considered
to build the sedimentary framework of gravel bars as recent flume
experiments have shown (MacKenzie and Eaton, 2017; MacKenzie et al., 2018).
We thus considered the mobilization of the *D*_{84} grain size as a priori
condition – and thus as a threshold – for a change in the sedimentary
arrangement of the target gravels bars.

The braided character of streams in Peru, however, complicates the calculation of sediment transport probabilities mainly because water flows frequently in multiple active channels, and channel widths vary over short distances. For these streams, we selected reaches (ca. 100 m long) where several active braided channels merge to a single one, before branching again. We are aware that this could eventually bias the results towards a greater material mobility, mainly because flows in single-thread segments are likely to have a greater shear stress than in braided reaches where the same water runoff is shared by multiple channels.

## 2.1 Entrainment of bedload material

Sediment mobilization is considered to occur when bed shear stress *τ*
exceeds a grain-size-specific threshold *τ*_{c} (e.g., Paola et al.,
1992) as follows:

Threshold shear stress *τ*_{c} for the dislocation of grains with size
*D*_{x} (see Sect. 2.3.1 for further specifications) can be obtained using Shields
(1936) criteria *ϕ* for the entrainment of sediment particles:

where *g* denotes the gravitational acceleration, and *ρ*_{s}
(2700 kg m^{−s}) and *ρ* denote the sediment and water densities, respectively.

Bed shear stress *τ* is computed through (e.g., Slingerland et al.,
1993; Tucker and Slingerland, 1997)

Here, *S* denotes the energy gradient, and *R* is the hydraulic radius, which is
approximated through water depth *d* where channel widths *W**>*20 *d* (Tucker and
Slingerland, 1997), which is the case here. The combination of expressions
for (i) the continuity of mass including flow velocity *V*, channel width *W*, and
water discharge *Q*,

(ii) the relationship between flow velocity and channel bed roughness *n*
(Manning, 1891),

and (iii) an equation for Manning's roughness number *n* (Jarrett, 1984),

yields a relationship where bed shear stress *τ* depends on reach
gradient, water discharge and channel width (Litty et al., 2017).

This equation is similar to the expression by Hancock and Anderson (2002),
Norton et al. (2016), and Wickert and Schildgen (2019) with minor differences
regarding the exponent on the channel gradient *S* and on the ratio *Q*∕*W*. These are
mainly based on the different ways of how bed roughness is considered. Note
that this equation does not consider a roughness length scale (both vertical
and horizontal) because we have no constraints on this variable.

We explored whether Eq. (5) could be solved using the Darcy–Weisbach
friction factor *f* instead of Manning's *n*. According to Ferguson (2007), the
friction factor *f* varies considerably between shallow- and deep-water flows
and depends on grain size *D*_{x} relative to water depth *d* and, thus, on the
relative roughness. Ferguson (2007) developed a solution referred to as the
variable power equation (VPE), which accounts for the dependency of *f* on the
relative importance of roughness-layer versus skin friction effects and thus
on the *D*_{x}∕*d* ratios (see also Bunte et al., 2013). Calculations where the
VPE was employed indeed revealed that roughness-layer effects have an impact
on flow regimes where ${D}_{\mathrm{84}}/d>\mathrm{0.2}$ (Schlunegger and Garefalakis, 2018), which
is likely to be the case in our streams. However, similar to Litty et al. (2016), we are faced with the problem that we have not sufficient
constraints to analytically solve Eq. (5) with the VPE. We therefore
selected Mannings's *n* instead, which allowed us to solve this equation
analytically.

## 2.2 Monte Carlo simulations

Predictions of sediment transport probability are calculated using Monte
Carlo simulations performed within a MATLAB computing environment. We
conducted 10 000 simulations, and the results are reported as the
probability (in percent) of *τ**>**τ*_{c} (Eq. 1). All variables
that are considered for the calculations of both shear and critical shear
stress (Eqs. 7 and 2, respectively) are randomly selected within their
possible ranges of variation (Table 1). Except for the Shields variable
*ϕ* that we consider to follow a uniform distribution between 0.03
and 0.06 (see Sect. 2.3.1 for justification), we infer that all other
variables follow normal distributions, defined by their means and
corresponding standard deviations.

To ensure that no negative values introduce a bias to these iterations, only
strictly positive values for channel widths and gradients are considered. In
the case of water discharge, both null and positive values are kept for
further calculations. Values excluded from the calculations, i.e., returning
negative water discharge or null or negative channel width–slope gradient,
yield “NaN” in the resulting vector. For each of the 10 000 iterations,
*τ* and *τ*_{c} are compared, which yields either “1” (*τ**>**τ*_{c}) or “0” (*τ*≤*τ*_{c}). The sediment transport
probability is then calculated as the sum of ones divided by the number of
draws, from which the number of “NaN” values was subtracted before. Note
that *<*2500 “NaN” were obtained for Rio Chico (PRC-ME17), which we
mainly explain by the ca. 150 % relative standard deviation of the mean
annual water discharge estimated for that river.

## 2.3 Parameters, datasets, uncertainties and sensitivity analyses

### 2.3.1 Shields variable *ϕ* and threshold grain size

Assignments of values to *ϕ* vary and diverge between flume
experiments (e.g., Carling et al., 1992; Ferguson, 2012; Powell et al.,
2016) and field observations (Mueller et al., 2005; Lamb et al., 2008).
Here, we considered that at the incipient motion of *D*_{84}, the Shields
variable *ϕ* is equally distributed between 0.03 and 0.06 (Dade and
Friend, 1998) during the 10 000 iterations. We also explored a
slope-dependency of *ϕ* (Lamb et al., 2008; Bunte et al., 2013;
Pfeiffer et al., 2017; Pfeiffer and Finnegan, 2018), where

However, applying this slope-dependent characterization of *ϕ* did not
change our overall finding that the transport probability is dependent on
the sorting of the material (Fig. S1 in the Supplement). In the same context, Turowski
et al. (2011) reported a larger variation in the threshold conditions for
the mobilization of clasts than those employed here. However, their streams
have energy gradients between 0.06 and 0.1, with the consequence that some
of the material is entrained during torrential floods. The related
conditions thus differ from those of the much flatter streams (reach gradients *<*0.02), which we explored in this paper. Finally, we did not
explicitly include a grain-size specific hiding (e.g., Eq. A8 in
Pfeiffer and Finnegan, 2018) or a protrusion function (e.g., Carling, 1983;
Sear, 1996; van der Berg and Schlunegger, 2012) in our analysis, but we
suggest that the selected range between 0.03 and 0.06 considers most of the
complexities and scatters of *ϕ* values that are related to the
hiding of small clasts and the protrusion of large constituents (Buffington
et al., 1992; Buffington and Montgomery, 1997; Kirchner et al., 1990;
Johnston et al., 1998). In summary, we infer that the selection of uniformly
distributed *ϕ* values between 0.03 and 0.06 does account for the
large variability of *ϕ* values that are commonly encountered in
experiments and field surveys where energy gradients range between 0.001 and
0.02, which is the case here.

We consider that the most important critical shear stress is that required
to move *D*_{84}, rather than any other grain size percentile. We acknowledge
that other authors preferentially selected the *D*_{50} grain size as
a threshold to quantify the minimum flow strengths *τ*_{c} to entrain
the bed material (e.g., Paola and Mohrig, 1996; Pfeiffer and Finnegan, 2018;
Chen et al., 2018). The selection of *D*_{50} results in a lower threshold
and a greater transport probability than the employment of *D*_{84}. However,
among the various grain sizes, *D*_{84} has been considered to best
characterize the sedimentary framework of a gravel bar (Howard, 1980; Hey
and Thorne, 1986; Grant et al., 1990), and more recent experiments have also
shown that *D*_{84} better characterizes the channel form stability than
*D*_{50} (MacKenzie et al., 2018). Accordingly, flows that dislocate the
*D*_{84} grain size are considered strong enough to alter the gravel bar
architecture. We therefore followed the recommendation by MacKenzie et al. (2018) and selected the *D*_{84} grain size to quantify the threshold
conditions in Eq. (2).

### 2.3.2 Grain size data

We collected grain size data from streams where water discharge has been
monitored during the past decades. These are the Kander, Lütschine,
Rhein, Sarine, Simme, Sitter and Thur rivers in the Swiss Alps (Fig. 1a).
The target gravel bars are situated close to a water gauging station. At
these sites, five to six digital photographs were taken with a Canon EOS PR. The
photos covered the entire lengths of these bars. A meter stick was placed on
the ground and photographed together with the grains. Grain sizes were then
measured with the Wolman (1954) method using the free software package
ImageJ 1.52n (https://imagej.nih.gov, last access: 7 August 2019). Following Wolman (1954), we used
intersecting points of a grid to randomly select the grains to measure. A
digital grid of 20 cm×20 cm was calibrated with the meter stick on each photo.
The size of the grid was selected so that the spacing between intersecting
points was larger than the *b* axis of most of the largest clasts (Tables 1 and
S2 in the Supplement). The grid was then placed on the photograph with its origin
at the lower-left corner of the photo. The intermediate or *b* axis of
approximately 250–300 grains (ca. 50 grains per photo; Table S2)
underneath a grid point was measured for each gravel bar. In this context,
we inferred that the shortest (*c* axis) was vertically oriented and that the
photos displayed the *a* and *b* axes only. In cases where more than half of the
grain appeared to be buried, the neighboring grain was measured instead. In
the few cases where the same grain lay beneath several grid points, then the
grain was only measured once. Only grains larger than a few millimeters
(*>* 4–5 mm, depending on the quality and resolution of the photos)
could be measured. While the limitation to precisely measure the
finest-grained particles potentially biases the determination of *D*_{50}, it
will not influence the measurements of the *D*_{84} and *D*_{96} grain
sizes, as the comparison between sieving and measuring of grains with the
Wolman (1954) method has disclosed (Watkins et al., 2020). In addition, as
will be shown below, the consideration of the *D*_{96}∕*D*_{84} instead of
the *D*_{96}∕*D*_{50} ratios yields a similar positive relationship to the
mobility of grains. We complemented the grain size datasets with published
information on the *D*_{50},*D*_{84} and *D*_{96} grain size (Litty and
Schlunegger, 2017; Litty et al., 2017) for further streams in Switzerland (six
additional rivers) and Peru (21 sites) (Fig. 1a and b; Table 1). For a few
streams in Switzerland, Hauser (2018) presented *D*_{84} grain size data from
the same gravel bars as Litty and Schlunegger (2017), but the photo was
taken 1 year later and possibly from a different site. For these five
locations, we took the arithmetic mean of both surveys (Table 1, data marked
with c). All authors used the same approach upon collecting
grain size data, which justifies the combination of the new with the
published datasets.

*Italic*: Swiss rivers; roman text: Peruvian rivers. Water discharge data from the Swiss rivers are taken from the Swiss Federal Office for the Environment (FOEN: https://www.hydrodaten.admin.ch/, last access: 10 September 2019). Reported values represent discharges monitored over the period 1990–2011, except for the Rhein (1977–1990). Water discharge and drainage basin size data from the Peruvian rivers are taken from Reber et al. (2017) and Litty et al. (2017). #The grain size data from the Peruvian streams are taken from Litty et al. (2017). ^{a} The grain size data, channel width and gradient data from the Emme, Landquart, Reuss, Maggia and Sense rivers are taken from Litty and Schlunegger (2017). ^{b} While standard deviation on annual water flow represents interannual variance for Switzerland, it represents intra- annual variance in Peru. ^{c} Mean values of measurement results by Hauser (2017) and Litty and Schlunegger (2017). ^{d} The results are possibly biased by an error in the slope, which appears too steep; the consideration of a flatter slope (0.013) still yields 99 %.

We finally assigned an uncertainty of 20 % to the *D*_{84} threshold grain
size, which considers the variability in *D*_{84} within a gravel bar as the
analysis of the intra-bar variation in *D*_{84} for selected gravel bars in
Switzerland shows (Table S2). The assignment of a 20 % uncertainty to
the *D*_{84} threshold grain size also considers a possible bias that could
be related to the grain size measuring technique (e.g., sieving in the field
versus grain size measurements using the Wolman method; Watkins et al.,
2020). However, it is likely to underestimate the temporal variability in
the grain size data. Indeed, a repeated measurement on some gravel bars in
Switzerland has revealed that up to 2-fold differences in grain size could
be possible (Hauser, 2018).

### 2.3.3 Water discharge data

The Federal Office for the Environment (FOEN) of Switzerland has measured
the runoff values of Swiss streams over several decades. We employed the
mean annual discharge values over 20 years for these streams (Table S3)
and calculated 1 standard deviation thereof (see Table 1). For the
Peruvian streams, we used the mean annual water discharge values
*Q*_{mean} reported by Litty et al. (2017) and Reber et al. (2017). These
authors obtained the mean annual water discharge (Table 1) through a
combination of hydrological data reported by the Sistema Nacional de
Información de Recursos Hídricos and the TRMM-V6.3B43.2
precipitation database (Huffman et al., 2007). They also considered the
intra-annual runoff variability as 1 standard deviation from *Q*_{mean} to
account for the strong seasonality in runoff for the Peruvian streams, which
we employed in this paper. For the Peruvian streams, the assigned
uncertainties to *Q*_{mean} are therefore significantly larger than for the
Swiss rivers (Table 1). A reassessment of the interannual variability in
water discharge for those streams in Peru where the gauging sites are close
to the grain size sampling location (distance of a few kilometers) yields a
1 standard deviation of ca. 50 %, which is still much larger than for the
Swiss rivers (Table S3). We therefore run sensitivity tests where we
considered scenarios with different relative values for 1*σ* standard
deviations of *Q*_{mean}.

We additionally ran sensitivity tests to explore how the mobility
probability changes if discharge quantiles instead of *Q*_{mean} are
considered (Table S4). We ran a series of Monte Carlo simulations for
various discharge quantiles and then calculated the resulting probability of
sediment mobilization for each of these quantiles. We then multiplied the
occurrence probability of each discharge quantiles (listed by the Swiss
authorities and calculated for the Peruvian streams based on 4 to 98 years
equivalent daily records) with the corresponding transport probability and
summed the values. This integration provides an alternative estimate of
transport probability (Table S4).

### 2.3.4 Channel width data

For the Swiss streams, channel widths and gradients (Table 1) were measured on orthophotos and lidar DEMs with a 2 m resolution provided by Swisstopo. From this database, gradients were measured over a reach of ca. 250 to 500 m. All selected Swiss rivers are single-thread streams following the classification scheme of Eaton et al. (2010), and flows are constrained by artificial banks where channel widths are constant over several kilometers. For these streams, we therefore measured the cross-sectional widths between the channel banks, similar to Litty and Schlunegger (2017).

We complemented this information with channel width (wetted perimeter) and energy gradient data for 21 Peruvian streams that were collected by Litty et al. (2017) in the field and on orthophotos taken between March and June. This period also corresponds to the season when the digital photos for the grain size analyses were made (May 2015). We acknowledge that widths of active channels in Peru vary greatly on an annual basis because of the strong seasonality of discharge (see above and large intra-annual variability in discharge in Table 1). We therefore considered scenarios where channel widths are twice as large as those reported in Table 1.

The uncertainties on reach gradient and channel width largely depend on the resolution of the digital elevation models underlying the orthophotos (2 m lidar DEM for Switzerland and 30 m ASTER DEM for Peru). It is not possible to precisely determine the uncertainties on the gradient values. Nevertheless, we anticipate that these will be smaller for the Swiss rivers than for the Peruvian streams mainly because of the higher resolution of the DEM. We ran sensitivity models where we explored how the probability of material transport changes in the Swiss rivers for various uncertainties on channel widths, energy gradients and mean annual discharge values.

## 3.1 Grain size data, critical and bed shear stress, and transport probability

The grain sizes range from 8 to 70 mm for *D*_{50}, 29 to 128 mm for
*D*_{84} and 52 to 263 mm for *D*_{96}. The smallest and largest
*D*_{50} values were determined for the Maggia and Rhein rivers in the Swiss
Alps, respectively (Table 1). The grain sizes in the Swiss rivers also
reveal the largest spread where the ratio between the *D*_{96} and *D*_{50}
grain size ranges between 2.2 (Sarine) and 17.7 (Maggia Losone I), while the
corresponding ratios in the Peruvian streams are between 2.1 (PRC-ME9) and
5.8 (PRC-ME17). In the Swiss Alps, the critical shear stress values *τ*_{c} (median) for entraining the *D*_{84} grain size range from ca. 20 Pa (Emme river) to ca. 90 Pa (Rhein and Simme rivers). In the Peruvian
Andes, the largest critical shear values are *<*80 Pa (PRC-ME39). The
shear stress values related to the mean annual water discharge *Q*_{mean}
range from ca. 15 to 100 Pa in the Alps and from 20 to *>*400 Pa in the Andes. Considering the strength of a mean annual flow and the
*D*_{84} grain size as threshold, the probability of sediment transport
occurrence in the Peruvian Andes and in the Swiss Alps comprises the full
range between 0 % and 100 %.

Rivers that are not affected by recurrent high-magnitude events (e.g.,
debris flows) and where the grain size distribution is not perturbed by
lateral material supply are expected to display a self-similar grain size
distribution (Whittaker et al., 2011; D'Arcy et al., 2017; Harries et al.,
2018), characterized by a linear relationship between the
*D*_{84}∕*D*_{50} and *D*_{96}∕*D*_{50} ratios. In case of the Maggia river,
the largest grains are oversized if *D*_{50} and the grain size distribution
of the other streams are considered as reference (Fig. 2a). This could
reflect a response to the supply of coarse-grained material by a tributary
stream where the confluence is *<*1 km upstream of the Maggia sites.
Alternatively – and possibly more likely – it reflects the response to the
high-magnitude floods in this stream (Brönnimann et al., 2018) that
could explain why the largest grains tend to be oversized. In particular,
while the ratio between the last and first quantiles of discharge is
*<*150 in the Swiss streams on the northern side of the Alps, the
ratio is 860 in the Maggia river. Such ratios are not rare in Peru. However,
the Peruvian streams appear to have adapted to such a large discharge
variability through their network of braided channels that are not confined
by artificial banks along most of the streams. In either case, because the
grains in the Maggia river have a different size composition than the other
streams (Fig. 2a), we excluded the Maggia data from further analyses. From
this dataset, we then explored whether *D*_{84} depends on the sorting of the
grains (*D*_{96}∕*D*_{50} ratio), but a possible negative correlation between
these variables is very weak (*R*^{2}=0.2) with a *p*-value of 0.011 (Fig. 2b).

## 3.2 Correlations between channel metrics, water discharge, material sorting and transport probability

The probability of sediment transport occurrence correlates positively with
the reach gradient (*R*^{2}=0.46, *p*-value =0.016 for Swiss rivers;
*R*^{2}=0.34, *p*-value =0.0056 for streams in Peru; Fig. 3a). For
the Peruvian rivers, the probability of occurrence also scales negatively
with channel width (*R*^{2}=0.37, *p*-value =0.0033; Fig. 3b) and
critical shear stress *τ*_{c} (*R*^{2}=0.48, *p*-value =0.00047;
Fig. 3d), which itself depends on the threshold grain size *D*_{84}. No
significant correlations are found between the transport probability and
mean annual water discharge for the Swiss and Peruvian rivers (Fig. 3c).

Notably, the probability of material transport correlates positively and
linearly with the *D*_{96}∕*D*_{50} ratio (Fig. 4a). The observed relationship
appears stronger for the Swiss rivers (*R*^{2}=0.76), than for the
Peruvian streams (*R*^{2}=0.36), and both correlations are significant
with *p*-values of 0.00022 and 0.0041, respectively. These correlations
suggest that poorer-sorted bed material, here expressed by a high
*D*_{96}∕*D*_{50} ratio, has a greater transport probability than
better-sorted sediments. If the normalized residuals are plotted against the
sorting, then they do not show any specific and significant patterns and,
therefore, appear independent of the sorting (Fig. 4b). This suggests that
the inferred linear relationships between the transport probability and the
*D*_{96}∕*D*_{50} ratio are statistically robust. Although Fig. 4a implies
that the regression for the Swiss rivers (slope: 0.16±0.06;
intercept: $-\mathrm{0.34}\pm \mathrm{0.31}$) differs from that of the Peruvian streams
(slope: 0.18±0.11; intercept: $-\mathrm{0.02}\pm \mathrm{0.46}$), the regression
parameters do not significantly differ when considering them within their
95 % confidence intervals. In addition, sites with larger values of
*D*_{96}∕*D*_{50} tend to be associated with smaller values of *D*_{84} (albeit
with a weak correlation), but as noted earlier the influence of *D*_{84} on
the critical shear stress is not sufficient to explain the observed
relationship between mobility and *D*_{96}∕*D*_{50} in the Swiss streams.

Because the sorting itself could potentially depend on channel metrics and
water discharge, we explored possible correlations between these variables.
We find that the *D*_{96}∕*D*_{50} ratio negatively correlates with channel
widths for Peruvian streams (*R*^{2}=0.20, *p*-value =0.040) but not
with any of the other variables in both mountain ranges (e.g., reach
gradient, mean annual discharge and discharge variability; see Fig. S5). In the same sense, the positive relationship between the
*D*_{96}∕*D*_{50} ratio and the reach gradient in the Swiss rivers (*R*^{2}=0.23) is statistically not significant (*p*-value =0.12). As mentioned
above, the *D*_{96}∕*D*_{50} ratio negatively correlates with *D*_{84}, but the
correlation is weak and explains 20 % of the observations only (Fig. 2b).

## 3.3 Discharge quantiles, uncertainties on reach slopes and channel widths

The use of discharge quantiles yields sediment transport probabilities that
are positively and linearly correlated with the transport probability
estimated with *Q*_{mean} (Fig. S4). In addition, the
correlations are very similar between the Swiss (slope: 0.74±0.02;
intercept: 0.05±0.01) and Peruvian streams (slope: 0.73±0.19;
intercept: 0.03±0.14). The mean annual discharge estimates
*Q*_{mean} are likely biased by infrequent but large-magnitude floods, which
could explain the 25 % larger transport probabilities if *Q*_{mean} is used
as reference discharge.

The assignments of different uncertainties on reach gradients, channel
widths and discharge have no major influence on the inferred relationships
between transport probability and sorting (Tables S6 and S7). For the
Peruvian streams, however, assignments of 2-fold-larger values to channel
widths will decrease the transport probability for a given sorting by ca.
10 %–15 %, consistent with Figs. 3b and S5 that illustrate negative
correlations between channel width, *D*_{96}∕*D*_{50} ratio and transport
probability. The inferred linear relationship between both variables,
however, will remain (Table S7).

## 4.1 Controls of channel metrics on the transport probability

Our analysis documents a slope dependency of sediment transport probability
for the Swiss and Peruvian streams. Such a relationship has been documented
before for mountainous rivers in the USA (Torizzo and Pitlick, 2004;
Pfeiffer and Finnegan, 2018) and for other sites including the Alps (Van den
Berg and Schlunegger, 2012). Pfeiffer and Finnegan (2018) reported transport
probabilities, at conditions of an annual flow, that range between 8% and
nearly 100 % for the west coast in the USA, 1 % and 12 % for the Rocky
Mountains, and *<*10 % for the Appalachian Mountains. These
estimates are generally lower than the probabilities reported here. This
most likely reflects the effect of the low channel gradients of the US
streams that are ca. 3 times flatter than the rivers analyzed here (Table 1). These differences thus emphasize the controls of the reach gradient on
the entrainment probability of coarse-grained bed material.

The regression analysis also documents that channel widths and critical
shear stress have an influence on the transport probability of clasts. This
is particularly the case for the braided streams in Peru where wider
channels and greater critical shear stresses tend to lower the transport
probability (Fig. 3b, d). Since braided streams dynamically adjust their
channel widths to changes in the caliber and the rates of the supplied
material (Church, 2006), a dependency of transport probability on channel
width and grain-size specific threshold (including *D*_{84}) was expected.
The absence of corresponding relationships in the Swiss streams is probably
due to the managed geometry of these streams where artificial banks
constrain the channel widths over tens of kilometers.

## 4.2 Controls of material sorting on the transport probability

Interestingly, our regression analysis of the variables disclosed a positive
correlation between the *D*_{96}∕*D*_{50} ratio of the bed material and the
transport probability. This relationship maintains if transport
probabilities are calculated based on discharge quantiles (Table S4)
and if larger channel width and discharge variability particularly for
Peruvian streams are considered (Table S7). Such a dependency will also
remain if critical shear stress is calculated using a different grain size
percentile. This is because grain size *D*_{x} linearly propagates into the
Eq. (2) and thus into the probability of *τ**>**τ*_{c}.
Therefore, although the resulting probabilities will adjust according to the
threshold grain size, the relationships between the *D*_{96}∕*D*_{50} ratio
and the mobilization probability will not change. Furthermore, because of
the linear relationship between the *D*_{84}/*D*_{50} and
*D*_{96}∕*D*_{50} ratios (Fig. 2), the same dependency of transport
probability on the sorting will also emerge if the *D*_{96}∕*D*_{84} ratios
are used. This suggests that the sorting of the bed material has a
measurable impact on the mobility of gravel bars and thus on the frequency
of sediment mobilization irrespective of the selection of a threshold grain
size. In addition, there appears to be a feedback where a poorer sorting
(large *D*_{96}∕*D*_{50} ratio) tends to be associated with a lower *D*_{84}
(Fig. 2), which additionally increases the sediment transport probability.
We note that while the data are relatively scarce and scattered (i.e., the
same transport probability for a ca. 2-fold difference in the
*D*_{96}∕*D*_{50} ratio), the relationships observed between the probability
of transport occurrence and the degree of material sorting are significant
with *p*-values ≪0.05. Finally, for a given
*D*_{96}∕*D*_{50} ratio, the probability of material transport tends to be
greater in the Peruvian than in the Swiss rivers (Fig. 4a). We tentatively
explain the apparent small divergence in the transport probability between
both settings (i.e., regression parameters overlap within their 95 %
confidence interval) by the differences in the flow patterns (braided versus
single-thread artificial channels).

## 4.3 Controls on the sorting of the bed material

None of the possible variables such as channel reach gradient, mean water
discharge and discharge variability are significantly correlated with the
bed material sorting (Fig. S5). Exceptions are the Peruvian streams
where wider channels tend to be associated with a better sorting (i.e.,
lower *D*_{96}∕*D*_{50} ratio). We lack further quantitative information to
properly interpret these patterns, but it appears that material sorting
represents an additional yet independent variable that influences the
probability of transport, at least for the sites we have investigated in
this paper. Because the sorting of the bed material in the analyzed streams
appears not to strongly depend on the hydrological conditions at the reach
scale, it could possibly reflect an inherited supply signal from further
upstream (Pfeiffer et al., 2017). Indeed, detailed grain size analyses along
fluvial gorges in the Swiss Alps have shown that the hillslope-derived
supply of large volumes of sediment perturbs the granulometric composition
of the bed material (van den Berg and Schlunegger, 2012; Bekaddour et al.,
2013). Using the results of flume and numerical experiments, Jerolmack and
Paola (2010) suggested that these source signals are likely to be shredded
during sediment transport as a consequence of what they considered as
ubiquitous thresholds in sediment transport systems. However, based on a
detailed analysis of downstream fining trends in alluvial fan deposits,
Whittaker et al. (2011), D'Arcy et al. (2017) and Brooke et al. (2018)
proposed that primary source signals of grain size compositions are likely
to propagate farther downstream in a self-similar way. Accordingly, the
original grain-size sorting of the supplied material could be maintained
although a general fining of the sediments along the sedimentary routing
system would be observed. This idea could offer an explanation of why the
*D*_{96}∕*D*_{50} ratios are to a large extent independent from other variables.
It also points to the importance of sediment supply not only for controlling
the bankfull hydraulic geometry of channels (Pfeiffer et al., 2017) but
also for the sorting of the material. Finally, a supply control could
possibly explain why sorting appears to vary with *D*_{84}, albeit with a
poor correlation, where a better sorting tends to be associated with larger
*D*_{84}.

## 4.4 Relative importance of sorting versus gradient on the transport probability

Because gradient and sorting are independent variables and since the
transport probability depends linearly on both variables, the transport
probability can be described as a linear but weighted combination of
gradient and sorting. We therefore assess whether the transport probability
(Tp) in both the Swiss (*i*=1) and the Peruvian (*i*=2) rivers can be
predicted using a multiple linear regression: $T{p}_{i}={\mathit{\alpha}}_{i}{S}_{\mathrm{n}}+{\mathit{\beta}}_{i}{G}_{\mathrm{n}}+{\mathit{\delta}}_{i}$, where *S*_{n} and *G*_{n} are the
sorting and gradient normalized to their respective maximum, and *α*, *β*, and *δ* are the regression parameters. We decided to normalize
both the sorting and gradient to their maximum values so that both variables
vary on a similar [0–1] range, and the inferred linear coefficients *α*, *β* and *δ* can be directly compared between the Swiss and the
Peruvian rivers. The model outputs show that when sorting and gradient are
combined, then the predictions of the transport probability in both the
Swiss (*R*^{2}=0.85, $p=\mathrm{2.24}\times {\mathrm{10}}^{-\mathrm{4}}$) and the Peruvian (*R*^{2}=0.61, $p=\mathrm{1.9}\times {\mathrm{10}}^{-\mathrm{4}}$) rivers are significantly improved compared to simple
linear regressions. The results also reveal that the relative importance of
sorting on the transport probability is greater (*α* is 1.22±0.26 for the Swiss streams and 1.46±0.41 for the Peruvian streams) than
the relative controls of reach gradient (*β* is 0.62±0.27 for
the Swiss streams and 0.67±0.20 for the Peruvian rivers). The comparison
of the estimated factors thus suggests that the relative importance of
sorting on the transport probability could be twice as large as the controls
of gradient, although our estimation is associated with large uncertainties
(2.0±1.0 in Switzerland; 2.2±0.9 in Peru). Interestingly, we
also note that the apparent greater probability of transport in the Peruvian
rivers, as we infer based on all simple linear regressions reported in
Figs. 3 and 4, remains with our multiple linear regression analysis
(*δ* is $-\mathrm{0.42}\pm \mathrm{0.12}$ for the Swiss streams and $-\mathrm{0.28}\pm \mathrm{0.19}$ for
the Peruvian streams; Fig. 5). Again, this suggests that an additional
component (intrinsic geomorphic setting such as, e.g., braided vs.
single thread) may contribute to the observed higher probability of sediment
transport in the Peruvian rivers than in the Swiss ones. The search for an
answer to this questions, however, is beyond the scope of this paper and
would require additional research.

We confirm the results of previous research that the transport probability
of coarse-grained material in mountainous streams largely depends on the
reach gradient. We also find a positive correlation between the
*D*_{96}∕*D*_{50} ratio of the bed material and the transport probability where
a poorer sorting of the material results in a larger probability of material
entrainment. Despite the large scatter in the dataset, this relationship is
statistically significant with *p*-values ≪0.05, which
suggests that the sorting of coarse-grained bed sediments has a measurable
impact on the mobility of the bedload material. Regression analyses
additionally reveal that sorting exerts a greater control on the transport
probability than reach gradient. Furthermore, the lack of a significant
correlation between reach gradient and sorting implies that both variables
are largely independent from each other, at least for the investigated
rivers in Switzerland and Peru. We therefore propose that the sorting of the
bed material represents an additional, yet important variable that
influences the mobility of material on gravel bars. Finally, we identify two
main open questions that we cannot resolve with our dataset. First, Fig. 5
illustrates that 15 % of the transport probability observations in
Switzerland and 40 % of the data in Peru cannot be fully explained by a
combination of sorting and reach gradient, and interpretations thereof most
likely require the consideration of the anthropogenic management of the
streams (braided and free flow in Peru versus engineered single-thread
channels in Switzerland). Second, we have not identified a significant
correlation between the sorting and the other variables such as reach
gradient, water discharge and discharge variability. This led us to propose
that material supply need to be included in the discussion as well.
Furthermore, the critical shear stress, through its dependency on
*D*_{84}, is also a function of the sorting (albeit with a weak correlation;
Fig. 2b). This could invoke a possible feedback where a poorer sorting
tends to be associated with lower *D*_{84} and thus with a lower critical
shear stress, which further promotes the mobility of grains. We do not have
the required data to fully address these latter two questions and suggest
that it could serve as a topic in future research.

All data that have been used in this paper are listed in Table 1 and in the Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/esurf-8-717-2020-supplement.

FS and RD designed the study. RD conducted the Monte Carlo simulation. PG provided the grain size data in the Supplement. FS wrote the paper with input from RD and PG. All authors discussed the article.

The authors declare that they have no conflict of interest.

The Federal Office for the Environment (FOEN) is kindly acknowledged for providing runoff data for the Swiss streams. This research was supported by SNF (no. 155892). The constructive comments by Georgios Maniatis, two anonymous reviewers and the handling editor (Rebecca Hodge) are kindly acknowledged and significantly improved the science of this paper.

This research has been supported by the Swiss National Science Foundation (grant no. 155892).

This paper was edited by Rebecca Hodge and reviewed by Georgios Maniatis and two anonymous referees.

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