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

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

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.

Sediment mobilization is considered to occur when bed shear stress

Bed shear stress

We explored whether Eq. (5) could be solved using the Darcy–Weisbach
friction factor

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

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,

Assignments of values to

We consider that the most important critical shear stress is that required
to move

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 (

Channel morphometry (width and gradient), grain size and water discharge
measured at the research sites. The table also shows the results of the
various calculations (critical shear stress

We finally assigned an uncertainty of 20 % to the

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

We additionally ran sensitivity tests to explore how the mobility
probability changes if discharge quantiles instead of

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.

The grain sizes range from 8 to 70 mm for

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

The probability of sediment transport occurrence correlates positively with
the reach gradient (

Relationships between transport probability and

Notably, the probability of material transport correlates positively and
linearly with the

Because the sorting itself could potentially depend on channel metrics and
water discharge, we explored possible correlations between these variables.
We find that the

The use of discharge quantiles yields sediment transport probabilities that
are positively and linearly correlated with the transport probability
estimated with

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,

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

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

Interestingly, our regression analysis of the variables disclosed a positive
correlation between the

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

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 (

Transport probability for the Swiss and Peruvian rivers plotted as a function of the combined response to gradient and sorting. Blue diamonds correspond to the Swiss rivers, while grey circles are Peruvian ones. Both best multiple linear regression fits (solid line) and their 95 % confidence intervals (dashed curves) are presented. Note that the variables on the axis are adjusted as a result of projecting the multiple linear regression models onto a bivariate plot.

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

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:

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.