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<?xml version="1.0" encoding="utf-8" ?>
<!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Publishing DTD v1.2 20190208//EN"
"JATS-publishing1.dtd">
<article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" dtd-version="1.2" article-type="other">
<front>
<journal-meta>
<journal-id></journal-id>
<journal-title-group>
<journal-title>Journal of Open Source Software</journal-title>
<abbrev-journal-title>JOSS</abbrev-journal-title>
</journal-title-group>
<issn publication-format="electronic">2475-9066</issn>
<publisher>
<publisher-name>Open Journals</publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="publisher-id">4282</article-id>
<article-id pub-id-type="doi">10.21105/joss.04282</article-id>
<title-group>
<article-title>pySBeLT: A Python software package for stochastic
sediment transport under rarefied conditions</article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<contrib-id contrib-id-type="orcid">0000-0002-0812-9509</contrib-id>
<name>
<surname>Zwiep</surname>
<given-names>Sarah</given-names>
</name>
<xref ref-type="aff" rid="aff-1"/>
</contrib>
<contrib contrib-type="author">
<contrib-id contrib-id-type="orcid">0000-0002-9309-1137</contrib-id>
<name>
<surname>Chartrand</surname>
<given-names>Shawn M.</given-names>
</name>
<xref ref-type="aff" rid="aff-1"/>
<xref ref-type="aff" rid="aff-2"/>
</contrib>
<aff id="aff-1">
<institution-wrap>
<institution>School of Environmental Science, Simon Fraser
University</institution>
</institution-wrap>
</aff>
<aff id="aff-2">
<institution-wrap>
<institution>Department of Earth Sciences, Simon Fraser
University</institution>
</institution-wrap>
</aff>
</contrib-group>
<pub-date date-type="pub" publication-format="electronic" iso-8601-date="2022-01-12">
<day>12</day>
<month>1</month>
<year>2022</year>
</pub-date>
<volume>7</volume>
<issue>74</issue>
<fpage>4282</fpage>
<permissions>
<copyright-statement>Authors of papers retain copyright and release the
work under a Creative Commons Attribution 4.0 International License (CC
BY 4.0)</copyright-statement>
<copyright-year>2022</copyright-year>
<copyright-holder>The article authors</copyright-holder>
<license license-type="open-access" xlink:href="https://creativecommons.org/licenses/by/4.0/">
<license-p>Authors of papers retain copyright and release the work under
a Creative Commons Attribution 4.0 International License (CC BY
4.0)</license-p>
</license>
</permissions>
<kwd-group kwd-group-type="author">
<kwd>Python</kwd>
<kwd>geomorphology</kwd>
<kwd>sediment transport</kwd>
<kwd>stochastic</kwd>
<kwd>Poisson</kwd>
</kwd-group>
</article-meta>
</front>
<body>
<sec id="summary">
<title>Summary</title>
<p>Granular sediment of various sizes moves downstream along river
beds when water flow is capable of entraining particles from the bed
surface. This process is known as bed load sediment transport because
the particles travel close to the boundary. It is common to treat the
transport process as a predictive problem in which the mean transport
rate past a stationary observation point is a function of local water
flow conditions and the grain size distribution of the bed material
(<xref alt="Ancey, 2020" rid="ref-AnceyU003A2020" ref-type="bibr">Ancey,
2020</xref>;
<xref alt="Parker, 2008" rid="ref-ParkerU003A2008" ref-type="bibr">Parker,
2008</xref>;
<xref alt="Wainwright et al., 2015" rid="ref-WainwrightU003A2015" ref-type="bibr">Wainwright
et al., 2015</xref>). However, a predictive approach to the bed load
problem neglects the stochastic nature of transport due to the
movements of individual particles
(<xref alt="Einstein, 1937" rid="ref-EinsteinU003A1937" ref-type="bibr">Einstein,
1937</xref>;
<xref alt="Furbish &amp; Doane, 2021" rid="ref-FurbDoaneU003A2021" ref-type="bibr">Furbish
&amp; Doane, 2021</xref>), and interactions between moving particles
and those on the bed surface
(<xref alt="Ancey et al., 2006" rid="ref-AnceyU003A2006" ref-type="bibr">Ancey
et al., 2006</xref>,
<xref alt="2008" rid="ref-AnceyU003A2008" ref-type="bibr">2008</xref>;
<xref alt="Lee &amp; Jerolmack, 2018" rid="ref-LeeJerolU003A2018" ref-type="bibr">Lee
&amp; Jerolmack, 2018</xref>). Here, we present an open-source Python
model, <monospace>pySBeLT</monospace>, which simulates the kinematics
of rarefied particle transport (low rates) as a stochastic process
along a riverbed profile. <monospace>pySBeLT</monospace> is short for
<italic>Stochastic Bed Load Transport</italic>. The primary aim of
<monospace>pySBeLT</monospace> is to offer an efficient and reasonable
numerical means to probe connections between individual particle
motions and local transport rates, or the flux. We suggest that
<monospace>pySBeLT</monospace> is a suitable teaching tool to help
introduce bed load transport to advanced undergraduate and graduate
students alike.</p>
</sec>
<sec id="statement-of-need">
<title>Statement of need</title>
<p>Research at the intersection of geomorphology, geophysics and
hydraulics is increasingly focused on building a theoretical
foundation for the treatment of bed load transport as a stochastic
phenomenon
(<xref alt="Ancey, 2020" rid="ref-AnceyU003A2020" ref-type="bibr">Ancey,
2020</xref>;
<xref alt="Furbish &amp; Doane, 2021" rid="ref-FurbDoaneU003A2021" ref-type="bibr">Furbish
&amp; Doane, 2021</xref>). Associated theories are commonly tested
against laboratory data from “rarefied” transport conditions
(<xref alt="Furbish et al., 2016" rid="ref-FurbU003A2016" ref-type="bibr">Furbish
et al., 2016</xref>), where transport rates are low to moderate,
interactions between two or more moving particles are rare, and a
relatively small fraction of particles on the bed surface participate
in transport
(<xref alt="Ancey, 2010" rid="ref-AnceyU003A2010" ref-type="bibr">Ancey,
2010</xref>;
<xref alt="Fathel et al., 2015" rid="ref-FathelU003A2015" ref-type="bibr">Fathel
et al., 2015</xref>;
<xref alt="Roseberry et al., 2012" rid="ref-RoseberryU003A2012" ref-type="bibr">Roseberry
et al., 2012</xref>;
<xref alt="Wu et al., 2020" rid="ref-WuU003A2020" ref-type="bibr">Wu
et al., 2020</xref>). For example, laboratory experiments using a
downstream light table counting device and conducted at roughly twice
the shear stress threshold for particle motion involve the transport
of less than approximately 12% of particles on the upstream bed
surface
(<xref alt="Chartrand, 2017" rid="ref-ChartrandU003A2017" ref-type="bibr">Chartrand,
2017</xref>). This result suggests that the flux measured across a
boundary or within an area of bed surface is directly linked to the
motions of individual particles arriving from upstream locations
(<xref alt="Furbish et al., 2012" rid="ref-FurbishU003A2012" ref-type="bibr">Furbish
et al., 2012</xref>).</p>
<p>Particle motions are controlled by several influencing factors
including fluid turbulence, the irregular bed surface, and collective
effects
(<xref alt="Ancey et al., 2006" rid="ref-AnceyU003A2006" ref-type="bibr">Ancey
et al., 2006</xref>,
<xref alt="2008" rid="ref-AnceyU003A2008" ref-type="bibr">2008</xref>;
<xref alt="Lee &amp; Jerolmack, 2018" rid="ref-LeeJerolU003A2018" ref-type="bibr">Lee
&amp; Jerolmack, 2018</xref>). As a result, the connection between
particle movements and the bed load transport rate has been difficult
to formulate mathematically. <monospace>pySBeLT</monospace> provides
an extensible framework within Python to numerically examine
correlations between upstream particle entrainment rates and travel
distances, with downstream flux. <monospace>pySBeLT</monospace> was
motivated by a birth-death, immigration-emigration Markov model for
bed load transport
(<xref alt="Ancey, 2010" rid="ref-AnceyU003A2010" ref-type="bibr">Ancey,
2010</xref>;
<xref alt="Ancey et al., 2008" rid="ref-AnceyU003A2008" ref-type="bibr">Ancey
et al., 2008</xref>). Here, the movements of individual particles are
represented by stochastic entrainment, motion, and deposition
processes, and sediment flux is represented as a counting phenomenon
where the number of particles in motion above the bed surface is a
random variable
(<xref alt="Ancey et al., 2008" rid="ref-AnceyU003A2008" ref-type="bibr">Ancey
et al., 2008</xref>). The model supports ensemble simulations so that
repeat numerical experiments can be conducted efficiently, or the
problem can be probed across a range of input parameter values
(discussed below).</p>
<p><monospace>pySBeLT</monospace> is run forward in time according to
default or user specified parameter values. After initialization,
<monospace>pySBeLT</monospace> first constructs a bed of fixed
particles of <bold>‘set_diam’</bold> in both the downstream and
cross-stream dimensions (one particle wide in the present build), and
over a downstream domain length <bold>‘bed_length’</bold>. Bed surface
particles of <bold>‘particle_diam’</bold> are then randomly placed at
vertices between fixed bed particles until
the<bold>‘particle_pack_frac’</bold> is met. Vertices are defined by a
contact point between two adjacent particles. The bed of surface
particles is then separated into <bold>‘num_subregions’</bold> set by
the user. Subregion boundaries occur at domain locations set by a
distance = <bold>‘bed_length’</bold> / <bold>‘num_subregions’</bold>.
Following construction of the bed surface the forward simulations are
ready to commence.</p>
<p>Simulation iterations involve three steps (Fig. 1): (1) the number
of particle entrainment events per <bold>‘num_subregions’</bold> are
drawn from a Poisson pmf, and this is done randomly for each numerical
step up to <bold>‘iterations’</bold>; (2) surface particles from each
subregion are randomly selected for entrainment, and if there are
insufficient surface particles available for entrainment, then all
available particles are moved; (3) each entrained particle moves a
distance according to a randomly sampled value from either the normal
or lognormal distribution (see THEORY.md for more details), and is
placed at the nearest vertex between two particles that is available
for placement. Placed particles are permitted to stack up to the
<bold>‘level_limit’</bold> in height. Particles are not permitted to
travel to the same available vertex. To stop this from occuring the
entrained particles are moved in random order and once a particle is
placed on a vertex, that vertex is no longer considered available for
the present iteration. Travel distances of particles that exceed
<bold>‘bed_length’</bold> are returned and queued at the upstream
boundary, and are introduced back into the domain at the next
numerical step according to travel distance sampling described above.
This specifially means that the particle travel distance which
resulted in crossing of the downstream domain does not influence the
travel distance of the particle when queued at the upstream domain–a
new travel distance for such particles will be sampled during the next
numerical step. This overall process repeats for the specified
<bold>‘iterations’</bold>.</p>
<p><monospace>pySBeLT</monospace> tracks a number of different
parameters through a simulation: the vertical and horizontal positions
of every particle center, the randomly sampled number of entrainment
events, the number of particles actually entrained, the actual
particle travel distance, the particle ‘age’, or the number of
numerical steps since last entrainment for every particle, and the
number of particles which cross all boundaries, i.e. sub-region and
downstream at x_max. All values, or the values needed to derive this
information, are stored in HDF5 data files using the
<monospace>h5py</monospace> package
(<xref alt="Collette, 2013" rid="ref-ColletteU003A2014" ref-type="bibr">Collette,
2013</xref>).</p>
<p><monospace>pySBeLT</monospace> produces a time varying signal of
particle flux counted at the downstream domain (as well as internal
subregion domains), with a particle bed that changes through particle
stacking and pile removal, and downstream motions of travel distance
(Fig. 2). An implication of particle stacking within the context of
the <monospace>pySBeLT</monospace> stochastic framework is a time
varying signal of the average “particle age”, as well as the average
“particle age range”, defined as the difference of the maximum and
minimum particle ages. The model can be readily modified to simulate
kinematics using different probability distributions (see THEORY.md
for more details), or examining particle age dynamics for deeper beds
of particles available for transport. The relatively simple
parameterization of <monospace>pySBeLT</monospace> execution also
makes it suitable for use as a teaching tool within advanced
undergraduate and graduate courses emphasizing bed load sediment
transport.</p>
</sec>
<sec id="figures">
<title>Figures</title>
<table-wrap>
<table>
<colgroup>
<col width="100%" />
</colgroup>
<thead>
<tr>
<th align="center"><inline-graphic mimetype="image" mime-subtype="png" xlink:href="../paper/figures/Figure1.png" /></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><italic>Figure 1. Graphic illustrating the
three steps of particle transport modelling by
<monospace>py_SBeLT</monospace>. The
<bold>‘level_limit’</bold> in height is set to 3 in the
graphic.</italic></td>
</tr>
</tbody>
</table>
</table-wrap>
<table-wrap>
<table>
<colgroup>
<col width="100%" />
</colgroup>
<thead>
<tr>
<th align="center"><inline-graphic mimetype="image" mime-subtype="png" xlink:href="../paper/figures/Figure2.png" /></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><italic>Figure 2. Example
<monospace>py_SBeLT</monospace> output of particle flux at
downstream boundary and particle bed configuration at
numerical step 100</italic></td>
</tr>
</tbody>
</table>
</table-wrap>
</sec>
<sec id="acknowledgements">
<title>Acknowledgements</title>
<p>S. Zwiep was funded in part through an Undergraduate Student
Research Award from the National Science and Engineering Research
Council of Canada (NSERC). S.M. Chartrand was funded through a
Postdoctoral Fellowship awarded by NSERC, and through internal
research funding provided by Simon Fraser University. The model was
inspired by discussions with D.J. Furbish, who also provided useful
input and critical feedback at various stages of model development and
testing. K. Pierce also provided helpful feedback during model
development. G. Baker provided insightful mentorship for S. Zwiep
during improvements to the model.</p>
</sec>
</body>
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