Entrainment, motion and deposition of coarse particles transported by

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1029/,

Dynamics of bed load particles in supercritical flows close to the onset of motion. J. Heyman1, P. Bohorquez2 and C. Ancey3 The fair understanding of bed load dynamics in established transport conditions contrasts with the relatively poor knowledge and the rich variety of phenomena occurring close to the initiation of transport. Steep streams are also known to resist most of the existing predictive theories. In order to gain knowledge of the principal mechanisms involved in such transport cases, it seems necessary to refine our vision of the transport process down to the individual grain dynamics. Here, we present 10 experiments carried out in a steep, shallow water flume made of an erodible bed of natural, uniform gravel at low transport conditions. We simultaneously recorded bed load particle motions and bed and water elevations using two high-speed cameras. 8 km of particle trajectories were reconstructed with the help of a robust and original tracking algorithm. We propose a statistical analysis of this data in order to estimate how dependent particle velocity, particle acceleration, particle diffusivity, particle entrainment, and deposition rates are upon the variations of local flow and bed conditions. Among other results, we found that: (i) the average particle streamwise acceleration was non-linearly dependent on particle velocity, and increased with downward slopes; (ii) the particle velocity distribution could be separated between an exponential close to the bed and a Gaussian at higher elevations; (iii) the particle diffusivity grew linearly with the shear velocity; (iv) the particle deposition rate was proportional to the inverse of the shear stress, and it decreased linearly with downward slopes; and (v) the entrainment rate was strongly correlated to the immediate proximity of other moving particles, but only weakly to the shear stress and bed slope. These results confirm the peculiar characteristics of bed load particles dynamics over steep slopes at low transport rates. The relationships between flow and sediment transport found here may serve as closure formula for bed load transport modeling.

arXiv:1604.00769v1 [physics.flu-dyn] 4 Apr 2016

Abstract.

1. Introduction

correlations in the position of moving grains and attributed them to the interaction between moving and resting particles. At these low transport rates, interactions between moving particles and the bed may indeed induce hysteretical behaviors [Clark et al., 2015]. Houssais et al. [2015] recently observed in their experiments that close to the onset of motion, transport rates needed to be computed over increasingly long time periods to get unbiased averages. At low transport rates, when only few particles are moving and do not modify significantly the flow, it is convenient to model bed load transport with the help of a single equation for sediment mass conservation, expressing the local balance between particle entrainment, transport and deposition [Charru and Hinch, 2006a, b; Ancey et al., 2008]. This strategy has been shown to yield accurate predictions of the sediment flux in mildly sloping beds [Lajeunesse et al., 2010] and its probabilistic equivalent was proven to accurately describe statistical moments in transport rates over steep slopes [Ancey et al., 2008; Heyman et al., 2014; Ma et al., 2014]. These models rely on the separate quantification of several processes, such as the particle entrainment and deposition rates [Papanicolaou et al., 2002; Valyrakis et al., 2010; Lajeunesse et al., 2010] and the particle transport down a slope (velocity, diffusivity) [Nikora et al., 2002; Ancey et al., 2003, 2008; Lajeunesse et al., 2010; Martin et al., 2012]. Experiments and simulations agree that bed load particles are advected with an average velocity increasing linearly with shear velocity—that is, to the square root of the shear stress [Ni˜ no et al., 1994; Abbott and Francis, 1977; Lajeunesse et al., 2010; Dur´ an et al., 2012; Shim and Duan, 2015]. Areal entrainment rates (e.g., the number of particles entrained per unit time and unit bed surface) have been

The inherent complexity of bed load transport makes the prediction of sediment transport rates difficult in both natural rivers and the laboratory. This complexity is particularly obvious when bed load transport occurs on steep bed slopes (θ > 2%) and/or close to incipient motion [Recking, 2013]. Bed load transport occurring in one-dimensional, shallow water flows over steep slopes is generally characterized by : (1) the significant fluctuations in bed load transport rates; (2) the effects of slope angle on particle dynamics; (3) and the rich variety of upper flow regime bedforms, such as standing waves, antidunes, breaking antidunes, cyclic steps, chutes and pools, and slug flows [Fernandez Luque and van Beek , 1976; Ni˜ no et al., 1994; Ancey et al., 2006; Church and Zimmermann, 2007; Ancey et al., 2008; Ferguson, 2012; Heyman et al., 2013, 2014]. When bed load occurs close to the onset of motion, fluctuations of the transport rates are predominant. In their experiments, Heyman et al. [2014] observed persistent spatial

1 Groupe Milieux Divis´ es, UMR CNRS 6626, Institut de Physique de Rennes, Universit Rennes I, Campus de Beaulieu, F-35042 Rennes, France 2 Area ´ de Mec´ anica de Fluidos, Departamento de Ingenier´ıa Mec´ anica y Minera, CEACTierra, Universidad de Ja´ en, Campus de las Lagunillas, 23071, Ja´ en, Spain 3 Laboratory of Environmental Hydraulics, School of ´ Architecture, Civil and Environmental Engineering, Ecole Polytechnique F´ ed´ erale de Lausanne, Switzerland

Copyright 2016 by the American Geophysical Union. 0148-0227/16/$9.00

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HEYMAN ET AL.: DYNAMICS OF BED LOAD PARTICLES

found to be linearly dependent on shear stress close to incipient motion [Dur´ an et al., 2012; Lajeunesse et al., 2010], whereas particle deposition rates (e.g., the rate at which a moving particle deposits onto the bed) have generally be found to be independent of shear stress [Lajeunesse et al., 2010; Ancey et al., 2008]. How dependent the particle diffusivity (defined in section 3.2.3)—are on flow variation has not been investigated deeply as yet [Drake et al., 1988; Furbish et al., 2012a; Heyman et al., 2014; Ramos et al., 2015; Bialik et al., 2015]. These processes were mainly studied in uniform and steady flows, under stationary sediment transport conditions. The effects of local and short-lived variations in flow conditions and sediment supply, common in steep flows, have not been investigated yet. The precise estimation of these processes intimately relies on the ability to measure the dynamics of particles at the scale of the grain diameter d50 . A convenient way to measure the transport process at such scale was found in high-speed photography and video recording [Francis, 1973; Abbott and Francis, 1977; Drake et al., 1988; Ni˜ no et al., 1994; B¨ ohm et al., 2004; Radice et al., 2006; Lajeunesse et al., 2010; Roseberry et al., 2012; Campagnol et al., 2013; Heyman et al., 2014; Heays et al., 2014; Frey, 2014]. Technological limitations have meant that until recently, filmed samples of bed load particle motion remained short and only covered small bed areas; this may bias statistical results [Fathel et al., 2015]. As described above, at the onset of motion, the necessary film acquisition time and window length may be considerable. The presence of bedforms exacerbate these experimental requirements, since averages need to be computed over several wavelengths. Today, digital image analysis allows an efficient post-treatment of large video samples in order to gather informations about grain dynamics [B¨ ohm et al., 2004; Radice et al., 2006; Campagnol et al., 2013; Heyman and Ancey, 2014]. In this article, we present an original experimental setup capable of quantifying sediment transport rates at the grain scale (with a precision of about 0.1d50 ) over lengths of the order of 150d50 , together with the local flow and bed characteristics. These were obtained by taking long sequences of high-speed films of sediment transport in a steep flume, and subsequently digitally processing the images using an efficient, automatic and original tracking algorithm. In total, 8 km of particle trajectories were obtained under various flow and sediment transport conditions. To the best of our knowledge, no equivalent data in terms of precision and resolution have been previously reported in the literature. We provide access to the tracking algorithm and the complete raw data (see Supplementary Informations and the database https://goo.gl/p4GbsR). The statistics of particle dynamics—particle acceleration, velocity, diffusivity, entrainment and deposition rates—depending on the local flow and topography characteristics are thoroughly explored and compared to previous findings. The article is organized as follows. We begin by describing the experimental apparatus used and the data available. For the sake of concision, the digital processing was entirely detailed in the Supporting Information. The experimental results are divided in two parts. First, the kinematics of moving particles are explored (e.g., moving particle accelerations and velocities) and second, the specificity of the exchange between moving and resting particles is studied (e.g., entrainment and deposition processes). These results are further analyzed and compare to the existing literature body in the discussion section and eventually summarized in the conclusion.

pump brought water in a tank, and later entered the channel through an honeycomb filled hole located at the bottom of the tank. The flow was supercritical, and thus flow depth was only controlled by the input water discharge. Water and sediment were thus simply left to flow out at the downstream end of the channel. The sediment constituting the granular bed was made up of natural gravel particles with a narrow grain-size distribution: median diameter was d50 = 6.4 mm, and d90 = 6.7 mm. Choosing a narrow grain-size distribution facilitated particle tracking and limited grain sorting effects. Sediment was fed into the top of the flume at a constant but adjustable rate via conveyor belt. Two cameras with 1280×200 pixel resolution were placed side by side at the downstream end of the channel and took pictures through the transparent side wall at a rate of 200 frames per second (fps). The total length of the observation window was approximately 1 m (giving a resolution of about 0.4 mm/pixel). Ten experiments were performed using various mean bed slopes, mean water discharge rates, and mean sediment inputs (Table 1). For each experiment, we waited a few hours until that the flow and sediment rates stabilized. Then, we filmed 2 to 8 sequences of 150 s at 200 fps (i.e., 30,000 frames). This corresponded to the maximum random access memory available on the computer used. In some experiments, we acquired more sequences of only 4,000 frames (i.e., 20 s) at a time. 2.2. Tracking Procedure Image processing and automatic particle tracking were subsequently performed on the images. The processing steps from the raw images to particle trajectories are thoroughly described in the Supporting Information and we only summarized them here. First, moving and resting pixels were separated via the background subtraction technique. Moving particles were detected from the clusters formed by moving pixels and their centroids saved at each frame. The centroids at consecutive frames were associated owing to simple association rules (e.g., nearest neighbor). This simple association gave parts of particle trajectories referred to as “tracklets”. Any ambiguity in the particle association process leads to the end of the current tracklet and the creation of a new one. Bed and water elevation were also determined at all frames. In a second pass, tracklets were associated into trajectories by applying a Kalman filter to each missing measurements. The algorithm was coded in Matlab and is available on-line together with a sample raw video. This technique was compared with acoustic methods (transport rates measurement by an acoustic sensor [Heyman et al., 2013]). Both methods gave comparable average sediment discharges (see Supporting Information). 2.3. Tracking variables

2.1. Setup

For each experiment, the data available consisted of a set of particle positions ~ xp (t), where p is the particle index and t is the frame number, projected on a two-dimensional (~ x, ~z) plane parallel to the flume walls. Particle velocities and accelerations were computed using second order finite differences

Experiments were carried out in a steep, narrow flume: 2.5 m long and 3.5 cm wide. The slope and water discharge were manually adjustable and the water discharge was monitored using an electromagnetic flow meter. The

~ xp (t + 1) − ~ xp (t − 1) , 2∆t ~ xp (t + 1) + ~ xp (t − 1) − 2~ xp (t) ~ap (t) = , ∆t2

2. Experiments

~vp (t) =

(1) (2)

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HEYMAN ET AL.: DYNAMICS OF BED LOAD PARTICLES Table 1. Average experimental parameters: tan θ (%), bed slope; Sf (%), friction slope; u∗ , shear velocity;

ws , settling velocity; τb∗ ; Shields number; Fr, Froude number; u ¯, depth-averaged velocity; h, water depth; qs∗ , ∗ , dimensionless sediment discharge at the flume inlet; dimensionless sediment discharge (tracking algorithm); qs,in γ ∗ , dimensionless particle activity; up , average particle velocity; r↓p ∗ , dimensionless particle deposition rate; r↑ ∗ , dimensionless areal entrainment rate.

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (Pool)

− tan θ 3.5 3.4 3.5 4.5 4.9 4.7 4.9 5.1 5.1 5.2 4.8

−Sf 3.8 2.9 3.2 4.7 4.8 4.4 4.4 5.2 4.9 4.9 5.0

u∗ /ws 0.170 0.172 0.182 0.189 0.188 0.176 0.180 0.190 0.194 0.175 0.185

τb∗ 0.093 0.095 0.106 0.115 0.114 0.099 0.104 0.116 0.121 0.098 0.110

Fr 1.18 1.22 1.28 1.28 1.32 1.28 1.31 1.37 1.39 1.32 1.29

u ¯/ws 1.37 1.42 1.47 1.49 1.37 1.31 1.34 1.34 1.40 1.25 1.35

¯ 50 h/d 4.1 4.2 4.0 4.2 3.3 3.2 3.2 3.0 3.1 2.7 3.3

∗ qs∗ /qs,in (10−3 ) 3.2/2.4 3.4/4.9 4.2/4.9 2.6/2.4 2.0/2.4 2.9/3.0 6.7/6.6 2.6/3.6 5.0/4.9 4.5/4.8 3.9/4.8

where ∆t = 0.005s. The horizontal and vertical components of the state vectors are denoted as ~ xp = (xp , zp ), ~vp = (up , wp ) and ~ap = (ax,p , az,p ). The cumulative distance traveled by moving particles was about 8 km. Given the camera resolution of 0.4 mm/pixel, the minimum observable change in particle velocity was about 0.002 m/s. In practice, it was even smaller since each centroid was estimated from the barycenter of pixels belonging to the particle, giving a sub-pixel precision. It is worth pointing out that, to avoid any bias, ~vp (t) and ~ap (t) were computed from the original tracklets, ignoring the interpolated part of the trajectories built during the second pass of the algorithm. Local particle activity γ(x, y, t) (i.e., the number of moving particles per unit bed area), was estimated from the position of moving particles. The cross-stream position of particles y was not resolved in our experimental setup, thus the particle activity is expressed per unit flume width and denoted as γ(x, t) [particles/m2 ]. Because the tracked particles are points in the space plane, γ(x, t) must be computed using a kernel estimation technique γ(x, t) =

N (t) 1 X Kw [x − xp (t)] , B p=1

(3)

where B is the channel width, N (t) is the number of moving particle at time t and Kw is a smoothing kernel of bandwidth w [m] [Diggle, 2014]. Similarly, local sediment discharge [m2 /s] is obtained with the product of particle activity and particle velocity: N (t) Vp X qs (x, t) = Kw [x − xp (t)] up (t), B p=1

(4)

with Vp as the particle volume. Taking Kw as a box filter (e.g., 0 everywhere except on a interval where it is constant), Eqs. (3)–(4) reduce to a classical volume averaging [Ancey et al., 2008; Ancey and Heyman, 2014]. Note that if w → 0, the traditional expression of the sediment flux as the product of a surface times a velocity is recovered [Heyman, 2014]. In addition, the bed elevation b(x, t), the local bed slope tan θ(x, t) = ∂x b(x, t), the water surface elevation w(x, t), and its local slope tan α(x, t) = ∂x w(x, t) were also automatically extracted from the images. The details of the numerical procedure to extract the flow and bed variables is given in the Supporting Information. Note that, in contrast to the usual conventions in bed load transport, we took the angles to be negative for downward slopes and positive for adverse slopes. The definition of “local” quantities, such as bed slope or bed elevation, is neither obvious nor unique. They depend

γ ∗ (10−3 ) 6.05 6.46 7.34 4.96 3.71 5.13 11.99 4.60 8.43 7.77 7.03

up /ws 0.56 0.57 0.61 0.56 0.56 0.59 0.60 0.60 0.63 0.62 0.60

r↓ ∗ (10−2 ) p 1.63 1.69 1.38 1.57 1.84 2.42 1.47 1.47 1.32 1.50 1.48

r↑ ∗ (10−4 ) 0.97 0.89 0.92 0.77 0.68 0.84 1.76 0.68 1.14 1.10 1.04

Frames 4 × 3 · 104 2 × 3 · 104 14 × 4 · 103 4 × 3 · 104 4 × 3 · 104 23 × 4 · 103 8 × 3 · 104 8 × 3 · 104 4 × 3 · 104 8 × 3 · 104 1.408 · 106

intimately upon the scale at which they are observed, in a similar manner to which the length of Britain’s coast depends on the size of the ruler used to measure it. As the bed is made of rough gravel, at the limit of an infinitely small ruler, bed slopes alternating between + and -90◦ are likely to be observed, though bed load may not be influenced much by such small asperities. As a consequence, some of the results obtained may depend on the spatial averaging scale chosen. The exact averaging procedure is detailed in the Supporting Information. An example of the typical data obtained is shown in Fig. 1 for experiment (g). 2.4. Additional variables In addition to the variables previously defined, we estimated the water depth, the depth-averaged flow velocity, and the bed shear stress as follows. Water depth is simply h(x, t) = w(x, t) − b(x, t). For stationary flows, the depthaveraged flow velocity reads u ¯(x, t) = Qw /(Bh(x, t)), with Qw as the prescribed water discharge and B as the channel width. Bed shear stress is more difficult to estimate precisely. In steady, uniform conditions for infinite-width channels, the balance of forces gives directly τ = %gh sin −θ, where h is the water depth. However, in the present case, the bed slope and flow depth change locally so that these forces may not balance everywhere. Moreover, the channel’s lateral glass walls and the relatively small channel-width-over-water-depth ratio (0.5 to 2) precluded any use of the infinite channel width

Table 2. Dimensionless numbers governing flow and sediment transport. Rh , hydraulic radius; u ¯, depth-average water velocity; ν, fluid kinematic viscosity; g gravity acceleration; h, water depth; d50 , particle mean diameter; s = %s /%, density ratio between sediment and fluid; u∗ , shear velocity; B, channel width; ks , bed equivalent roughness; κ, von Karman constant; ws = 0.55m/s, settling velocity determined from Eq. (37) in Brown and Lawler [2003]. Variable Name Expression Re Reynolds number 4Rh√u ¯/ν Fr Froude number u ¯/ gh
1/3 d∗ dimensionless grain d50 (s − 1)g/ν 2 diameter Rep flow particle d50 u∗ /ν Reynolds number Res settling particle d50 ws /ν Reynolds number channel aspect raB/h tio Relative roughness ks /h ws ∗ dimensionless setws /((s − 1)gν)1/3 tling velocity P Rouse number ws /(κu∗ ) St Stokes number d50 ws s/(9ν)

Value 3.5 · 104 1.1–1.3 157 630 3520 0.5–1.5 0.25–0.65 22.5 13 977

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HEYMAN ET AL.: DYNAMICS OF BED LOAD PARTICLES

1

.6 8 m /s

0.6

u¯ = 0

Distance (m)

0.8

0.4 0.2 0 80

Time (s)

85

95

Elevation (m)

0.04 0

0.03

0.2

0.4

0.6

0.8

1 ||~vp ||

100 Bed (10d50 ) Bed (4d50 ) Water

0.02 0.01 0 -0.01 0

0.2

0.4 Distance (m)

0.6

0.8

Figure 1. (a) Spatio-temporal tracklets obtained by the tracking algorithm. Depth-average flow velocity and direction is indicated by the black arrow. (b) The same tracklets plotted in the place (x, z) together with the estimation of bed and water elevation. Bed elevation was shown for spatial averages of 4 and 10d50 . assumption. In such extremely narrow channels, secondary currents are strong and can significantly modify the shear stress repartition between the lateral walls and the bed [Guo and Julien, 2005]. Alternatively, bed shear stress may be estimated using dimensional analysis : τ (x, t) f = u ¯(x, t)2 , % 8

(5)

where f is a function of dimensionless numbers characterizing the flow. For instance, f = 8κ2 / ln(11Rh /ks )2 in the −1/3 Keulegan formula, and f = 8gRh /K 2 with a Manning– Strickler formula, where κ is the von Karman constant, Rh is the hydraulic radius, ks is the equivalent roughness, and K is the Strickler coefficient. The difference in flow resistance between the lateral glass walls and the granular bed can be modeled by separating the hydraulic radius Rh into a bed hydraulic radius (Rb ) and a “wall” hydraulic radius (Rw ), with each of them bearing a part of the total shear stress [Guo, 2014]. Semi-analytical expressions for bed shear stress have also been proposed specially for such narrow flumes [Guo and Julien, 2005]. The applicability of different bed friction laws is studied in the Supporting Information. Their predictions gave similar results in our channel. We chose a friction factor f computed from Keulegan’s formula. A constant ks ≈ 2.0d50 was chosen so that, the average friction slope balanced the average bed slope. The bed- and wall-borne shear stress were distinguished and only the bed-borne shear stress was considered to be relevant for bed load transport. In addition, when bedforms are present, form drag may alter the value of the bed shear stress available for sediment transport. However, we believe that, in the supercritical flow conditions studied, wake effects were relatively limited and

we further assumed that bottom shear stress could be sufficiently well estimated by flow resistance laws, regardless of the local topography. 2.5. Orders of magnitude and dimensions The problem of sediment transport can be conveniently studied using dimensional analysis. Some of the dimensionless numbers useful in the study of sediment transport are summarized in Table 2, along with their average values in our experiments. In addition to the numbers defined in Table 2, the sediment discharge qs is made dimensionless by qs∗ = p 3 qs / gd50 (s − 1) ≈ 2 − 7 · 10−3 , the bed shear stress [kg/(ms2 )] by τ ∗ = τ /(%(s − 1)gd50 ) ≈ 0.09 − 0.12, also known as the Shields number. Particle activity γ [particles/m2 ] is made dimensionless by γ ∗ = γd250 ≈ 4 − 12 · 10−3 . With reference to Brown and Lawler [2003], we found that the settling velocity of particles was ws ≈ 0.55 m/s. By using the orders of magnitude obtained, we were able to refine the characterization of the sediment transport in our experiments. First, the particle Reynolds number suggested a rough hydrodynamic bed boundary, common to gravel bed streams and the Froude number indicated supercritical flow conditions in all experiments. On rare occasions, hydraulic jumps were observed to form over mature antidunes but rapidly died out. Second, the fairly high value of the Rouse number allowed us to exclude transport by suspension. The sediment in these experiments was thus only transported as bed load. Third, the relatively high Stokes number suggested that, similarly to aeolian bedload transport, particle-particle interactions through collisions are not fully damped by fluid viscosity [Joseph et al., 2001].

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HEYMAN ET AL.: DYNAMICS OF BED LOAD PARTICLES

Distance (m)

Distance (m)

1 0.8

tan θ 0.1

0.6

0

0.4

-0.1

0.2

-0.2

0 1

-0.3

0.8

0.2

0.6

0.15

0.4

0.1

0.2

0.05

τ∗

0

0 0

50

Time (s)

100

150

Figure 2. Spatio-temporal variations in bed slope tan θ (top) and Shields number (bottom). Flow direction is from bottom to top. The local averages are performed over a scale k = 10d50 using the averaging method described in the Supporting Information. Particle entrainment and particle deposition events are indicated by white and black dots respectively. Critical Shields numbers for incipient particle motion ranging between 0.02 and 0.08 are reported in the literature for particle Reynolds number and dimensionless grain diameter similar to our experiments [Buffington and Montgomery, 1997]. This would correspond to a transport stage ∗ (τ ∗ /τcr ) of 1 to 6. In practice, we observed that the transport stages in all the experiments were very close to, if not ∗ below, 1 (τ ∗ < τcr ). Most of time, when the sediment feeding system was stopped at the end of an experiment, sediment transport in the channel also ceased rapidly, proving that the flow alone was often not sufficiently strong to entrain new particles during the experimental time scales considered. Conversely, when sediment particles were feed at the input of the channel, sediment transport started spontaneously and was maintained without net bed aggradation nor erosion. In other words, the initiation threshold appeared to be significantly higher than the cessation threshold in all our experiments. This particular conditions were also reported in previous experimental studies [Francis, 1973; Drake et al., 1988; Ancey et al., 2002], in-situ [Reid et al., 1985], and more recently in numerical models [Clark et al., 2015]. Critical shear stress is indeed ill-defined in gravelbed streams: the gravel size and shape distribution is wide, particle exposure to the flow is varied, and turbulent flow is intermittent; thus particles are not entrained systematically past a given shear stress threshold. On the contrary, the rate of entrainment of particles by the flow slowly increases from 0 at low shear stress to larger values at higher shear stress. In spite of this, the concept of critical shear stress still gives a useful reference point for the characterization of the transport regime. Sediment transport rates were fairly low, as shown by the small dimensionless particle activity and discharge. On average, only 0.4% to 1.2% of the bed was covered by moving particles. In comparison, Lajeunesse et al. [2010] experimentally explored regimes where at least 5% of the bed was moving (− tan θ ≈ 0 − 11 %, d50 ≈ 1 − 5mm); Dur´ an et al. [2012] numerically simulated bed load regimes where at least 8% of the bed was moving (tan θ = 0, various d50 ). In the range explored in this study, a bed load sheet layer did not developed and particles moved sporadically by jumping. This observation was confirmed by the relatively low sediment discharge qs∗ recorded in all experiments, confirming ∗ that the transport stage was τ ∗ /τcr ≈ 1.

3. Results 3.1. Preliminary remarks Bed and water elevations varied considerably in time and space, even though the water and sediment inputs were kept constant (Fig. 1). Indeed, antidunes developed naturally and migrated upstream in the experimental flume, locally modifying bed topography, flow, and sediment transport. The typical length of these structures ranged between 5h and 10h (about 10–20 cm). Their celerities depended upon the transport rates but were vanishingly small compared to sediment and water velocities. An example of such migration can be observed through the spatio-temporal variation of bed slope and bed shear stress in Fig. 2. Flow and bed conditions being continuously evolving, it made little sense to try to relate the statistics of transport processes to averages made over the whole experimental window. In contrast, we envisioned the whole set of experiments as representative of various local and instantaneous boundary conditions that may affect sediment transport: we consequently ignored the individual particularities of each experiment (mainly the prescribed flume angle, sediment input, and water discharge). Statistics concerning particle transport were estimated by pooling the whole experimental dataset (about 5 million values), without regard to specific run conditions. First, we analyze the motion of bed load particles without taking into account their entrainment and deposition. Second, we focus on the mass exchanges between the granular bed and the moving particles. 3.2. Kinematics Once entrained, bed load particles were transported downstream by the water flow. Their motion consisted of various alternating stages: free flight in the fluid column, rebound, and rolling over the granular bed. Particle were subject to various fluctuating forces: turbulent flow drag, collision with other particles, and/or granular friction. As a result, particle velocities fluctuated considerably during transport and moving particles tend to disperse in space while they are transported, resulting in a diffusive effect on particle activity [Furbish et al., 2012a; Ancey and Heyman, 2014; Heyman et al., 2014]. In the following, we successively

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3

up /u∗ 4 5 6

7

8

9 10

1

zp /h

hax,p i /g

0.05 0 up /¯ u up /u∗

-0.05 -0.1

-0.2

2

0

0.2 0.4 0.6 0.8 up /¯ u

1

1.2 1.4

(b)

0.8

4

0.6

3

0.4

2 P (zp )

0.2

1 0

-0.2

-1

-0.4 -1 0.2

-0.5 0 0.5 hax,p i /g haz,p i /g

(d)

-2 1

0.09(sin θ4 − µ cos θ4 ) 0.20(sin θ10 − µ cos θ10 ) 0.92(sin θ40 − µ cos θ40 )

0.1 hax,p i /g

tan −2.5◦

 ( a2x,p − hax,p i2 )/g 2

0.6d50

0

1.5

1

5

(c)

zp /d50

0.1

2

1.15¯ u

0.15

1 1.0u∗

-1 0 (a)

0

0.5

0

-0.1 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 up /¯ u

k = 4d50 k = 10d50 k = 40d50 -0.3

-0.2

-0.1 0 tan(θk )

0.1

0.2

Figure 3. (a) First and (b) second moments of streamwise particle acceleration as a function of particle velocity. (c) Average streamwise and instantaneous vertical acceleration as a function of particle elevation zp /h (circles) and zp /d50 (triangles). Arrows indicate the direction of the average momentum change depending on particle velocity. (d) Streamwise particle acceleration as a function of bed slope computed with various averaging scales k: 4, 10 and 40d50 . analyze particle acceleration, velocity and finally their apparent diffusivity. 3.2.1. Acceleration The study of particle acceleration provides some insight into the nature and magnitude of forces driving bed load particles. In Figs. 3(a) and (b), we plotted the first and second moments of longitudinal particle acceleration as functions of the particle velocity. The longitudinal acceleration varied non-linearly with up (Fig. 3(a)). Scaling particle velocity by u∗ or u ¯ gave similar results. Within the velocity ranges [0, u∗ ] and [1.15¯ u, ∞[, hax,p i < 0, and thus particle velocity tended to decrease: forces acted against their motion. In contrast, for velocities ranging from u∗ to 1.15¯ u, the particles accelerated on average: the mean forces acted in the direction of their motion. When up < 0, the wide scatter of points makes the analysis difficult but it can be safely admitted that hax,p i > 0 as particles moving temporarily backward rapidly turned back in the flow direction. Longitudinal particle accelerations showed large variations around the mean (Fig. 3(b)). Indeed, the standard deviation of longitudinal acceleration was always close to g while the mean never exceeded 0.1g (Fig. 3 (a) and (b)). Variations are the strongest at relatively low (up < 0) or

high particle velocities (up > 1.15¯ u). We can distinguished two local minima, at up = 0 and up ≈ 1.15¯ u, as well as a local maximum near u ¯/2. We plotted the dependence of longitudinal and vertical accelerations upon particle elevation in Fig. 3(c). The change of sign in vertical particle acceleration at about zp = 0.6d50 tells us that most particles located below 0.6d50 experienced upward acceleration, whereas above this elevation, they were subject to negative acceleration of the order of −g/2. As previously noted with the distribution of particle elevations, the particles were thus mainly transported at an elevation of zp ≈ 0.6d50 . The interesting feature in Fig. 3(c) is that both vertical and streamwise particle accelerations changed sign at the same elevation. When particles were moving below 0.6d50 , their vertical accelerations were positive but their streamwise accelerations negative on average. In contrast, when particles were above 0.6d50 , their vertical accelerations were negative and their streamwise accelerations positive on average. The longitudinal particle acceleration shows a quasilinear dependence on local bed slope (Fig. 3(d)); the largest accelerations being observed on the steepest slopes. Acceleration becomes negative for values of tan θ larger than tan(−2.5◦ ) = −0.043, e.g., for weak and adverse bed slopes. It is interesting to note that the scale at which the bed slope is computed influences the relationship: the larger the ruler, the strongest the dependence.

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HEYMAN ET AL.: DYNAMICS OF BED LOAD PARTICLES hup i /u∗ 4

5

6

7

8

9

1.4

0.8

4

1.2

0.6

3

0.4

2 1 P (zp /d50 ) -1

0.4

-2 1.2

0.2

(b)

0.6 hup i /¯ u

0.8

1

2.5

P (up ) P (up |zp ≤ 0) P (up |zp > 2d50 )

τL σu2 p /(d50 ws )

100

0.4

2 1.5

0.6

10−3 0

2

4

0.05 0.1

0.8

1

8

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12

u∗ /ws u¯/ws 3.63u∗ − 0.06ws 0.57¯ u − 0.16ws 1.2 1.4 u ¯/ws

1.6

1.8

2

(d)

u∗ /ws

1

0.3

0.15 0.1 u∗ /ws hDu i = 9.8u∗ /ws − 0.63

0.3

∗

0 6 up /u∗

2

0.1

0.5

N (3.3, 2.2) N (6.9, 1.5) Exp(1.5)

0.35 (c)

Fr

∗

10−2

0.3

0.25

hDu i =

pdf

10−1

0.25

0.6

-0.2 0.2

1

0.8

0

0

0.2

0.73Fr − 0.36

0

0

-0.4

u∗ /ws

0.15 1

hup i /ws

0.2

1 zp /d50

(a)

0.1

σup (m/s)

3

√ hup i / gh

2

τL (s)

1

5

1

zp /h

0

0.1

0.15

0.2 u∗ /ws

0.25

0.3

Figure 4. (a) Average particle velocity function of particle elevation. Blue triangles: hup i /u∗ function of zp /d50 ; Black circles: hup i /¯ u function of zp /h; Green line: distribution of moving particle elevations function of zp /d50 . (b) Distribution of particle velocity and theoretical fits. N (µ, σ) stands for the Gaussian distribution of mean µ and standard deviation σ and Exp(λ) for the exponential distribution with mean λ. (c) Particle velocity function of local shear velocity (blue triangles) and local depth-average flow velocity (black circles). Inset: particle Froude number function of local flow Froude number. (d) Dimensionless particle diffusivity function of shear velocity. Insets: Lagrangian time scale function of shear velocity (top) and standard deviation of particle velocities function of shear velocity (bottom). 3.2.2. Velocity In Fig. 4(a), we consider the average particle velocity hup i as function of the particle elevation zp above the gravel bed. Note that the instantaneous particle elevation is not known exactly: first, because the gravel bed level cannot be defined exactly; and second, because the particle’s vertical position is not uniquely defined in the image projection plane (the camera was at a slight angle to the flume). This is why, as shown in Fig. 4(a), particles were observed moving at elevations as low as −2d50 . zp still provides a rough estimate of the particle’s vertical position in the fluid column. When averaged over numerous trajectories, individual elevation errors should cancel each other out. Particle velocity can be scaled either by using the depthaveraged velocity u ¯ or the shear velocity u∗ , and the particle elevation by using either the flow depth h or the grain diameter d50 . As shown in Fig. 4(a), both options gave similar results. However, retaining both scaling options allowed us to distinguish two motion regimes: at low elevations (zp ≤ 0), particle velocity was close to the shear velocity, whereas at higher elevations (zp ≥ 0.5h), average particle velocity reached the depth-averaged flow velocity. In other words, the rolling and sliding motion of particles

in contact with the bed occurred at a velocity close to the shear velocity, whereas the particles saltating high enough in the fluid column reached the depth-averaged flow velocity. The distribution of moving particle elevations P (zp ), also plotted in Fig. 4(a), suggested that, most of the time, particles move between −1 and 2 particle diameters under/above the zero bed level. At those heights, particles travel at very different velocities. The presence of these two motion regimes was also apparent when looking at the distribution of velocities, as shown in Fig. 4(b). The distribution of particle velocities located at or under the bed level was approximately exponential, whereas the distribution of particle velocities located two grain diameters or above the bed level was clearly Gaussian. The overall distribution was a composite of both, and may be fairly well approximated by a truncated Gaussian distribution. A linear relationship is observed between particle velocity and both u∗ and u ¯ (Fig. 4(c)). Here, all velocities were made dimensionless by using settling velocity ws . The parti√ cle Froude number (up / gh) is also linearly dependent upon the flow Froude number (inset of Fig. 4(c)). 3.2.3. Diffusivity

HEYMAN ET AL.: DYNAMICS OF BED LOAD PARTICLES

Particle velocity fluctuations favored the spread of particles through space, with some moving faster or slower than the bulk average velocity, resulting in a bulk diffusive effect on particle concentrations. The average particle diffusivity may be obtained by computing the mean squared displacement of particles. In the case of normal diffusion (we do not consider here anomalous diffusion), the latter can be shown to grow asymptotically linearly with time at a rate proportional to Du [Taylor , 1922]. The diffusivity is also related to the Lagrangian integral time scale of particle velocity time series τL (i.e., the integral of the particle velocity autocorrelation ρup (t)) and the variance of particle velocity σu2 p [Taylor , 1922; Furbish et al., 2012a]: Du = τL σu2 p .

0.015

3.3. Mass exchanges The experiments being carried out at the incipient stage of bed load transport, particle motion was intermittent. Resting bed particles were occasionally set in motion (entrainment) and subsequently came to a halt further downstream, where a bed asperity was sufficiently deep to retain them (deposition). No bed creeping was observed at the onset of motion: sediment motion occurred only by successive particle jumps. In addition, no developed bed load sheet layer formed at the low transport stages investigated here, so that each particle trajectory, from rest to rest, was clearly identifiable. This said, the entrainment of a resting particle into the flow was not always instantaneous. Some particles were first oscillating, slowly sliding or rolling on top of the granular bed before being fully dislodged and entrained in the flow. Similarly, some of the moving particles remained unstable long after being deposited. Thus, the exact time of particle entrainment or deposition was not always obvious to distinguish. The criteria used to detect entrainment and deposition events in practice are explained in the following. 3.3.1. Definitions and averages The entrainment of a particle (E) was assumed to be a random event that occurred if two conditions were met: (i) the particle velocity magnitude exceeded a given velocity threshold vth ; and (ii) the distance from the particle mass center to the estimated bed elevation did not exceed zmax . The first condition is usual in tracking experiments [Radice et al., 2006; Ancey et al., 2008; Campagnol et al., 2013], and the second aimed at reducing overestimations of particle entrainment rates due to the existence of broken trajectories. In a similar manner, the deposition of a particle (D) is assumed to be a random event that occurs if: (i) the particle velocity magnitude fell below a given velocity threshold vth ; and (ii) the distance from the particle mass center to the estimated bed elevation did not exceed zmax . In practice, we fixed vth = 0.01ws ≈ 5.5 mm/s and zmax = d50 = 6.4 mm. It

0.01 i

0.005

hd e f

a b

Pc j

0 0

(6)

An estimation of τL in non-uniform environments is possible if we assume that the autocorrelation of particle velocity is well modeled by a decaying exponential: ρup (t) ≈ exp(−t/τL ) [Martin et al., 2012]. Consequently, τL could be computed based on the very first lags of the particle velocity autocorrelation function. The estimation of σup from the trajectories is straightforward. Values of τL were roughly constant and close to 0.08 s while σup increased with shear velocity (inset of Fig. 3(d)). We introduce the dimensionless particle diffusivity Du ∗ = Du /(d50 ws ) and plot its variations against the shear velocity in Fig. 3(d). Du ∗ increased approximately linearly with u∗ /ws , from 0.5 at low shear to 2 at higher shear velocities. A saturation of diffusivity values at high shear velocity appears in Fig. 3(d). However, the wide scattering of points precludes jumping to conclusions.

1:1 g

hr↑ ∗ i / hr↓p ∗ i

X-8

0.005

hγ ∗ i

0.01

0.015

Figure 5. Average particle activity compared to average bed exchange rate (r↑ ∗ /r↓p ∗ ). is worth mentioning that these definitions are purely operational and the entrainment and deposition rates may depend upon the chosen velocity and elevation thresholds. We denote P (D, ∆t) as the probability of a moving particle being deposited during the short time span ∆t. The particle deposition rate r↓p [s−1 ] is then directly r↓p = P (D, ∆t)/∆t. Note the use of the subscript p to indicate that the rate refers to a single particle. Assuming that all particles are similar, the areal deposition rate (e.g., the number of particles deposited per square meter per second) is simply r↓ = r↓p γ. Similarly, P (E, ∆t) is the probability of a resting particle being entrained during ∆t. This gives the particle entrainment rate r↑p = P (E, ∆t)/∆t [s−1 ]. Assuming a constant areal density of particles resting on the bed ψ, the probability of any resting particle being entrained in the infinitesimal time interval ∆t, in a bed surface A, is ˜ ∆t) = ψAP (E, ∆t). Thus, the areal entrainment rate P (E, of particles per unit bed area, denoted by r↑ [particles m−2 s−1 ], is r↑ = ψr↑p . We used the dimensionless entrainment and deposition rates, r↑ ∗ = r↑ d350 /ws , r↑p ∗ = r↑p d50 /ws , r↓ ∗ = r↓ d350 /ws and r↓p ∗ = r↓p d50 /ws , respectively. Entrainment and deposition rates are linked to bed elevation changes via the mass conservation equation (Exner Equation) [Einstein, 1950] (1 − ξp )

∂b = Vs (r↓ − r↑ ) = Vs (r↓p γ − r↑ ) , ∂t

(7)

where ξp is the bed porosity. In a stationary, uniform case in which the bed is neither aggradating nor eroding (∂b/∂t = 0), eq. (7) implies that hr↑ i / hr↓p i = γeq . As can be verified in Fig. 5, the present study’s experiments showed no net aggradation or erosion. We found average pool values of hr↑ ∗ i = 1.04 · 10−4 and hr↓p ∗ i = 0.0148. The areal deposition is thus hr↓ ∗ i = hr↓p ∗ i hγ ∗ i = 1.04 · 10−4 , confirming the absence of net bed aggradation or erosion. 3.3.2. Local estimates A difficulty arises when estimating the effects of local flow and bed conditions on particle entrainment and deposition rates. Indeed, the local flow characteristics are by definition continuous spatio-temporal fields, whereas entrainment or deposition occur at discrete locations (Fig. 2). Bayes’ theorem provides a simple solution to this estimation problem (see details in the Supporting Information). For instance, to estimate the dependence of the deposition rate on bed slope, Bayes’ theorem gives r↓p (θ) = hr↓p i fΘ|D /fΘ , with fΘ|D the probability density function of bed slopes where particle deposited and fΘ the probability density function of all bed slopes encountered by moving particles.

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HEYMAN ET AL.: DYNAMICS OF BED LOAD PARTICLES

2

(a)

∗ 10−4 [3.3(τ ∗ − τcr ) + 0.8]

(c)

(10−4 )

1.5

r↑ ∗

1

0.5

0 0.04

r↓p ∗ (4d50 ) r↓p ∗ (10d50 ) r↓p ∗ (40d50 )

(b)

0.03

r↓p ∗

r↑ ∗ (4d50 ) r↑ ∗ (10d50 ) r↑ ∗ (40d50 )

r↑ ∗ (4d50 ) r↑ ∗ (10d50 ) r↑ ∗ (40d50 ) (d)

0.035 tan θ4 + 0.016 0.046 tan θ10 + 0.017 0.099 tan θ40 + 0.019

0.02

0.01 r↓p ∗ (4d50 ) r↓p ∗ (10d50 ) r↓p ∗ (40d50 )

0 0

0.05

0.1 ∗ τ ∗ − τcr

0.15

0.2 -0.2

-0.15

-0.1

-0.05 tan θk

0

0.05

0.1

Figure 6. (a) Particle entrainment rate and (b) deposition rate functions of the excess Shields number ∗ ∗ τ ∗ − τcr , with τcr = τ0∗ sin (θ + α) / sin α, α = 47◦ and τ0∗ = 0.035. (c) Particle entrainment rate and (d) deposition rate functions of the local bed slope tan θk , where k is the averaging scale. The same formula applies for local areal entrainment rates within the difference that the probability fΘ should not be computed with slope samples taken only at moving particle sites, but rather with all the possible bed slopes. The difference lies in the fact that particles may be entrained from anywhere on the bed surface while they are only able to deposit on the locations they visit. In Fig. 6(a) and (b), we present the dependence of r↑ ∗ ∗ and r↓p on the excess Shields number τ ∗ − τcr , where τcr is computed according to the Fernandez Luque and van Beek ∗ [1976] formula: τcr = τ0∗ sin (θ + α) / sin α, with α = 47 and τ0∗ = 0.035, a reference Shields number at zero slope. The averaging scale chosen played no significant role here. The areal entrainment rate did not increase significantly ∗ with τ ∗ − τcr ; for excess shear stresses between 0.05 to 0.1, ∗ r↑ showed even a slight decrease. It is also interesting to see that at low excess Shields, the areal entrainment rate did not become zero (as one would expect at the motion threshold). Fig. 6(b) shows a strong inverse correlation between the deposition rate and the excess Shields number. At high excess Shields numbers, the deposition rate is approximately equal to 0.01, whereas at low excess Shields number, its value constantly increases. The dependence of r↑ ∗ and r↓p ∗ on local bed slope θ is presented in Fig. 6(c) and (d). The averaging scale of local bed slope influenced the results. This is particularly apparent for the deposition rate in Fig. 6(d): the larger the averaging scale, the stronger the dependence of r↓p ∗ on θ.

Fig. 6(c) shows no clear relationship between r↑ ∗ and bed slope: at small averaging scales (k ≤ 10d50 ) particle entrainment rates were larger on smaller bed slopes but at large averaging scales (k ≥ 40d50 ), the relationship disappeared. In contrast, particle deposition rates follow clearer trends. Deposition linearly increased with bed slope at all averaging scales: the steepest the slope, the smaller the deposition rate. By definition, the areal particle deposition rate depends on local particle activity. In contrast, the areal entrainment rate dependence on local particle activity is not established nor obvious. We estimated this dependence owing to Bayes’ theorem. First, the local particle activity γ(x, t) needs to be estimated from Eq. (3) and is shown in Fig. 7. We chose a Gaussian kernel of bandwidth 5d50 , in the idea that if a relationship between entrainment and activity exists through particle collisions for instance, it shall be well localized in space and time. The results are shown in Fig. 8. Surprisingly, r↑ shows the same strong dependence on γ, in a similar way as r↓ does. A tangent fitted for values of γ ∗ < 0.03 gives a slope of approximately 0.016 for both areal deposition and entrainment rates. The strong correlation of r↑ with γ, opposed to the weak correlation with τ ∗ and θ suggests that particle entrainment is mostly triggered by surrounding moving particles rather than by the destabilizing action of the flow and bed slope. We develop this point in the discussion section.

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HEYMAN ET AL.: DYNAMICS OF BED LOAD PARTICLES

Figure 7. Estimated local activity γ ∗ (color), particle entrainment (white dot) and particle deposition (black dot). Flow direction is from bottom to top. Estimation of the local density of particles is made using a Gaussian kernel with a fixed spatial bandwidth of size 5d50 . 6

r↑ ∗ , r↓ ∗

(10−4 )

5 4 3 2 r↓ ∗ r↑ ∗ 0.016γ ∗

1 0

0

0.01

0.02

γ∗

0.03

0.04

0.05

Figure 8. Entrainment and deposition rates as a function of local particle activity. The latter is estimated with Eq. (3) and a Gaussian kernel of bandwidth 5d50 .

4. Discussion 4.1. Particle kinematics Longitudinal particle motion (in the streamwise direction) may be modeled with Newton law: dxp = up , dt dup mp = f1 (up , zp , u ¯, . . . ) + f2 (up , zp , u ¯, . . . )ξ(t), dt

(8) (9)

where f1 and f2 ξ(t) are an average and a fluctuating force respectively, with ξ a white noise[Furbish et al., 2012a; Ancey and Heyman, 2014; Fan et al., 2014]. f1 is the average drift force felt by any particle in motion and is the result of both the average flow drag and the average friction over the bed, whereas f2 ξ(t) simultaneously represents all the force fluctuations arising during particle motion, mainly the randomness of collision angles and velocities with the bed and fluctuations in the turbulent flow drag. 4.1.1. Particle acceleration As shown by Eq. (9), the first and second moments of longitudinal particle acceleration is related to the average and fluctuating forces on a moving particle:

acting  hax,p (t)i = hf1 i /mp and a2x,p − hax,p i2 = hf2 i2 δ(0)/m2p , respectively. In our experiments, the average particle acceleration hax,p i changed sign depending on the instantaneous particle velocity. Deceleration dominated for up ∈ [0, u∗ ] (forces

acted against motion) and up ∈ [1.15¯ u, ∞[ while particle acceleration dominated when up ∈ [u∗ , 1.15¯ u] (forces acted in favor of motion). Particles with velocities of u∗ accelerated or decelerated in equal proportions. Large fluctuations around these averages favored the mixing of particle velocities. Looking at Fig. 3(a), the history of a typical particle flight may be reconstructed as follows. Starting from rest at up = 0, a particle is entrained by a sufficiently strong fluctuating force (the second moment of acceleration at up = 0). If particle velocity do not reach u∗ , it will probably stop rapidly. In contrast, if the fluctuating forces succeeded to bring the particle to a velocity larger than u∗ , the particle will be further accelerated to a velocity of about u ¯, presumably by the average flow drag. The factor 1.15 found experimentally may have two origins. First, the maximum flow velocity is higher than the depth-average velocity so that, in high saltation, particles are advected at a velocity slightly larger than u ¯. Second, a systematic bias in the value of u ¯ may exist since the bed elevation was hard to determine exactly, so that the factor may be 1.0 instead. Now, if a fluctuating force accelerates a particle to velocities larger than about u ¯, the average flow drag is now acting in opposition of motion. Meanwhile, the particle will eventually collide with bed and loose part of its streamwise momentum. If the momentum loss is enough to bring particle velocity below u∗ , the particle will probably stop and return to the state up = 0. Fig. 3(c) further suggests that at the bed, part of the stream-wise momentum is converted to vertical momentum by particle impacts. In all the motion history, the fluctuation of forces insures that all physical velocity states are explored. The momentum state up ≈ u∗ appears as a turning point in particle motion history. At this velocity, a particle may, with equal probability, return to rest or reach a velocity close to the depth-average flow velocity. Interestingly, Quartier et al. [2000] showed that for certain slopes, a cylinder rolling down a rough plane may either come to rest or reach a constant velocity depending of its initial kinetic energy, and presents an acceleration diagram very similar to the one showed in Fig. 3(a). This hysteretic behavior is well documented in the literature of granular media and comes from the fact that a moving grain must always rise above the grains beneath it in order to maintain its motion. The bed roughness forms thus an area of potential trapping wells for a moving particle without sufficient kinetic energy [Riguidel et al., 1994; C. Ancey et al., 1996; Dippel et al., 1996]. Looking at 3(a), the “equilibrium” velocities are found for up = 0 and up ≈ u ¯, where acceleration changes sign (from positive to negative values). The null velocity must be an attractor since moving particles all came to rest after

HEYMAN ET AL.: DYNAMICS OF BED LOAD PARTICLES a while. In addition, it is clearly identified by the peak in the distribution of particle velocities at up = 0 (Fig.4(b)). The existence of a second attractor is better seen in Fig. 1, where many particle trajectories in the time-space plane are approximately parallel to the mean flow velocity. The dependence of longitudinal particle acceleration on bed slope is better understood when considering the longitudinal imbalance between two forces: the particle weight and a Coulomb like friction force modeling bed-particle contacts (we ignore the flow drag contribution here). The mean acceleration of a particle would then read hax,p i = g(sin(−θ) − µ cos(−θ)),

(10)

where µ is the friction coefficient. This equation fits well the experimental data of Fig. 3(d) if gravity is reduced by a factor ranging between 0.1 to 0.9, for bed slopes averaged over scales between 4d50 and 40d50 respectively, and with µ = tan −2.5◦ . Two remarks can be made from this crude model. First, the friction coefficient is relatively small and second, its application is better in the case of large averaging scales, the gravity approaching its normal value. The first point may be explained by the presence of a mean positive flow drag force that reduced the apparent friction. The reduction of the effective gravity for small averaging scales is less clear. A plausible explanation is that, when a particle is saltating, it “feels” the effect of local bed slope after several contacts with the bed, so that the effective bed slope needs to be computed over large scales to be relevant for particle motion. 4.1.2. Particle velocity Getting back to Eq. (9), providing expressions for f1 and f2 , and integrating leads, for simple cases, to the theoretical expressions of particle velocity density function [Ancey and Heyman, 2014; Fan et al., 2014]. In the case of a uniform flow, Ancey and Heyman [2014] showed that the distribution of particle velocities was Gaussian if f1 was proportional to the velocity difference between flow and particles, and f2 was constant. Furbish et al. [2012b] showed that with both f1 and f2 proportional to up , particle velocities were exponentially distributed. Fan et al. [2014] also got exponentially distributed velocities based on a piecewise constant f1 and a constant f2 . It is worth pointing that the previous force models are not fully compatible with the dynamics of particles observed in our experiments, since both f1 and f2 were non-linear functions of up (Fig. 3(a) and (b)). Experimentally, some authors have reported exponential distributions of particle velocity [Lajeunesse et al., 2010; Furbish et al., 2012b] for relatively small settling particle Reynolds number (Res ≈ 420 for Lajeunesse et al. [2010] and Res = 50 for [Furbish et al., 2012b]). Others have reported Gaussian and truncated Gaussian distributions over fixed beds [Martin et al., 2012] and erodible beds [Ancey and Heyman, 2014], generally at larger settling particle Reynolds number (respectively Res ≈ 2400 and Res ≈ 2700). Fig. 4(b) suggests that the distribution function of particle velocity is also of Gaussian shape in our experiments (Res ≈ 3500). A classification of particle velocity distributions may thus be proposed in terms of settling Reynolds numbers or Stokes numbers: at low Res (low St), velocities are exponentially distributed while at large Res (large St), velocity distribution is Gaussian. Indeed, in both limiting cases, particle dynamics are different: at low St, particle trajectories are strongly perturbed by the near bed turbulent flow velocities while at high St, their ballistic trajectories are less altered by turbulent flow fluctuations. Fig. 4(b) also suggests that depending on the dominant type of particle motion—low rolling/sliding or high saltation—the shape of the particle velocity distributions may differ: exponentially distributed velocities at low elevations and Gaussian distribution for high particle flights (Fig. 4(b)). Depending on particle elevation, the nature of

X - 11

forces governing particle motion changes (Fig. 3(b)), and changes in return the statistical characteristics of their velocities. As is often reported in literature, the average particle velocity increased linearly with the shear velocity of the flow. A linear fit to the data gave hup i ≈ 3.6u∗ − 0.06ws .

(11)

This scaling is consistent with Bagnold [1973] law and also widely used in sedimentation engineering (Yalin and Ferreira da Silva [2001]). For comparison, Abbott and Francis [1977] and Martin et al. [2012] reported coefficients between 13 and 14 over fixed beds, whereas Lajeunesse et al. [2010] and Ni˜ no et al. [1994] reported coefficients of 4.4 and 5.6, respectively, for flows over erodible beds—this is closer to our findings. Using direct numerical simulations, Dur´ an et al. [2012] also reported a linear relationship between hup i and u∗ , with a coefficient of approximately 3. At high velocities, the linear relationship is clearer when up is scaled by u ¯ than by u∗ . These particle velocities are obtained for high saltations in the water column, where the flow velocity is close to u ¯. This effect was also reported by Bagnold [1973]. Note that u∗ and u ¯ are related through the friction factor f , which in turn depends on the relative roughness, explaining the disparities between u∗ and u ¯ scaling. Chatanantavet et al. [2013] proposed that a linear relationship √ may also exist between a particle Froude number (hup i / g h) and the flow Froude number. Indeed, √ taking Eq. (11) and then using Eq. (5) and dividing by gh, we also have r hup i f ws √ Fr − 0.06 √ = 3.6 . (12) 8 gh gh √ Thus, hup i / gh ≈ βFr−c, where β depends of f and c of h. Taking the average pool values of f = 0.15 and h = 3.3cm, we should get β = 0.49 and c = 0.06. A linear trend is clearly seen in the inset of Fig. 4(c) and gives β = 0.73 and c = 0.36. The estimated β and c are larger than predicted, questioning the validity of (12). Nevertheless, Chatanantavet et al. [2013] obtained 0.6 ≤ β ≤ 0.8 for bedrock channels, which seems consistent with our experimental value of β = 0.73. 4.1.3. Diffusivity Particle velocity fluctuations were previously shown to result in a macroscopic diffusive effect on the particle flux [Furbish et al., 2012a]:   ∂γ qs = Vs up γ − Du , (13) ∂x with Du [m2 /s], the particle diffusivity. There is little previous experimental evidence of the key role of bed load particle diffusion involving weak and partial bed load transport [Drake et al., 1988; Nikora et al., 2002; Furbish et al., 2012a; Ancey and Heyman, 2014; Heyman et al., 2014]. By a separate estimation of the particle velocity autocorrelation time and the standard deviation of particle velocity (a method referred to as AC in the following), we found that the dimensionless particle diffusivity Du ∗ ranged between 0.2 and 2, depending on the shear velocity. By comparison, Furbish et al. [2012a] obtained Du ∗ ≈ 0.7 − 1.9 using a similar calculation. In contrast, using the linearity of the mean squared displacement (referred to as the MSD method), Heyman et al. [2014] found a larger effective diffusivity (Du ∗ = 2.8 − 5) as did Ramos et al. [2015] (Du ∗ ≈ 3.6).

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This may originate from the fact that the MSD method retains the large time limit while the AC method is computed at short time intervals. Particle motion is ballistic at short time intervals (typically time periods smaller than two consecutive particle collisions), resulting in sub-diffusion (the mean square displacement grows with t2 when t → 0 ) [Martin et al., 2012; Heyman, 2014; Bialik et al., 2015]. Thus, apparent diffusivity is smaller at short time scales. The variety of processes (and time scales involved) causing bed load particle diffusion may also potentially explains the differences between MSD and AC estimates [Bialik et al., 2015]. The primary source of velocity fluctuations comes from repeated impacts of particles against the bed. The time scale of these fluctuations depends thus upon the impact frequency and can be long for particles in saltation. Turbulent drag fluctuations and variations in contact forces also modify particle path. In addition to being less diffusive, these processes are potentially acting on shorter time scales than successive particle impacts, explaining thus MSD and AC divergences. 4.2. Mass exchanges Bed load models distinguishing particle entrainment from particle deposition are particularly suited for the characterization of particle transport at the onset of motion [Charru and Hinch, 2006a, b; Ancey et al., 2008; Ancey and Heyman, 2014]. Indeed, at these low transport stages, particle motion is rare and intermittent: between long periods of rest, particles are occasionally entrained into the flow, transported, and deposed further downstream. In our experiments, the non-spherical shape of particles precludes from any slow bed creeping, so that particle entrainment and deposition were the sole processes involved into sediment motion initiation and halt. In those conditions, particle activity obeys the following mass conservation equation: ∂γ 1 ∂qs + = r↑ − r↓ , ∂t Vs ∂x γ(0, t) = γ0 (t),

(14)

where qs is given by Eq. (13), and γ0 (t) is the particle activity at the upstream boundary condition [Ancey and Heyman, 2014]. Numerous experimental works studied the variations of qs with flow and sediment characteristics but fewer data exist on the individual behaviors of particle entrainment and deposition rates—r↑ and r↓ (or equivalently r↑p and r↓p )— probably because their individual quantification is cumbersome. Lajeunesse et al. [2010] found r↓p ∗ ≈ 0.094 (or 0.052 when using the settling velocity computed by Brown and Lawler [2003]) and r↑ ∗ ≈ 3.2 · 10−3 for a transport stage τ /τcr ≈ 2. Ancey et al. [2008] obtained r↓p ∗ ≈ 0.05 and r↑ ∗ ≈ 7 · 10−4 for the same transport stage. Lajeunesse et al. [2010] reported a clear positive linear dependence on ∗ τ ∗ − τcr for bed load transport over gentle slopes. Experiments by Ancey et al. [2008], also carried out over steep slopes, showed no strong correlation between flow strength and entrainment rates. In the present study, entrainment rates and deposition rates were smaller than those found in the literature (1·10−4 and 0.015 respectively), probably because the transport stage was also smaller. Surprisingly, we did not find any correlation between the areal entrainment rates and the excess local Shields number. The computation of a modified areal entrainment rate (e.g., an areal entrainment rate computed only on the locations visited by moving particles) was found to increase linearly with the local excess Shields number, with a slope of 3.3 · 10−4 . This value is much smaller than the slope found by Lajeunesse et al. [2010] (0.43). In contrast, the particle deposition rates where found to be strongly dependent on local bed shear stress and bed

slope. Deposition increased strongly at low excess Shields number and seemed to stabilize at a value close to 0.01 for high Shields number. The overall dependence was nonlinear. The consideration of two extreme cases helped us to understand this behavior. Under vanishingly weak flow forces, a moving particle will surely deposit within a time proportional to its settling time, corresponding to r↓p ∗ ≈ 1. In contrast, when dragged by flow forces, we have shown from the acceleration diagram previously that particles may stop if up ≤ u∗ . Thus, the larger u ¯, the smaller the probability for a particle to stop, and the smaller r↓p ∗ . The saturation of r↓p ∗ → 0.01 at large Shields cannot, however, be explained by these arguments. Einstein [1950] suggested that, independently of the flow conditions, bed load particles always travel an average distance of 100d50 before depositing. Taking an average particle velocity of hup i = 0.33 m/s, this corresponds to r↓p ∗ ≈ 0.006, a value close to the limiting particle deposition rate found in our experiments. By definition, we have r↓ ∗ = r↓p ∗ γ ∗ . A linear fit in Fig. 8 suggests that r↓p ∗ = 0.016, a value very close to the average particle deposition rate computed independently (hr↓p ∗ i ≈ 0.015). More interestingly, the areal entrainment rate showed a similar dependence to γ ∗ in Fig. 8. Recently, some studies have suggested that particle entrainment could depend on particle activity via a kind of feedback loop mechanism. This behavior was observed experimentally on steep slopes with low to moderate mobility regimes and was shown to favor the motion of particles in groups rather than individually [Heyman et al., 2014] and to generate high fluctuations in bed load discharge [Ancey et al., 2008; Heyman et al., 2013]. Ancey et al. [2008] proposed that r↑ ∗ = r0 + r1 γ ∗ ,

(15)

where r0 and r1 are two positive scalars (r1 ≡ µd50 /ws , where µ [s−1 ] is referred to as a collective entrainment rate in Ancey et al. [2008]). The high particle Stokes number suggests that the energy transfer during collisions between the moving and resting particles may be important. At the onset of particle motion, this energy may be sufficient to trigger the entrainment of an impacted particle and its subsequent transport by the flow. In this case the areal entrainment rate directly depends on particle activity (r1 > 0). Fig. 8 suggests that r1 ≈ r↓p ∗ ≈ 0.016 and that r0 ≈ 0, confirming the fact that particle entrainment was primarily triggered by the passage of other moving particles. Rewriting Eq. (14) together with Eq. (15) and the estimated values ε = r↓p ∗ − r1  1 and r0 ≈ 0, we get ∂γ ∗ ∂q ∗ + s = εγ ∗ , ∂t ∂x γ ∗ (0, t) = γ0∗ (t),

(16)

If ε < 0, γ ∗ explodes at finite time so that to insure a physical solution, we must have ε > 0, at least on average. At ∗ the limit ε → 0, the equilibrium value γeq is not uniquely determined by (16). In this limiting case, and for any value of γ ∗ , the source term in the Exner equation (7) is null, so that for any γ ∗ the bed neither aggrades nor erodes. γ ∗ thus depends solely on the imposed upstream boundary condition γ0∗ (t). As described in the experimental methods, a conveyor belt continuously injected moving particles into the channel, at a prescribed and constant rate, one meter upstream of the tracking window. After a short transition period, the average sediment discharge stabilized to a constant value. Meanwhile, at the end of an experiment, we noticed that

HEYMAN ET AL.: DYNAMICS OF BED LOAD PARTICLES bed load transport froze rapidly in the channel after stopping the feeding system. This corroborates previous findings that at low mobility regimes, sediment transport intensity (e.g., the particle activity) is solely determined by the upstream boundary condition and not by the flow intensity. Interestingly, this dependence was reported experimentally by Martin et al. [2014] in similar conditions (d50 = 12 − 16mm and − tan θ = 10%). In less extreme scenarios, ε is very small but larger than 0, so that it takes a very long distance for particle activity to relax to its equilibrium value (γeq = 0 if r0 = 0). This distance—also called the saturation length—was calculated theoretically by Heyman et al. [2014] for the case of a feedback entrainment mechanism and reads q −1 2`∗c −2 (17) 1 + 4Pec − 1 `sat = d50 Pec p with Pec = hup i `∗c /(ws Du∗ ) and `∗c = Du∗ /ε. `sat grows rapidly as ε−1 suggesting that the impact of boundary conditions may spread over very long distances.

5. Conclusions In the present article, we analyzed an original dataset of measurements taken from 10 bed load transport experiments carried out in steep, supercritical, shallow-water flows. Natural gravel of uniform diameters was used to ensure a physical similarity with steep mountainous torrents while maintaining simplified experimental conditions. Using two lateral cameras and a powerful, novel tracking algorithm, more than 8 km of bed load particle trajectories were reconstructed. To the best of our knowledge, this is the first time that such a large dataset of particle trajectories has been collected. Moreover, we simultaneously recorded bed and water elevations so that various flow and bed spatio-temporal variables could be reconstructed, such as the local bed shear stress and the local bed slope. The particle trajectory data, together with the tracking algorithm codes, are available on-line to any interested researchers. We presented some statistical characteristics of particle trajectories and described their dependence upon local flow and bed variations. As bed morphologies spontaneously developed in the experimental channel, flow and bed topography were continuously changing through time and space, influencing bed load transport in their turn. Because of these fluctuations, we chose to consider the group of experiments as a combined set of various topographic and flow boundary conditions and to study their impact on the sediment transport process. Some of our results are comparable to previous experimental findings related to bed load transport under water: 1. the average streamwise particle velocity scaled with the shear velocity no et al., 1994; Lajeunesse et al., 2010], √ [Ni˜ whereas hup i / gh was proportional to the channel Froude number [Chatanantavet et al., 2013], 2. the distribution function of particle velocities is Gaussian shaped [Martin et al., 2012; Ancey and Heyman, 2014], 3. the dimensionless particle diffusivity, characterizing the average spread of particles while they advected downstream, is of order one in bed load transport [Furbish et al., 2012a]. In addition, several new phenomena were observed in those specific experimental conditions. With regards to particle kinematics, the original results may be summarized as follows: 1. The average streamwise particle acceleration is a nonmonotonic function of particle velocity: two attractors are found at up = 0 and up ≈ u ¯. In the range of velocities up = [0, u∗ ] and up ≈ [¯ u, ∞[, particles were globally decelerating. In the range of up = [u∗ , u ¯], particles were globally accelerating.

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2. The average streamwise particle acceleration is a linear function of bed slope. This dependence can be simply modeled by a Coulomb friction like force resisting to particle motion. 3. The fluctuations of the streamwise particle acceleration were wide (of the order of g, that is about 10 times the mean). They were non-linearly dependent on particle velocity. Minimum fluctuations were observed close to the two velocity attractors (up = 0 and up ≈ u ¯). 4. The particle velocity distribution could be separated into two contributions: an exponential distribution for particles moving close to the bed, and a Gaussian distribution for particles transported higher in the fluid column. 5. The average streamwise particle diffusivity was found to be a linear function of the shear velocity. Concerning mass exchanges (e.g., entrainment and particle deposition), the low transport rates chosen in the experiments revealed several original characteristics: 4. The particle deposition rate was inversely correlated with the local Shields number: the lower the Shields number, the higher the deposition rate. It was also correlated with the bed slope: the steeper the bed slope, the smaller the deposition rate. 5. The areal entrainment rate did not show a simple linear dependence on either the bed shear stress or the bed slope. In contrast, it was strongly dependent on the local particle activity, in a similar manner to the areal deposition rate. The behavior of particle entrainment close to the incipient motion threshold is particularly interesting. We showed, that for these low mobility regimes (τ /τcr ≈ 1), the entrainment of bed particles was mainly triggered by other moving particles. As a consequence, the relationship between the flow power and sediment transport rates was not unique. In fact, the sediment transport rates depended strongly on the upstream boundary conditions. In other words, various sediment transport rates may be observed for the same bed slope and shear stress. In addition to the sediment supply “shortage” effect, reducing bed load rates in gravel bed rivers [Zimmermann et al., 2010; Recking, 2012; Travaglini et al., 2015], we conclude that there also exists a sediment supply “excess” effect that can increase transport rates over long distances. We also found that the averaging scale did not significantly influence the dependence of transport rates on bed shear stress. This result is important for numerical models of bed load transport. It suggests that the closure equations for bed load transport should be valid whatever the numerical grid size chosen. In contrast, we found that the transport rates are very sensitive to the averaging scale of bed slope. Generally, the longer the distance that bed slopes are constant, the more influence they have on transport rates; rapidly changing slopes have less influence on transport rates. These results were obtained owing to massive and automatic image analysis that allowed us to gather large trajectory samples in short time. Inevitably, automatic tracking introduces errors in the trajectory data that may bias results. Particle entrainment and deposition rates are particularly prone to trajectory defects since their estimations rely on the knowledge of the entire particle trajectory. For instance, it is likely that particle entrainment and deposition rates were artificially increased by the broken trajectories that the tracking algorithm could not reconstruct. On the contrary trajectories could have been erroneously merged in the reconstruction step, decreasing thus globally the entrainment and deposition rates. It is however unlikely that

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these errors modify the qualitative dependence on flow and bed conditions since they are expected to occur randomly in space and time.

Notation h•i •∗ θ d50 d90 Sf ws τb Re Fr u∗ u h qs qs,out qs,int γ up ~xp = (xp , zp ) ~vp = (up , vp ) ~ap = (ax,p , az,p ) B Vs % g R ks f %s Rep Res St t f1

averaging , dimensionless variable, bed slope, mean grain diameter, 90 percentile grain diameter, friction slope, particle settling velocity, bed shear stress, channel Reynolds number, Froude number, flow shear velocity, depth-averaged flow velocity, water depth, sediment discharge, sediment discharge at the flume outlet, sediment discharge at the flume inlet, particle activity, particle velocity, particle position vector, particle velocity vector, particle acceleration vector, channel width, particle volume, fluid density, gravity acceleration, hydraulic radius, characteristic roughness, friction coefficient, sediment density, flow particle Reynolds number, settling particle Reynolds number, Stokes number, time, average force,

f2 ξ(t) Du κ τL ρup σup zmax vth E D r↓p r↓ r↑p r↑ α τcr τ0 ξp r0 , r 1 ε `sat `c Pec

random force, particle diffusivity, von Karman constant, Lagrangian integral time scale of particle velocity, particle velocity autocorrelation function, standard deviation of particle velocity, max elevation for a particle to deposit or erode, velocity threshold for a particle to deposit or erode, particle entrainment event, particle deposition event, particle deposition rate, areal deposition rate, particle entrainment rate, areal entrainment rate, critical angle from Fernandez Luque and van Beek [1976], critical shear stress, critical shear stress at zero slope, bed porosity, entrainment rate coefficients, difference between particle deposition rate and “collective” entrainment rate, saturation length, correlation length, P´eclet number taken at the correlation length.

Acknowledgments. This work was supported by the Swiss National Science Foundation under grant number 200021 129538 (a project called “The Stochastic Torrent: stochastic model for bed load transport on steep slope”), by R’Equip grant number 206021 133831, and by the competence center in Mobile Information and Communication Systems (grant number 5005-67322, MICS project). We give great thanks to Bob De Graffenried for his technical support. Finally, we are particularly thankful to the 3 anonymous reviewer and to the editorial board, Amy East and John M. Buffington, for the considerable efforts and benevolence they showed in their revisions. All data and Matlab scripts are available online on https://goo.gl/p4GbsR.

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