Rarefied particle motions on rough hillslope surfaces are controlled by the balance between gravitational heating of particles due to conversion of potential to kinetic energy and frictional cooling of the particles due to collisions with the surface. Here we elaborate on how particle energy is partitioned between kinetic, rotational, and frictional forms during downslope travel using measurements of particle travel distances on a laboratory-scale hillslope, supplemented with high-speed imaging of drop–impact–rebound experiments. The drop–impact–rebound experiments indicate that particle shape has a dominant role in energy conversion during impact with a surface. Relative to spherical and natural rounded particles, angular particles give greater variability in rebound behavior, resulting in more effective conversion of translational to rotational energy. The effects of particle shape on energy conversion are especially pronounced on a sloping sand-roughened surface. Angular particles travel shorter distances downslope than rounded particles, though travel distance data for both groups are well fit by generalized Pareto distributions. Moreover, particle–surface collisions during downslope motion lead to a transverse random-walk behavior and transverse particle diffusion. Transverse spreading increases with surface slope as there is more available energy to be partitioned into the downslope or transverse directions during collision due to increased gravitational heating. Rounded particles exhibit greater transverse diffusion than angular particles, as less energy is lost during collision with the surface. Because the experimental surface is relatively smooth, this random-walk behavior represents a top-down control on the randomization of particle trajectories due to particle shape, which is in contrast to a bottom-up control on randomization of particle trajectories associated with motions over rough surfaces. Importantly, transverse particle diffusion during downslope motion may contribute to a cross-slope particle flux and likely contributes to topographic smoothing of irregular hillslope surfaces such as scree slopes.

Recent descriptions of sediment transport on hillslopes involving long-distance particle motions have focused on nonlocal transport, where the particle flux at a hillslope position

To date, probabilistic formulations have mostly involved kinematic descriptions of particle motions and transport with limited elucidation of the associated mechanics. Nonetheless, key elements of these formulations – the particle entrainment rate and the probability distribution of travel distances – provide the basis for connecting the probabilistic formulations with the associated mechanics of particle disentrainment. By explicitly including the distribution of particle travel distances – which depend on forces acting on the particles, particle characteristics such as size and shape, and the surface over which the particles move – nonlocal formulations formally acknowledge the probabilistic nature of sediment motions. Then, by considering the behavior of a great number, or cohort, of particles, it becomes possible to describe the probabilistic physics of sediment transport without necessarily considering the details of individual particle motions

Nonlocal formulations of transport on hillslopes to this point have focused on downslope travel of particles, neglecting particle motions in two dimensions. Herein we examine transverse (cross-slope) motions in relation to downslope particle motions, extending recent work by

The purpose of this paper therefore is to describe elements of particle motions on rough hillslopes as a setup for extending current probabilistic descriptions of these motions to two dimensions. We first focus on how particles interact with the surface over which they move and the effects of these interactions on one-dimensional travel distances. We then turn to transverse components of motion. In Sect. 2 we describe the essential elements of particle energy extraction during surface collisions, leading to deposition. In Sect. 3 we summarize the probabilistic theory

As a point of reference, below we refer to both particle disentrainment and particle deposition. Disentrainment is a probabilistic (mechanical) concept. Indeed, the disentrainment rate is a probability per unit distance and in essence represents a spatial Poisson rate constant with dimension [L

The initial phase of work presented here and in several companion papers

By focusing experiments on the rarefied transport of particles down hillslopes of unchanging characteristics (e.g., slope, surface roughness), we are able to observe the stochastic nature of particle–surface collisions and associated two-dimensional travel demonstrated by this simple system. In this problem, stochasticity is directly due to the physical characteristics of both the particle and the sloping surface. In natural systems, more complex topography with varying slopes and roughness features, including sediment capacitors which disrupt motions

Consider the motion of a sediment particle dropped onto a rough surface inclined at an angle

Diagram depicting angular particles dropped onto a surface inclined at angle

Let

Returning to the simple hillslope (Fig.

Each particle in the cohort has mass

The ratio of gravitational heating to frictional cooling is defined by the dimensionless Kirkby number

Focusing specifically on the rarefied particle motions described above and visualized in Fig.

Plot of the generalized Pareto distribution of particle travel distances

The specific form of this distribution is determined by the magnitude of the Kirkby number. A small value of

The quantity

The first set of experiments are aimed at demonstrating the basis for treating the proportion of energy extraction,

Representative samples of natural rounded particles (left), natural angular particles (center), and natural small particles of mixed angularity (right) used in experiments. Particles were hand selected, and care was taken to avoid picking only the “prettiest”, most-well-shaped particles. Glass marbles, of similar size to the rounded and angular particle groups, are not shown.

Cumulative probability plots of

Drop experiment results of

The percentage of recovered energy for each angularity and surface arrangement is recorded in Table

Average particle energy partitioning following first collision with a surface as a proportion of initial energy

To visualize this, consider a sphere with no initial rotation dropped onto a smooth surface (Fig.

Illustration of collinear and non-collinear collisions for particles with center of mass depicted in black and line of collision depicted in red. Particle shape and orientation affects likelihood of collinear collision as does surface roughness.

Particle shape together with characteristics of the surface and incident conditions of motion determine the rebound motion. For a sphere with acute incident angle to the surface, the collision is non-collinear, but the collision geometry is precisely determined by the incident angle, leading to only one outcome. In contrast, for a non-spherical (angular or rounded) particle with the same incident angle, a great number of outcomes are possible depending on the particle orientation, with each outcome involving a different extraction of incident kinetic energy. Several intriguing rebound paths were observed during experimentation although no perfect collinear collisions were observed, as to be expected. Files Rounded_colinear.avi and Angular_colinear.avi provided in

With this view of how particle shape likely influences the partitioning of kinetic energy into rotational energy during particle–surface collisions, we now turn to particle travel distances. As evident from Eq. (

Picture of setup for travel distance experiments.

High-speed imaging of particles launched from the catapult show that particles travel a small distance after launch before their first collision with the surface and experience negligible rotational motion during this flight. The length of this initial flight consistently increases with slope, and particle motion only starts to become randomized by surface collisions following this initial flight, often with the onset of rotational motion. The inflection in the initial exceedance probability plots (not shown) reflects the uniformity of the launch velocities followed by a finite distance over which randomization of the motions occurs. The inflection does not necessarily directly correspond with the flight distance, and the details of the physics prior to randomization are unclear

Plots of exceedance probabilities versus downslope travel distance for experiments over six values of slope

The modified experimental travel distance data are fit with the generalized Pareto distribution and plotted using exceedance probability plots (Fig.

Fitted and estimated parameter values for travel distance experiment data shown in Fig.

The fits and parameter values presented in Fig.

Slope-parallel velocities leaving the launcher cradle for rounded, angular, and small particles. Velocities were calculated from high-speed videos of particles launched at varied slopes.

The angularity of particles launched on the roughened experimental hillslope directly affects the downslope travel distances. High-speed imaging of initial particle impacts with the surface shows a variety of impact geometries which appear to influence the motions during the subsequent travel of the particles (see

Following initial contact with the surface, downslope motions appear to transition to bouncing and rotational motions, as shown in files Angular_18%slope.avi and Angular_28%slope.avi. At higher slopes of 0.18, 0.25, and 0.28, the rounded particle distributions transition to a heavy-tailed behavior, and eventually particles do not stop on the concrete surface due to net heating (

Plot of modified exceedance probability

The small particles, which were not separated by angularity, suggest that size does not directly affect downslope travel distance. These small particles are fit with a single distribution, although a mixed distribution may be more appropriate. The small particles experience a behavior in between those of the larger angular and rounded particles, which may be more aptly described by two or more distributions. This possibility suggests that angularity influences travel distance more than size or mass. This is consistent with the idea that mass does not appear in formulation of

Particle diffusion is a key element of sediment transport

Straight-on view illustration of a Galton board with fixed pegs through which particles move illustrating random walks that result a binomial distribution.

This surface roughness is only a part of the problem of particle motions on hillslopes, where particle characteristics – in particular angularity – provide a top-down influence on randomizing motion. In our experiments, particles are not interacting with set roughness elements like equally spaced pegs but rather are experiencing stochastic motions from the outset due to collisions with the randomly roughened surface in concert with the influence of particle angularity. Natural angularity plays into this randomization of motion, as evident from the results of Sect. 4.2, as the particles do not mimic the spheres normally used in Galton boards. The decreasing likelihood of collinear collisions with increased angularity and roughness increases the degrees of freedom available to the particles. The particle is free to move in any direction, although it is still influenced by gravity. Such variation in forces applied to a particle and additional degrees of freedom thus lead to spreading behavior that is more complex than that on a Galton board.

Transverse

Scatter plots of pooled final resting positions of

By examining the deposition data for individual slopes, we can see the difference in spreading behavior for angular and rounded particles over the range of observed slopes (Fig.

Scatter plots of final resting positions of angular (dark gray) and rounded (medium gray) particles on slopes

Net transverse displacements to the final

Plots of exceedance probabilities versus the absolute value of transverse slope travel distance for experiments over six values of slope

To further quantify these interrelated spreading behaviors, we calculate the cumulative variance

As described above, transverse spreading as measured by the variance

Plots in log–log space of transverse position variance

Both rounded and angular particles spatially exhibit highly sub-diffusive behavior where the fitted lines have slopes less than 1. This indicates a change in the effectiveness of conversion of downslope translational energy into lateral motion. Under conditions of net heating, enough energy exists in the system to be partitioned into both downslope and lateral translational motions. Anomalous diffusion may be characteristic of systems such as these, where there is no additional compelling force driving motion other than gravity. On a hillslope, there is a fixed amount of potential energy that can be partitioned before the particle reaches the bottom of the slope. Particles have maximum potential energy available to be converted to kinetic energy at the crest of the hillslope and have a maximum kinetic energy at some point between this initial position and deposition location.

Conceptual illustration of lateral diffusion of a line of particles delivered to a slope with a gully-like indentation. Transverse spreading of particles as they travel downslope result in filling of an the indentation leading to surface smoothing. Inset is a rendering of eroded crater with gullies in HiRISE image PSP_007456_2140 as an example of smoothing in landscapes due to particle diffusion. Original Image credit: NASA/JPL/University of Arizona.

Energy may be lost immediately due to collision with the surface resulting in rotational motion downslope and/or laterally, translational motion downslope and/or laterally, or energy lost to some other form. As the effectiveness of gravitational heating increases with slope, the effectiveness of frictional cooling remains approximately constant. Particles that lose energy immediately are unlikely to travel far in either direction, which is apparent in the magnitude of lateral spread of angular particles on high slopes compared to rounded particles (Fig.

The laboratory experiments presented above demonstrate that particle motions are distinctly probabilistic in behavior due to inherent variability in energy extracted when moving over a surface. Particle properties, especially angularity, influence particle travel dynamics in a top-down manner. The angularity of particles directly affects the energy extraction during particle–surface collisions and in turn the distances that particles travel downslope and in the transverse direction. Even on a relatively smooth surface, angular particles lose energy to friction or rotation more readily than spheres or comparably sized rounded particles. Surface characteristics, such as relative surface roughness, also influence travel dynamics but in a bottom-up manner, as in the manner of a Galton board. Angularity, in addition to surface roughness, introduces randomization of motions and is thus an important element of the transport problem. An important, open question concerns the extent to which top-down effects of particle shape (rounded versus angular) on energy extraction and travel distances are discernible from those associated with a bottom-up control due to surface roughness.

Lateral particle diffusion is a key component of transport in many settings and a better understanding of the observable kinematics and underlying mechanics is needed to fully describe sediment transport. Transverse spreading, though neglected in hillslope literature, has been described in a few fluvial bed load transport projects, but analysis is largely limited to kinematic interpretation

Consider an idealized cliff that supplies particles as a line source to a hillslope with transverse variations in elevation reminiscent of swales or gullies formed by transverse variations in particles delivery rates or perhaps formed by channelized flow (Fig.

Particle shape plays a dominant role in how energy is partitioned during impact with a surface. Relative to spherical and natural rounded particles, angular particles give greater variability in rebound behavior.

The effects of particle shape on energy conversion are especially pronounced when considering multiple collision events during downslope travel. Angular particles travel shorter distances than rounded particles for the same surface slope although downslope travel distance data for both angularity groups are well fit by generalized Pareto distributions.

Consecutive particle–surface collisions during downslope travel lead to transverse particle diffusion, the magnitude of which depends on particle shape and surface slope. Transverse particle diffusion during downslope motion may contribute to a cross-slope particle flux and likely contributes to topographic smoothing of irregular hillslope surfaces.

This random-walk behavior represents a top-down control on the randomization of particle trajectories due to particle shape, which is in contrast to a bottom-up control on randomization of particle trajectories associated with surface topography. Surface roughness is not the only factor that influences downslope and transverse travel distances, at least in rarefied systems.

For completeness with reference to future work, here we provide the two-dimensional versions of the entrainment forms of the particle flux and the Exner equation. The latter has been described previously in relation to bed load transport

Let

Consider the position

The probability that a particle beginning motion at

The next task is to integrate over all possible starting positions

The formulation shows that in order to calculate the downslope flux through an interval

In turn, let us consider the local elevation of a surface denoted as

Data, including video and audio files, are archived and readily accessible via the Vanderbilt University Institutional
Repository (

The reported work represents an intellectual co-conspiracy between the authors. SGWW led the experiments and wrote the paper with critical review and input from DJF.

The authors declare that they have no conflict of interest.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We greatly appreciate Brandt Gibson for helping us set up the experimental slope. We thank Rachel Glade for her preliminary thoughts on lateral motion and Kristen Fauria and Shawn Chartrand for their comments during editing.

This research has been supported by the National Science Foundation (grant nos. EAR-1420831 and EAR-1735992).

This paper was edited by Eric Lajeunesse and reviewed by Irene Ippolito and one anonymous referee.