Abrasion of sedimentary particles in fluvial and eolian environments is widely associated with collisions encountered by the particle. Although the physics of abrasion is complex, purely geometric models recover the course of mass and shape evolution of individual particles in low- and middle-energy environments (in the absence of fragmentation) remarkably well. In this paper, we introduce the first model for the collision-driven collective mass evolution of sedimentary particles. The model utilizes results of the individual, geometric abrasion theory as a

Probably the most fundamental observation on pebbles is that they appear to be segregated both by size and shape, and it is broadly accepted that the dynamics are driven by two physical processes: transport and abrasion. Which of these processes dominates may depend on the geological location and also on timescales; however, geologists appear to agree that, in general, neither process should be ignored.

In coastal environments, one of the most remarkable accounts of pebble size and shape distribution is provided by

In fluvial environments, while downstream fining of sediment has been often attributed to transport

As we can see, both in coastal and fluvial environments it is a generally accepted fact that the two processes (transport and attrition) appear to compete in shaping the evolution of pebble shape and pebble mass distributions. How exactly this competition may play out and in what manner attrition may contribute to this process is the subject of our paper.

We also remark that while all available observations indicate that attrition could be a relevant factor in the evolution of shape and mass distributions, so far, in the absence of any predictive theory, no datasets have been collected which would allow verifying any theoretical predictions. We will point out potential strategies for verification in Sect.

Since the seminal papers by

The first stepping stone between the theory of individual abrasion and collective abrasion is the model for

size evolution should follow Sternberg's law,

mass loss in a collision should be a monotonically increasing function of collision energy, and

the model should be fully compatible with the geometric evolution model.

Schemes for

The unified theory in

We also note that in the case of two identical particles (e.g., two particles with identical masses

Independently of individual (and binary) abrasion theory there exists broad interest in collective shape and size evolution models tracking mutually colliding populations of

The collision kernel can be derived from the binary equations (the physical model of the

Once the kernel has been established, we make the assumption that for large

The above-outlined structure is characteristic of coagulation fragmentation models

we only consider collisions where the relative mass loss is small (i.e., the particles lose only fragments with small relative mass), and

the small fragments generated in the collisions are not considered further in the evolution.

By implementing these two assumptions into the statistical model based on the
collision kernel (Eqs.

To describe our construction we will need to address both the size evolution of individual particles (under the collision kernel) as well as the evolution of size distributions. While particle size appears in both settings, we need to distinguish carefully: in individual and binary models particle size evolves in time; in collective models size distribution evolves in time. As a consequence, in the individual setting the variable denoting size may be differentiated with respect to time; in the collective setting this is not the case. We will use

Schematic description of the evolution of mass distribution of a pebble population: in a dispersing process the relative size variation

The collision kernel (Eqs.

For

For

As collisional abrasion may occur within a broad range of energies, these two basic scenarios of the model (illustrated in Fig.

In general, the evolution equations generated by Eqs. (

We approximate the kernel (Eqs.

We regard the full kernel; however, we only investigate density functions obtained as a small perturbation of the Dirac delta (i.e., populations of almost identical particles). This is done in Appendices

We numerically compute both the discrete and the continuum models. For details see Sect.

We will briefly refer to the first two approximations as the continuum model. In the case of the third approximation we do direct, discrete simulations of finite particle populations; we use the full kernel and we call this the discrete model. One startling feature of the latter (as compared with the former) is the appearance of outliers, i.e., particles substantially larger than the vast majority (illustrated in Fig.

As outlined above, our model is defined on two levels: the collision kernel (Eqs.

One may test the model at the input level, by fitting the kernel (Eqs.

One may test the model at the output level by measuring the time evolution of full mass distributions and fitting the respective material and environmental parameters

The above simple procedures apply only for homogeneous populations. We lay out the procedures for the testing of the model for heterogeneous populations in Appendix

The first simplification described in Sect.

The second simplification in Sect.

Using these considerations, for our problem the master equation is found to be

We aim to understand some scenarios characteristic of pebble populations by investigating the Cauchy-type initial value problem associated with Eq. (

Detailed physical modeling of the collisional event can make the interaction kernel highly complex; for a recent review on kernels see

We first consider two simple kernels which satisfy the mathematical requirement of leading to analytically soluble Fokker–Planck equations. However, as we will show, these very analytical results highlight that these kernels are physically not admissible. Next, we investigate the parameter-dependent compound kernel suggested in

First, we consider the

In the very same manner let us investigate the

According to Appendix

In order to be in accordance with Sternberg's law and to have a control on the evolution of the relative variance, following the lead of

Evolution of an initial (

In natural events, both velocity and collision probability (cross section) may depend on particle size: in laminar flows relative velocity and collision probability is proportional to linear size, while in a turbulent flows velocity could be inversely proportional to linear size and collision probability could be proportional to projected area. In the collision kernel (Eq.

To have a global view, it may be of interest to estimate the parameter

In order to examine the validity of these assumptions we made discrete element simulations using the event-driven method

Here we interpret the intuitive picture of fluvial abrasion in the context of our statistical model. In our model a fluvial environment may be represented by a fluvial population, consisting of

To make a meaningful characterization of geologically relevant scenarios, we will regard two extreme cases which represent brackets on geological processes. In both cases we assume that the mass evolution is driven by binary collisions and we regard the limit as

In the first extreme scenario we assume that particles are chosen uniformly from the full fluvial population: i.e., the riverbed has no special role. In this case almost all collisions will happen among a pair of small particles (

In the second extreme scenario we assume that the small particles exclusively collide with the riverbed (large particle); i.e., we only have (

Intuitively it is clear that any geologically relevant process is in between the above two extreme cases, and, although we do not deliver a rigorous proof, it appears plausible that in a geologically relevant setting

Here we perform computations to illustrate the main results presented in Sect.

The evolution of a pebble population under the compound kernel was simulated both in the frame of discrete and the continuum model, i.e., by direct event-based simulation and by discretizing the partial differential equation. (see Sect.

Evolution of a lognormal PDF in the compound kernel under the continuous (c) and discrete (d) models at the parameter values

Although the lognormal distribution is certainly

Parameters

The continuum model describes the

Simulation of a finite sample with

In order to estimate the robustness of this scenario we use a simple approximation by assuming that all but one particle have volume

Critical curve

In this paper we presented the first statistical model for the collective mass evolution of pebble populations under collisional abrasion. While our model is certainly not unique, it is compatible with

existing geological observations,

existing geometrical theory of individual and binary abrasion of pebbles,

existing theory for individual mass evolution of pebbles (Sternberg's law), and

exiting statistical theory of coagulation and fragmentation.

In the spirit of standard statistical theory for collective evolution, our model is based on two components: (i) the binary collision kernel and, based on that, (ii) the governing equation for the evolution of probability density functions for mass distribution. Regarding (i) we used the model from

Our collision kernel includes the single scalar parameter

We investigated our model on two levels: (i) as a continuum model by regarding the evolution of the Fokker–Planck equation and (ii) as a discrete model by running discrete event-based simulations. In the case of the continuum model we derived our results analytically and also from numerical simulation of the Fokker–Planck equation, while in the discrete model we relied on numerical computations. With regard to the existence of the critical parameter

Among many small pebbles, large boulders are often visible in mountain ranges or rivers. While this phenomenon is commonly attributed to transport, our model suggests that under some conditions, here again transport and abrasion may act in unison: we identified a curios phenomenon

While our paper only dealt with size distributions, there exist also related observations on shape: sharp peaks in distributions of axis ratios (also referred to as equilibrium shapes) are mentioned in

In the context of the binary evolution model (Eqs.

In the statistical setting the binary abrasion parameters can be organized into an

Based on the above considerations, the statistical model is controlled by the

One may test the model at the

One may test the model at the

Next we show an example for testing the model at the input level by using the data obtained in

Abrasion rate

Differential equations governing the time evolution of the first and second moments can be readily obtained; hence the mean

In the case of the product kernel the IVPs describing the evolution of the mean

The truncated compound kernel is obtained from the compound kernel as the truncated Taylor polynomial computed at

Here we show that a population of identical particles, characterized by a Dirac-delta function as input PDF is preserved in the model with the compound kernel regardless of the value of parameter

This shows that the variance of the distribution is constant, and it started at

As the model lacks diffusion, the behavior of a degenerate distribution with all the mass concentrated at a single value is worth studying because long-term existence of a set consisting of identical particles can take place in the model. In Appendix

In the paper we assumed that the particle collision probability depends on the volume of the particles as

The first is the smooth gradient flow. In such a case the driving fluid has a strong, but on a particle size scale constant, velocity gradient in one of the spatial directions. Such situations may arise, e.g., in shallow water layers. Here the relative velocity of the particles grows with the distance. If we are at distance

The other extreme case is a fully chaotic motion where equipartition takes place

Let us have a sample with

The archived code, referred to in this paper, is available at the following public repository:

GD proposed the problem and supervised the research. AAS carried out the analytical and numerical study of the continuous model. TJ developed the discrete numerical model. GD, AAS, and JT wrote the paper.

The authors declare that they have no conflict of interest.

The authors are indebted to David J. Furbish, Sebastien Carretier, and Duccio Bertoni for reviews and feedback. Responding to their comments helped to improve the paper substantially. The authors sincerely thank Jérôme Lavé and Mikaël Attal for sharing the original dataset of their abrasion experiments. The research reported in this paper was supported by the BME Water Sciences & Disaster Prevention TKP2020 Institution Excellence Subprogram, grant no. TKP2020 BME-IKA-VIZ and the NKFIH grant K134199.

This research has been supported by the Nemzeti Kutatási Fejlesztési és Innovációs Hivatal (grant no. K134199) and the Institution Excellence Subprogram (grant no. TKP2020 BME-IKA-VIZ).

This paper was edited by Simon Mudd and reviewed by Duccio Bertoni and Sebastien Carretier.