In the process leading to the proof, some fascinating properties of these equations were identified, which allow us to make useful predictions about shape evolution of particles under chipping. Synthesizing existing mathematical theory 28 , 36 , 41 , 42 see Methods for details , we can make several predictions about how circularity increases with mass loss due to chipping. These predictions are summarized in Fig. Considering the restricted case of curvature-driven erosion Eq.
We empirically test this idea next, using data from a wide range of environments where chipping by bed-load saltation is expected to occur. Although quantitative measurements of particle shape evolution by collisions in laboratory experiments have been reported for decades, there are few data sets from natural systems 1 , 6 , 7.
Further, direct comparison of field data requires that the same shape parameters, measured at comparable resolution, be available for all of them. In the field, however, attrition mass is rarely known.
We measured particle shape using identical image techniques see Methods in three different systems for which the source location and initial shape of particles could be determined Fig. River data were reported by Litwin et al. The desert dune data set comes from samples collected along an 8-km transect in White Sands National Monument that was reported by Jerolmack et al. The resulting circularity values for all systems are plotted as a function of transport distance, with the distance axes linearly scaled for each such that the data lie on top of each other Fig.
It must be seen as merely a coincidence that the distance axis for the river and desert data is the same scale. For the river and laboratory data, this is likely because the initial particles are fragments having a similar shape The White Sands initial particles are crystals formed by precipitation, not fragments; it is likely coincidence that the initial circularity of these grains is comparable to those of the other systems.
This result indicates that transport distance is linearly related to mass loss, as has been suggested in earlier studies 5 , The former is connected to the latter by particle attrition rate; however, direct conversion between them is difficult because attrition rate depends on the velocity and frequency of bed-load collisions and also material properties The agreement in shape evolution among these systems is encouraging, given that they involve particles of very different sizes and material strengths, transported in different fluids and collided in different ways.
The addition here of aeolian sand data to the previous river and drum results 20 provides empirical support that rounding by chipping due to bed load is general. Field data were collected from different environments: A gravel-bed river in Puerto Rico, B gypsum dune field in New Mexico, C pebble beach in Marina di Pisa, and D experiment in a rotating drum. All illustrated environments select for the conditions of bed-load transport right side where impacts from saltation drive chipping.
Field, experimental, and model data showing the evolution of circularity with A transport distance from source and B mass loss. Data points represent the median value of R. Note in A that axes have been linearly rescaled so that curves fall on each other; the curves of circularity naturally collapse when replotted against mass loss B.
We now rescale the data to test for generality in the curve of circularity as a function of attrition mass. The latter was measured for the drum experiment and inferred from a model fit to the river data [see the study of Litwin et al. We also point out that rotating drum experiments of volcanic clasts reported by Manga et al. The results therefore consist of average shape and mass for each of the four sets of pebbles recovered at two points in time.
Qualitatively, we see that initially angular pebbles experienced more rounding than initially rounded pebbles Fig. Data indicate that the change in R measured for each recovered pebble set follows the expected curve quite well. Red and blue show initially angular and rounded pebbles, respectively. The legend in B applies to both panels.
C Examples of an initially angular top and initially rounded bottom pebble before left and after right the experiment. We would like to test whether the observed patterns for particle shape match predictions from a purely geometric chipping model. Solving the partial differential Eq. Here, we implement a stochastic, discrete event—based chipping model that has been shown to reproduce the shape evolution behavior of Eq.
Moreover, the discrete model reflects more directly the way that impact chipping actually occurs. Simulation parameters are chosen to represent binary collisions among identical particles, the self-dual case 17 , as an idealized model for bed-load transport of like-sized grains. The initial particle shape for each simulation was determined from a 3D topographic scan of an actual gypsum fragment see Methods.
In total, 21 simulation runs were carried out using a different gypsum fragment each time, and results averaged together to produce the shape evolution curves presented. Vertex and edge removal events are associated with the Gaussian and mean curvature—dependent terms in Eq. For the self-dual case, the probabilities can be determined uniquely. Initial fragments experience 67, collisions over the course of a run.
Although model results for R plot slightly below the data, this may be the result of either i a smaller initial value of R for the fragments used in the simulations or ii the presence of some fragmentation in the data see below.
It is important to emphasize that no parameters in the model were tuned to fit the data. If one needed to relax the assumption of self-duality, then parameters could be tuned to account for this. It appears that most—if not all—of the common rounding pattern of bed-load particles can be explained by chipping of initial fragments due to collisions with same-sized particles.
The pure chipping model reproduces the major features of the evolution of circularity in the data; however, there are aspects of shape evolution that are not captured by this simplest model.
We examine this in detail for the rotating drum experiment, because this is the system for which the most information is known. However, model values are systematically offset below the experimental data fig. The mismatch is more severe for axis ratio for which model and experimental trends are qualitatively different.
The rapid increase in axis ratio for the experiment hints that fragmentation may be playing a role. That pattern indicates that moderate fragmentation of the limestone particles occurred at the start of the experiment, but this effect became negligible with continuing mass loss. Trends in axis ratio and equilibrium points for the river data are similar to experiments, although, unsurprisingly, are much more erratic; fragmentation may also play a role in the initial attrition of river pebbles in the upper reaches of the profile.
The initial increase is evidence of fragmentation occurring in the beginning of the experiment. To test whether fragmentation can explain deviations between model and experiment, we added a simple fragmentation component to the model that is motivated by recent simulations of Domokos et al. The model is predicated on the notion that fragmentation happens preferentially to particles that are i elongated low axis ratios and ii at the start of their evolution. The model randomly chooses particles and times for fragmentation, and each fragmented particle is split in two with both products retained for further attrition.
This is implemented probabilistically using an axis ratio—dependent parameterization of the Weibull distribution—a widely used model for predicting the time of failure 44 —to determine the time of fragmentation for each particle if it happens at all. The numerical chipping model with moderate fragmentation is able to reproduce empirical trends for both axis ratio and equilibrium points Fig. Shape data examined are A roundness, B axis ratio, C number of stable points, and D number of unstable points; the latter two were not available for White Sands data.
Data points represent average values. The data also suggest that circularity in different systems may not asymptotically approach a value of 1; rather, rounding in some systems appears to saturate at sufficiently large mass loss ratios Fig. Previous studies have indicated that frictional abrasion becomes increasingly important in the lower-gradient portions of rivers where collision energy decreases 5 , It has also been suggested from field observations that wave-worked pebbles initially evolve toward spherical shape but that later, in their trajectory, they become flat 3.
Our results show that particle shape evolves along a common curve when chipping is the dominant mode of attrition and that fragmentation and friction cause deviations from this curve. Here, we perform a scale analysis to first delineate the upper bound for chipping, that is, fragmentation, due to bed-load transport in water. Bed load is a kind of dense granular flow, where it has been shown that the relevant velocity scale for collisions is the terminal fall velocity w s Defining a single critical impact energy for the transition from chipping to fragmentation is an oversimplification, because even repetitive low-energy collisions may eventually lead to fragmentation by fatigue failure Nevertheless, a useful upper bound is the one associated with the onset of intense cracking for which a single collision may produce fragmentation.
This critical energy E c , is a property of not only the material but also particle size 48 , 49 and shape for example, long and thin particles fragment more easily 25 , The control of size is reasonably well described from data, whereas shape is less explored. The range of natural pebble shapes is limited, however, such that this influence may be secondary; one study found that the difference in critical impact energy between rounded river quartz pebbles and crushed fragments was negligible For simplicity, we neglect shape controls here.
Above this size, particles of a given material will always fragment under transport, whereas below this size, we expect that chipping and rounding will occur. Large, immobile boulders in streams will resist fragmentation because they are not transported. Limestone particles had comparable but reduced impact energies—around half the value of quartz—over the same range and so should have moderately smaller values for D c.
These results are consistent with laboratory experiments 10 , 20 , 26 and observations from rivers 5 , Impact energy for aeolian transport of sand grains is well below the critical value so fragmentation is negligible, although weak dust aggregates have been proposed to experience breakup by fragmentation Bed-load transport causes frequent collisions among particles due to saltation.
The results above allow us to demonstrate that these collisions are in the range where chipping should dominate attrition, in water and air. It has also been observed that fragmentation of river rocks becomes significant in steep, mountain headwaters 10 , This slope corresponds to the observed transition in sediment transport mode from bed load to debris flows 53 in rivers. Results suggest that highly energetic debris flows produce fractured rocks, whereas the transition to bed load is associated with a transition to chipping and rounding of these rock fragments.
We propose that the lower boundary of chipping, and the transition to frictional abrasion, may be cast in terms of the minimum energy required for a particle to transition from rolling to saltation, E s. Bed-load particles underwater move not only by saltation but also by rolling and sliding, which should contribute to frictional abrasion rather than chipping. Rolling and sliding are negligible for aeolian bed-load transport, where entrainment by impact collision is always dominant We posit that the large difference in attrition rates between frictional abrasion and chipping leads to a dominance of the latter, unless saltation is negligible.
For the range of fluid stresses associated with bed-load transport in rivers, some saltation is likely always present; pure rolling and sliding only occur in the immediate vicinity of the threshold of motion For particles with kinetic energy smaller than this value, they should move by rolling and sliding but not saltation.
The presence of disk-shaped pebbles on the landward margins of beaches 3 suggests that these may be environments in which saltation is minimal, and where we hypothesize that transport occurs very close to the threshold of motion. We expect a narrow range in phase space, associated with transport right at the threshold of motion, where the shape evolution of underwater pebbles will be dominated by frictional abrasion—a condition found on some beaches and in some portions of rivers.
There are two relevant characteristic energy scales in the phase diagram: i The critical energy associated with fragmentation E c is the upper bound for chipping and is a property of the pebble material and size and potentially shape , and ii the energy associated with the onset of saltation E s is the boundary between chipping and frictional abrasion and is a property of pebble size or mass. However, there is a critical particle size, D c , related to the critical energy E c through settling velocity , associated with grains large enough that they will always fragment if they are subject to collision.
Although qualitative and somewhat speculative at this point, the phase diagram serves to place the chipping and rounding results developed above in the context of the full range of attrition processes, as well as to guide future research toward assessing the boundaries among these processes. The zone corresponding to chipping and self-dual collisions is highlighted in gray; this is the region associated with bed-load transport.
This region of phase space associated with bed load, where we expect geometric rounding, is highlighted in Fig. In principle, the shape evolution of particles undergoing attrition is sensitive to the ratio of impactor to target particle size, the initial shape, and the energy of collisions. However, nature selects for a subset of conditions that lead to a universal rounding behavior. Pebbles in rivers and coasts, sand grains in a desert dune field, and limestone rocks in a drum all exhibit a remarkably consistent trend in the evolution of circularity when cast as function of attrition mass.
What these different systems have in common is that particle attrition occurs because of low-energy collisions. The fragmentation by physical and chemical weathering of parent rocks 25 produces fragments having a universal size-shape relationship, which serve as the initial material for bed-load transport.
Size-selective sorting of particles by bed load ensures that collisions are typically among like-sized particles. Finally, the energy of collisions associated with the bed-load transport in air and water is small enough relative to rock strength that fragmentation is typically rare, but large enough to overcome viscous damping and to dominate over frictional abrasion, leading to chipping.
These findings may also extend to attrition in pyroclastic flows, where rounding consistent with chipping has been observed The grain-size distribution and lithology of the river-bed material were examined at nine sites.
There are two principal results. Andesite boulders or large cobbles make up the framework sizes in the upstream part of the study reach, while chert pebbles make up the framework sizes in the downstream part. There are few andesite pebbles or chert boulders in the river bed. Because the mobility of gravel depends mainly on its size, hydraulic sorting by lithology does not occur within the same size class.
These results indicate clearly that particle abrasion occurs in the Watarase River and is responsible, at least in part, for the downstream decrease in particle size of bed material. Shibboleth Sign In. OpenAthens Sign In. Institutional Sign In. Sign In or Create an Account. User Tools.
Sign In. Advanced Search. Skip Nav Destination Article Navigation. Close mobile search navigation Article navigation. Volume 64, Number 1a. Previous Article Next Article. Article Navigation. Total Cards Subject Geography. Level Not Applicable. Create your own flash cards! Sign up here. Supporting users have an ad free experience! Flashcard Library Browse Search Browse. Create Account. Details Title Bradshaw Model Description An explanation of changes in channel characteristics with distance downstream.
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