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Cited by 163Hydrology and Sediment Transport ProcessesGeological formations and processesGroundwater flow and contamination studies

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How River Rocks Round: Resolving the Shape-Size Paradox

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Domokos, Gabor, Jerolmack, Douglas J., Sipos, Andras Á., Török, Ákos·2014·PLoS ONE
DOI 10.1371/journal.pone.0088657PMC PMC3922984
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Abstract

River-bed sediments display two universal downstream trends: fining, in which particle size decreases; and rounding, where pebble shapes evolve toward ellipsoids. Rounding is known to result from transport-induced abrasion; however many researchers argue that the contribution of abrasion to downstream fining is negligible. This presents a paradox: downstream shape change indicates substantial abrasion, while size change apparently rules it out. Here we use laboratory experiments and numerical modeling to show quantitatively that pebble abrasion is a curvature-driven flow problem. As a consequence, abrasion occurs in two well-separated phases: first, pebble edges rapidly round without any change in axis dimensions until the shape becomes entirely convex; and second, axis dimensions are then slowly reduced while the particle remains convex. Explicit study of pebble shape evolution helps resolve the shape-size paradox by reconciling discrepancies between laboratory and field studies, and enhances our ability to decipher the transport history of a river rock.

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Figure 1 Definition sketch. (a) 2D schematic of the physical situation studied, showing an abrading cuboid colliding with a flat plane. Zone of positive curvature on the colliding corner is highlighted with arrows indicating surface-normal abrasion. (b) Three scenarios of the chopping model: Vertex chopping (Event A) corresponding to Gaussian-curvature-driven abrasion, edge chopping (Event B) corresponding to Mean-Curvature-driven abrasion, and face chopping (Event C) corresponding to uniform (Eikonal) abrasion.
Figure 1 Definition sketch. (a) 2D schematic of the physical situation studied, showing an abrading cuboid colliding with a flat plane. Zone of positive curvature on the colliding corner is highlighted with arrows indicating surface-normal abrasion. (b) Three scenarios of the chopping model: Vertex chopping (Event A) corresponding to Gaussian-curvature-driven abrasion, edge chopping (Event B) corresponding to Mean-Curvature-driven abrasion, and face chopping (Event C) corresponding to uniform (Eikonal) abrasion.CC BY · PubMed Central
Figure 2 Two-phase abrasion illustration. The 2D schematic shows two well-separated phases emerging spontaneously from Gaussian-curvature-driven abrasion: In Phase I edges abrade but axis ratios remain constant; in Phase II, axis ratios evolve towards the sphere. Accompanying perspective images are topographic laser scans that illustrate the two phases in 3D; they were performed for a separate experiment with a smaller cuboid having similar axis ratios.
Figure 2 Two-phase abrasion illustration. The 2D schematic shows two well-separated phases emerging spontaneously from Gaussian-curvature-driven abrasion: In Phase I edges abrade but axis ratios remain constant; in Phase II, axis ratios evolve towards the sphere. Accompanying perspective images are topographic laser scans that illustrate the two phases in 3D; they were performed for a separate experiment with a smaller cuboid having similar axis ratios.CC BY · PubMed Central
Figure 3 Experimental images of abrasion. Three rows correspond to three orthogonal views of the specimen, and columns show time evolution in terms of number of drum rotations. Separation of Phases I and II can be observed by visual inspection.
Figure 3 Experimental images of abrasion. Three rows correspond to three orthogonal views of the specimen, and columns show time evolution in terms of number of drum rotations. Separation of Phases I and II can be observed by visual inspection.CC BY · PubMed Central
Figure 4 Comparison of experimental and numerical results . (a–e) Evolution of shape parameters versus volume, V. Shown are: axis ratios (a) y1 and (b) y2, (c) convexity index β, (d) Wadell sphericity, r, and (e) superellipsoid exponent, n. (f) Evolution of V versus the rotation number (rot), a proxy for time. Gray line: experimental data. Black solid line: level-set method approximation of the PDE (Eq. 2). Dashed line: chopping model approximation (Eq. 3). Best fit coefficients correspond to pure Gaussian flow. Note abrupt change for all shape parameters (a–e) at transition from Phase I to Phase II, shown with vertical dashed line. Pebble volume exhibits no abrupt change through time (f); the fitted exponential trend is identical in Phase I and Phase II. Data used to generate this figure are contained in File S1.
Figure 4 Comparison of experimental and numerical results . (a–e) Evolution of shape parameters versus volume, V. Shown are: axis ratios (a) y1 and (b) y2, (c) convexity index β, (d) Wadell sphericity, r, and (e) superellipsoid exponent, n. (f) Evolution of V versus the rotation number (rot), a proxy for time. Gray line: experimental data. Black solid line: level-set method approximation of the PDE (Eq. 2). Dashed line: chopping model approximation (Eq. 3). Best fit coefficients correspond to pure Gaussian flow. Note abrupt change for all shape parameters (a–e) at transition from Phase I to Phase II, shown with vertical dashed line. Pebble volume exhibits no abrupt change through time (f); the fitted exponential trend is identical in Phase I and Phase II. Data used to generate this figure are contained in File S1.CC BY · PubMed Central
Figure 5 Shape evolution of a tetrahedron under the purely Gaussian chopping model in Eq. 3 with p = 1. (a) Initial facets are shown in red; the transition to Phase II abrasion occurs when these facets have been entirely removed. (b) As with the cuboid, surface convexity β increases during Phase I and stabilizes at β≈1 in Phase II.
Figure 5 Shape evolution of a tetrahedron under the purely Gaussian chopping model in Eq. 3 with p = 1. (a) Initial facets are shown in red; the transition to Phase II abrasion occurs when these facets have been entirely removed. (b) As with the cuboid, surface convexity β increases during Phase I and stabilizes at β≈1 in Phase II.CC BY · PubMed Central

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References 47 · Cited by 163

Research works printed but not linked: 43

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Metadata

Authors
Domokos, G., Jerolmack, D. J., Sipos, A. Á., & Török, Á.
Affiliations
  • Budapest University of Technology and Economics (HU)
  • University of Pennsylvania (US)
Corresponding author
Jerolmack, Douglas J.
ORCID
Venue
PLoS ONE
Publisher
Public Library of Science (PLoS)
Published
2014-02-12
Volume
9
Issue
2
Pages
e88657
ISSN
1932-6203 (ISSN-L)
DOI
10.1371/journal.pone.0088657
PMC
PMC3922984
OpenAlex
W1997823684
Cited by
163
References
47
Citations by year
2026: 7 · 2025: 8 · 2024: 15 · 2023: 12 · 2022: 5 · 2021: 14 · 2020: 34 · 2019: 10 · 2018: 33 · 2017: 5 · 2016: 7 · 2015: 4 · 2014: 8
FWCI
6.95
Type
journal article
Language
en
Open access
Licence
Locations
  • PLoS ONE (publishedVersion, cc-by)
  • Europe PMC (PubMed Central) (publishedVersion, cc-by)
  • PubMed Central (publishedVersion, cc-by)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • ScholarlyCommons (University of Pennsylvania) (submittedVersion, other-oa)
  • ScholarlyCommons (University of Pennsylvania) (submittedVersion)
  • RePEc: Research Papers in Economics (submittedVersion)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
  • arXiv (Cornell University)
Funders
  • National Science Foundation (1224943)
  • Hungarian Scientific Research Fund (T104601)
Topics
  • Physical Sciences › Earth and Planetary Sciences › Earth-Surface Processes › Hydrology and Sediment Transport Processes
  • Physical Sciences › Earth and Planetary Sciences › Earth-Surface Processes › Geological formations and processes
  • Physical Sciences › Environmental Science › Environmental Engineering › Groundwater flow and contamination studies
Keywords
riverbed sediments, downstream fining, curvature-driven flow, fluvial sedimentology
MeSH terms
Algorithms, Geology, Minerals, Models, Theoretical, Probability, Normal Distribution, Geologic Sediments, Imaging, Three-Dimensional, Rivers

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Domokos, G., Jerolmack, D. J., Sipos, A. Á., & Török, Á. (2014). How River Rocks Round: Resolving the Shape-Size Paradox. PLoS ONE. https://doi.org/10.1371/journal.pone.0088657