### Abstract

We propose a mathematical model which suggests that the two main geological observations about shingle beaches, i.e. the emergence of predominant pebble size ratios and strong segregation by size, are interrelated. Our model is based on a system of ordinary differential equations (ODEs) called the box equations that describe the evolution of pebble ratios. We derive these ODEs as a heuristic approximation of Bloore's partial differential equation (PDE) describing collisional abrasion and verify them by simple experiments and by direct simulation of the PDE. Although representing a radical simplification of the latter, our system admits the inclusion of additional terms related to frictional abrasion. We show that non-trivial attractors (corresponding to predominant pebble size ratios) only exist in the presence of friction. By interpreting our equations as a Markov process, we illustrate by direct simulation that these attractors may only be stabilized by the ongoing segregation process.

Original language | English |
---|---|

Pages (from-to) | 3059-3079 |

Number of pages | 21 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 468 |

Issue number | 2146 |

DOIs | |

Publication status | Published - Oct 8 2012 |

### Fingerprint

### Keywords

- Collisional abrasion
- Friction
- Pebble-shape evolution
- Segregation
- Transport

### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)
- Physics and Astronomy(all)

### Cite this

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*,

*468*(2146), 3059-3079. https://doi.org/10.1098/rspa.2011.0562

**The evolution of pebble size and shape in space and time.** / Domokos, G.; Gibbons, G. W.

Research output: Contribution to journal › Article

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, vol. 468, no. 2146, pp. 3059-3079. https://doi.org/10.1098/rspa.2011.0562

}

TY - JOUR

T1 - The evolution of pebble size and shape in space and time

AU - Domokos, G.

AU - Gibbons, G. W.

PY - 2012/10/8

Y1 - 2012/10/8

N2 - We propose a mathematical model which suggests that the two main geological observations about shingle beaches, i.e. the emergence of predominant pebble size ratios and strong segregation by size, are interrelated. Our model is based on a system of ordinary differential equations (ODEs) called the box equations that describe the evolution of pebble ratios. We derive these ODEs as a heuristic approximation of Bloore's partial differential equation (PDE) describing collisional abrasion and verify them by simple experiments and by direct simulation of the PDE. Although representing a radical simplification of the latter, our system admits the inclusion of additional terms related to frictional abrasion. We show that non-trivial attractors (corresponding to predominant pebble size ratios) only exist in the presence of friction. By interpreting our equations as a Markov process, we illustrate by direct simulation that these attractors may only be stabilized by the ongoing segregation process.

AB - We propose a mathematical model which suggests that the two main geological observations about shingle beaches, i.e. the emergence of predominant pebble size ratios and strong segregation by size, are interrelated. Our model is based on a system of ordinary differential equations (ODEs) called the box equations that describe the evolution of pebble ratios. We derive these ODEs as a heuristic approximation of Bloore's partial differential equation (PDE) describing collisional abrasion and verify them by simple experiments and by direct simulation of the PDE. Although representing a radical simplification of the latter, our system admits the inclusion of additional terms related to frictional abrasion. We show that non-trivial attractors (corresponding to predominant pebble size ratios) only exist in the presence of friction. By interpreting our equations as a Markov process, we illustrate by direct simulation that these attractors may only be stabilized by the ongoing segregation process.

KW - Collisional abrasion

KW - Friction

KW - Pebble-shape evolution

KW - Segregation

KW - Transport

UR - http://www.scopus.com/inward/record.url?scp=84866426357&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84866426357&partnerID=8YFLogxK

U2 - 10.1098/rspa.2011.0562

DO - 10.1098/rspa.2011.0562

M3 - Article

AN - SCOPUS:84866426357

VL - 468

SP - 3059

EP - 3079

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 0080-4630

IS - 2146

ER -