### Abstract

The ideas of advection and diffusion of sediment particles are probabilistic constructs that emerge when the Master equation, a precise, probabilistic description of particle conservation, is approximated as a Fokker–Planck equation. The diffusive term approximates nonlocal transport. It ‘looks’ upstream and downstream for variations in particle activity and velocities, whose effects modify the advective term. High-resolution measurements of bedload particle motions indicate that the mean squared displacement of tracer particles, when treated as a virtual plume, primarily reflects a nonlinear increase in the variance in hop distances with increasing travel time, manifest as apparent anomalous diffusion. In contrast, an ensemble calculation of the mean squared displacement involving paired coordinate positions independent of starting time indicates a transition from correlated random walks to normal (Fickian) diffusion. This normal behavior also is reflected in the particle velocity autocorrelation function. Spatial variations in particle entrainment produce a flux from sites of high entrainment toward sites of low entrainment. In the case of rain splash transport, this leads to topographic roughening, where differential rain splash beneath the canopy of a desert shrub contributes to the growth of a soil mound beneath the shrub. With uniform entrainment, rain splash transport, often described as a diffusive process, actually represents an advective particle flux that is proportional to the land surface slope. Particle diffusion during both bedload and rain splash transport involves motions that mostly are patchy, intermittent and rarefied. The probabilistic framework of the Master equation reveals that continuous formulations of the flux and its divergence (the Exner equation) represent statistically expected behavior, analogous to Reynolds-averaged conditions. Key topics meriting clarification include the mechanical basis of particle diffusion, effects of rarefied conditions involving patchy, intermittent motions, and effects of rest times on diffusion of tracer particles and particle-borne substances.

Original language | English (US) |
---|---|

Pages (from-to) | 214-237 |

Number of pages | 24 |

Journal | Earth Surface Processes and Landforms |

Volume | 42 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2017 |

### Fingerprint

### Keywords

- Exner equation
- master equation
- particle diffusion
- sediment transport

### ASJC Scopus subject areas

- Geography, Planning and Development
- Earth-Surface Processes
- Earth and Planetary Sciences (miscellaneous)

### Cite this

*Earth Surface Processes and Landforms*,

*42*(1), 214-237. https://doi.org/10.1002/esp.4084

**The elements and richness of particle diffusion during sediment transport at small timescales.** / Furbish, David Jon; Fathel, Siobhan L.; Schmeeckle, Mark; Jerolmack, Douglas J.; Schumer, Rina.

Research output: Contribution to journal › Article

*Earth Surface Processes and Landforms*, vol. 42, no. 1, pp. 214-237. https://doi.org/10.1002/esp.4084

}

TY - JOUR

T1 - The elements and richness of particle diffusion during sediment transport at small timescales

AU - Furbish, David Jon

AU - Fathel, Siobhan L.

AU - Schmeeckle, Mark

AU - Jerolmack, Douglas J.

AU - Schumer, Rina

PY - 2017/1/1

Y1 - 2017/1/1

N2 - The ideas of advection and diffusion of sediment particles are probabilistic constructs that emerge when the Master equation, a precise, probabilistic description of particle conservation, is approximated as a Fokker–Planck equation. The diffusive term approximates nonlocal transport. It ‘looks’ upstream and downstream for variations in particle activity and velocities, whose effects modify the advective term. High-resolution measurements of bedload particle motions indicate that the mean squared displacement of tracer particles, when treated as a virtual plume, primarily reflects a nonlinear increase in the variance in hop distances with increasing travel time, manifest as apparent anomalous diffusion. In contrast, an ensemble calculation of the mean squared displacement involving paired coordinate positions independent of starting time indicates a transition from correlated random walks to normal (Fickian) diffusion. This normal behavior also is reflected in the particle velocity autocorrelation function. Spatial variations in particle entrainment produce a flux from sites of high entrainment toward sites of low entrainment. In the case of rain splash transport, this leads to topographic roughening, where differential rain splash beneath the canopy of a desert shrub contributes to the growth of a soil mound beneath the shrub. With uniform entrainment, rain splash transport, often described as a diffusive process, actually represents an advective particle flux that is proportional to the land surface slope. Particle diffusion during both bedload and rain splash transport involves motions that mostly are patchy, intermittent and rarefied. The probabilistic framework of the Master equation reveals that continuous formulations of the flux and its divergence (the Exner equation) represent statistically expected behavior, analogous to Reynolds-averaged conditions. Key topics meriting clarification include the mechanical basis of particle diffusion, effects of rarefied conditions involving patchy, intermittent motions, and effects of rest times on diffusion of tracer particles and particle-borne substances.

AB - The ideas of advection and diffusion of sediment particles are probabilistic constructs that emerge when the Master equation, a precise, probabilistic description of particle conservation, is approximated as a Fokker–Planck equation. The diffusive term approximates nonlocal transport. It ‘looks’ upstream and downstream for variations in particle activity and velocities, whose effects modify the advective term. High-resolution measurements of bedload particle motions indicate that the mean squared displacement of tracer particles, when treated as a virtual plume, primarily reflects a nonlinear increase in the variance in hop distances with increasing travel time, manifest as apparent anomalous diffusion. In contrast, an ensemble calculation of the mean squared displacement involving paired coordinate positions independent of starting time indicates a transition from correlated random walks to normal (Fickian) diffusion. This normal behavior also is reflected in the particle velocity autocorrelation function. Spatial variations in particle entrainment produce a flux from sites of high entrainment toward sites of low entrainment. In the case of rain splash transport, this leads to topographic roughening, where differential rain splash beneath the canopy of a desert shrub contributes to the growth of a soil mound beneath the shrub. With uniform entrainment, rain splash transport, often described as a diffusive process, actually represents an advective particle flux that is proportional to the land surface slope. Particle diffusion during both bedload and rain splash transport involves motions that mostly are patchy, intermittent and rarefied. The probabilistic framework of the Master equation reveals that continuous formulations of the flux and its divergence (the Exner equation) represent statistically expected behavior, analogous to Reynolds-averaged conditions. Key topics meriting clarification include the mechanical basis of particle diffusion, effects of rarefied conditions involving patchy, intermittent motions, and effects of rest times on diffusion of tracer particles and particle-borne substances.

KW - Exner equation

KW - master equation

KW - particle diffusion

KW - sediment transport

UR - http://www.scopus.com/inward/record.url?scp=85007247424&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85007247424&partnerID=8YFLogxK

U2 - 10.1002/esp.4084

DO - 10.1002/esp.4084

M3 - Article

AN - SCOPUS:85007247424

VL - 42

SP - 214

EP - 237

JO - Earth Surface Processes and Landforms

JF - Earth Surface Processes and Landforms

SN - 0197-9337

IS - 1

ER -