### Abstract

We provide a probabilistic definition of the bed load sediment flux. In treating particle positions and motions as stochastic quantities, a flux form of the Master equation (a general expression of conservation) reveals that the volumetric flux involves an advective part equal to the product of an average particle velocity and the particle activity (the solid volume of particles in motion per unit streambed area), and a diffusive part involving the gradient of the product of the particle activity and a diffusivity that arises from the second moment of the probability density function of particle displacements. Gradients in the activity, instantaneous or time-averaged, therefore effect a particle flux. Time-averaged descriptions of the flux involve averaged products of the particle activity, the particle velocity and the diffusivity; the significance of these products depends on the scale of averaging. The flux form of the Exner equation looks like a Fokker-Planck equation (an advection-diffusion form of the Master equation). The entrainment form of the Exner equation similarly involves advective and diffusive terms, but because it is based on the joint probability density function of particle hop distances and associated travel times, this form involves a time derivative term that represents a lag effect associated with the exchange of particles between the static and active states. The formulation is consistent with experimental measurements and simulations of particle motions reported in companion papers.

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

Article number | F03031 |

Journal | Journal of Geophysical Research: Earth Surface |

Volume | 117 |

Issue number | 3 |

DOIs | |

State | Published - 2012 |

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### ASJC Scopus subject areas

- Earth-Surface Processes
- Geophysics

### Cite this

*Journal of Geophysical Research: Earth Surface*,

*117*(3), [F03031]. https://doi.org/10.1029/2012JF002352

**A probabilistic description of the bed load sediment flux : 1. Theory.** / Furbish, David Jon; Haff, Peter K.; Roseberry, John C.; Schmeeckle, Mark.

Research output: Contribution to journal › Article

*Journal of Geophysical Research: Earth Surface*, vol. 117, no. 3, F03031. https://doi.org/10.1029/2012JF002352

}

TY - JOUR

T1 - A probabilistic description of the bed load sediment flux

T2 - 1. Theory

AU - Furbish, David Jon

AU - Haff, Peter K.

AU - Roseberry, John C.

AU - Schmeeckle, Mark

PY - 2012

Y1 - 2012

N2 - We provide a probabilistic definition of the bed load sediment flux. In treating particle positions and motions as stochastic quantities, a flux form of the Master equation (a general expression of conservation) reveals that the volumetric flux involves an advective part equal to the product of an average particle velocity and the particle activity (the solid volume of particles in motion per unit streambed area), and a diffusive part involving the gradient of the product of the particle activity and a diffusivity that arises from the second moment of the probability density function of particle displacements. Gradients in the activity, instantaneous or time-averaged, therefore effect a particle flux. Time-averaged descriptions of the flux involve averaged products of the particle activity, the particle velocity and the diffusivity; the significance of these products depends on the scale of averaging. The flux form of the Exner equation looks like a Fokker-Planck equation (an advection-diffusion form of the Master equation). The entrainment form of the Exner equation similarly involves advective and diffusive terms, but because it is based on the joint probability density function of particle hop distances and associated travel times, this form involves a time derivative term that represents a lag effect associated with the exchange of particles between the static and active states. The formulation is consistent with experimental measurements and simulations of particle motions reported in companion papers.

AB - We provide a probabilistic definition of the bed load sediment flux. In treating particle positions and motions as stochastic quantities, a flux form of the Master equation (a general expression of conservation) reveals that the volumetric flux involves an advective part equal to the product of an average particle velocity and the particle activity (the solid volume of particles in motion per unit streambed area), and a diffusive part involving the gradient of the product of the particle activity and a diffusivity that arises from the second moment of the probability density function of particle displacements. Gradients in the activity, instantaneous or time-averaged, therefore effect a particle flux. Time-averaged descriptions of the flux involve averaged products of the particle activity, the particle velocity and the diffusivity; the significance of these products depends on the scale of averaging. The flux form of the Exner equation looks like a Fokker-Planck equation (an advection-diffusion form of the Master equation). The entrainment form of the Exner equation similarly involves advective and diffusive terms, but because it is based on the joint probability density function of particle hop distances and associated travel times, this form involves a time derivative term that represents a lag effect associated with the exchange of particles between the static and active states. The formulation is consistent with experimental measurements and simulations of particle motions reported in companion papers.

UR - http://www.scopus.com/inward/record.url?scp=84868322270&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84868322270&partnerID=8YFLogxK

U2 - 10.1029/2012JF002352

DO - 10.1029/2012JF002352

M3 - Article

VL - 117

JO - Journal of Geophysical Research: Atmospheres

JF - Journal of Geophysical Research: Atmospheres

SN - 2169-897X

IS - 3

M1 - F03031

ER -