A probabilistic description of the bed load sediment flux: 1. Theory

• Published in: Journal of Geophysical Research: Earth Surface Vol. 117; no. F3
• Main Authors: Furbish, David Jon, Haff, Peter K., Roseberry, John C., Schmeeckle, Mark W.
• Format: Journal Article
• Language: English
• Published: Washington, DC Blackwell Publishing Ltd 01.09.2012; American Geophysical Union
• Subjects: advection; Atoms & subatomic particles; Bed load; bed load sediment; diffusion; Earth sciences; Earth, ocean, space; Exact sciences and technology; Fluctuations; Fokker-Planck equation; Sediment load; Streambeds; advection; Master equation; Travel time; Entrainment (atmosphere, ocean); Position; joints; Particle motion; simulation; velocity; Particle flux; conservation; displacements; Fokker Planck equation; diffusivity; Diffusion; particles; theory; probability density function
• ISSN: 0148-0227; 2169-9003; 2156-2202; 2169-9011
• DOI: 10.1029/2012JF002352

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. Key Points The bed load flux involves advective and diffusive parts Gradients in particle activity induce a diffusive flux Particle positions and velocities define a statistical ensemble