TY - JOUR

T1 - A mechanistic-stochastic formulation of bed load particle motions

T2 - From individual particle forces to the Fokker-Planck equation under low transport rates

AU - Fan, Niannian

AU - Zhong, Deyu

AU - Wu, Baosheng

AU - Foufoula-Georgiou, Efi

AU - Guala, Michele

PY - 2014/3

Y1 - 2014/3

N2 - Bed load transport is a highly complex process. The probability density function (PDF) of particle velocities results from the local particle momentum variability in response to fluid drag and interactions with the bed. Starting from the forces exerted on a single particle under low transport rates (i.e., rolling and sliding regimes), we derive here the nonlinear stochastic Langevin equation (LE) to describe the dynamics of a single particle, accounting for both the deterministic and the stochastic components of such forces. Then, the Fokker-Planck equation (FPE), which describes the evolution of the PDF of the ensemble particle velocities, is derived from the LE. We show that the theoretical PDFs of both streamwise and cross-stream velocities obtained by solving the FPE under equilibrium conditions have exponential form (PDFs of both positive and negative velocities decay exponentially), consistent with the experimental data by Roseberry et al. (). Moreover, we theoretically show how the exponential-like PDF of an ensemble of particle velocities results from the forces exerted on a single particle. We also show that the simulated particle motions using the proposed Langevin model exhibit an emergent nonlinear relationship between hop distances and travel times (power law with exponent 5/3), in agreement with the experimental data, providing a statistical description of the particles' random motion in the context of a stochastic transport process. Finally, our study emphasizes that the motion of individual particles, described by the LE, and the behavior of the ensemble, described by the FPE, are connected within a statistical mechanics framework.

AB - Bed load transport is a highly complex process. The probability density function (PDF) of particle velocities results from the local particle momentum variability in response to fluid drag and interactions with the bed. Starting from the forces exerted on a single particle under low transport rates (i.e., rolling and sliding regimes), we derive here the nonlinear stochastic Langevin equation (LE) to describe the dynamics of a single particle, accounting for both the deterministic and the stochastic components of such forces. Then, the Fokker-Planck equation (FPE), which describes the evolution of the PDF of the ensemble particle velocities, is derived from the LE. We show that the theoretical PDFs of both streamwise and cross-stream velocities obtained by solving the FPE under equilibrium conditions have exponential form (PDFs of both positive and negative velocities decay exponentially), consistent with the experimental data by Roseberry et al. (). Moreover, we theoretically show how the exponential-like PDF of an ensemble of particle velocities results from the forces exerted on a single particle. We also show that the simulated particle motions using the proposed Langevin model exhibit an emergent nonlinear relationship between hop distances and travel times (power law with exponent 5/3), in agreement with the experimental data, providing a statistical description of the particles' random motion in the context of a stochastic transport process. Finally, our study emphasizes that the motion of individual particles, described by the LE, and the behavior of the ensemble, described by the FPE, are connected within a statistical mechanics framework.

KW - Fokker-Planck equation

KW - Langevin equation

KW - bed load motion

KW - microscale and macroscale

KW - stochastic

UR - http://www.scopus.com/inward/record.url?scp=84898677978&partnerID=8YFLogxK

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U2 - 10.1002/2013JF002823

DO - 10.1002/2013JF002823

M3 - Article

AN - SCOPUS:84898677978

SN - 2169-9003

VL - 119

SP - 464

EP - 482

JO - Journal of Geophysical Research: Earth Surface

JF - Journal of Geophysical Research: Earth Surface

IS - 3

ER -