Exner-based master equation for transport and dispersion of river pebble tracers: Derivation, asymptotic forms, and quantification of nonlocal vertical dispersion

A. Pelosi, G. Parker, R. Schumer, H. B. Ma

Research output: Contribution to journalArticle

Abstract

Ideas deriving from the standard formulation for continuous time random walk (CTRW) based on the Montroll-Weiss Master Equation (ME) have been recently applied to transport and diffusion of river tracer pebbles. CTRW, accompanied by appropriate probability density functions (PDFs) for walker step length and waiting time, yields asymptotically the standard advection-diffusion equation (ADE) for thin-tailed PDFs and the fractional advection-diffusion equation (fADE) for heavy-tailed PDFs, the latter allowing the possibilities of subdiffusion or superdiffusion. Here we show that the CTRW ME is inappropriate for river pebbles moving as bed load: a deposited particle raises local bed elevation, and an entrained particle lowers it so that particles interact with the "lattice" of the sediment-water interface. We use the Parker-Paola-Leclair framework, which is a probabilistic formulation of the Exner equation of sediment conservation, to develop a new ME for tracer transport and dispersion for alluvial morphodynamics. The formulation is based on the existence of a mean bed elevation averaged over fluctuations. The new ME yields asymptotic forms for ADE and fADE that differ significantly from CTRW. It allows vertical as well as streamwise advection-diffusion. Vertical dispersion is nonlocal but cannot be expressed with fractional derivatives. In order to illustrate the new model, we apply it to the restricted case of vertical dispersion only, with both thin and heavy tails for relevant PDFs. Vertical dispersion shows a subdiffusive behavior.

Original languageEnglish (US)
Pages (from-to)1818-1832
Number of pages15
JournalJournal of Geophysical Research F: Earth Surface
Volume119
Issue number9
DOIs
StatePublished - Sep 17 2014

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advection-diffusion equation
pebble
rivers
Advection
tracers
tracer techniques
probability density function
derivation
Rivers
tracer
Probability density function
advection
river
probability density functions
random walk
beds
Sediments
morphodynamics
formulations
sediment-water interface

ASJC Scopus subject areas

  • Geophysics
  • Forestry
  • Oceanography
  • Aquatic Science
  • Ecology
  • Water Science and Technology
  • Soil Science
  • Geochemistry and Petrology
  • Earth-Surface Processes
  • Atmospheric Science
  • Earth and Planetary Sciences (miscellaneous)
  • Space and Planetary Science
  • Palaeontology

Cite this

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abstract = "Ideas deriving from the standard formulation for continuous time random walk (CTRW) based on the Montroll-Weiss Master Equation (ME) have been recently applied to transport and diffusion of river tracer pebbles. CTRW, accompanied by appropriate probability density functions (PDFs) for walker step length and waiting time, yields asymptotically the standard advection-diffusion equation (ADE) for thin-tailed PDFs and the fractional advection-diffusion equation (fADE) for heavy-tailed PDFs, the latter allowing the possibilities of subdiffusion or superdiffusion. Here we show that the CTRW ME is inappropriate for river pebbles moving as bed load: a deposited particle raises local bed elevation, and an entrained particle lowers it so that particles interact with the {"}lattice{"} of the sediment-water interface. We use the Parker-Paola-Leclair framework, which is a probabilistic formulation of the Exner equation of sediment conservation, to develop a new ME for tracer transport and dispersion for alluvial morphodynamics. The formulation is based on the existence of a mean bed elevation averaged over fluctuations. The new ME yields asymptotic forms for ADE and fADE that differ significantly from CTRW. It allows vertical as well as streamwise advection-diffusion. Vertical dispersion is nonlocal but cannot be expressed with fractional derivatives. In order to illustrate the new model, we apply it to the restricted case of vertical dispersion only, with both thin and heavy tails for relevant PDFs. Vertical dispersion shows a subdiffusive behavior.",
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