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

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.