Numerical modeling of traces in gravel-bed rivers

Abstract

The erosion, transport and deposition of pebbles in rivers have often been studied by considering the motion of tracer particles. There are reports of bedload tracing programs in field and laboratory since the late 1930s. The theoretical basis for the study of the dispersal of sediment tracer particles was delineated for the first time in 1950 by Einstein, who formulated the problem in terms of a standard 1D random walk in which each particle moves in a series of steps punctuated by waiting times. Subsequent to Einstein’s original work on tracers, the study of random walks has been extended to the case of continuous time random walks (CTRW). CTRW, accompanied by appropriate probability distribution functions (PDFs) for walker step length and waiting time, yields asymptotically the standard advectiondiffusion 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 of particles, which is often referred as non-local behavior or anomalous diffusion. In latest years, considerable emphasis has been placed on non-locality associated with heavy-tailed PDFs for particle step length. This appears to be in part motivated by the desire to construct fractional advective-diffusive equations for pebble tracer dispersion corresponding to the now-classical fADE model. Regardless of the thin tail of the PDF, the degree of non-locality nevertheless increases with increasing mean step length. In the thesis, we firstly consider the general case of 1D morphodynamics of an erodible bed subject to bedload transport analysing the effects of non-locality mediated by both heavyand thin-tailed PDFs for particle step length on transient aggradational- degradational bed profiles. Then, we focus on tracers. (i) We show that the CTRW Master Equation is inappropriate for river pebbles moving as bed material load and (ii) by using the Parker-Paola-Leclair (PPL) framework for the Exner equation of sediment conservation, which captures the vertical structure of bed elevation variation as particles erode and deposit, we develop a new ME for tracer transport and dispersion for alluvial morphodynamics. The new ME is derived from the Exner equation of sediment continuity and it yields asymptotic forms for ADE and fADE that differ significantly from CTRW. It allows a) vertical dispersion, as well as streamwise advection-diffusion, and b) mean waiting time to vary in the vertical. We also show that vertical dispersion is nonlocal (subdiffuive), but cannot be expressed with fractional derivatives. Vertical dispersion is the likely reason for the slowdown of streamwise advection of tracer pebbles observed in the field, which is the key result of our modeling when co-evolution of vertical and streamwise dispersion are considered. [edited by author]