Abstract

In November 1960, Sayre and Hubbell (1965) monitored the propagation of a plume of radioactive sand in a Nebraskan stream. Using a scintillator detector, they observed that the plume gradually spread as it was entrained downstream. Tracking pebbles in gravel-bed rivers, or glass beads in a laboratory channel, reveals a similar behavior: sediments disperse as they propagate downstream. To investigate this process, we derive the equations governing the evolution of a plume of tracers from the erosion-deposition model introduced by Charru et al. (2004). Neglecting velocity fluctuations, we find that the exchange of particles between the bedload layer and the sediment bed disperses the tracers. Furthermore, we find a transition between two asymptotic regimes. The tracers, initially at rest, are progressively set into motion by the flow. During this regime, the plume continuously accelerates, spreads non-linearly with time and becomes increasingly skewed in the direction of propagation. With time, the skewness of the plume eventually reaches a maximum before decreasing. This marks the transition to an advection-diffusion regime in which the plume advances at constant velocity, spreads linearly, and becomes symmetrical. We express analytically the position, the variance and the skewness of the plume, and investigate their asymptotic regimes. In the field, bedload transport is intermittent. We show that these asymptotic regimes become insensitive to this intermittency when expressed in terms of the position of the plume, instead of time.