Master Equation for Streamwise and Vertical Advection and Dispersion of Tracer Pebbles with no Active Layer: Age-based Waiting Time Formulation

Abstract:

Much of the recent work on advection-dispersion of sediment in rivers has been motivated by, and based on the Master Equation of the Continuous Time Random Walk model. The model, which considers a particle moving through a lattice, has provided the basis for the study of asymptotic forms that include one kind of nonlocality associated with heavy-tailed PDF for particle step length and waiting time. In the case of pebbles moving as bed material load (and thus interacting with the bed), however, the lattice itself varies according to the entrainment and deposition of pebbles, so that CTRW does not formally apply. Here we outline the development of an Exner-based Master Equation for the dispersal of bedload particles. The Master Equation has no active layer, and describes both streamwise and vertical advection-dispersion. Of particular importance is the treatment of waiting time. The formulation is based on the evolution of age structure in population dynamics. As a particle is deposited and buried at some elevation, its age is set to zero; it continues to age in place until the bed erodes down to that level and re-entrains the particle.