A new mixing equation for bed material composition in bed form dominant conditions

Vertical mixing beneath the bed surface and its effect on sediment composition has long been underestimated. This paper proposes a mixing equation to illustrate the temporal and spatial variations of bed material composition for bed form dominant rivers. A continuous mass conservation equation for elevation-specific nonuniform bed material composition is initially presented. A vertical mixing equation is finally derived as a diffusion-type partial differential equation with a source term characterized by a vertical diffusion coefficient and the geometric dimensions of bed forms. Coupled with boundary conditions, this equation is further developed in a vertical mixing model at the scale of bed forms, in which the variations of bed material composition are clearly illustrated. The equation accounts for vertical exchange fluxes driven by bed form migration and their effect on bed material composition, which helps elucidate the mixing mechanism. In application to riverbed degradation, the proposed equation is capable to describe armoring development in consideration of the interior mixing, which provides more satisfying predictions for bed surface materials. The new mixing equation helps gain insight into predicting the variations of bed material composition during morphological changes.