A mathematical model for the density-augmented spreading of a particle-laden discharge in a turbulent shallow-water flow

A two-dimensional horizontal dispersion equation is derived for the many ways that buoyancy modifies the spreading of a mono-disperse discharge of particles in a shallow-water flow. The particle-induced lateral density gradient gives rise to a secondary transverse flow which tends to carry particles outwards. Stratification with the consequent reduced mixing, changes the local vertical profile of the original flow and gives rise to a Burgers nonlinearity (slowing down of sinking particles or a speeding up of buoyant particles). Stratification also modifies the non-local horizontal distribution of the current, with an inflow towards particle-laden regions where the drag is reduced. A simple eddy-diffusivity turbulence model is used which permits explicit evaluation of most of the linear and nonlinear coefficients. Nonlinear effects are shown to be significant for particle volume concentrations of only 10 parts per million.