Analytical model of two-dimensional dispersion in laterally nonuniform axial velocity distributions

An analytical model of the advection-dispersion phenomenon in rivers with a laterally nonuniform axial velocity distribution is presented. The dispersion phenomenon is assumed to be governed by the two-dimensional advection-diffusion equation with constant but anisotropic turbulent diffusion coefficients. An infinitely long river with a prismatic cross section bounded laterally by parallel nondispersive banks is assumed. The velocity distribution is allowed to vary in an arbitrary functional form, but was restricted in this work to a family of power-law velocity distributions. The method of moments is used to derive the important statistical parameters of the concentration distribution. The concentration moment equations are solved analytically using the method of Greens function coupled with the method of images. The moments are then used to construct an approximate model of two-dimensional dispersion for an arbitrary velocity function, an initial distribution, and source injection scenarios. A one-dimensional simplification of the two-dimensional dispersion model with a time-dependent dispersion coefficient is also outlined. The dependence of the concentration moments on the velocity distribution and the shape of the source distribution is analyzed, and numerical results for a plane source and vertical line sources at the centerline and at the side are compared. The effect of the asymmetry of the velocity profile on the mixing length and time and on the skewness of the concentration distribution is shown to be significant, which might partly explain the persistence of the skewness observed in the field. However, the effect of the source injection scenario was not significant at large times. The analytical results can be used to model the fate and movement of pollutants and to better assess the effect of discharge siting on the dispersion of a contaminant. The model can be also used as a simple practical tool in simulating transport in a nearly prismatic river system.