Mass diffusivity estimation for shallow waters using exact solutions for the Korteweg-De-Vries equation

Abstract

Pollutant dispersion in rivers occurs as a consequence of diffusion and advection both transversal and longitudinal dretions of the flow. The contribution related to the diffusion process, ruled by Fick´s law, is proportional to diffusion coefficient, which must be estimated. In this work a new analytical solution to the Korteweg-de-Vries equation is obtained, in order to evaluate the increase in the mass diffusivity due to the action of gravity waves along water bodies. The proposed method consists in mapping the original KdV equation into an ordinary differential one whose solution is obtained by integration. When a soliton or a wave packet is produced on the surface, a certain amount of water is transferred from the neighborhoods, carrying the pollutants by means of advection transport. However, since the oscillations are alternant along the water body, and the typical wavelength of the packets is much smaller than the distance between margins, this advection process can be regarded as an isotropic diffusion mechanism, when observed at a geographic scale. Hence, the mass diffusivity due to the gravity waves can be estimated from the local values for the laplacian and the time derivative of the concentration distribution, obtained through a mass balance in a region around the soliton. Numerical solutions are presented.