Theoretical Investigation into Longitudinal Dispersion in Natural Rivers — A Scaling Dispersion Model

Sammanfattning: Using a broad variety of scientific methods this dissertation systematically investigates the fundamental mechanisms underlying the dispersion processes in natural rivers and their mathematical descriptions in a geometrical framework. Emphasis is on the geometrical variations and irregularities of a natural river and their scales. Analysis starts with the integral scale and the Fickian dispersion in a straight channel by simply considering the influence of channel width to depth ratio on the dispersion and then a longitudinal dispersion coefficient equation containing the transverse mixing coefficient is derived. This equation is extended for meandering rivers by incorporating channel sinuosity. By taking account of the influence of smaller and smaller scale irregularities on the dispersion, a new dispersion mechanism, scaling dispersion, is presented to elucidate the physics underlying the non-Fickian longitudinal dispersion processes of pollutants in natural rivers and thereby a scaling dispersion model is developed. The model is comprised of (1) a fractional advection-dispersion-reaction equation (FRADRE), (2) methods for estimation of the parameters involved in the FRADRE, and (3) a numerical scheme devised to evaluate the solution of the FRADRE. FRADRE is characterized by a fractional differential operator parameter F, called as fractor, reflecting the heterogeneity of natural media and acting on the dispersion term. The simplest form of the FRADRE is an integer-order diffusion equation and its solution is a semi-infinite Fourier probability distribution, leading to a new approach for estimation of the initial mixing length. FRADRE entails four parameters which can be estimated using a moment-based method developed by means of the Laplace Transform of fractional derivatives if field observations of dispersion processes are available. Results of parameter estimation show that the fractor F varies in the range of 1.4 – 2.0 with 1.65 possessing the maximum occurring frequency. In case of no detailed dye concentration test data, the longitudinal dispersion coefficient can be determined by the two methods derived from a channel shape equation for straight streams and from a more versatile channel shape equation for meandering rivers, respectively. A numerical scheme, designated as F.3 Central Finite-Difference Scheme, is proposed for solution of the fractional partial differential equation FRADRE. The F.3 scheme recovers the conventional central finite-difference scheme when fractor F = 2. FRADRE can be numerically solved by applying the F.3 scheme for the fractional dispersion equation and the semi-Lagrangian method in conjunction with the natural cubic-spline interpolation for the pure advection equation. Dispersion characteristics computed by the scaling dispersion model were in excellent agreement with those observed by 20 dye tests conducted on the U.S. rivers, Mississippi, Missouri, Red, and Monocacy, thereby demonstrating the efficacy of the scaling dispersion model for prediction of the non-Fickian dispersion processes in natural rivers.