Dispersion resulting from aggregating hydrodynamic properties in water quality modeling

Abstract

Long-term water quality modeling often requires the use of different modeling modules, including hydrodynamics and water quality, to simulate processes that are dominant in different spatial and temporal scales. Averaging procedures, often used to aggregate fine-grid hydrodynamic model results as input to water quality modeling, produce a cross-correlation term, which serves as a function similar to dispersion that cannot be defined a priori. The present study shows that under the assumption of smoothly varying flow field, the sub-grid transport processes (the cross-correlation term) can be expressed analytically as a source/sink term which takes the form of a Lagrangian time-convolution integral, not in a Fickian form as suggested in many previous studies. This Lagrangian time-convolution integral resembles the dispersion term obtained from an analytical study by G.I. Taylor (Proceedings of the London Mathematical Society A 20 (1921) 196–211) in which the memory of velocity field is used to define the dispersion coefficient. Information needed in the dispersion integral include memories of fine-grid velocity field and mean concentrations. While Fickian dispersion accounts for the dispersion processes only at the present time, the Lagrangian time-convolution correctly accounts for the memory of detailed dispersion effects introduced by the averaged transport processes. The theoretical results for the Lagrangian dispersion equation are tested in two analytical experiments. Results show that the mean concentration profiles obtained from direct fine-grid 2-D simulations are recoverable from the mean transport equation for the coarse grid only when the Lagrangian time-convolution term is included.