Scaling of governing equations in hydrology

A simple scaling method for hydrological governing equations has been developed. This scaling method is intended to adapt the scale of the governing equation from one grid- to another grid-scale. Prior to applying this scaling method, traditional point-scale equations must be unsealed to a grid-scale equation. The developed scaling method is applied to two different hydrological processes; (1) 1-D advection dispersion problem in a single channel and (2) flood routing problem in a large channel network.; A Previous work by Kavvas (2001) created an ensemble averaged advection dispersion equation (EAADE). Consequently, only the developed scaling technique is applied to this readily available EAADE after some modifications. The scaled-EAADE is used to compute the evolution of solute concentration with time in a 1-D open channel. In this scenario, the point mean velocity values are given at a larger scale than the desired smaller scale, and the scaled equation is used to estimate these smaller scale values of solute concentration. For comparison purposes the problem is also solved with the unsealed-EAADE when given the full velocity profile to determine the concentration values at the smaller scale. Scaled equation solution results fit quite well with actual solution results. For a flood routing problem in large channel networks a new ensemble averaged channel flow equation (EACFE) is derived. This derivation is done by ensemble averaging the Taylor expansion of the point-scale channel flow equation around the mean value of the parameters of the questioned equation. Utilizing this EACFE, ensemble averaged solution for the same ordered channels (according to Strahler-Horton ordering) are obtained. Instead of solving each channel flow one by one, the ensemble averaged solution is computed for the ensemble of channels that are of same order within the channel network. Using this methodology, the computational load and parameter data requirement is reduced substantially. In order to further reduce the computational load and data requirement, scaling and ensemble averaging methods are coupled. Besides the obtained economy in computations and data requirement, this methodology also provided excellent results when compared to numerical solutions of the flows within the actual, detailed channel network.