A conservative flow routing formulation: Déjà vu and the variable-parameter Muskingum method revisited

- We provide the mathematical and numerical proof of the mass conservation and the theoretical proof of the momentum conservation of the Muskingum-Cunge-Todini-Method.
- We show why the original Muskingum-Cunge and non-linear diffusion-wave routing is not mass/momentum conservative.
- We demonstrate the effectiveness of the method in reproducing correct stage-discharge relations.
- We apply the Muskingum-Cunge-Todini method to a real-world watershed.
- We perform multi-year hydrologic/hydraulic simulations.

SummaryA wide range of approaches are used for flow routing in hydrological models. One of the most attractive solutions is the variable-parameter Muskingum (VPM) method. Its major advantage consists in the fact that (i) it can be applied to poorly-gauged basins with unknown channel geometries, (ii) it requires short execution time and (iii) it adequately captures, also in the presence of mild slopes, the most salient features of a dynamic wave such as the looped rating curve and the steepening of the rising limb of the hydrograph. In addition, the method offers the possibility to derive average water levels for a reach segment, a quantity which is essential in flood forecasting and flood risk assessment. For reasons of computational economy the method is also appropriate for applications, in which hydrological and global circulation models (GCM) are coupled, and where computational effort becomes an issue. The VPM approach is presented from a philosophical and conceptual perspective, by showing the derivation of its mass and momentum balance properties from the point to the finite scale, and by demonstrating its strengths by means of an application in an operational context. The principal novel contributions of the article relate to (a) the extension of the Muskingum-Cunge-Todini approach to accept uniformly distributed lateral inflow, (b) the use of power law cross sections and (c) the validation of the method through a long-term simulation of a real-world case, including the comparison of results to those obtained using a full Saint Venant equations model.