Scaling of peak flows with constant flow velocity in random self-similar networks

Abstract

A methodology is presented to understand the role of the statistical self-similar topology of real river networks on scaling, or power law, in peak flows for rainfall-runoff events. We created Monte Carlo generated sets of ensembles of 1000 random self-similar networks (RSNs) with geometrically distributed interior and exterior generators having parameters <i>p</i>_i and <i>p</i>_e, respectively. The parameter values were chosen to replicate the observed topology of real river networks. We calculated flow hydrographs in each of these networks by numerically solving the link-based mass and momentum conservation equation under the assumption of constant flow velocity. From these simulated RSNs and hydrographs, the scaling exponents β and &phi; characterizing power laws with respect to drainage area, and corresponding to the width functions and flow hydrographs respectively, were estimated. We found that, in general, &phi; > &beta;, which supports a similar finding first reported for simulations in the river network of the Walnut Gulch basin, Arizona. Theoretical estimation of β and &phi; in RSNs is a complex open problem. Therefore, using results for a simpler problem associated with the expected width function and expected hydrograph for an ensemble of RSNs, we give heuristic arguments for theoretical derivations of the scaling exponents &beta;^<i>(E)</i> and &phi;^<i>(E)</i> that depend on the Horton ratios for stream lengths and areas. These ratios in turn have a known dependence on the parameters of the geometric distributions of RSN generators. Good agreement was found between the analytically conjectured values of &beta;^<i>(E)</i> and &phi;^<i>(E)</i> and the values estimated by the simulated ensembles of RSNs and hydrographs. The independence of the scaling exponents &phi;^<i>(E)</i> and &phi; with respect to the value of flow velocity and runoff intensity implies an interesting connection between unit hydrograph theory and flow dynamics. Our results provide a reference framework to study scaling exponents under more complex scenarios of flow dynamics and runoff generation processes using ensembles of RSNs.