Scaling of peak flows with constant flow velocity in random self-similar networks R. Mantilla^{1}, V. K. Gupta^{2}, and B. M. Troutman^{3}^{1}IIHR-Hydroscience & Engineering, The University of Iowa, Iowa City, IA, 52242, USA

^{2}Department of Civil, Environmental and Architectural Engineering, Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, CO,~80309, USA

^{3}US Geological Survey, Denver Federal Center, Lakewood, CO, 80225, USA

Received: 18 January 2010 – Revised: 15 October 2010 – Accepted: 17 June 2011 – Published: 22 July 2011

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 *p*_{i}
and *p*_{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 φ 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, φ > β, which supports a
similar finding first reported for simulations in the river network of the
Walnut Gulch basin, Arizona. Theoretical estimation of β and φ 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 β^{(E)} and φ^{(E)} 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 β^{(E)} and φ^{(E)} and the values estimated
by the simulated ensembles of RSNs and hydrographs. The independence of the
scaling exponents φ^{(E)} and φ 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.

**Citation:** Mantilla, R., Gupta, V. K., and Troutman, B. M.: Scaling of peak flows with constant flow velocity in random self-similar networks, Nonlin. Processes Geophys., 18, 489-502, doi:10.5194/npg-18-489-2011, 2011.