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

Abstract

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 Correspondence to: R. Mantilla (ricardo-mantilla@uiowa.edu) 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.

The Abundance, Size, and Spacing of Lakes and Reservoirs Connected to River Networks

“…By coupling a new understanding of river network topology with geospatial data sets and network modeling, we can examine how different types of river networks with lakes/reservoirs transport sediment, propagate geomorphic adjustment (Benda et al, ; Czuba & Foufoula‐Georgiou, ; Czuba et al, ; Gran & Czuba, ), process carbon and nutrients (Bertuzzo et al, ; Helton et al, ; Wollheim et al, ), and disperse species (Fuller et al, ). Previous descriptions of river networks and their scaling laws have been instrumental for understanding of geomorphic patterns (Dietrich et al, ; Tarboton et al, ), timing of discharge (Kirkby, ; Mantilla et al, ), and dispersal and production of aquatic insects (Sabo & Hagen, ). Similarly, the scaling patterns of network topology with lakes/reservoirs and the distributions of lake/reservoir size and spacing provide simple rules for generating theoretical river networks with varying numbers, sizes, and spacings of lakes/reservoirs across stream orders.…”

Can floods in large river basins be predicted from floods observed in small subbasins?

“…This hypothesis is validated through a host of empirical‐based studies (Merz and Bloschl, ; Ogden and Dawdy, ; Furey and Gupta, ; Gupta et al, ; Ayalew et al, ) and numerical rainfall‐runoff simulations in synthetic and natural river basins (e.g. Gupta et al, , ; Gupta and Waymire, ; Menabde and Sivapalan, ; Menabde et al, ; Mantilla et al, , ; Furey and Gupta, ; Mandapaka et al, ; Ayalew et al ., , ).…”

On the propagation of diel signals in river networks using analytic solutions of flow equations

“…(2), accordingly. For our calculation, we assume the function K(q i ) to be constant, K(q i ) = v i / l, where v i is the velocity of link i and l is the length of the link, which is assumed to be uniform over all links in the network (Mantilla et al, 2011). For simplicity, K(q i ) will be called k i .…”