Scaling of peak flows with constant flow velocity in random self-similar networks
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 dist...
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Copernicus Publications
2011-07-01
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Series: | Nonlinear Processes in Geophysics |
Online Access: | http://www.nonlin-processes-geophys.net/18/489/2011/npg-18-489-2011.pdf |
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doaj-97de65a664d145d8842a833328ce6d2e2020-11-24T21:39:06ZengCopernicus PublicationsNonlinear Processes in Geophysics1023-58091607-79462011-07-0118448950210.5194/npg-18-489-2011Scaling of peak flows with constant flow velocity in random self-similar networksR. MantillaV. K. GuptaB. M. TroutmanA 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><sub>i</sub> and <i>p</i><sub>e</sub>, 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;<sup><i>(E)</i></sup> and &phi;<sup><i>(E)</i></sup> 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;<sup><i>(E)</i></sup> and &phi;<sup><i>(E)</i></sup> and the values estimated by the simulated ensembles of RSNs and hydrographs. The independence of the scaling exponents &phi;<sup><i>(E)</i></sup> 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.http://www.nonlin-processes-geophys.net/18/489/2011/npg-18-489-2011.pdf |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
R. Mantilla V. K. Gupta B. M. Troutman |
spellingShingle |
R. Mantilla V. K. Gupta B. M. Troutman Scaling of peak flows with constant flow velocity in random self-similar networks Nonlinear Processes in Geophysics |
author_facet |
R. Mantilla V. K. Gupta B. M. Troutman |
author_sort |
R. Mantilla |
title |
Scaling of peak flows with constant flow velocity in random self-similar networks |
title_short |
Scaling of peak flows with constant flow velocity in random self-similar networks |
title_full |
Scaling of peak flows with constant flow velocity in random self-similar networks |
title_fullStr |
Scaling of peak flows with constant flow velocity in random self-similar networks |
title_full_unstemmed |
Scaling of peak flows with constant flow velocity in random self-similar networks |
title_sort |
scaling of peak flows with constant flow velocity in random self-similar networks |
publisher |
Copernicus Publications |
series |
Nonlinear Processes in Geophysics |
issn |
1023-5809 1607-7946 |
publishDate |
2011-07-01 |
description |
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><sub>i</sub> and <i>p</i><sub>e</sub>, 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;<sup><i>(E)</i></sup> and &phi;<sup><i>(E)</i></sup> 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;<sup><i>(E)</i></sup> and &phi;<sup><i>(E)</i></sup> and the values estimated by the simulated ensembles of RSNs and hydrographs. The independence of the scaling exponents &phi;<sup><i>(E)</i></sup> 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. |
url |
http://www.nonlin-processes-geophys.net/18/489/2011/npg-18-489-2011.pdf |
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