Tests of peak ¯ow scaling in simulated self-similar river networks Merab Menabde a , Seth Veitzer b,c , Vijay Gupta b,d , Murugesu Sivapalan a, * a Department of Environmental Engineering, Centre for Water Research, University of Western Australia, Crawley, WA 6009, Australia b Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, CO, USA c US Geological Survey, Denver Federal Center, Lakewood, CO, USA d Department of Civil and Environmental Engineering, University of Colorado, Boulder, CO, USA Received 20 July 2000; received in revised form 21 February 2001; accepted 30 March 2001 Abstract The eect of linear ¯ow routing incorporating attenuation and network topology on peak ¯ow scaling exponent is investigated for an instantaneously applied uniform runo on simulated deterministic and random self-similar channel networks. The ¯ow routing is modelled by a linear mass conservation equation for a discrete set of channel links connected in parallel and series, and having the same topology as the channel network. A quasi-analytical solution for the unit hydrograph is obtained in terms of recursionrelations.Theanalysisofthissolutionshowsthatthepeak¯owhasanasymptoticallyscalingdependenceonthedrainage area for deterministic Mandelbrot±Vicsek MV) and Peano networks, as well as for a subclass of random self-similar channel networks.However,thescalingexponentisshowntobedierentfromthatpredictedbythescalingpropertiesofthemaximaofthe width functions. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Self-similar networks; Peak ¯ow; Scaling 1. Introduction Empirically observed scaling or power-law relations between regional ¯ood quantiles and drainage areas [3,20] are the result of complex interactions among nu- merous physical processes governing runo generation and transport. These include the generation of surface and subsurface runo involving precipitation, in®ltra- tion,andevapotranspiration,andthetransportofwater across hills and through a river network. A physical understandingofregionalstatisticalscalingrelationships inpeak¯ows,andpredictionsfromungaugedbasins,isa long-standing open problem. Recent progress towards solving this problem is based on applications and de- velopment of new theories and models that involve no- tions of self-similarity and scale invariance [6]. Building onthisrecentwork,weinvestigatehowthetransportof waterobeyingmassconservation,andattenuationdueto storage in channel networks, combine with their self- similar topologic or branching structure in determining thespatialscalingstructureofpeak¯ows. Our focus here is on routing rather than on the spatial and temporal variability of precipitation and runo generation; see Gupta et al. [5], Menabde and Sivapalan [13], and Troutman and Over [24] for an in- vestigation concerning the eect of precipitation on scaling exponents in idealised deterministic self-similar networks. For the present purpose, we simplify the runo input into a network to be spatially uniform and instantaneous, and the ¯ow is routed by a link-based massconservationequation[6].Atime-varyingsolution of this equation requires that a physical relationship be knownbetweenstorageanddischargeforeachlinkina network. To better understand the analytical structure of this complex equation, we ignore the empirically observed downstream hydraulic-geometric variations in velocity [7,9], which makes this relationship linear. Therefore, one can formally view this formulation as a network of linear reservoirs in the sense of Nash [14], which are topologically connected in series and parallel according to the topology of a channel network. Theproposedapproachleadstoresultsdierentfrom those based on the analysis of scaling properties of the width function maxima assuming a constant velocity approximation [5,27]. The latter assumes that every drop of water in the channel network travels to the Advances in Water Resources 24 2001) 991±999 www.elsevier.com/locate/advwatres * Corresponding author. Tel.: +61-9-380-2320; fax: +61-9-380-1015. E-mail address: sivapalan@cwr.uwa.edu.au M. Sivapalan). 0309-1708/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII:S0309-170801)00043-4