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The central hypothesis of a nonlinear geophysical flood theory postulates that, given space-time rainfall intensity for a rainfall-runoff event, solutions of coupled mass and momentum conservation differential equations governing runoff generation and transport in a self-similar river network produce spatial scaling, or a power law, relation between peak discharge and drainage area in the limit of large area. The excellent fit of a power law for the destructive flood event of June 2008 in the 32,400-km2 Iowa River basin over four orders of magnitude variation in drainage areas supports the central hypothesis. The challenge of predicting observed scaling exponent and intercept from physical processes is explained....
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Key results in the last 20 years have established the theoretical and observational foundations for developing a new nonlinear geophysical theory of floods in river basins. This theory, henceforth called the scaling theory, has the explicit goal to link the physics of runoff generating processes with spatial power-law statistical relations between floods and drainage areas across multiple scales of space and time. Published results have shown that the spatial power law statistical relations emerge asymptotically from conservation equations and physical processes as drainage area goes to infinity. These results have led to a key hypothesis that the physical basis of power laws in floods has its origin in the self-similarity...
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We examine the appearance of power-law behavior in rooted tree graphs in the context of river networks. It has long been observed that the tails of statistical distributions of upstream areas in river networks, measured above every link, obey a power-law relationship over a range of scales. We examine this behavior by considering a subset of all links, defined as those links which drain complete Strahler basins, where the Strahler order defines a discrete measure of scale, for self-similar networks with both deterministic and random topologies. We find an excellent power-law structure in the tail probabilities for complete Strahler basin areas, over many ranges of scale. We show analytically that the tail probabilities...
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Recent work has demonstrated that the topological properties of real river networks deviate significantly from predictions of Shreve's random model. At the same time the property of mean self-similarity postulated by Tokunaga's model is well supported by data. Recently, a new class of network model called random self-similar networks (RSN) that combines self-similarity and randomness has been introduced to replicate important topological features observed in real river networks. We investigate if the hypothesis of statistical self-similarity in the RSN model is supported by data on a set of 30 basins located across the continental United States that encompass a wide range of hydroclimatic variability. We demonstrate...
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River networks in the landscape can be described as topologic rooted trees embedded in a three-dimensional surface. We examine the problem of embedding topologic binary rooted trees (BRTs) by investigating two space-filling embedding procedures: Top-Down, previously developed in the context of random self-similar networks (RSNs), and Bottom-Up, a new procedure developed here. We extend the concept of generalized Horton laws to interior sub catchments and create a new set of scaling laws that are used to test the embedding algorithms. We compare the two embedding strategies with respect to the scaling properties of the distribution of accumulated areas Aω and network magnitude Mω for complete order streams ω. The...
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