Generalizing a nonlinear geophysical flood theory to medium‐sized river networks

Abstract

[1] 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-km 2 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. We show scaling in mean annual peak discharges, and briefly discuss that it is physically connected with scaling in multiple rainfall-runoff events. Scaling in peak discharges would hold in a nonstationary climate due to global warming but its slope and intercept would change. Citation:

Analyzing the effects of excess rainfall properties on the scaling structure of peak discharges: Insights from a mesoscale river basin

“…While these studies highlighted the respective dependence of the exponent and the intercept on the excess rainfall duration and intensity, more studies must be conducted to determine whether or not these findings hold true in larger catchments and also to investigate the effect of the interplay of these parameters with other catchment physical variables. Although Gupta et al [2010] showed that the historical flood event of June 2008 that devastated the Iowa River basin in Eastern Iowa (A % 32,400 km 2 ) obeys scaling invariance with drainage area, no study, to the best of our knowledge, has demonstrated whether or not the findings from the 21 km 2 GCEW Gupta, 2005, 2007] hold true in larger watersheds. Apart from demonstrating the existence of scale-invariant peak discharges in a mesoscale river basin at the rainfall-runoff event scale, such a study would help establish the physical connection between the flood scaling parameters and rainfall and catchment physical properties that vary from event to event.…”

“…The width function is calculated as the total number of channel-links at a given distance from the outlet of a catchment. Gupta et al [2010] define the width function as being equivalent to the streamflow response to an instantaneous rainfall that is instantaneously injected into channel-links and moves along the drainage network with constant velocity and without attenuation. Recent theoretical studies show that the scaling exponent of the width function maxima is the lower bound of the flood scaling exponent [Ayalew et al, 2014a[Ayalew et al, , 2014bMandapaka et al, 2009;Mantilla et al, 2006].…”

“…In a major departure from the peak discharge quantile based analysis used in the regional flood-frequency methodology, a number of rainfallrunoff event based studies were undertaken in an effort to describe the flood scaling parameters in terms of rainfall and catchment physical processes that transform rainfall to peak discharge [Ayalew et al, 2014a[Ayalew et al, , 2014bDi Lazzaro and Volpi, 2011;Furey and Gupta, 2005;Gupta, 2004;Gupta et al, 1996Gupta et al, , 2007Gupta et al, , 2010Mandapaka et al, 2009;Mantilla et al, 2006Mantilla et al, , 2011Morrison and Smith, 2001;Ogden and Dawdy, 2003]. These studies, which cover a range of theoretical and empirical studies, revealed that peak discharges from nested catchments exhibit scale-invariance with drainage area at the rainfall-runoff event scale and the relationship between the two exhibits the following power law relation:…”

Horton laws for hydraulic–geometric variables and their scaling exponents in self-similar Tokunaga river networks

“…Consequently, it opens a new door to understanding how the geometry, statistics and dynamics in river networks are mutually coupled on many spatial scales, which has far-reaching implications for understanding and modeling river flows, as explained above, and transport of sediments and pollutants in river networks in the future. A few discrete research efforts have been made to understand the nature of the flood scaling from physical processes on an annual timescale (Poveda et al, 2007;Lima and Lall, 2010), as well as on the event timescale (Ogden and Dawdy, 2003), but connecting the body of work on the annual scale to flood scaling for events remains an important open problem (Gupta et al, 2010).…”

Estimation of Peak Flow Rates for Small Drainage Areas

“…Gupta et al (2007) determined that θ was 1 for small basins dominated by rainfall-runoff variability and 0.5 for large basins dominated by network structure and flow dynamics. Gupta et al (2010) used mean annual peak flow data for the Iowa River basin (6.6 km 2 to 32,374 km 2 ) and obtained a θ value of 0.54. Di Lazzaro and Volpi (2011) determined a θ value of 0.52 using data for a watershed area located in the Tiber River region, Central Italy, that varied from 218 km 2 to 4116 km 2 .…”