## 1. Introduction

[2] Significant insight gained toward the understanding of the impact of the network geomorphology on the streamflow response has been achieved using the theory of geomorphologic instantaneous unit hydrograph (GIUH) [*Rodriguez-Iturbe and Valdes*, 1979; *Gupta et al.*, 1980] in which the basin's hydrologic response was for the first time coupled to the geomorphologic structure of the river network. During the last two decades, several contributions linking the network structure and the flow dynamics have appeared in the literature [*Wang et al.*, 1981; *Mesa and Mifflin*, 1986; *van der Tak and Bras*, 1990; *Rinaldo et al.*, 1991; *Jin*, 1992; *Naden*, 1992; *Snell and Sivapalan*, 1994; *Robinson et al.*, 1995; *Yen and Lee*, 1997]. *Rinaldo et al.* [1991] derived an analytical expression, in the form of a dispersion coefficient, which quantifies the portion of the basin's hydrograph variance that is due to the influence of the river network organization. They called this influence as geomorphologic dispersion.

[3] These developments provide a significant leap forward in our understanding of the physical basis of hydrologic response. However, the characteristics of the network geometry have not been effectively coupled to hydraulic geometry relations which implicitly characterize the nonlinearity of flow dynamics. As a result simplified assumptions of spatially invariant hydrodynamic characteristics over the entire network are invoked. For example, in the work of *Rinaldo et al.* [1991] an analytical expression for the travel time distribution for individual reaches is obtained by assuming that the mean travel time within each reach can be computed using a velocity that is invariant within the basin boundaries. However, flow through the channel network is inherently nonlinear and shows a strong dependence on scale, i.e., the size of the basin. *Pilgrim* [1976], using tracer studies, showed that the average flow velocities are a nonlinear function of the discharge, but reach an asymptotic value at high flows. *Carlston* [1969] empirically investigated *Leopold and Maddock*'s [1953] hydraulic geometry relationships, for streams with natural channels, and concluded that width (*w*), depth (*d*) and velocity (*v*) are proportional to powers of the discharge *Q*, i.e., *w* ∝ *Q*^{ζ}, ϑ*h* ∝ *Q*^{ϑ}, and *v* ∝ *Q*^{η}. He found that the mean values of ζ, ϑ and η were 0.46, 0.38 and 0.16 indicating a slowly increasing velocity. This was an indication that the increase in discharge in the downstream direction is accommodated through either an increase in flow depth and/or an increase in width. However, most studies linking flow and network geometry do not account for this dependence on hydraulic geometry. In addition, although these studies support the contention that flow through a stream network is generally nonlinear, the argument of slowly increasing velocity has been used to support the development of linear or quasi-linear models (i.e., models applicable to a certain range of flow conditions) by using a spatially uniform celerity and hydrodynamic dispersion over the entire basin. Consequently these models are valid for small basins thereby limiting their applicability.

[4] Nonlinearity has been investigated by relaxing the hypothesis of constant velocity through the channel network in the works of *Valdes et al.* [1979], *Wang et al.* [1981], *Robinson et al.* [1995], and *Yen and Lee* [1997]. However, none of these studies deals specifically with the coupling of the hydraulic geometry with the stream network organization which is the subject of this paper.

[5] In this research we show that the presence of spatially varying celerities induces a dispersion effect, referred to as kinematic dispersion, on the network travel time distribution. Its contribution to the total dispersion is comparable to that due to the heterogeneity of path lengths, that is geomorphologic dispersion, and significantly larger than the hydrodynamic dispersion. If this contribution is ignored the hydrograph shows a higher peak flow, a shorter time to peak and shorter duration.

[6] The rest of the paper is organized as follows. Section 2 provides a brief review of the current state of the art on which this research is based. Analytical equations for the various dispersion mechanisms that arise in this study are derived in section 3. In section 4 we present two different approximations that can be used to estimate the network instantaneous response function. Section 5 summarizes the hydraulic geometry relations used in this research. A case study is presented in section 6, in which the relative contributions of the various dispersion mechanisms is analyzed. Summary and conclusions are given in section 7. Appendix A describes some technical results pertinent to this research.