The hydrologic response of a channel network is defined by decomposing the process of runoff formation into two distinct contributions, one accounting for the mechanisms of travel time within individual reaches (hydrodynamic dispersion), and the other accounting for the morphology of the network structure (geomorphological dispersion). Exact Laplace transforms of first passage time distributions at the outlet of a network are obtained by a consistent approximation of travel time distributions through individual reaches. The moments of such distributions are obtained analytically in the general case. Closed-form first-passage distributions are obtained in the particular case of basin-constant hydrodynamic dispersion. The variance of the resulting travel time distributions is shown in this paper to be made up of two additive contributions corresponding to the two dispersion mechanisms considered. The geomorpho}ogic dispersion coefficient is shown to depend on the ratios of bifurcation, length and area of the network suggesting that, at the scale of an organized network, heterogeneities other than those related to the convection field shape the dispersive character of transport. In particular, a significant application of the general solution to Hortonian channel networks suggests that models based on accurate specification of the geometry and the topology of the network and a simplified dynamics capture the foremost features of the travel time distributions in a broad range of dispersivities within individual reaches. channel networks [e.g., Shreve, 1966; Smart, !972; Mandelbrot, 1983; Abrahams, 1984; Tarboton et al., 1988; La Barbera and Rosso, 1989; A. Marani et al., A note on fractal channel networks, Paper number 90WR02501, •3-t 397/91/90WR-0250 ! $05.00 1990 (hereinafter Marani et al. (submitted manuscript, 1990))]. Similar ideas have been explored with reference to random networks (modeling, somewhat arbitrarily, porous media as a random resistor network [De Arcangelis et al., 1986]), whereas we propose, in the framework of studies on transport by travel time distributions, the geomorpho!ogic analysis of channel networks. Recent contributions have provided new significant inroads toward a unifying approach for transport processes based on travel (first-passage, arrival or residence) time distributions. It is granted in this study that (1) the arrival time distribution at the outlet of a channel network after an instantaneous pulse is the geomorphologic unit hydrograph (GUH) which is the core of the hydrologic response [Rodriguez-!turbe and Valdes, 1979; Gupta et al., 1980; Gupta and Waymire, 1983]; and (2) travel time distributions may be related in a rational manner to approaches based on the solution of the mass and momentum balance equations in an Eulerian framework, are of general nature and robust in characterizing the transport process, and blend all sources of uncertainty into a unique curve [Rinaldo and Marani, 1987; Shapiro and Cvetkovic, 1988; Dagan and Nguyen, 1989; Rinaldo et al., 1989; Dagan, 198...

This paper is the outcome of a community initiative to identify major unsolved scientific problems in hydrology motivated by a need for stronger harmonisation of research efforts. The procedure involved a public consultation through online media, followed by two workshops through which a large number of potential science questions were collated, prioritised, and synthesised. In spite of the diversity of the participants (230 scientists in total), the process revealed much about community priorities and the state of our science: a preference for continuity in research questions rather than radical departures or redirections from past and current work. Questions remain focused on the process-based understanding of hydrological variability and causality at all space and time scales. Increased attention to environmental change drives a new emphasis on understanding how change propagates across interfaces within the hydrological system and across disciplinary boundaries. In particular, the expansion of the human footprint raises a new set of questions related to human interactions with nature and water cycle feedbacks in the context of complex water management problems. We hope that this reflection and synthesis of the 23 unsolved problems in hydrology will help guide research efforts for some years to come. ARTICLE HISTORY

46Process-based hydrological models have a long history dating back to the 1960s. 47Criticized by some as over-parameterized, overly complex, and difficult to use, a more 48 nuanced view is that these tools are necessary in many situations and, in a certain class of 49 problems, they are the most appropriate type of hydrological model. This is especially the 50 case in situations where knowledge of flow paths or distributed state variables and/or 51 preservation of physical constraints is important. Examples of this include: spatiotemporal 52 variability of soil moisture, groundwater flow and runoff generation, sediment and 53 contaminant transport, or when feedbacks among various Earth's system processes or 54 understanding the impacts of climate non-stationarity are of primary concern. These are 55 situations where process-based models excel and other models are unverifiable. This article 56 presents this pragmatic view in the context of existing literature to justify the approach where 57 applicable and necessary. We review how improvements in data availability, computational 58 resources and algorithms have made detailed hydrological simulations a reality. Avenues for 59 the future of process-based hydrological models are presented suggesting their use as virtual 60 laboratories, for design purposes, and with a powerful treatment of uncertainty. 61

Three principles of optimal energy expenditure are used to derive the most important structural characteristics observed in drainage networks: (I) the principle of minimum energy expenditure in any link of the network, (2) the principle of equal energy expenditure per unit area of channel anywhere in the network, and (3) the principle of minimum total energy expenditure in the network as a whole. Their joint applica,tion results in a unified picture of the most important empirical facts which have been observed in the dynamics of the network and its three-dimensional structure. They also link the process of runoff production in the basin with the characteris.tics of the network.

Seemingly unrelated empirical hydrologic laws and several experimental facts related to the fractal geometry of the river basin are shown to find a natural explanation into a simple finite-size scaling ansatz for the power laws exhibited by cumulative distributions of river basin areas. Our theoretical predictions suggest that the exponent of the power law is directly related to a suitable fractal dimension of the boundaries, to the elongation of the basin, and to the scaling exponent of mainstream lengths. Observational evidence from digital elevation maps of natural basins and numerical simulations for optimal channel networks are found to be in good agreement with the theoretical predictions. Analytical results for Scheidegger's trees are exactly reproduced