Abstract:The water path from any point of a basin to the outlet through the self-similar river network was considered. This hydraulic path was split into components within the Strahler ordering scheme. For the entire basin, we assumed the probability density functions of the lengths of these components, reduced by the scaling factor, to be independent and isotropic. As with these assumptions, we propose a statistical physics reasoning (similar to Maxwell's reasoning) that considers a hydraulic length symbolic space, built on the self-similar lengths of the components. Theoretical expressions of the probability density functions of the hydraulic length and of the lengths of all the components were derived. These expressions are gamma laws expressed in terms of simple geomorphological parameters. We validated our theory with experimental observations from two French basins, which are different in terms of size and relief. From the comparisons, we discuss the relevance of the assumptions and show how a gamma law structure underlies the river network organization, but under the influence of a strong hierarchy constraint. These geomorphological results have been translated into travel time probability density functions, through the hydraulic linear hypothesis. This translation provides deterministic explanations of some famous a priori assumptions of the unit hydrograph and the geomorphological unit hydrograph theories, such as the gamma law general shape and the exponential distribution of residence time in Strahler states.

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Cudennec et al. (2004) focused on the organization of river networks, from the point of view of water paths down to the outlet, in order to link geomorphological and hydrological aspects. With this purpose, the 'hydraulic path' was identified and split into 'components' within the Strahler ordering scheme, which appear to be newly defined geomorphological objects. Indeed, the 'ith-order component' of a given hydraulic path is neither necessarily a single i-order link nor necessarily a whole i-order stream (see Figure 1 in Cudennec et al. (2004)). From the identification of these objects, metric variables were considered: (i) the length of a generic hydraulic path L, called 'hydraulic length'; (ii) the length of its ith-order component l i .These variables are linked through Equation (6) Then the self-similarity of the river network was assumed, and the probability density functions (p.d.f.s) of the lengths of these components-reduced by the ratio r l used as a scaling factor-were supposed to be independent and isotropic. This allowed a statistical physics reasoning (similar to Maxwell's) to be proposed, that considers a hydraulic length symbolic space, built on the self-similar lengths of the components. Theoretical expressions of the p.d.f.s of the lengths of all the components and of the hydraulic length were derived [Equations (29) and (45), respectively in Cudennec et al. (2004)]. These theoretical proposals were applied to two actual river basins. The comparisons of theoretical and experimental results showed a gradient of decreasing relevance of the theoretical proposals through the Strahler