Abstract
The water path from any point of a basin to the outlet through the selfβsimilar 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 selfβsimilar 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. Copyright Β© 2004 John Wiley & Sons, Ltd.