J. Phys. A: Math. Gen. 29 (1996) 6701–6708. Printed in the UK Exact analysis of the Peano basin A Flammini and F Colaiori Istituto Nazionale di Fisica della Materia (INFM), International School for Advanced Studies (SISSA), via Beirut 2-4, 34014 Trieste, Italy Received 21 May 1996 Abstract. The Peano basin is analysed as a deterministic model for river networks. The fundamental distributions characterizing the real basins morphology can be explicitly calculated. The recently proposed finite size-scaling ansatz is tested apart from oscillatory amplitudes typical of deterministic fractals. 1. Introduction In recent years a variety of approaches towards the statistical characterization of river networks have been proposed, in the attempt to describe a basin’s morphology [1–6]. In real drainage networks, rainfall collected by the basin flows downhill through channels, which, by means of erosion, self-organizes in a spatial tree-like structure. Such networks are known to exhibit power-law behaviour typical of fractal structures [7–9] in drainage sub-basin areas and mainstream-lengths distributions. A river basin can be described giving a scalar field of elevations and defining drainage directionsbysteepestdescents. Thusinasimplelatticepicturearivernetworkisrepresented by an oriented spanning tree over a two-dimensional lattice. To each site i one can associate a local injection of mass (the average annual rainfall at site i ) that can be taken equal to 1. Then the flow A i at site i , or equivalently the drained area at site i can be defined as the sum of the injection over all points upstream with respect to i (we say that site j is upstream with respect to site i if drainage directions go from j to i ). Variables A i are thus related by: A i = � j w i,j A j � r i (1) where w i,j is 1 if site j is a nearest neighbour of i upstream with respect to site i and 0 otherwise. In natural basins these areas can be investigated through experimental analysis of digital elevation maps (DEM’s) [3]. Another relevant quantity in a basin’s morphology is the upstream length relative to a site, defined as the length of the stream obtained starting from the site and moving in the upstream direction towards the nearest neighbour with biggest area A (the one leading to the outlet excluded), since a source, i.e. a site with no incoming links, is reached. If two or more equal areas are encountered, one is randomly selected. Recently a simple finite size-scaling ansatz has been proposed [10] leading to natural explanation of scaling properties. For a lattice of given linear size L call p�A,L) the probability density distribution of cumulated areas A and π�l,L) the probability density distribution of the upstream lengths l , i.e. the fraction of sites with, respectively, area A or 0305-4470/96/216701+08$19.50 c � 1996 IOP Publishing Ltd 6701