Simple computer model of a fractal river network with fractal individual watercourses

A random walk computer model of river network is proposed. It is shown that the length and the width of both network and of individual streams that constitute it exhibit scaling sigma /sub /// approximately Lv// and sigma perpendicular to approximately Lv ( sigma // and sigma perpendicular to are the longitudinal and lateral sizes of the object, L is the overall length of the object). The simulated individual streams display self-similar behaviour at small L (v/sub ///=vperpendicular to =0.80+or-0.03) and self-affine behaviour at large L (v/sub ///=0.99+or-0.03, vperpendicular to =0.50+or-0.03). Similar behaviour is observed for simulated river networks too: v/sub ///=vperpendicular to =0.66+or-0.03 correspond to these in the self-similarity region, while in the self-affinity region v/sub ///=0.74+or-0.03 and vperpendicular to =0.43+or-0.03. Proceeding from the self-affinity of individual rivers and river networks Hack's empirical law L approximately Fbeta has been substantiated (L is the length of the main river, F the catchment area), where beta =1/(1+H), H=vperpendicular to /v/sub ///, Hurst's exponent for river networks. The scaling for the water mass distribution over the river network in the self-affine region is also revealed: sigma m// approximately Lv, sigma m perpendicular to approximately Lv, vm//=0.72+or-0.03, vm perpendicular to =0.38+or-0.03. It is shown that in this region the water mass M depends upon the network total length and upon the catchment area as a power law: M approximately L1.67 approximately F1.43.