River channel networks created by Poisson Equation and Inhomogeneous Permeability Models

This paper presents original mathematical models to simulate formation of dendritic tree networks such as natural river channels. These are referred to as the Poisson Equation Model (PEM) and as the Inhomogeneous Permeability Model (IPM), respectively, where the two-dimensional Poisson equations are solved under the homogeneous or the inhomogeneous condition. Particularly important is the IPM, which assumes that permeability varies depending on the site, which reflects regional fluctuations of geographical properties such as soil, precipitation, and so on. Natural river basins are supposed to realize a kind of optimization principle such that total energy expenditure or flow resistance is minimal, as stated by the Optimal Channel Network (OCN) theory or the Constructal law. From a viewpoint of optimization, possible structures of area-to-point flow systems are explored by use of PEMs and IPMs. According to our numerical simulations, it is supposed that the origin of the fractal structure that characterizes natural fluvial landscapes is heterogeneity in geomorphic conditions. This study indicates that the permeability randomization method can be utilized not only as a practical but also as an instructive tool to reproduce naturally disturbed river network systems.