River channel networks created by Poisson Equation and Inhomogeneous Permeability Models (II): Horton's law and fractality

Abstract

In the previous paper, we prepared the Poisson Equation and the Inhomogeneous Permeability Models (PEM & IPM) that can create tree-shaped networks under the conditions of homogeneous or inhomogeneous permeability. The driving force of channel network formation is derived from two-dimensional Poisson equations in both models, the solutions of which are supposed to represent a gravitational pressure field. Particularly important is the latter IPM that succeeds to simulate seemingly natural and realistic river channel basins, in which regional fluctuations of geographical properties concerning soil, precipitation and vegetation are reflected by inhomogeneous permeability. However, we did not refer to the relationship with the Horton's law and the identification of fractal dimensions. This paper examines the consistency with the Horton's law and the measurement of fractal dimensions in tree network systems generated by IPMs under the more improved resolution. Our numerical simulations show good accordance with the Horton's law, however, the calculation of fractal dimensions using the bifurcation and length ratios is not satisfactory because of a large uncertainty. Then, we originally propose an alternative method, referred to as the “extended cluster dimension”, which makes possible to identify the exact value of fractal dimensions in river network systems.