Minimum energy dissipation river networks with fractal boundaries

A minimalist model for coevolving supply and drainage networks

Numerous complex systems, both natural and artificial, are characterized by the presence of intertwined supply and/or drainage networks. Here, we present a minimalist model of such coevolving networks in a spatially continuous domain, where the obtained networks can be interpreted as a part of either the counter-flowing drainage or co-flowing supply and drainage mechanisms. The model consists of three coupled, nonlinear partial differential equations that describe spatial density patterns of input and output materials by modifying a mediating scalar field, on which supply and drainage networks are carved. In the two-dimensional case, the scalar field can be viewed as the elevation of a hypothetical landscape, of which supply and drainage networks are ridges and valleys, respectively. In the three-dimensional case, the scalar field serves the role of a chemical signal, according to which vascularization of the supply and drainage networks occurs above a critical 'erosion' strength. The steady-state solutions are presented as a function of non-dimensional channelization indices for both materials. The spatial patterns of the emerging networks are classified within the branched and congested extreme regimes, within which the resulting networks are characterized based on the absolute as well as the relative values of two non-dimensional indices.

Simple stochastic lattice gas automaton model for formation of river networks

A stochastic lattice gas automata model for formation of river networks is proposed. The model is based on two-dimensional lattice gas automata with three fundamental principles at each node. The water source is regarded as a fixed point where a drop of water drips every time step. This system can be treated as a memory network: the probability of water moving along a direction relies on the history of the channel segment along which water drops have moved. Last, we find that the width of the river channel and the number of channels with this width meet a scaling law when the system reaches a critical status.

Geometry of river networks. I. Scaling, fluctuations, and deviations

This paper is the first in a series of three papers investigating the detailed geometry of river networks. Branching networks are a universal structure employed in the distribution and collection of material. Large-scale river networks mark an important class of two-dimensional branching networks, being not only of intrinsic interest but also a pervasive natural phenomenon. In the description of river network structure, scaling laws are uniformly observed. Reported values of scaling exponents vary, suggesting that no unique set of scaling exponents exists. To improve this current understanding of scaling in river networks and to provide a fuller description of branching network structure, here we report a theoretical and empirical study of fluctuations about and deviations from scaling. We examine data for continent-scale river networks such as the Mississippi and the Amazon and draw inspiration from a simple model of directed, random networks. We center our investigations on the scaling of the length of a subbasin's dominant stream with its area, a characterization of basin shape known as Hack's law. We generalize this relationship to a joint probability density, and provide observations and explanations of deviations from scaling. We show that fluctuations about scaling are substantial, and grow with system size. We find strong deviations from scaling at small scales which can be explained by the existence of a linear network structure. At intermediate scales, we find slow drifts in exponent values, indicating that scaling is only approximately obeyed and that universality remains indeterminate. At large scales, we observe a breakdown in scaling due to decreasing sample space and correlations with overall basin shape. The extent of approximate scaling is significantly restricted by these deviations, and will not be improved by increases in network resolution.

River networks on the slope-correlated landscape

We study the morphologies of river networks on various landscapes. In general, the probability density distribution of drainage area a of the river network scales as P(a) approximately a(-tau). We consider a slope-slope correlation function G(r) and define the persistent length R where G(r=R) becomes zero. In our restricted solid on solid network model, R is independent of the system size L and tau is close to 4/3, which is the value of the Scheidegger's river network model with random walk process. We also consider an avalanche model, where R is proportional to L. There is a large slope-slope correlation length and the river network does not follow the directed random walk process with tau approximately 1.42.

Universality Classes of Optimal Channel Networks

Energy minimization of both homogeneous and heterogeneous river networks shows that, over a range of parameter values, there are only three distinct universality classes. The exponents for all three classes of behavior are calculated.