Water erosion as a fractal growth process

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.

Area-slope relation in a simple erosion model

We discuss area-slope relation of a landform evolved by a simple lattice model of an erosion process. Observing a steady state of the model, where river networks on the surface are stationary, we present the relation between the power exponent of the area-slope relation and the parameters of the erosion process of the model. We show how to determine the parameters of the erosion process from observations of landform by making use of the area-slope relation. This method may be useful for morphological simulations of the landforms.

Self-organized critical drainage networks

We introduce time-dependent boundary conditions in a model of drainage network evolution based on local erosion rules. The changing boundary conditions prevent the model from becoming stationary; it approaches a state where fluctuations of all sizes occur. The fluctuations in the sizes of the drainage areas show power law behavior with an exponent that differs significantly from that of the static distribution of the drainage areas. Thus, the model exhibits self-organized criticality and proposes a novel concept for predicting fractal properties of drainage networks.

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.

Scaling in geology: landforms and earthquakes

Landforms and earthquakes appear to be extremely complex; yet, there is order in the complexity. Both satisfy fractal statistics in a variety of ways. A basic question is whether the fractal behavior is due to scale invariance or is the signature of a broadly applicable class of physical processes. Both landscape evolution and regional seismicity appear to be examples of self-organized critical phenomena. A variety of statistical models have been proposed to model landforms, including diffusion-limited aggregation, self-avoiding percolation, and cellular automata. Many authors have studied the behavior of multiple slider-block models, both in terms of the rupture of a fault to generate an earthquake and in terms of the interactions between faults associated with regional seismicity. The slider-block models exhibit a remarkably rich spectrum of behavior; two slider blocks can exhibit low-order chaotic behavior. Large numbers of slider blocks clearly exhibit self-organized critical behavior.