New type of self-organized criticality in a model of erosion.

Mechanisms of creating fractals have been clarified in various fields of science [1-3]. A remarkable success may be the discovery of so-called Laplacian fractals which involve pattern formations in diffusion-limited ag gregation, viscous fingering, crystallization, electrodeposition, dielectric breakdown, chemical dissolution, and bac terial colony growth. Fractal geometry is especially powerful for characteriz ing random surfaces such as the Earth's relief [4] and porous solids' surfaces [2]. A lot of attention has been paid to random surface growth phenomena which exhibit fractal scalings [5]. Recently, the model of self-organized criticality [6] (SOC), which was introduced as a model of a sandpile. has been attracting much attention because it automati cally converges to a statistically steady state where criti cal or fractal behaviors are found both in space and time. It is anticipated that a large portion of the fractals in na ture may be created by this kind of self-organization. In this paper we analyze surfaces under water erosion. We will show that a surface spontaneously evolves into a kind of critical state characterized by fractal scalings. The critical state is very different from that of the SOC model in that the patterns of water flows (i.e., river pat terns) on the surface are frozen, namely, they do not change after the system reaches the critical state. Thus we have critical behavior only in space. Landscapes and river patterns are familiar to every body but they have not attracted many physicists' interest so far. Mandelbrot first showed that coastlines can be characterized by fractal dimensions [5]. He also graphi cally demonstrated that the Earth's relief is a fractal by proposing fractional Brownian surfaces as landscapes. Although the resulting surfaces look natural at first sight, there is an obvious flaw, that is, no river exists on the sur face. River patterns are also known to be typical fractals [3,4]. Scheidegger proposed a lattice model of rivers which is defined on a slope where water on a site flows randomly to either the left down site or the right down site [7]. His river model was shown to be identical to the one-dimensional random particle aggregation model with uniform injection by regarding the direction of the slope as the time axis [8]. The model is known to show critical behavior automatically like the SOC model: for example, it was proved that the system converges to a steady state where the distribution of the drainage basin area or parti cle mass rigorously follows a power law [9]. For the creation of real river patterns and landscapes we believe that the effect of water erosion should plav the central role. By this motivation we model the erosion process on a lattice and simulate the formation of river patterns. The model is defined on a two-dimensional triangular lattice. Each site has two variables, h(x,y), the height of the Earth's surface, and s(x,y), the water flow intensity. For a given initial configuration time evolution is per formed according to the following procedures. 0) Rain fall.—We assume that rain falls constantly on every site in the amount of s0 (= 1). (2) Water flow.— For every site (x,v) we find the lowest value of height in the six nearest neighbors, mmihix',/)}, where ix'J) denotes a nearest-neighbor site. The water at (x,y) flows to the lowest neighbor if the destination's height is lower than h(x,y). Applying this procedure for all sites we can draw a global water flow pattern as shown in Fig. 1, which we call a river pat tern. The flow intensity s(x,y) is defined by the sum of water flows from neighbors and the rain fall on the site; so, when the river pattern becomes stationary all rain to the upstream of (x,y) gathers at (x,y), and s(x,v) is equal to the size of the drainage basin area for the site (x,y). (We treat local minimum sites in a separate way as described later.) (3) Water erosion.— By the effect of erosion the height at (x,y) is decreased by 8h(x,y) = F(J(.x,y))[h(x,y) min [h(x',/)}], where J(x,y) is the water power which