Continuum model for river networks.

The effects of erosion, avalanching and random precipitation are captured in a simple stochastic partial differential equation for modelling the evolution of river networks. Our model leads to a self-organized structured landscape and to abstraction and piracy of the smaller tributaries as the evolution proceeds. An algebraic distribution of the average basin areas and a power law relationship between the drainage basin area and the river length are found. 64.60.Ht,68.70.+w,92.40.Fb,92.40.Gc Typeset using REVTEX 1 A fractal river network is a striking example of self-organized criticality. The physics of river network evolution arises from an interplay of the structured landscape governing the water flow with the erosional effects of the water feeding back into further sculpting of the landscape. Extensive studies of the fractal characteristics of real river networks have been carried out. [1–7] Hack [2] has studied the relationship between the length of a river l and the area of a drainage basin s. s is a measure of the total area of the land covered by the principal stream and its tributaries that feed into the network. Hack’s measurements indicate that for basin areas s ranging over almost five decades (up to 375 square miles), s ∼ l with the exponent 1/φ ∼ 0.57 . Other measurements of the distribution of drainage basin areas suggest a power law scaling of the form P (s) ∼ s with τ = 1.45± 0.03 . [1] Most of the models of river networks fall into two categories. The first is restricted to reproducing the statistical properties of networks [8]. More recently, models for the evolution of river networks have been developed. Based on careful studies of river data [1], Rinaldo and coworkers [9] have suggested that the effects of local optimal rules equivalent to critical erosion parameters lead to statistical characteristics for the networks similar to global constraints of minimum energy dissipation. Leheny and Nagel [10] introduced a lattice model that incorporated erosion and showed a competition in growth between neighboring river basins relevant for late stages of evolution. Kramer and Marder [11] have also constructed a lattice model that allows for the elucidation of the scaling properties of the large scale features of river networks. In addition, they have proposed coupled differential equations for two scalar fields, the height of the soil and the depth of the water flowing over the soil. An analysis of these equations has led to an understanding of the shape and stability of individual river channels. Our principal goal is to introduce and numerically study a simple stochastic partial differential equation for the evolution of the landscape of a river network. Our model is a field theory for the soil height h(~x, t) and takes into account the effects of random precipitation, erosion and the avalanching of soil. An initially smooth landscape evolves into a nontrivial spatially self-organized state in which Hack’s law and the algebraic distribution of drainage 2 basin areas are obtained. The evolution equation may be written in the compact form: ∂ ∂t h(~x, t) = D · h(~x, t)− k∇h(~x, t) + η(~x, t) (1) where D· ≡ D1∂ y + D̃2(| ∇h |)∂2 x and ~x ≡ (x, y). The coefficient D̃2(| ∇h |) is defined as: D̃2[| ∇h |] = 