Evolution of river networks

Abstract

Why does a sheet of water flowing over an initially featureless surface spontaneously form a river network? To address this question, we construct a simple model which enables us to examine the shape and stability of individual river channels. We compare predictions for the geometry of fluvial channels with experimental data. In addition, we construct a lattice model which allows us to look at large-scale features of river networks and calculate their scaling relations. PACS numbers: 68.70.4-w, 92.40.Fb, 92.40.GcIf fractals are the geometry of nature, one must still ask how nature produces them. Branched river networks are among nature's most common patterns, spontaneously producing structure over a huge range of length scales [lj. At the heart of this problem is the question of why and how individual river channels are formed. This Letter presents a nonlinear model which describes the evolution of an arbitrary initial landscape covered by a distribution of water, and has the goal of understanding instabilities which lead to the coalescence of the water into channels and, later, river networks. Rivers have been studied extensively by a wide variety of researchers with an equally wide variety of techniques and goals. Geomorphologists have found scaling relationships among various combinations of basin statistics from field data, such as drainage density and branching ratios [2]. Hydrologists have likewise extracted power laws for channel parameters such as width, depth, velocity, and slope as functions of total channel discharge [3]. Other investigators have examined the shape [4-6] of individual equilibrium channels in erodible material while somehave constructed models for the evolution of an entire drainage network [7-9]. However, we are unaware of a previous approach which predicts the shapes spontaneously formed when water flows over an initially featureless surface and allows one to answer questions about selection and stability. We differ from previous authors because we do not try to find closed equations for height of soil alone; we treat water explicitly as well. The aim is to find the simplest model which reveals the essential features of river formation. We pose our model in terms of two scalar fields, b(x,y,t) and d(x,y,t), where b(x,y,t) is the height of soil above some arbitrary horizontal level, dix,y,t)is the depth of water flowing over the soil, x and y are spatial coordinates, and / is time. It will be convenient to introduce an auxiliary field s(x,yj)=b(x,yj) + d(x,yj) which defines the overall surface of the land plus water. The time evolution of the system is given by

The Formation of River Channels

“…Progress in the first of these aims was made by Kramer and Marder (1992), who developed a nonlinear evolution equation for channel depth by seeking particular solutions of their hillslope model, which was similar, but by no means identical to, the Smith-Bretherton model. The main difference between their result and that of the present paper is that their model is partially empirical, and the derivation of the channel model is not placed in the context of a formal asymptotic approximation to the full model.…”

Sediment Transport Dynamics in River Networks Induced by Water Diversion

“…This feedback mechanism indicates that river network is a self-organized system (Rodriguez-Iturbe, 1997), and some dynamical laws lead its evolution and some statistic laws dominate its steady state (Leopold, 1953;Leopold & Maddock, 1953;. So, river networks have attracted, in decades past, a good much attention of physicists and geophysicists (Banavar, et al, 1997;Manna & Subramanian, 1996;Manna, 1998;Sinclair & Ball, 1996;Kramer & Marder, 1992;Takayasu & Inaoka, 1996;Caldarelli, et al, 1997;Giacometti, 2000;Somfai & Sander, 1997;Rinaldo, et al, 1996, and so on). They focused mainly on the distributions of river parameters or the scaling relations between them, as well as the evolutionary mechanism, that is, what creates the distributions and scaling relations?…”

Distributed soil erosion simulation for effective erosion prevention

“…Impact of topography on erosion processes in relation to channel initiation was studied by Dietrich et al [1993] and, in relation to colluvial deposits, by Reneau et al [1989]. Several distributed landscape evolution models also describe a number of aspects of erosion processes [e.g., Kirkby, 1986;Willgoose et al, 1991;Kramer and Marder, 1992;Howard, 1994]. Most of these models are designed for geologic timescales and focus on the evolution of topography and stream networks rather than on detailed erosion risk prediction and prevention related to recent changes of cover or short-term antropogenic factors (see, however, recent studies of mining lands rehabilitation by I,l•llgoose and Gyasi-Agyei [1995] and Willgoose and Riley [1998]).…”