The Emergence of Channelized Drainage Patterns and Scaling in a Class of Landscape Evolution Models
Bjorn Birnir 
Terence R. Smith 
George E. Merchant
Department of Computer Science,
University of California at Santa Barbara,
Santa Barbara, California 93106, USA
Department of Geography,
University of California at Santa Barbara,
Santa Barbara, California 93106, USA
Department of Mathematics,
University of California at Santa Barbara,
Santa Barbara, California 93106, USA
A simple system of equations describing landscape evolution
was studied analytically and numerically by Smith, Birnir, and Merchant [1997] and Smith, Merchant, and Birnir [1997]. They showed that an initially flat two-dimensional surface is unstable when eroded by water and that channels will form. The area of greatest channelization first occurs towards the bottom of the slope and gradually moves upward as the surface erodes. The process of channelization, which is closely associated with the development of concave landscape profiles characteristic of the controlling sediment transport laws, involves the merging of ``rivulets'' and the formation of larger ``channels'' until a pattern of valleys separated by ridges emerges. The emergent landscapes of valleys and ridges are well-described by solutions to the original nonlinear partial differential equations (PDE's). These solutions depend on the characteristics of the water and sediment transport laws and on the boundary conditions for the PDE's. They emerge after characteristic periods of time; they are separable in time and space
and they possess stable solutions. They are also characterized by a simple variational principle.
In this paper we show that the process of channel formation that occurs before the emergence of the separable solutions is controlled by a class of self-similar solutions
to the PDE's. Our theory for these solutions predicts
that the autocorrelation functions of both the water
surface and the water depth should scale with the spatial variable according
to a polynomial scaling law [see Bak et al.], with an
invariant value 
over
variations in the values of the exponents
of the sediment transport law.
Numerical experiments validate these predictions.
This implies that the self-similar solutions exhibiting such scaling
act as a "transient attractor" controlling the channelization.
Furthermore, the invariance of the results over
a range of parameter values indicates that our equations belong
to a large universality class.
Hence the process of channelization is described by a scaling process in which a concavity and an area of maximal channelization is initiated on the originally linear surface in a region in which the locus of maximal downstream erosion meets a fixed lower boundary. This concavity, and the associated area of maximal channelization, then travels upstream, creating a snake-like region in which the channelization process is at a maximum and in which the self-similar scaling takes place. The water surface and the water depth scale with the same exponents which are positive because both increase due local departures from the global conservation of water and sediment. The snake travels upstream with a sinous motion and is eventually destroyed in a collision with the upper boundary. At this point, the evolution of the system is controlled by a longer-term ``transient attractor'' involving the separable solutions.
