The scaling of fluvial landscapes

Abstract

The analysis of a family of physically-based landscape models leads to the analysis of two stochastic processes that seem to determine the shape and structure of river basins. The partial differential equation determine the scaling invariances of the landscape through these processes. The models bridge the gap between the stochastic and deterministic approach to landscape evolution because they produce noise by sediment divergences seeded by instabilities in the water flow. The first process is a channelization process corresponding to Brownian motion of the initial slopes. It is driven by white noise and characterized by the spatial roughness coefficient of 0¢ 5. The second process, driven by colored noise, is a maturation process where the landscape moves closer to a mature landscape determined by separable solutions. This process is characterized by the spatial roughness coefficient of 0¢ 75 and is analogous to an interface driven through random media with quenched noise. The values of the two scaling exponents, which are interpreted as reflecting universal, but distinct, physical mechanisms involving diffusion driven by noise, correspond well with field measurements from areas for which the advective sediment transport processes of our models are applicable. Various other scaling laws, such as Hack's law and the Law of Exceedence Probabilities, are shown to result from the two scalings, and Horton's Laws for a river network are derived from the first one.

Mathematical Models for Erosion and the Optimal Transportation of Sediment

“…For a detailed derivation of these equations we refer to [45] and [8]. When many simulations are performed and an ensemble average over these simulations is taken, a statistically stationary equilibrium water depth emerges.…”

“…On the top of the slope, at x ¼ 0 in Figure 2, we assume that there is no sediment flowing over the top of the Figure 1, from [8].…”

“…Our focus is the equations developed and studied by the first author and his collaborators in [44], [45], [8], [9]. These equations improve the original model in [42] by including a pressure term (in addition to the gravitational and friction terms) that prevents water from accumulating in an unbounded manner in surface concavities; see [28].…”

“…points where rH is defined and non-zero, then at these points ru ¼ rH jrHj : ð1:2Þ Fig. 1: A desert landscape consisting of a pattern of valleys separated by mountain ridges, from [8].…”

Mathematical Analysis of a Model of River Channel Formation

“…WILLGOOSE et al (1991) reverted to an artificial channel indicator variable, and KRAMER and MARDER (1992) used cellular lattice models, a development which has formed the thrust of simulation models since, e.g., those of HOWARD (1994) and TUCKER and SLINGERLAND (1994). There have been efforts to solve the Smith-Bretherton model directly (e.g., SMITH et al, 1997;BIRNIR et al, 2001), although these are problematical, unsurprisingly since the original Smith-Bretherton model is actually ill-posed.…”