The Stochastic Theory of Fluvial Landsurfaces

Abstract

A stochastic theory of fluvial landsurfaces is developed for transport-limited erosion, using well-established models for the water and sediment fluxes. The mathematical models and analysis is developed showing that some aspects of landsurface evolution can be described by Markovian stochastic processes. The landsurfaces are described by nondeterministic stochastic processes, characterized by a statistical quantity the variogram, that exhibits characteristic scalings. Thus the landsurfaces are shown to be SOC (Selforganized-critical) systems, possessing both an initial transient state and a stationary state, characterized by respectively temporal and spatial scalings. The mathematical theory of SOC systems is developed and used to identify three stochastic processes that shape the * and the University of Iceland, 107 Reykjavík 1 surface. The SOC theory of landsurfaces reproduces established numerical results and measurements from DEMs (digital elevation models).

Mathematical Models for Erosion and the Optimal Transportation of Sediment

“…The theoretical results can then be applied to actual modeling and simulations by distinguishing those solutions which predict the optimal flow of sediment, and therefore produce accurate long-term models for the partial differential equation. Simulations and observations of real landsurface shapes that retain their form for a long time but decrease in elevation were studied extensively in [8] and [9]. In [45], separable solutions of the equations (2.1) and (2.1) were discovered; these solutions exhibit the same behavior as the simulations and observations in [8] and [9] for some constant λ.…”

“…The collapsing hill function is a strong solution to (2.2) under these conditions, and as with the mountain ridge functions, it may be piecewise defined to ensure the boundary conditions are satisfied. We are most interested in the mountain ridges, because they are observed both empirically and in simulations for significant time intervals; see [8] and [9]. The empirically observed mountain ridges are in fact more complicated than the ridges modeled by our mountain ridge functions.…”

“…The initial surface is unstable, but in spite of this it is possible to solve the two equations (2.1) and (2.2) numerically with modern numerical methods. The first author and his collaborators did this in [44], [45], [8], [9] and [50]. Thus they gained considerable insight into the properties of the solutions, and the main purpose of this paper is to develop the full nonlinear analysis based on these insights.…”

“…The second group has produced remarkable simulations of evolving channel networks; see [52,53], [18], [48] and [35]. The third group has lead to an increasing understanding of the physical mechanisms that underlie erosion and channel formation; see [40], [42], [30], [36], [28], [27], [29], [43], [22,23,24,25,21], [44,45,46], [39], [50], [9], [15], [7], [41].…”

“…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].…”