UC Santa Barbara

The Smith and Bretherton model for fluvial erosion consists of a pair of partial differential equations: one governing water flow and one governing sediment flow. Numerical solutions of these equations have been shown to provide realistic models of the evolution of fluvial landscapes. Further analysis of these equations and their numerical solutions show that they possess scaling laws that are known to exist in nature. The preservation of these scaling laws in simulations is highly dependent on the numerical method used. Two numerical methods, both optimized for overland flow, have been used to simulate these surfaces. The implicit method exhibits the correct scaling laws, but the explicit method fails to do so. These equations, and the resulting models, help bridge the gap between the deterministic and stochastic theories of landscape evolution. Despite current advances in processing power and parallelism, numerical simulations of these surfaces take months of computation time. Some alterations can be made to code parameters to decrease computation time, but sacrifice accuracy of the resulting surfaces. Using the known deterministic and stochastic theories of these equations, the large scales of the model are generated from a series of elementary functions. The small scales are generated using Hurst fractal interpolation; a modified version of fractal interpolation functions, a relatively recent technique of interpolation explained herein. The generated surfaces provide great insight into the scaling laws satisfied by these surfaces.