Evolution of drainage networks and hillslopes

Abstract

Drainage networks are an integral part of the dynamics of almost every hydrologic system. We, as scientists, are currently in an extremely exciting period in the study of drainage networks. Over the past decade we have seen a dramatic paradigm shift from a largely empirical to a more fundamental understanding and quantitative descriptions of the inherent forms and processes that create drainage networks. During this period of time, much of the work has focused on river networks and has generally only considered changes in length scales. This dissertation focuses on the influence o f time scales in drainage network development.

One of the more prominent current theories of drainage network development is that the three-dimensional form of the drainage networks is largely dominated by optim ality in energy dissipation. A relationship between a characteristic discharge or drainage area and elevation gradient has been presented in the literature. S = gQZ = S = gAZ. Originally it was suggested that z = -0.5 for optim ality in the above sense; however, more recently some real world systems have been shown to be near optimal while deviating from this value of z. Field data shows that this relationship varies dramatically from z = -0.5. An explanation for this deviation is presented using a simulation approach by accounting for differences in flow distributions based on drainage area and accounting for the relative effectiveness of each flow.

There is very little quantitative data in the literature on how networks grow with time. A physical experiment was conducted in an experimental hillslope facility 10 [m ] x 3 [m] on two different slopes, 9° and 5°, to study the structure of drainage networks on a slope void of vegetation as they develop through time under constant rainfall. Two new quantitative measures are proposed and tested as a means to describe the network growth as a function of time and slope. Space filling characteristics of the networks, specifically the fractal dimension Df, are calculated at 1-hour intervals for 4- to 5-hour rainfall simulations. Statistical analysis shows that the network becomes more space filling with time and that this occurs more rapidly on the steeper slope. Fourier series fits to width functions at 1-hour intervals over the duration of the rainfall simulation are shown to be very accurate describing the frequency characteristics of the networks. The signal strength associated with each frequency in the Fourier series is analyzed statistically. The analysis shows that low frequencies become relatively more important with time and that the bifurcation characteristics remain constant through time. Results are also presented which show that while the geomorphic characteristics of the networks, like Horton’s bifurcation and length ratios, do not vary for the two different slopes, the width, depth, and width-to-depth ratio do depend on slope.

A two-dimensional hillslope model is presented which solves the coupled full hydrodynamic equations for overland flow, Richards equation for infiltration in one-dimension, and a physically based sediment detachment and transport equation. This is the most advanced hillslope model yet to be developed. The use of Richards equation allows continuous simulations of discontinuous rainfall events, and the coupling of the sediment detachment and transport algorithm with the overland flow algorithm allows the modeling of hillslope topographic evolution. The energy expenditure characteristics o f an evolving hillslope are found to possess attributes, which have been characterized as optimal in the past and are subject to the scales at which the measurements are made.