|This dissertation explores signatures and consequences of different mathematical descriptions of hillslope sediment transport. Three mathematical models are common in geomorphology – local linear diffusion, local nonlinear diffusion, and nonlocal transport. There is a long tradition of local models, both liner and nonlinear; however, recent work suggests value in exploring applications of nonlocal sediment transport models. This dissertation is divided into three chapters that (1) demonstrate nonlocal transport and compare it to linear models, (2) identifies signatures of linear and nonlinear elements of transport models, and (3) explores detailed consequences of nonlocal models.
Chapter Two presents the first demonstration of nonlocal sediment transport in a natural setting. We numerically simulate the evolution of several lateral moraines according to linear, nonlinear, and nonlocal models, and evaluate the model output against the observed moraine form. In doing so, we demonstrate the nonlocal models can explain the morphology of the lateral moraines more accurately than linear models. We show that nonlocal models and nonlinear models share mathematical properties and for this reason, nonlinear models also reproduce moraine profiles that nearly match the observed.
Chapter Three explores possible signatures for local linear, nonlinear, and nonlocal models that we can expect to find in the land-surface form. In particular we place the evolution of certain landforms with specified boundary conditions into wavenumber domain via the Fourier Transform. Theory and observation show that linear diffusion is manifest in wavenumber domain as vertical spectral decay whereas nonlinear processes are manifest as compressional spectral decay. Despite linear and nonlinear processes appearing very similar in land-surface evolution in the spatial domain, they are distinctly different in wavenumber domain. We also show that insofar as nonlocal models can have linear and nonlinear components there is a range of behaviors in wavenumber domain for nonlocal processes.
Sediment transport occurs by instances of particle motion separated by periods of rest. Chapter Four of this dissertation is a probabilistic approach to characterizing the rest times of particles on hillslopes. Rest times are one of two components that describe the fundamental diffusive behavior of a transport process which can be either normal or anomalous diffusion. Normal diffusion can result when the distribution of particle rest times has a thin tail. Anomalous diffusion can result when particle rest times have heavy tails. This chapter demonstrates that, in the absence of temporary dams, particle rest times on hillslopes are a mixture of a power law and exponential distributions but have thin tails. However, when temporary blockages are added to a hillslope, the rest time distributions can become heavy-tailed if the distribution of blockage ages is also heavy-tailed. We further suggest that such behavior could be observed on hillslopes with 137Cs. This would be the first demonstration of particle spreading behavior on a hillslope.