Modeling sediment transport with random walks

The motion of a sediment grain along the bed of a river is fundamentally a random walk. A grain moves intermittently, with an episode of motion followed by an interval of rest. The factors that control the erosion, transport, and deposition of sediment are so complex that the length of a particular step or the duration of the subsequent rest should be considered random variables, as opposed to deterministically predictable quantities. The sum of these random steps and rests is the trajectory of the sediment grain and a random walk. The power of the random walk as a mathematical tool is that the distribution of the sum tends towards one of two limiting forms. The distribution of outcome of a random walk describes both the uncertainty in the trajectory of a single grain and the expected spatial distribution of a population of sediment grains. Its limiting form depends on the moments of the underlying probability distributions of step length and resting time. This work is an attempt to constrain the form of those distributions for fluvial sediment, and in particular, to look for evidence that step lengths and resting times follow probability distributions that are "heavy-tailed." Chapter 1 finds evidence for heavy-tailed step lengths in the data from a 50-year-old tracer experiment. Chapter 2 proposes a mechanism by which heavy-tailed resting times could arise on the floodplain of a meandering river. Chapter 3 describes the early results from a large tracer experiment designed to measure steps and rests directly in a natural stream.