How Fractal are Coastlines Really? Observation and Theory

Rocky coastlines have been held up as a prime example of fractal geometry since Mandelbrot introduced the concept. However, we will present a map of the fractal dimensions measured for the contiguous United States coastline which shows that many open-ocean sand--and even rocky--coastlines have fractal dimensions close to one; i.e. they tend to not be very fractal. The fractal nature of rocky coastlines likely represents an inherited fluvial or glacial signature that tends to be erased by coastal processes. Recent theoretical and numerical-modeling developments indicate that wave-driven coastal processes on sandy shores tend to produce one-dimensional coastlines. Gradients in alongshore sediment flux tend to smooth a shoreline, as long as the local wave climate is dominated by 'low-angle' waves (waves that approach the coastline in deep water from angles, relative to the coastline orientation, that are lower than the sediment-flux- maximizing angle). Even when a regional wave climate is dominated by high-angle waves--which produce an instability in plan-view shoreline shape--on the large scale, coastlines self organize in a way that produces locally low-angle-dominated wave climates almost everywhere. These processes explain why wave-dominated sandy coastlines, such as the Carolina and Texas coasts, exhibit fractal dimensions barely above one; wave- driven alongshore transport is an anti-fractal landsculpting agent over a range of scales greater than 0.2 km. In contrast, fluvial landsculpting produces famously fractal topography. When rapid sea-level rise causes the approximately horizontal plane of sea level to intersect a fractal fluvial topography, a fractal coastline results. Where wave energy is low, relative to rock erodibility, the fluvial fractal signature can persist. However, on the rocky West Coast of the US, fractal dimensions are relatively low (1.1 - 1.2), suggesting modification by wave-driven processes; that the production and rearrangement of sediment into ever-expanding pocket beaches has been reducing the fractality of this high-wave-energy, relatively easily eroded coastline. Glacially carved coastlines, such as that of Maine (and some parts of western Britain and Norway), exhibit high fractal dimensions (approximately 1.5), where erodibility is low enough the self-similarity of the intersection of sea-level with a glacially sculpted topography remains. Although wave-driven coastal processes tend to generate low-fractal-dimension shorelines, on sandy coastlines dominated by tidal currents, coastal processes also etch a fractal dendritic network of channels into the coastline. Tidally dominated coastlines, such as those in the Georgia Bight (Southeastern US), sport highly fractal shapes as a result (fractal dimensions approximately 1.5).