Conference on Mathematical Geophysics: Pattern and Form in Earth Dynamics
(Turin, Italy, June 2002)

Pattern formation from finite-amplitude interactions: The example of sandy coastlines

A. Brad Murray, Andrew Ashton
Div. of Earth and Ocean Sciences
Nicholas School of the Environment and Earth Science/ Center for Nonlinear and Complex Systems
Duke University

Pattern formation in geophysical systems can be divided into two categories. In the first, instabilities cause infinitesimal perturbations to grow, but negative feedbacks relatively quickly stabilize their amplitude, preserving a pattern that strongly resembles the predictions of linear or weakly nonlinear stability analysis. Fluid convection patterns (driven by buoyancy and/or surface tension) provide the classic example.

In the second kind of pattern formation, perturbations that grow because of an instability develop an amplitude and shape that significantly affects their surroundings. The growing features then interact with each other in ways that cause the pattern to further evolve. The pattern that develops after an extended time results from these finite-amplitude interactions, and no longer resembles the initially fastest-growing instabilities. Many of the patterns visible in nature fall into this second category. For example, eolian ripples first emerge as small disorganized bumps, but interactions between these bumps lead to a pattern that exhibits continual increases in wavelength and organization. Other geomorphological examples in which finite-amplitude interactions determine pattern characteristics span the range from fluvial and nearshore morphodynamics to periglacial sorted patterns.

The self-organization in a numerical model of large-scale, plan-view features on sedimentary coastlines provides striking examples of this second type of pattern formation. With some wave climates, an instability inherent in the relationship between alongshore sediment flux and shoreline orientation causes the growth of perturbations on an otherwise straight shoreline. With an extended spatial domain, growing features interact with each other in ways that lead to coastline shapes and behaviors that could not be predicted by examining the initial instability. The mechanisms of the interactions and the resulting emergent structures and behaviors depend on the wave climate (the distribution of angles that waves approach from), but the scale of coastline features increases with time in any case. Possible natural examples include the 100-km-scale capes and 'cuspate forelands' that constitute much of the southeastern coast of the United States, and the 'cuspate spits' that protrude from the coast of the Sea of Azov (off of the Black Sea).