"Pattern Formation from Emergent, Finite Amplitude Interactions: the Example of Sandy-Coastline Evolution"
Researchers are increasingly studying morphodynamic instabilities and self-organization as possible explanations for pattern formation in nature. Often, such research concentrates on the initial instability; how a morphological perturbation can cause a change in the patterns of flow and sediment transport that in turn cause the perturbation to grow. These investigations often employ linear (or weakly nonlinear) stability analyses that address whether an instability exits, and if so what scales of perturbations will grow most rapidly, starting from a homogeneous 'basic state.' Numerical models designed to examine the effects of infinitesimal-amplitude perturbations on flow and sediment transport are also used to address these questions. Such investigations involve the explicit or implicit assumption that the wavelengths and shapes of the fastest-growing perturbations can be related to the characteristics of natural patterns. Examinations of how a pattern evolves after perturbations reach finite amplitude are less common. However, in many cases the interactions between finite-amplitude features cause a pattern to change both quantitatively and qualitatively after the initial stage of perturbation growth. For example, natural and modeled eolian-ripple patterns increase in wavelength and plan-view organization as smaller and faster ripples overtake and merge with larger and slower ones [e.g. Anderson, 1990; Landry and Werner, 1994]. Laboratory and model braided-stream patterns evolve from the initially developing, fairly regular checkerboard pattern of thalwegs and shoals predicted by linear stability analyses into a chaotic assemblage of fully developed and dynamic bars and channels with a wide range of scales [e.g. Murray and Paola, 1994; Murray and Paola, 1997]. In such cases, the characteristics of the fully developed patterns observed in nature are created by the interactions between emergent structures-features with scales and shapes not predicted by examinations of the equations governing slight variations from a simple basic state-and do not reflect the scale or arrangement of the initially fastest growing perturbations. Investigating a finite-amplitude mode of pattern formation cannot be accomplished analytically, but requires experimentation with physical models or numerical models designed to treat geometrically complex and evolving, finite-amplitude morphological features and associated boundary conditions. The dynamics of plan-view coastline patterns in a recently developed model [Ashton et al., 2001] provide an extreme example of self-organization resulting from interactions between emergent structures. Ashton et al.  pointed out that considering alongshore sediment flux as a function of the relative angle between deep-water waves and local shoreline orientation leads to the conclusion that an instability in plan-view shoreline shape is a robust possibility; perturbations in an otherwise smooth shoreline will grow when waves in deep water approach from angles that deviate sufficiently from shore-normal. Treating the case of a single angle of wave attack, we perform a linear stability analysis. This analysis, while verifying that for wave-angle distributions weighted toward high-angle waves perturbations will grow, is not relevant to shoreline evolution in nature. It cannot be used to address: 1) the characteristics of perturbations growing to finite-amplitude; 2) the results of interactions between different growing features on an extended length of shoreline; or 3) the more-realistic case of waves approaching from different angles at different times. A simple numerical model designed to explore the long-term evolution of an extended shoreline domain [Ashton et al., 2001] shows several modes of interactions that lead to large-scale coastline features including cuspate forelands, cuspate spits, and alongshore 'sandwaves.' (This model is similar to '1-line' models commonly used in coastal engineering, but employs an algorithm designed to treat arbitrarily complex shoreline configurations and the full range of wave-approach angles. This model is also specifically designed to address large-scale morphodynamics; it implicitly averages over relatively small-scale and short-term variations in shoreline, surf-zone, and inner shelf morphology and the resultant variations in wave conditions.) Here we highlight the various ways that large-scale features can emerge and interact with each other in the model under various wave climates, leading to the organization of the different patterns. The changes in shoreline orientation resulting from the growth of large-scale features change the sediment fluxes into neighboring shoreline segments. In addition, a feature can alter the local wave climates felt in other shoreline regions. When features develop large cross-shelf amplitudes, this affect can extend great distances. The resulting self-organization of coastline shapes involves the interplay of local and non-local, nonlinear interactions, and could not be readily predicted by examining the equations governing local sediment flux or the initial growth of shoreline perturbations. References Anderson, R., Eolian ripples as examples of self-organization in geomorphological systems, Earth Sci. Reviews, 28 (29), 77-96, 1990. Ashton, A., A.B. Murray, and O. Arnoult, Formation of coastline features by large-scale instabilities induced by high-angle waves, Nature, 414, 296-300, 2001. Landry, W., and B.T. Werner, Computer simulations of self-organized wind ripple patterns, Physica D, 77, 238-260, 1994. Murray, A.B., and Paola, C., 1994, A cellular model of braided streams: Nature, v. 371, 54-57. Murray, A.B., and C. Paola, Properties of a cellular braided stream model, Earth Surf. Proc. Landf., 22, 1001-1025, 1997.