Self-Stabilized Fractality of Seacoasts through Damped Erosion

Abstract

Erosion of rocky coasts spontaneously creates irregular seashores. But the geometrical irregularity, in turn, damps the sea-waves, decreasing the average wave amplitude. There may then exist a mutual self-stabilisation of the waves amplitude together with the irregular morphology of the coast. A simple model of such stabilisation is studied. It leads, through a complex dynamics of the earth-sea interface, to the appearance of a stationary fractal seacoast with dimension close to 4/3. Fractal geometry plays here the role of a morphological attractor directly related to percolation geometry.PACS numbers: 64.60. Ak,92.40.Gc Coastline morphology is of current interest in geophysical research [1], and coastline erosion has important economic consequences [2]. Also, the recent concern about global warming has increased the demand for a better understanding of coastal evolution. At the same time, although the geometry of seacoasts is often used as an introductory archetype of fractal morphology in nature [3,4], there has been no explanation about which physical mechanism could justify that empirical observation. In the field litterature[5] one can read: "As a matter of some urgency, researchers concerned with coastal evolution should consider the alternative models, even if there are few supporting data. The ideas of ... stochastic development, ... and criticality, all deserve investigation." The present work propose a minimal, but robust, model of evolution of rocky coasts towards fractality.The model describes how a stationary fractal geometry can arises spontaneously from the mutual selfstabilization of coast morphology and sea eroding power [6]. If, on one hand, erosion generally increases the geometrical irregularity of the coast, on the other hand this increase creates a stronger damping of the sea and a consequent diminution of its eroding power. The increased damping argument relies on the studies of fractal acoustical cavities, which have shown that viscous damping is augmented on a longer, irregular, surface [7,8]. In the following, a minimal two-dimensional model of erosion is introduced which leads to the spontaneous evolution of a smooth seashore towards fractality as shown in Fig. 1.Rocky coasts erosion is the product of marine and atmospheric causes [9]. There exist many different erosion processes: wave quarrying, abrasion, wetting and drying, frost shattering, thermal expansion, salt water corrosion, carbonation, hydrolysis. A simplified picture is used here by assuming that the different processes can be separated into two categories: "rapid" mechanical erosion (namely wave quarrying) and "slow" chemical weakening. The justification is that mechanical erosion generally occurs rapidly, mainly during storms, after rock has been al- tered and weakened by the slow weathering processes. We first study the supposedly rapid erosion mechanisms. Then we show that the full complex dynamics, involving fast and slow processes, build a dynamic equilibrium that preserves the fractal shape of the coast.The sea, t...

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