A numerical retro-action model relates rocky coast erosion to percolation theory

We discuss various situations where the formation of rocky coast morphology can be attributed to the retro-action of the coast morphology itself on the erosive power of the sea. Destroying the weaker elements of the coast, erosion can creates irregular seashores. In turn, the geometrical irregularity participates in the damping of sea-waves, decreasing their erosive power. There may then exist a mutual self-stabilization of the wave amplitude together with the irregular morphology of the coast. A simple model of this type of stabilization is discussed. The resulting coastline morphologies are diverse, depending mainly on the morphology/damping coupling. In the limit case of weak coupling, the process spontaneously builds fractal morphologies with a dimension close to 4/3. This provides a direct connection between the coastal erosion problem and the theory of percolation. For strong coupling, rugged but non-fractal coasts may emerge during the erosion process, and we investigate a geometrical characterization in these cases. The model is minimal, but can be extended to take into account heterogeneity in the rock lithology and various initial conditions. This allows to mimic coastline complexity, well beyond simple fractality. Our results suggest that the irregular morphology of coastlines as well as the stochastic nature of erosion are deeply connected with the critical aspects of percolation phenomena.

💡 Research Summary

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The paper presents a minimal yet powerful numerical model that captures the feedback loop between rocky coastline morphology and the erosive power of sea waves. The authors argue that the irregular geometry of a coast increases the viscous damping of incoming waves, which in turn reduces the wave amplitude and therefore the erosive force acting on the shoreline. This retro‑action creates a self‑stabilizing system: as erosion removes weaker rock elements, the coastline becomes longer and more irregular, enhancing damping and limiting further erosion.

The model treats the sea‑coast system as a damped resonator. An average wave power P₀ supplies energy, and the wave amplitude Ψ satisfies Ψ² ∝ P₀ Q, where Q is the quality factor. Q is decomposed into a morphology‑dependent component Q_coast (due to viscous losses along the shoreline) and a morphology‑independent component Q_other. Empirical studies of acoustic cavities with fractal boundaries suggest that viscous losses scale roughly with the perimeter; the authors adopt Q_coast ∝ 1/L_p(t), where L_p(t) is the total coastline length at time t. Consequently, the erosive force per unit length is modeled as

 f(t) = f₀ /