The size of large cliff failures may be described in several ways, for instance considering the horizontal eroded area at the cliff top and the maximum local retreat of the coastline. Field studies suggest that, for large failures, the frequencies of
these two quantities decrease as power laws of the respective magnitudes, defining two different decay exponents. Moreover, the horizontal area increases as a power law of the maximum local retreat, identifying a third exponent. Such observation suggests that the geometry of cliff failures are statistically similar for different magnitudes. Power laws are familiar in the physics of critical systems. The corresponding exponents satisfy precise relations and are proven to be universal features, common to very different systems. Following the approach typical of statistical physics, we propose a scaling hypothesis resulting in a relation between the three above exponents: there is a precise, mathematical relation between the distributions of magnitudes of erosion events and their geometry. Beyond its theoretical value, such relation could be useful for the validation of field catalogs analysis. Pushing the statistical physics approach further, we develop a numerical model of marine erosion that reproduces the observed failure statistics. Despite the minimality of the model, the exponents resulting from extensive numerical simulations fairly agree with those measured on the field. These results suggest that the mathematical theory of percolation, which lies behind our simple model, can possibly be used as a guide to decipher the physics of rocky coast erosion and could provide precise predictions to the statistics of cliff collapses.
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 irregul
ar 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.
