Self-stabilised fractality of sea-coasts through damped erosion.

B. Sapoval, 
Laboratoire de Physique de la Matière Condensée, C.N.R.S. Ecole Polytechnique, 91128 Palaiseau, France.

Centre de Mathématiques et de leurs Applications, Ecole Normale Supèrieure, 94235 Cachan, France.

A. Baldassarri,
INFM, UdR Roma 1, Dipartimento di Fisica, Università di Roma "La Sapienza", P.le Aldo Moro 2, 00185 Rome, Italy.

A. Gabrielli
"Enrico Fermi" Center Via Panisperna 89 A, Compendio del Viminale, Palaz. F, 00184 Rome, Italy.

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.

Time evolution of the coastline morphology. Case of a weak coupling between the damping of the sea-waves and the coast morphology.
Left : Beginning of the erosion process, the coast is irregular but not yet fractal.
Right : Fractal coastline at the end of the erosion process. The fractal dimension is equal to 4/3. This is due to the close link between the erosion mechanism and gradient percolation. Here the resulting gradient is small.

Snapshots taken during long term erosion for a small system  with a moderate coupling. Note that the effective fractal dimension fluctuates around the universal value 4/3 corresponding to very small gradient.  For a stronger damping coupling with the sea force, the observed dimension due to the same mechanism would be smaller than 4/3.

"Artistic" view of a self-stabilized fractal sea-coast