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Fractals

Jagged coastlines

Physicists try to explain why coastlines vary so much

NINA MATTHEWS / CREATIVE COMMONSA model developed by physicists from the Federal University of Ceará (UFC) and from ETH, the Swiss Federal Institute of Technology, is the first to produce computer simulations of a considerable variety of possible coastline designs. The authors of the study, published in July in Physical Review E, are the first to admit that this is a simplified approach to a complex phenomenon. However, they hope that the model, which exploits the use of known geometrical figures, such as fractals, might in the future help to monitor maritime erosion, a constant concern for coastal cities.

“Clouds are not spheres, mountains are not cones and shores are not circles,” the French mathematician Benoit Mandelbrot once said. He was the person who created the term “fractal” in 1975, referring to the inability of conventional geometry to depict the forms of nature. Fractals, the geometric forms with a rough appearance, full of hollows, do the job much better. One can observe this in the satellite pictures of Google Earth, the online tool. Several stretches of coast in the world – especially in Norway, but also in Brazil, particularly along the border between the states of Pará and Maranhão, and São Paulo and Rio de Janeiro, seem not to change their appearance, regardless of how high they are seen from. Parts of the coast just a few kilometers long look like miniature versions of stretches hundreds of kilometers long. This is the chief property of fractals: the similarity between the details of the parts and the complete figure.

Certain parts of the coast, however, have more tortuous contours,  seeming to have a sharper “fractality,” a notion mathematically captured by the fractal dimension concept. The value of this quantity can range from 1, for a long beach with a smooth coastline, such as certain parts of the northeastern shore near a perfectly straight line, up to, in theory, 1, for a very irregular coastline, full of bays and capes within more capes and bays, that measuring its perimeter with maximum resolution is almost impossible. Analyses suggest that real coastlines do not go beyond 1.6 – most lie between 1 and 1.4.

Models
Although coastlines have been cited as examples of fractals since the 1960’s, it was only in 2004 that the first explanation about how nature sculpts them arose. The French physicist Bernard Sapoval and his Italian colleagues Andrea Baldassari and Andrea Gabrielli created a simple model of the erosive force of the sea upon rocky coasts. According to the proposed mechanism, the tortuousness of the shore results from a balance between the force of the waves and the capacity of the coastline to attenuate it.

As the bays and capes carved by the sea become more jagged, the power of these geological features to trap and dissipate the power of the waves rises. The mechanism is assumed to be the same as that which makes walls with rough surfaces excellent sound insulators. Though they look realistic, the coastlines that emerged from the simulations always had a fractal dimension of 1.33, a figure that describes the east coast of the United States and parts of the Rio de Janeiro southern coast well. However, this does not account for the full variety of shore contours.

048-049_Erosao_187After Sapoval presented this work at an UFC seminar, the physicist José Soares de Andrade Junior and his doctoral students Pablo Morais and Erneson Oliveira started to think about how to produce virtual coastlines with different fractal dimensions. Together with a Portuguese scientism, Nuno Araújo, and a German one, both of whom are physicists at ETH, they created a model that despite simplifying the impact of the sea substantially, deals more realistically with the spatial distribution of rocks. Whereas the preceding model randomly distributed the rocks that were more resistant to erosion or less so along the coast, in a non correlated manner, the new model tries to simulate the affinities that exist between neighboring rocks, by introducing far reaching statistical correlations in space.

“These correlations can change the coast’s fractal dimension,” says Andrade. Thus, the researchers were able to generate coastlines with fractal dimensions that ranged from 1 to 1.33, depending on the distribution of resistance to erosion of the rocks.

Moreover, the new model suggests that the coast only takes on fractal shapes when the power of the sea is offset by the hardness of the rocks. If the resistance of the rocks is far greater than the power of the waves, the coast acquires a rough shape, but it is not fractal. When the power of the waves outdoes, by far, the resistance of the rocks, the coast is eroded continuously and the coastline acquires the shape of a special type of fractal, called self-affine. “This fractal has unequal contraction and dilation properties in different directions,” explains Andrada. He expects to identify these different coastal geometries in real satellite images soon. With the aid of geologists from the UFC Sea Sciences Institute, he plans to find out whether the dynamics of the erosion occur in the way that the model forecasts. “If, for example, we identify accelerated erosion,” he says, “some type of protection measure could be taken.”

As with all models , those of the French-Italian group and of the UFC team are a simplified representation of reality and disregard a factor that oceanographers and coastal engineers consider essential for determining the coastline: the submarine relief, which determines the direction in which the waves are propagated and how they hit the coast.

The rivers of sand, the flow of sediments raised by the waves and carried by the ocean currents, are another important detail that the models disregard. Dieter Muehe, a coastal geomorphology specialist from the Federal University of Rio de Janeiro, explains that the areas with the greatest risk of erosion lie at the points where the waves converge  and where the occurrence of the phenomenon involves greater energy.

Andrade admits that he does not know exactly how to include the movement of sand in his model, which he regards as a first approximation of what actually occurs on coastlines on a continental scale. The movement of sediment, by comparison, should be on a smaller scale, approximately the size of a beach. The UFC group that is also studying the movement of sand dunes has started investigating how sand is carried by water. Andrade states: “It is necessary to study the phenomenon from the physical standpoint, on a smaller scale, before transposing it to a larger one.”

Scientific article
MORAIS, P.A. et al. Fractality of eroded coastlines of correlated landscapes. Physical Review E. July 7, 2011.

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