Mathematical frontiersSpanish mathematicians model submarine avalanches and tsunamis

A team of Andalusian and French scientists has put forward a mathematical model that enables submarine avalanches and certain types of tsunamis to be studied using equations, according to a recent article in the Journal of Computational Physics. Mathematicians are already applying the model to analyze landslides on the island of Alborón (Almería).

“El modelo que hemos desarrollado puede servir para predecir aquellos tsunamis que se forman tras la caída brusca de sedimentos cerca de la costa” (“The model we have developed can be used to predict tsunamis that form following the sudden fall of sediments near the coast”), Enrique Fernández Nieto explains. Fernández is one of the authors of the study and is from the Department of Applied Mathematics from the University of Seville. Rock-fall can be favored by the existence of an abrupt slope at the bottom of the rock-fall, by an accumulation of material, or by a strong surge that destabilizes the layer of sediments and causes it to fall.

The study uses equations known as Savage-Hutter equations, named after the two scientists who proposed these in order to study rock avalanches. “However, for the first time we are addressing the need to take into account the coupling between the two layers implicated in the processes of submarine avalanches: water and rocks,” says Fernández. He considers that studying this two-layer coupling “is complicated, but fundamental to the total dynamic, because rock movements cause the water to move, and possibly the tides of water that can displace granular material.”

To derive the model, the mathematicians took into account the porosity of the sediments, the forces that interact in the process, and the “Coulomb friction term” (“fricción de tipo Coulomb”), which refers to the parameters of the equation that are opposed to the movement of the mass of rock when it falls.

To explain this concept, Fernández points to the example of a column of water in a container, the door of which is opened: the liquid spills out and is directed toward a horizontal, constant, and still surface. When the same experiment is performed with a column of sand grains, the final state has the shape of a bell. The terms that produce this resulting slope, which is no longer horizontal due to the friction between the particles, is what is referred to as the “Coulomb friction term.”

In addition to the internal friction angles and at the bottom, the mathematicians evaluate other parameters such as the flotability of the submerged material, the topography of the land, the initial direction of the water and its height. Once all these data have been obtained, the mathematicians introduce these into a computer program and by using animations analyze and visualize the evolution of the submarine avalanches and tsunamis.

The researchers have established the effectiveness of their equations using data from a very well documented tsunami that occurred in Papua New Guinea in 1998. In partnership with geologists from the Spanish Institute of Oceanography (IEO) and by means of a joint project financed by the Junta de Andalucia, the researchers are studying the period during which certain landslides occurred in the past in the vicinity of the Island of Alborán (Almería), and are also evaluating the likelihood of their recurring in the future in the Alborán marine basin.

Scientists from the University of Seville, the University of Malaga, the Escuela Normal Superior in Paris, the University of Savoie, also in France, and from the Seismology Team from the Institute of Physics from the Globe of Paris have all participated in the study.