June 14, 2021

# Mathematics to respond to a tsunami | Science

Every moment, the nodes of the international seismic network collect movement data at the bottom of the sea, while thousands of buoys, strategically located, detect irregularities on the surface. This information is sent continuously to the alert centers, where seismic events that exceed a certain intensity are identified and occur in critical areas. When this happens, an alert is generated, and the protocols are launched. Given the risk of a tsunami, the important thing is to act quickly. Mathematics is key to predicting the magnitude of the catastrophe and mitigating, as far as possible, the damage.

After five minutes enough data is available (at least the location and intensity of the earthquake) to make the first numerical simulations. The mathematical models predict in real time the propagation of the wave train and the impact on the coast. The model chosen in each center will depend on the type of information they want to obtain, seeking the best balance between being descriptive, on the one hand, and manageable, on the other. The more precision you look for, the more physical description should be entered into the model.

In a tsunami there are several physical phenomena, governed by different partial differential equations. A general idea of ​​the height of the waves or their arrival time to the coast can be deduced from the wave propagation equations. They are linear equations (the variables have no exponents or operators complicated), simple to formulate and solve. However, to obtain more refined information it is necessary to consider non-linear equations. The most elementary are the shallow water equations. They were proposed at the end of the 19th century by the French mathematician and engineer Adhémar Jean Claude Barré de Saint-Venant, who obtained them from the Navier Stokes equations (which describe the movement of a fluid and whose resolution is one of the millennium problems), simplifying the complexity.

The shallow water system is somewhat simpler, although it also has its analytical difficulties. In any case, to obtain predictions it is necessary to perform numerical approximations. It is important that they are good estimates but also that they can be calculated (using high performance computing techniques) in a short period of time. If you do not have the results before the wave arrives, the model would be useless. The group EDANYA, at the University of Malaga, has managed to perform a real simulation of eight hours of propagation of a tsunami in the Mediterranean in 30 seconds. For now, his is the best solution for tsunami modeling.

This group works, among other topics, in the design of new models for the simulation of tsunamis by adding additional terms to the equations of shallow water, as well as in efficient techniques for their discretization, which determine the resolution that will have the result. "In propagation in the middle of the ocean you can use tights -Retils that are used to find approximate solutions- of kilometers, but as the wave approaches the coast, higher resolution will be needed ", explained Manuel Castro Díaz, member of the group. Afterwards, the team determines how to efficiently implement this type of algorithm, using computer graphics cards to reduce computing time. The result is an accurate prediction (with errors less than 5% when compared to data obtained in laboratory experiments) on the shape and height of waves in near-shore areas, and flood maps (ie , how far the wave will reach in the affected territory).

But could the model fail, and not get solutions (or not give them in time)? Yes, although it is unlikely. For the moment, just in case that happens, bases of pre-calculated solutions are used, catalogs of hundreds of thousands of events that have previously been simulated. "It is a lot of stored information and it is complicated to process, so many problems are generated. Although at first the alert centers were reluctant to use the simulation tool in real time, because it is new, they realize that it is more than enough: it provides more flexibility and precision, "concludes Castro. Your models already used in the Tsunami Early Warning Center, with headquarters in Rome and Pisa, for example, which monitors the most seismic areas of the Mediterranean, off the Greek coasts, in Turkey and the coast of Algeria.

Beyond this application, the approximation and numerical simulation of the EDANYA group is used to model other phenomena governed by similar physical laws: the littoral and fluvial dynamics; the formation of planets or black holes; the cardiovascular system … The models allow studying the evolution of environmental and health fluids in all these cases, with little cost and great precision.

Agate Rudder (ICMAT).

Coffee and Theorems is a section dedicated to mathematics and the environment in which they are created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share points of contact between the mathematics and other social and cultural expressions and remind those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: "A mathematician is a machine that transforms coffee into theorems".

Editing and coordination:Agate Rudder (ICMAT).