Predicting waves of large amplitude for tsunami warning systems

Funded by the Vienna Science and Technology Fund (WWTF), the project 'The flow beneath a surface water wave' aims to investigate the effect of currents on ocean waves. The results could be of importance for tsunami warning systems.

How can one distinguish mathematics from physics? While mathematics investigates properties of man-made abstract structures, physics seeks the general rules of nature by using mathematics. Adrian Constantin, who regards himself a 'pure mathematician', is entering unknown territory. The professor at the Faculty of Mathematics at the University of Vienna is investigating mathematical models for wave-current interaction. The aim of the four year project 'The flow beneath a surface water wave' is to study the effect of an underlying current on a surface water wave. 'Seaman claim that certain currents could double the amplitude of incoming waves. It is a big challenge to prove this assertion.'

When the sea retreats

The main reason for Constantin's study was the catastrophic tsunami in 2004 which resulted in more than 230,000 victims. Many aspects related to the disaster still remain vague. For example, prior to the arrival of the tsunami in Thailand the sea retreated, while in India there was no such warning. 'Satellite measurements show that in the direction of Thailand there was a first wave of depression, while in India the opposite direction was first confronted with a wave of elevation,' says Constantin.

Danger on beaches with a mild slope

The initial shape of the wave determines the behavior of tsunami waves as they approach the shore. A related question is the number of waves that hit a particular location. Mathematical considerations show that this number is related to the wave profile at its source. 'The number of tsunami waves close to the shore is never greater than the initial number of waves,' he explains. The amplitude of these waves can only be predicted roughly since this aspect is closely related to the often very complicated topography of the sea bed.

If we know the shape of the tsunami waves at an earlier point in time, we could predict some of its features as it approaches the shore. Constantin says, 'Tsunami waves are very dangerous at mildly sloped beaches, whereas in steep regions reflection occurs. The speed of the tsunami wave is proportional to the square root of the water depth.'

Pressure on the seabed

Constantin, an occasional diver, finds the pressure deep down in the sea to be of special interest. 'The pressure beneath provides information about incoming waves. As the wave crest approaches overhead one notices an increase of pressure, while a decrease in pressure signals the arrival of a wave trough'.

Data from the flume

In collaboration with the Franzius Institute of Hydraulics, Waterways and Coastal Engineering at the Leibniz University of Hannover Constantin is investigating whether the conclusions derived from mathematical models are in agreement with field data. 'The cooperation with the Ocean engineers is important to us. Because of their involvement in city planning for coastal regions in South East Asia threatened by tsunami events, they have access to a lot of tsunami specific data.' Moreover, the longest water wave tank in the world is located in Hannover and is being used by Constantin and his team for experiments.