Nonlinear hydroelastic waves with applications to ice sheets

Free boundary problems are challenging mathematical problems which involve solving nonlinear problems in domains whose shapes have to be found as part of the solution. They occur in many industrial, environmental and biological applications. Examples are coating problems, ocean waves and tumors. In this proposal we concentrate on hydroelastic waves and in particular on applications to ice sheets. The research is motivated by the understanding of man-made large floating structures (especially airports) and by Antarctic exploration where often heavy equipment will travel over roads on floating ice and aircraft will operate on floating ice-sheet runways. Waves under ice sheets have also recently attracted attention because they are one of the many factors that need to be considered as having an impact in climate change.
The mathematical formulation of hydroelastic waves involve complicated systems of nonlinear integro-differential equations for which most studies have been restricted to linear models. Although linear approximations are often adequate, there are many situations in which nonlinearities and large defections of the ice sheet cannot be neglected. Therefore we propose to develop fully nonlinear numerical theories and weakly nonlinear asymptotic models to tackle these problems. Solitary waves, dark solitons, internal waves and three dimensional waves are among the topics to be studied. The previous experience of the PI, CIs and visiting researcher with nonlinear waves will be very instrumental for the success of the project.

The intended RA is Dr Leonardo Xavier Epsin. His strong background in modelling, asymptotics and numerical simulations
makes him very well suited to work on the proposed problems. In case he were not able to take the position, other qualified candidates are known to the Pi and CIs.

Water waves in general have far reaching impact on the environment and on many aspects of human activity. Their mathematical study has a broad range of impacts beyond the field of mathematics: from interpretations of remote sensing of the ocean's surface to atmosphere-ocean interactions in general circulation models (GCM) and tsunami predictions. In the case of water waves under ice sheets, the questions we pose and shall answer during this research, in particular, have impact on environmental and climate science, on the use of surface ice in polar regions for vehicles and aircraft landing, and on the behavior of other large scale floating structures.

Waves under ice sheets have attracted renewed interest, given that they are one of the many factors that need to be considered as having an impact in climate change. In general it is understood that waves propagating under ice affect sea-ice extent. Most pertinent to the current modelling effort may be the recent implication of infra-gravity water waves (a term used to denote waves generated by various processes and with period longer than 30 seconds) in the breakup of large portions of the floating Ross ice shelf in Antarctica (Brominski (2010)). The ocean swell, which ultimately creates these infra-gravity waves is generated by large storms in the mid-oceans. Whether a feedback occurs between heightened storm activity due to climate change and a hastening of the breakup of ice shelves due to wave-ice interaction is an open question.

A direct human impact of the research is in the understanding man-made large floating structures (especially airports) and in Antarctic exploration where often heavy equipment will travel over roads on floating ice, or aircraft will operate on floating ice-sheet runways. In all these cases, it is important to understand the behavior of waves generated by moving loads (heavy equipment or aircraft) in order to assert the safety of such operations. Indeed there have been extensive experimental campaigns (Squire et al (2011) and references therein) to understand the response of floating ice to moving forces and particular attention needs to be payed precisely in the resonant regime we propose to study. Man-made Very Large Floating Structures (VLFS) have been proposed in certain areas where land is at a premium or for military purposes. In particular a prototype 1000 metre runway - named Megafloat - was built in the bay of Tokyo in 2000 to test the feasibility of such projects. A comprehensive review of the use of VLFS and their hydroelastic response can be found in Andrianov (2005).

REFERENCES:

P. D. Bromirski, O. V. Sergienko, and D. R. MacAyeal (2010), Geophys. Res. Lett., 37
L02502.

V.A. Squire (2011), Phil. Trans. R. Soc. A, 369,
2813-2831.

A.I. Andrianov (2005), Hydroelastic analysis of vert large floating structures,
Ph.D Thesis, Mathematics and Computer Science, Delft University of Technology.