Data-driven reduced order prediction of extreme waves
Will Cousins, Themistoklis P. Sapsis
Rogue or freak waves are ocean waves whose height is abnormally large for a particular sea state. Often described an enormous "wall of water," (e.g. the New Year wave in Fig. 1) such waves have caused catastrophic damage to ships and coastal structures. For example, in 1978 the German super-tanker München vanished, along with her 26 crew members. Searches for the ship recovered little, but a lifeboat was recovered whose attachment pins showed evidence of being subjected to a great force. As this lifeboat was stowed 20m above the water line, some have conjectured that the München may have been struck by an extremely large wave [Liu, Geofizika 24, 2007]. Here we describe our method for reliably predicting these rogue waves before they occur while expending minimal computational effort, which we term Reduced Order Prediction of Extremes (ROPE).
The large, steep nature of these rogue waves, combined with recent evidence that they can occur more likely than Gaussian statistics would suggest, imply that nonlinear models are necessary to fully understand their dynamics. Thus, we focus our attention on models that incorporate this nonlinearity while remaining simple enough to be tractable. Two such examples are the equation of Majda, McLaughlin, and Tabak (MMT) and the Nonlinear Schrodinger Equation (NLS). The MMT model is similar to NLS, differing in that it includes a "fractional" differential operator in place of the 2nd derivative in NLS [Majda et. al., J. Nonlinear Sci. 6, 1997]. Although NLS and MMT are quite simple, they have attracted attention due to an inherent mechanism for generating rogue waves. Specifically, simple periodic solutions to both equations are unstable, with small perturbations initiating "focusing" events, where the nearby wave field is soaked up to produce a large, localized wave. This process, known as the Benjamin-Feir instability, is well understood theoretically and has been reproduced in experiment [Chabchoub et. al. PRL 106, 2011].
Figure 1: The New Year wave recorded at the Draupner platform in the North Sea on January 1, 1995.