Prediction and quantification of rare events in nonlinear water waves
Themis Sapsis

Abstract: The scope of this work is the development, application, and demonstration of probabilistic methods for the prediction and quantification of extreme events occurring in complex nonlinear systems involving water waves. Although rare these transitions can occur frequently enough so that they can be considered of critical importance. We are interested to address two specific issues related to rare events: i) short term prediction given measurements of specific quantities about the current system state (Rare Event Prediction Problem); and ii) quantification of the probability of occurrence of a rare event for a given energetic regime of the system (Rare Event Quantification Problem). We first analytically quantify the role of spatial energy localization on the development of nonlinear instabilities and the subsequent formation of rare events in water waves. We then prove that these localized instabilities are triggered through the dispersive ‘heat bath’ of random waves that propagate in the nonlinear wave field. The interaction of uncertainty induced through the dispersive wave mixing and nonlinear wave-wave instability defines a critical length-scale for the formation of rare events. To tackle the first problem we rely on this property and show that by merely tracking the energy of the wave field over this critical length-scale allows for the robust, inexpensive prediction
of the location of intense waves with a prediction window of 25 wave periods.
For the second problem, we also utilize the nonlinear stability analysis to decompose the state space into regions where rare events is unlikely to occur and regions that lead with high probability to the occurrence of a rare event. The two regions are treated differently and the information of the two regimes is merged through a total probability argument, allowing for the efficient quantification of rare events.