Statistical Phase Transitions and Extreme Events in Shallow Water Waves with an Abrupt Depth Change

Predicting extreme events in complex nonlinear systems is an extremely challenging and important area in climate science. Important examples include extreme events near coastal continental shelves for shallow and deep water waves. Recent laboratory experiments reveal a remarkable transition from near Gaussian to highly skewed anomalous statistics with extreme events by measuring the surface water wave displacements in shallow water across an abrupt depth change (ADC). A statistical dynamical model has been proposed and used to accurately predict the representative statistical transition using Gibbs invariant measures for the truncated KdV equation at low and high inverse temperatures representing flows before and after the ADC. In this paper, we use much lower dimensional truncated paradigm models to understand the statistical phase transition and the creation of extreme events. Especially, a two-mode interacting model with exact integrable dynamics is adopted to characterize the core transition mechanism as the model parameter varies. The choice of the radically truncated two-mode model is motivated by the self-similar solution structures with reducing numbers of truncated modes. A clear separation of distinct dynamics in the phase space is discovered for the Gibbs ensembles sampled from different inverse temperatures. Direct numerical tests with various model truncation sizes are presented to illustrate the statistical transition in parameter regimes. The analysis here can also provide a theoretical guideline for a wider variety of models concerning the generation of extreme events and anomalous statistics.