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Recent studies on free-surface gravity waves over uneven bathymetries have shown that “rogue” waves can be triggered by strong depth variations. This phenomenon is here studied by means of spectral simulations of the free-surface Euler equations. We focus on the case of a random, one-directional wave field with prescribed statistics propagating over a submerged step, and consider different depth variations, up to an almost deep-to-shallow transition ( ≈ 1.8 − 0.78, where is the characteristic wavenumber and the water depth). Strongly non-Gaussian statistics are observed in a region localized around the depth transition, beyond which they settle rapidly on the steady statistical state of finite-depth random wave fields. Extreme fluctuations are enhanced by stronger depth variations. Freak-wave formation is interpreted as a general signature of out-of-equilibrium dynamics, associated with the spectral settling from the deep-water to the finite-depth equilibrium. We also document that during such a transition the wave spectrum shows remarkable similarity with Phillips ω−5-law for the strongest depth variations considered.


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