Integrated linear and nonlinear spectral analysis of extreme ocean waves

This study develops a unified linear–nonlinear framework to identify and quantify the spectral structure associated with extreme ocean waves in wind-sea conditions. Using a 17-year record of 1 h surface-elevation time series from the Ekofisk platform (central North Sea) and a large ensemble of parametrically generated JONSWAP (the JOint North Sea WAve Project) seas, we combine conventional spectral diagnostics (significant wave height, peak period, peak enhancement factor, bandwidth, and the Benjamin–Feir index, BFI) with a direct scattering transform (DST) of the nonlinear Schrödinger equation. The DST decomposes each realization into stable and unstable (modulational) modes, providing a physically interpretable measure of nonlinear wave-group activity. Across both measured and synthetic datasets, the fitted JONSWAP peak enhancement factor increases with BFI but exhibits clear saturation, indicating that spectral peakedness does not grow without bound as instability strengthens. The expected number of large-amplitude modulations, unstable modes whose amplitudes exceed the carrier, shows an approximately exponential dependence on BFI, with percentile curves revealing a transition near BFI ≈ 0.04 beyond which modulationally driven wave packets become frequent. Linear Rayleigh-based predictions of 1 h maximum wave height remain nearly unbiased and essentially insensitive to BFI, whereas a nonlinear upper-bound estimate derived from the most energetic unstable mode increases with BFI and saturates slightly above twice the linear prediction. These results provide a practical route to incorporate BFI-dependent nonlinear amplification into extreme-wave assessment.