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The Alber equation is a phase-averaged second-moment model for the statisticsof a sea state, which recently has been attracting renewed attention. We extendit in two ways: firstly, we derive a generalized Alber system starting from asystem of nonlinear Schr\"odinger equations, which contains the classical Alberequation as a special case but can also describe crossing seas, i.e. twowavesystems with different wavenumbers crossing. (These can be two completelyindependent wavenumbers, i.e. in general different directions and differentmoduli.) We also derive the associated 2-dimensional scalar instabilitycondition. This is the first time that a modulation instability conditionapplicable to crossing seas has been systematically derived for generalspectra. Secondly, we use the classical Alber equation and its associatedinstability condition to quantify how close a given non-parametric spectrum isto being modulationally unstable. We apply this to a dataset of 100non-parametric spectra provided by the Norwegian Meteorological Institute, andfind the vast majority of realistic spectra turn out to be stable, but threeextreme sea states are found to be unstable (out of 20 sea states chosen fortheir severity). Moreover, we introduce a novel "proximity to instability"(PTI) metric, inspired by the stability analysis. This is seen to correlatestrongly with the steepness and Benjamin-Feir Index (BFI) for the sea states inour dataset (>85% Spearman rank correlation). Furthermore, upon comparing withphase-resolved broadband Monte Carlo simulations, the kurtosis and probabilityof rogue waves for each sea state are also seen to correlate well with its PTI(>85% Spearman rank correlation).

Modelling of ocean waves with the Alber equation: application to non-parametric spectra and generalization to crossing seas