Abstract
The Alber equation is a phase-averaged second-moment model used to study the statistics of a sea state, which has recently been attracting renewed attention. We extend it in two ways: firstly, we derive a generalized Alber system starting from a system of nonlinear Schrödinger equations, which contains the classical Alber equation as a special case but can also describe crossing seas, i.e., two wavesystems with different wavenumbers crossing. (These can be two completely independent wavenumbers, i.e., in general different directions and different moduli.) We also derive the associated two-dimensional scalar instability condition. This is the first time that a modulation instability condition applicable to crossing seas has been systematically derived for general spectra. Secondly, we use the classical Alber equation and its associated instability condition to quantify how close a given nonparametric spectrum is to being modulationally unstable. We apply this to a dataset of 100 nonparametric spectra provided by the Norwegian Meteorological Institute and find that the vast majority of realistic spectra turn out to be stable, but three extreme sea states are found to be unstable (out of 20 sea states chosen for their severity). Moreover, we introduce a novel “proximity to instability” (PTI) metric, inspired by the stability analysis. This is seen to correlate strongly with the steepness and Benjamin–Feir Index (BFI) for the sea states in our dataset (>85% Spearman rank correlation). Furthermore, upon comparing with phase-resolved broadband Monte Carlo simulations, the kurtosis and probability of rogue waves for each sea state are also seen to correlate well with the PTI (>85% Spearman rank correlation).
Highlights
Ocean waves are a very active field of mathematical modelling and analysis
An array of approximate models are well established and widely used, including the nonlinear Schrödinger equation (NLS) and its variants [4,5], the Zakharov equation [6], the coupled-mode systems [7,8,9], the High Order Spectral Method (HOSM) [10,11,12] and others. (In shallow water, an even larger collection of models is being used, but here we focus on deep water.) One reason for the wide use of approximate models in oceanography is the need to study large wavefields, with hundreds or thousands of individual wavelengths
It is thought that modulation instability and rogue waves may be more prominent in crossing seas, but this is still very far from fully understood [32,33,34,41,52]
Summary
Ocean waves are a very active field of mathematical modelling and analysis. The first principles of hydrodynamic models for gravity waves are well understood [1,2,3]. For a recent review of various stochastic models, one can see [24] Speaking, they are moment equations, starting from phase-resolved equations for the sea surface (such as Zakharov’s equation or the NLS) as an approximation for deterministic wave dynamics. A stochastic approach can directly answer this question e.g., by a phase-resolved Monte Carlo approach [27]; this has a number of advantages, but is clearly expensive to apply indiscriminately Another possibility would be using phase-averaged stochastic models, such as the moment equations discussed above, to directly investigate whether a sea state could likely support the rapid concentrations of energy. In this way, sea states of interest could be selected and computational resources focused on them. It appears that the sea states highlighted as more unstable by the Alber equation turn out to exhibit a higher probability of extreme events in a phase-resolved Monte Carlo simulation (details in Section 3.2.3 and Figure 5)
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