Fourth-order stability analysis of random weakly nonlinear unidirectional deep water waves

The modulational instability of a homogeneous random field of weakly nonlinear surface gravity waves is considered via the Wigner function approach. Starting from Dysthe's [K. B. Dysthe, Proc. R. Soc. Lond. A. 369, 105–114 (1979)] equation, a fourth-order extension of the cubic nonlinear Schrödinger equation that allows describing waves with broader bandwidth and larger steepness, we derive a modified Albers' equation governing the evolution of the Wigner distribution function for a Gaussian random inhomogeneous field of weakly nonlinear surface gravity waves. Carrying out the linear stability analysis of a stationary homogeneous spectrum to small inhomogeneous disturbances, we obtain a new integral dispersion relation for the modulation wavenumber and frequency of the disturbances, which takes into account the higher-order contributions (e.g., higher-order dispersion and wave-induced mean-flow effects). Approximating the solutions of this dispersion relation analytically for a symmetric (Lorentzian) spectrum suggests that mean-flow effects on the growth rate of instability are more important than higher-order dispersion effects. Additionally, the new dispersion relation is solved numerically, first, to verify the approximate fourth-order results and, second, to investigate the instability properties of the more realistic unidirectional JONSWAP (Joint North Sea Wave Project) spectra. We show that the previous stability criteria obtained with the Alber equation for the JONSWAP spectrum need to be revisited.