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We study statistical properties after a sudden episode of wind for waterwaves propagating in one direction. A wave with random initial conditions ispropagated using a forced-damped higher order Nonlinear Schr\"odinger equation(NLS). During the wind episode, the wave action increases, the spectrumbroadens, the spectral mean shifts up and the Benjamin-Feir index (BFI) and thekurtosis increase. Conversely, after the wind episode, the opposite occurs foreach quantity. The kurtosis of the wave height distribution is considered themain parameter that can indicate whether rogue waves are likely to occur in asea state, and the BFI is often mentioned as a means to predict the kurtosis.However, we find that while there is indeed a quadratic relation between thesetwo, this relationship is dependent on the details of the forcing and damping.Instead, a simple and robust quadratic relation does exist between the kurtosisand the bandwidth. This could allow for a single-spectrum assessment of thelikelihood of rogue waves in a given sea state. In addition, as the kurtosisdepends strongly on the damping and forcing coefficients, by combining thebandwidth measurement with the damping coefficient, the evolution of thekurtosis after the wind episode can be predicted.

Single-spectrum prediction of kurtosis of water waves in a non-conservative model