Evaluation of a spectral line width for the Phillips spectrum by means of numerical simulation

The work aims to check one of the assumptions under which the kinetic equation for water waves was derived in order to understand whether it can be applied to the situations described by the Phillips spectrum. We evaluate a spectral line width of the spectrum from the simulations in the framework of primordial dynamical equations at different levels of nonlinearity in the system, corresponding to the weakly turbulent Kolmogorov-Zakharov spectra ω^-4, Phillips spectra ω^-5, and intermediate cases. The original motivation of the work was to check one of the assumptions under which the kinetic equation for water waves was derived in order to understand whether it can be applied to the Phillips spectrum. It is shown that, even in the case of relatively high average steepness, when the Phillips spectrum is present in the system, the spectral lines are still very narrow, at least in the region of the direct cascade spectrum. It allows us to state that, even in the case of the Phillips spectrum, one of the assumptions used for the derivation of the Hasselmann kinetic equation is still valid, at least in the case of moderate whitecapping.