Diffusion approximation for a nonlinear kinetic integral: Validation of derivation and properties

The derivation of the diffusion approximation for the Hasselmann kinetic in tegral describing nonlinear interactions of gravity waves in water [1] is v alidated. It is shown that the diffusion approximation, which contains seco nd derivatives with respect to frequency and angle (or wave-number componen ts), follows from a successive analytical integration of a six-dimensional Hasselmann integral even without invoking the hypothesis of quasi-local int eractions of waves. An analysis of the integrand indicates that this approx imation is actually the approximation of small scattering angles. It is thi s term that was used in work [2]. However, in deriving the diffusion approx imation, there is no need to obtain the terms containing fourth derivatives (as was done in [2]), and one can restrict oneself to second derivatives, in accordance with the recent work [3]. The version of the diffusion approx imation proposed in the latter work is tested numerically, and it is shown that this version can be used to solve practical problems.