Validity of the WT theory relies on persistence of the RPA property of the wavefield over the nonlinear evolution time. Such persistence was demonstrated in [5] based on the evolution equation for the multi-mode PDF, where the product factorization of PDF was shown to hold with an accuracy sufficient for the WT closure. It was also emphasized in [5] that RPA must use independent phase factors rather than the phases independence of which does not survive over the nonlinear time. The theoretical prediction of persistent independence of 's and 's and about the growth of correlations of 's will be checked in this paper numerically.

Further, the WT approach predicts that the mean value of the phase grows with a rate given by the nonlinear frequency correction and that the r.m.s. fluctuation of the phase also grows in time [5]. In this paper we will see that in reality the time evolution of the phase is more complicated than this WT prediction: the phase exhibits quasi-periodic fluctuations intermittent by rare ``phase runs'', - monotonic changes over several linear periods by large values which can significantly exceed .