Wavefields with long spatial correlations.

Often in WT, studies are restricted to wavefields with fastly decaying spatial correlations [2]. For such fields, the statistics of the Fourier modes is close to being Gaussian. Indeed, it the correlation length is much smaller than the size of the box, then this box can be divided into many smaller boxes, each larger than the correlation length. The Fourier transform over the big box will be equal to the sum of the Fourier transforms over the smaller boxes which are statistically independent quantities. Therefore, by the Central Limit Theorem, the big-box Fourier transform has a Gaussian distribution. Corrections to Gaussianity are small as the ratio of the correlation volume to the box volume. On the other hand, in the RPA defined above the amplitude PDF is not specified and can significantly deviate from the Rayleigh distribution (corresponding to Gaussian wavefields). Such fields correspond to long correlations of order or greater than the box size. In fact, long correlated fields are quite typical for WT because, due to weak nonlinearity, wavepackets can propagate over long distance preserving their identity. Moreover, restricting ourselves to short-correlated fields would render our study of the PDF evolution meaningless because the later would be fixed at the Gaussian state. Note that long correlations modify the usual Wick's rule for the correlator splitting by adding a singular cumulant, e.g. for the forth-order correlator,

$$\langle \bar a_j \bar a_\alpha a_\mu a_\nu\rangle_\psi = A_j^2 A_... ...ta^\alpha_\mu) + Q_\alpha \delta^\alpha_\nu \delta^\alpha_\mu \delta^\alpha_j,$$

where $Q_\alpha = (A_\alpha^4 - 2A_\alpha^2)$. These issues were discussed in detail in [22].