Let us consider a wavefield in a periodic square basin of side and let the Fourier representation of this field be where index marks the mode with wavenumber on the grid in the -dimensional Fourier space. Discrete -space is important for formulating the statistical problem. For simplicity let us assume that there is a cut-off wavenumber so that thee is no modes with wavenumber components greater than , which is always the case in numerical simulation. In this case, the total number of modes is and index will only take values in a finite box, which is centered at 0 and all sides of which are equal to . To consider homogeneous turbulence, the large box (i.e. continuous ) limit, , will have to be taken later.

Let us write the complex as where is a real positive amplitude and is a phase factor which takes values on , a unit circle centered at zero in the complex plane. The most general statistical object in WT [5] is the -mode joint PDF defined as the probability for the wave intensities to be in the range and for the phase factors to be on the unit-circle segment between and for all .

The fundamental statistical property of the wavefield in WT is that all the amplitudes and phase factors are independent statistical variables and that all 's are uniformly distributed on . This kind of statistics was introduced in [6,4,5] and called ``Random Phase and Amplitude'' (RPA) field. In terms of the PDF, we say that the field is of RPA type if it can be product-factorized,

(14) |

where is the one-mode PDF for variable .

Note that in this formulation the distributions of remain unspecified and, therefore, the amplitudes do not have to be deterministic (as in earlier works using RPA) nor do they have to correspond to Gaussianity,

(15) |

where is the waveaction spectrum.

Importantly, RPA formulation involves independent *phase factors*
and not *phases* . Firstly, the phases
would not be convenient because the mean value of the phases is
evolving with the rate equal to the nonlinear frequency correction
[5]. Thus one could not say that they are ``distributed
uniformly from to ''. Moreover the mean fluctuation of the
phase distribution is also growing and they quickly spread beyond
their initial -wide interval [5]. But perhaps even more
important, it was shown in [5] that 's build mutual
correlations on the nonlinear timescale whereas 's remain
independent. In the present paper we are going to check this
theoretical prediction numerically by directly measuring the
properties of 's and 's.

In [6,4] RPA was *assumed* to hold over the nonlinear
time. In [5] this assumption was examined *a posteriori*,
i.e. based on the evolution equation for the multi-point PDF. Note
that only the phase randomness is necessary for deriving this
equation, whereas both the phase and the amplitude randomness are
required for the WT closure for the one-point PDF or the kinetic
equation for the spectrum. This fact allows to prove that, if valid
initially, the RPA properties survive in the leading order in small
nonlinearity and in the large-box limit [5]. Such an
approximate leading-order RPA is sufficient for the WT closure.