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L. Shemer, H. Jiao, E. Kit and Y. Agnon J. Fluid Mech. 427, 107-129, (2001)

Letter to the editor:

Dear Editor,

we would like to resubmit our paper for PRE. As explained below, the main objectives of the referee's are coming from poor understanding of the issues involved in our paper. We ahve sligtly expanded the manuscript, and we hope that it could be published in PRE with out further signinficant delays. (kak skazat; eto pomagche?).

The main point of our paper - we present NEW results (not technical improvement of the old results), as we give a QUANTITATIVE and VERIFIABLE predictions for the quantity, which has been traditionally ignored in the theory of weakly interacting dispersive waves.

We answer referee's questions below:

We disagree with the referee A that there are mistakes in our work, - as you will see below this is a result of not too careful reading. We also disagree that our work is just a technical improvement and not an essentially new theory with respect to the classical weak turbulence. Indeed, the classical weak turbulence deals with short correlated wave fields close to gaussian whereas we consider the long correlated fields the deviation from gaussianity for which is of order one. These fields are widespread in numerical simulations, including the operational wave weather forecasts which always start with fields with random phases and deterministic amplitudes.

`` ... Indeed, the signature in Fourier space of long-range correlations in physical turbulence is generally a power law singularity for small values of wavenumber, rather than a delta-function singularity. If there is some reasonable model of turbulence which does have delta-function singularities in the cumulant spectrum, it should be cited in the manuscript; I am unaware of any such example.''

Examples of such wave fields are widespread in numerical simulations. Indeed, the typical start is a field with random phases and deterministic amplitudes, which is long correlated in the x-space, strongly non-gaussian and is a special case of the fileds we are dealing with. Note that the referee A wrongly calls such field deterministic and this is the main problem with his objections (see also below).

`` ... So if one stays instead within the context of random models of turbulence, what kind of experiments and numerical simulations are contemplated? Is weak turbulence non-Gaussian? I know strong turbulence has non-Gaussian features, but there should be some target experiment or numerical simulation cited which motivates the theory.''

The referee has not read our paper carefully. We are not talking about the deterministic initial data. Instead, we talk about deterministic absolute values and random phases. This is a typical statistical initial condition for numerics and it is obviously a special case of the fields we are dealing with (RPA requirements are described on the first page). This mix-up seems to be the main basis for the referee A objections.

``The discussion about the Fourier spectrum being pinned to the Gaussian value in the large-box limit is misleading. Indeed, the Fourier-space statistics are nearly Gaussian, but the small deviation (proportional to some inverse power of the volume of the box) plays a physically important role reflecting the deviations from Gaussianity in physical space, and the third order cumulant plays an important role in giving the correct kinetic equation even for the nearly-Gaussian standard weak turbulence theory!

Yes, in standard weak turbulence there are small but important deviations from gaussianity. However, our analysis includes fields which are order-one different from being gaussian and this is what we wanted to emphasize.

`` The claim (on p. 4) that the one-point moments contain information wiped out in the multiple-point moments is misleading. Indeed, the one-point moments are just the multiple-point moments evaluated at coalescing values of wavenumber. There is no ``fusion'' problem in computing the multiple-point moments and then evaluating them with coalescing arguments in wavenumber space; the ``fusion'' problem has to do with coalescing points in phsyical space within the context of a high Reynolds number theory (which is not weak turbulence!). The source of confusion is that the cumulants of non-coincident values of the wavenumber are simply assumed to be zero by an ad hoc invocation of RPA; so the loss of information is due to the RPA assumption and not any physical mechanism.''

This is indeed a classical "fusion" effect, although in the Fourier rather than the coordinate space. It also has classical roots of non-cummuting limits, in our case, these are the limits of large time and of zero phase correlation length (in the k-space). And of course there are deep physical reasons for this phase decorrelation to appear in the dispersive wave systems.

`` On p. 8, the time scale $T$ of interest is assumed to be much less than the nonlinear time $1/\omega \epsilon^2$. But the differential equations for the cumulants in (8) have the right hand sides proportional to $\varepsilon^2$, so with this restriction, not much can be said because the time scale is restricted so that the $Q_k^{(p)}$ change negligibly from their initial values. Now, one might object that when $\omega$ is small, there are nontrivial predictions, but often the coupling coefficients for nonlinear waves vanish along with $\omega$ so if $\omega$ is small, so will $\gamma_k$ and $\rho_k$ be small and it still seems like the restriction of the prediction in equation (8) to time scales $T \ll 1/\omega \varepsilon^2$ means simply that not much happens to the cumulants on these time scales, which is not surprising since one would expect the cumulants to evolve on the actual nonlinear time scale.''

Typical timescale in our paper is $1/epsilon^2$ and not $T$. $T$ is just a technical intermediate time needed for the time-scale separation technique (same as in the derivation of the usual Kinetic equation). Our final equations show that all qumulants evolve at the same $1/epsilon^2$ timescale as the spectrum itself and they can experience order 1 changes.

In summary, we