**Yuri V. Lvovtex2html_wrap_inline$^1$, Sergey Nazarenkotex2html_wrap_inline$^2$, Boris Pokornitex2html_wrap_inline$^1$
tex2html_wrap_inline$^1$
Department of Mathematical Sciences, Rensselaer Polytechnic Institute,Troy NY 12180
tex2html_wrap_inline$^2$ Mathematics Institute, The University of Warwick, Coventry CV4 7AL, UK**

*Date:* March 20, 2006

We perform numerical simulations of the dynamical equations for free
water surface in finite basin in presence of gravity. Wave Turbulence
(WT) is a theory derived for describing statistics of weakly nonlinear
waves in the infinite basin limit. Its formal applicability condition
on the minimal size of the computational basin is impossible to
satisfy in present numerical simulations, and the number of wave
resonances is significantly depleted due to the wavenumber
discreteness. The goal of this paper will be to examine which WT
predictions survive in such discrete systems with depleted resonances
and which properties arise specifically due to the discreteness
effects. As in [1,2,3], our results for the wave
spectrum agree with the Zakharov-Filonenko spectrum predicted within
WT. We also go beyond finding the spectra and compute probability
density function (PDF) of the wave amplitudes and observe an
anomalously large, with respect to Gaussian, probability of strong
waves which is consistent with recent theory [4,5]. Using a
simple model for quasi-resonances we predict an effect arising purely
due to discreteness: existence of a threshold wave intensity above
which turbulent cascade develops and proceeds to arbitrarily small
scales. Numerically, we observe that the energy cascade is very
``bursty'' in time and is somewhat similar to sporadic sandpile
avalanches. We explain this as a cycle: a cascade arrest due to
discreteness leads to accumulation of energy near the forcing scale
which, in turn, leads to widening of the nonlinear resonance and,
therefore, triggering of the cascade draining the turbulence levels
and returning the system to the beginning of the cycle.

- Introduction
- Equations for the free surface
- Statistical Quantities in Wave Turbulence
- Theoretical WT predictions
- Evolution equations for the PDF's, moments and spectrum.
- Preservation of the RPA property.
- Steady state solutions.

- Resonant interaction in discrete -space
- Numerical simulation
- Results
- Spectrum
- Wave-amplitude probability density function and its moments
- Frequency properties.
- Amplitude and phase evolution.
- Nonlinearly active modes and cascade ``avalanches''
- Correlations of phases, phase factors and amplitudes.

- Discussions
- Acknowledgments
- Bibliography
- About this document ...