Discreteness and its effect on water-wave turbulence

We perform numerical simulations of the dynamical equations for a free water surface in a finite basin in the presence of gravity. Wave Turbulence (WT) is a theory derived for describing the 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 [A.I. Dyachenko, A.O. Korotkevich, V.E. Zakharov, Weak turbulence of gravity waves, JETP Lett. 77 (10) (2003); Phys. Rev. Lett. 92 (13) (2004) 134501; M. Onorato et al., Freely decaying weak turbulence for sea surface gravity waves, Phys. Rev. L 89 (14) (2002); N. Yokoyama, Statistics of Gravity Waves obtained by direct numerical simulation, JFM 501 (2004) 169–178], our results for the wave spectrum agree with the Zakharov–Filonenko spectrum predicted within WT. We also go beyond finding the spectra and compute the 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 [Y. Choi, Y.V. Lvov, S. Nazarenko, B. Pokorni, Anomalous probability of large amplitudes in wave turbulence, Phys. Lett. A 339 (3–5) (2004) 361–369 (also on arXiv: math-ph/0404022 v1); Y. Choi, Y.V. Lvov, S. Nazarenko, Probability densities and preservation of randomness in wave turbulence, Phys. Lett. A 332 (2004) 230–238; Joint statistics of amplitudes and phases in wave turbulence, Physica D 201 (2005) 121–149; Y. Choi, Y.V. Lvov, S. Nazarenko, Wave turbulence, in: Recent Developments in Fluid Dynamics, vol. 5, 2004, Transworld Research Network, Kepala, India (also on arXiv.org: math-ph/0412045)]. Using a simple model for quasi-resonances we predict an effect arising purely due to discreteness: the existence of a threshold wave intensity above which a 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.