A numerical study on statistical characteristics of amplitude fluctuations in a wave turbulence

A series of numerical studies has been performed to investigate the statistical characteristics of fluctuations of energy s = | a k | 2 of each component wave and its time derivative s ̇ = d | a k | 2 / d t in a wave turbulence for a Hamiltonian system with a non-decay type dispersion relation. It has been found that the probability density functions (PDF) of s and s ̇ evaluated among independent realizations are almost equivalent to the PDF’s evaluated among nearby modes in the k space in a single realization. Therefore, the averaging over the realizations and that over the nearby modes in a single realization are interchangeable. It has also been found that the PDF of s evolves in time toward the χ 2 distribution with two degrees of freedom, while that of s ̇ evolves toward the Laplace distribution (or the two-sided exponential distribution) at any wavenumber k , both in the peripheral of the spectrum (large k region) and in the energy-containing range (small k region).