Ensemble dynamics and the emergence of correlations in one- and two-dimensional wave turbulence.

We investigate statistical properties of wave turbulence by monitoring the dynamics of ensembles of trajectories. The system under investigation is a simplified model for surface gravity waves in one and two dimensions with a square-root dispersion and a four-wave interaction term. The simulations of decaying turbulence confirm the Kolmogorov-Zakharov spectral power distribution of wave turbulence theory. Fourth-order correlations are computed numerically as ensemble averages of trajectories. The shape, scaling, and time evolution of the correlations agree with the predictions of wave turbulence theory.