Direct Numerical Simulation of Downshift and Inverse Cascade for Water Wave Turbulence

By means of direct numerical simulations (DNS) based on the integrodifferential Zakharov equation, we study the long-term evolution of nonlinear random water wave fields. For the first time, formation of powerlike Kolmogorov-type spectra corresponding to weak-turbulent inverse cascade is demonstrated by DNS, and the evolution in time of the resulting spectra is quantitatively investigated. The predictions of the statistical theory for water waves, both qualitative (formation of the direct and inverse cascades, self-similar behavior) and quantitative (the spectra exponents, specific shape of self-similar functions, the rate of time evolution) are found to be in good agreement with the DNS results, except for the initial part of the evolution, where the established statistical theory is not applicable yet and the evolution has a much faster time scale.