The Impact of Frequency Bandwidth on a One-Dimensional Model for Dispersive Wave Turbulence

We combine theory and high-resolution direct numerical simulation to resolve a long-standing puzzle concerning the forced–dissipative statistics of a one-dimensional model for dispersive wave turbulence, which was introduced by Majda et al. (J Nonlinear Sci 6:9–44, 1997. https://doi.org/10.1007/BF02679124 ) as a test bed for wave turbulence theory (WTT). Numerous earlier studies had indicated significant discrepancies between the inertial range power law predictions of WTT and those observed in direct numerical simulations of that model. Exactly why and when these discrepancies would arise had been an open question. On the theoretical side, we utilize an exact scaling symmetry of the model to derive a one-parameter family of exact self-similar power laws for the inertial range, which includes the WTT prediction as a special case. We follow this up by numerical simulations at unprecedented resolution, combining white-noise forcing, infrared and ultraviolet dissipation, and a novel averaging technique for the estimation of mean values. Our converged numerical results refute an earlier hypothesis that the discrepancies might be due to variations in wave amplitude. Instead, we find incontrovertible evidence that the WTT prediction is always achieved across a wide range of wave amplitudes, but only if the inertial range is wide enough when measured in the ratio of wave frequency at the forcing scales to wave frequency at the dissipation scales. For a concave dispersion relation (as for deep water waves), frequency bandwidth is a much more stringent criterion than wavenumber bandwidth. At finite resolution, the observed power law is always steeper than the WTT prediction, but to excellent approximation the discrepancy in the exponent is simply proportional to the aforementioned frequency ratio. The picture that emerges is that of a self-similar WTT inertial range that is robustly ‘frustrated’ by finite bandwidth effects in a predictable manner. We test our predictions by varying the differential character of the nonlinear terms in the model and excellent agreement is found at low wave amplitudes in all cases. At high amplitude, however, one case exhibits a novel ‘breakthrough’ turbulent regime that has not been observed before and for which no theory presently exists. Finally, we discuss the observable implications of our findings for other systems featuring wave turbulence, including the important case of oceanic inertia–gravity waves, for which the admissible frequency range is bounded above and below, thus limiting the achievable size of an inertial range a priori.