Numerical investigation of the time scales of nonlinear spectral evolution in surface gravity waves*

The wave kinetic equation predicts spectral evolution of surface gravity waves on a kinetic time scale of O(ε^-4), where ε denotes the wave steepness. However, observations in [1] and several other works have identified that when a surface gravity wave field is subjected to a sudden perturbation by external forcing, its spectrum evolves on a “fast” dynamic time scale of O(ε^-2), contradicting the prediction of the wave turbulence theory. This problem was revisited in [2], which utilized the Majda-McLaughlin-Tabak (MMT) model with gravity wave dispersion to investigate the relevant time scales of nonlinear spectral evolution. In the current work, we expand on the findings and apply the methodology developed in [2] to surface gravity waves within the framework of the Euler equations for a free surface. We seek to provide a deeper understanding of the mechanisms governing nonlinear spectral evolution in surface gravity waves, advancing the insights obtained via the simplified MMT model. This study focuses on the nonlinear spectral evolution in both inverse and direct cascade processes in an effort to shed new light on the discrepancy between theoretical predictions and observed phenomena.

[1] S. Y. Annenkov and V. I. Shrira. “Fast” Nonlinear Evolution in Wave Turbulence. Phys. Rev. Lett., 102:024502 (2009).

[2] A. Simonis, A. Hrabski, and Y. Pan. On the time scales of spectral evolution of nonlinear waves. Journal of Fluid Mechanics, 979:A33 (2024).