The Permanent Downshifting at Later Stages of Benjamin–Feir Instability of Waves

Abstract

Some of the prominent features in later stages of the Benjamin–Feir (BF) instability development are still not well clarified: What are the conditions for a permanent downshifting in the wave energy spectrum, is this process inevitable accompanied by wave breaking dissipation? What is the mechanism of multiple downshifting and discrete energy spreading to higher frequencies, how does it depend on the initial steepness and frequency space of waves? We conducted experimental and theoretical studies on this issue and revealed a number of new features of the development of instability in the late stages of wave’s evolution. We employ the reduced (truncated) version of Zakharov equations, the multi-wave near-neighbor resonance model (NN model), which takes into account the most effective quasi-resonances with minimum detuning from the exact resonance conditions. We show that the near-neighbor model of wave interactions can adequately describe the number of new characteristic features of BF instability. NN model is much simpler than Zakharov equation for computation and analysis. The dissipation version of Zakharov equations based on the Tulin’s semi-empirical model is employed for the description of breaking wave’s propagation. We verify the new characteristic features of BFI for various initial wave conditions by experiments conducted in Tainan Hydraulics Laboratory and confirm the theoretical predictions of NN model. A strong permanent downshifting of the spectral maximum for gentle waves without wave breaking is revealed for initially two-time narrower spectral width in comparison with the most unstable case. For steep waves, a multiple downshifting regime is detected, accompanied by a wave breaking. The discrete energy flow to higher spectral components takes place in the breaking and no-breaking regimes. Results of numerical simulations of Zakharov and NN models reasonably correspond to each other and to our experimental and field observations on wave modulation.