Evolution of the stokes wave side-band instability along a super tank: Experiments and threshold modification of Tulin NLS model

Evolution of Stokes wave side-band instability along a super tank was studied experimentally and theoretically. The initial exponential growth of the resonant sidebands is followed by asymmetrical growth rates for the sidebands: lower sideband growth is much faster, and finally the main energy is concentrated in it and the primary wave. An active breaking process increases the frequency downshift during the latter stages of the wave propagation. Some type of wave stabilization takes place during the final post-breaking stages of the process and can be characterized by a sharp decreasing of wave breaking activity and stabilization of wave modulations. Our calculations show that Tulin (1996) dissipative modification of NLS model can satisfactory describe the first several stages of the wave train evolution: wave instability, the side band asymmetry and wave breaking effects. On the other hand, continuous wave breaking dissipation presumed in the model gives significantly overestimated values of wave attenuation on the latter stages of wave propagation and can not describe the wave modulation and restabilization at sufficiently long distances of propagation. The adjusted dissipative model based on the Nonlinear SchrÖdinger Equation is suggested for adequate description the obtained experimental data. Sink/Source terms due to wave breaking processes in its right side correspond to well-known Tulin (1996) model. The wave dissipation function includes the wave steepness threshold function and applied only in the regions with active wave breaking. Permanent frequency downshift as a result of wave breaking process and post-breaking wave modulations described by the model have the satisfactory quantitative correspondence to results of experiments conducted along a super tank.