The dynamics of wave ensembles in shallow water is studied within the framework of the nonlinear dispersive Korteweg – de Vries (KdV) equation by numerical simulation. Bimodal wave systems whose energy is distributed over two spectral domains are considered: the “additional” lobe which corresponds to the system of longer or shorter waves is added to the “main” spectral peak. The concerned problem describes, for example, the interaction between wind waves and swell in shallow water. The case of the unimodal waves (considered in (Pelinovsky, Sergeeva, 2006) is used as the reference. The limitations of the implied assumptions and the relationship of the idealized model to the realistic conditions in the ocean were discussed in the recent paper (Wang et al, 2018).
Based on the detailed consideration of the 6 simulated cases, the following general conclusions may be formulated.
The transition from the initial state to the quasi-equilibrium one is accompanied by strong variations of the wave characteristics, when the waves exhibit the most extreme features. In particular, the wave kurtosis grows suddenly and the abnormal heavy tails in the wave amplitude probability distributions appear. These processes are observed in all the cases of the bimodal spectra and are quite similar to the single-mode regime. The coexistence of a long-wave system smoothens the rapid oscillations of the wave extremes and kurtosis which take place during the transition stage.
The presence of a short-wave system makes the waves on average more symmetric. Skewness attains the minimum value compared to the other cases. The co-existence of shorter waves practically does not change the wave kurtosis or the probability of the wave heights.
In contrast, the presence of a long-wave system makes the waves more asymmetric and more extreme. The probability of large waves increases in the bimodal systems with a low-frequency component.
The initial wave spectrum expands as a result of the wave interaction and tends to a quasistationary state. One may anticipate that the formulated conclusions are applicable beyond the limits of the Korteweg-de Vries equation to other kindred frameworks and corresponding phenomena.
This work was supported by the Russian Science Foundation (project No. 18-77-00063).
ESTIMATION OF BOUND AND RELEASED INFRAGRAVITY WAVES BASED ON WAVE OBSERVATION AND NUMERICAL SIMULATION IN SHALLOW WATER
