Serre-Green-Naghdi Dynamics under the Action of the Jeffreys' Wind-Wave Interaction

We derive the anti dissipative Serre-Green-Naghdi (SGN) equations in the context of nonlinear dynamics of surface water waves under wind forcing, in finite depth. The anti-dissipation occurs du to the continuos transfer of wind energy to water surface wave. We find the solitary wave solution of the system, with an increasing amplitude under the wind action. This leads to the blow-up of surface wave in finite time for infinitely large asymptotic space . This dispersive, anti-dissipative and fully nonlinear phenomenon is equivalent to the linear instability at infinite time. The theoretical blow-up time is calculated based on real experimental data. Naturally, the wave breaking takes place before the blow-up time. However, the amplitude's growth resulting in the blow-up could be observed. Our results show that, based on the particular type of wind-wave tank data used in this paper, for h = 0.14 m, the amplitude growth rate is of order 0.1 which experimentally, is at the measurability limit. But we think that by gradually increasing the wind speed U10, up to 10 m/s, it is possible to have the experimental confirmation of the present theory in existing experimental facilities.