Fully nonlinear investigation on energy transfer between long waves and short-wave groups over a reef

Long waves are amplified as short-wave groups shoal and break over reefs, therefore, having significant impacts on coastal inundation, structure stability, and sediment transport. This study investigated the cross-reef variation of long-wave energy exchange with short-wave group over a reef using fully nonlinear analysis of simulation results by the non-hydrostatic model SWASH. The objective was to elucidate the mechanisms of long-wave transformation under nonlinear short-wave group forcing over a reef, and to assess the consequences of simplifications in linear and weakly nonlinear analyses in this problem. The energy transfer between short and long waves is the work done by radiation stress on long-wave velocity. Unlike conventional linear and weakly nonlinear analysis, the Stokes transport and long-wave modulation of local water depth are included in the fully nonlinear analysis. It was found that only the long-wave energy flux gradient given by the fully nonlinear analysis was balanced by the work done by wave radiation stress over a shallow reef. The fully nonlinear analysis showed that strict mass conservation has to be used to extract long wave velocity properly. In contrast, in linear and weakly nonlinear analysis, the long-wave velocity is extracted from single-point velocity measurements. The fully nonlinear analysis demonstrated that the generation and growth of incoming breakpoint-forced long waves overcame the dissipation of bound long waves in the surf zone, leading to amplification of incoming long-wave energy flux. This phenomenon occurred even when short waves mainly broke over the horizontal reef flat with large submergence, indicating that long-wave evolution is not locally controlled but dependent on wave spatial evolution history. Outgoing breakpoint-forced long waves were dissipated considerably during de-shoaling over the forereef due to substantial energy transfer to incoming short waves, though both of them are free waves. The consistent phase coupling between outgoing long waves and incoming short-wave groups at all frequencies was found to be the primary driving mechanism for the energy transfer. According to the fully nonlinear analysis, the reef-flat submergence may affect the long wave in a complex fashion, i.e., reducing the submergence may enhance the energy transfer from short waves to long waves or suppress long-wave growth by increasing its frictional dissipation at the same time.