Generation of edge waves in shallow water

Theoretical growth rates for resonantly driven edge waves in the nearshore are estimated from the forced, shallow water equations of motion for the case of a plane sloping bed. The forcing mechanism arises from spatial and temporal variations in radiation stress gradients induced by a modulating incident wave field. Only the case of exact resonance is considered, where the difference frequencies and wavenumbers satisfy the edge wave dispersion relation (the specific carrier frequencies are not important, only the forced difference values). The forcing is examined in the region seaward of the breakpoint and also within the fluctuating region of surf zone width. In each region, the forcing is dominated by the cross-shore gradient of onshore directed momentum flux, except for large angles of incidence and the lowest edge wave modes. Outside the surf zone, the spatial and temporal variation of the forcing is determined by considering the interaction of two progressive shallow water waves approaching the beach obliquely. In the surf zone, incident wave amplitudes are assumed to be proportional to the water depth. Thus inside the breakpoint, radiation stress gradients are constant and no forcing occurs. However, at the breakpoint, gradients arising from breaking and nonbreaking waves are turned on and off (like a wave maker) with timescales and length scales determined by the modulation of the breaker position. The forcing in this region is stronger, with inviscid growth rates resulting in edge waves growing to the size of the incident waves of the order of about 10 edge wave periods, a factor of 2-10 times larger than in the offshore region. Using a simple parameterization for frictional damping, edge wave equilibrium amplitudes are found to depend linearly on the ratio tan beta/C-d, where beta is the beach slope and C-d is a bottom drag coefficient. For tan beta/C-d about 3-10, equilibrium amplitudes can be as much as 75% of the incident waves over most of the infragravity portion of the spectrum. When the forcing is turned off, these dissipation rates result in a half-life decay timescale of the order of 10-30 edge wave periods.