We present a new numerical method for studying the evolution of free and bound waves on the nonlinear ocean surface. The technique, based on a representation due to Watson and West (1975), uses a slope expansion of the velocity potential at the free surface and not an expansion about a reference surface. The numerical scheme is applied to a number of wave and wave train configurations including longwave-shortwave interactions and the three-dimensional instability of waves with finite slope. The results are consistent with those obtained in other studies. One strength of the technique is that it can be applied to a variety of wave train and spectral configurations without modifying the code.
INTRODUCTIONThe wind generation of waves on the ocean surface and their subsequent evolution has been described for over 2 decades in terms of a weakly interacting field of nonlinear waves whose equations of motion are determined by a Hamiltonian [Phillips, 1966;Zakharov, 1968] for an incompressible, inviscid, irrotational liquid. To describe the processes of wave generation, evolution, and the subse-The gravity wave field on the sea surface is a conservative Hamiltonian system so that Hamilton's equations of motion provide a deterministic description of its evolution [Hasselmann, 1968; Broer, 1974;Watson and West, 1975;Miles, 1977; Milder, 1977;West, 1981]. If we assume that this field is well represented by N degrees of freedom, where N may be large but finite, the system can be represented by N coupled, deterministic, nonlinear rate quent development of wave instabilities, it has been found equations for the mode amplitudes. Moser [1973] gives a convenient to express the observables at the ocean surface general mathematical discussion of the separation of the in series expansions of the eigenfunctions of the linearized interactions in such Hamiltonian systems into resonant and nonresonant groups. The nonresonant interactions provide system. The expansion coefficients in such series are con-for a stable evolution in the phase space of the system, stant in the linearized systems but are variable in the nonlinear system. Because the linear water wave field is har-whereas the resonant interactions lead to instabihties. In a monic, the eigenfunctions are simple sines and cosines, and qualitative way, a resonance is a matching between both the series expansion are just Fourier series. The expansion space and time scales of the wave of interest and the scale coefficients are interpreted as the amplitudes of indepen-of the nonlinear interactions among the other dent waves in a linear wave field. Correspondingly, the waves. The existence of such resonance in water wave nonlinear surface is referred to as a nonlinear wave field, fields was explicitly pointed out by Phillips [1960]. He showed that just as for resonance in a linear system, the and the nonlinearities are interpreted as couplings or scatterings of the once linear waves [Hasselmann, 1962, resonant nonlinear interactions among gravity waves pro-1963a,b]. The Hamiltonian fo...
