McLean's instability analysis (1982) and Lin and Perrie's numerical simulations (1997a and c) both show that 4-wave interactions (primarily two-dimensional) dominate in deep water. However, the three-dimensional wave-wave interactions can indeed be observed in deep water (Su, 1982a and b). Su and Green's (1984) experimental results further suggest that the two-dimensional instabilities can trigger the three-dimensional instabilities in deep water, as long as the initial waves are at least moderate steepness (ak = 0.1). The present numerical simulations in deep water show that no matter how steep the initial waves are, the 4-wave interactions always dominate as long as the wave spectrum is broad. Until the 4-wave interactions transform the initial wave spectrum into a narrow band spectrum, the 5-wave interactions do not dominate. The reason for this is that the nonlinear transfer rate of the 4-wave interactions is proportional to the third power of the spectrum width, while the resonant domain of 5-wave interactions is global. Hence the width of the spectrum should not significantly affect the nonlinear transfer rate of 5-wave interactions.
Phillips (1960) first suggested that at least 4-wave interactions are needed to generate a resonant gravity wave mode. Hasselmann (1962) used a small perturbations theory to obtain the formulation of 4-wave interactions. However, the 4-wave interactions are primarily undirectional. That is, the nonlinear transfer is confined mostly to the direction of mean wave propagation. Zakharov (1991) employed a Hamiltonian approach to yield an expression for the nonlinear wave-wave interactions, which is similar to that of Hasselmann (1962). Zakharov's formulation is simpler and has a higher degree of symmetry and a wider range of validity. Thus it gave a more versatile formulation for studying higher-order wavewave interactions. Krasitskii (1994) had used it to calculate the 5-wave interaction formulation.
