A simple formula for nonlinear wave-wave interaction

A simple expression is introduced as an approximation for the rate of chang e of the spectral energy density of surface gravity waves due to nonlinear wave-wave interaction. It has the Form of a second-order nonlinear diffusio n operator, and conserves wave energy, momentum and wave action. It is inde pendent of the details of the dispersion relation, so it can possibly be us ed for both deep and shallow water, although its application to shallow wat er is not explicitly considered. The directional dependence of the Formula is essential in permitting the wave momentum to be conserved, in addition t o the wave energy and action. The formula may be useful in discussing the q ualitative behavior of wave spectrum evolution without making elaborate cal culations. It is consistent with the observed and modelled result that nonl inear effects tend to cause the wave energy to be transferred to lower wave frequencies. However, when applied to a JONSWAP wave spectrum it behaves r ather diffusively, tending to directly reduce the amplitude of the spectral peak. In the absence of other wave energy source terms, the formula Leads to various time-independent wave spectra, whose dependence on scalar wave n umber is linked to the angular wave energy distribution. In general the dir ectional spreading of the spectrum tends to increase as the scalar wavenumb er increases. The limiting directionally-isotropic spectrum has the Kitaigo rodskii equilibrium-range behavior, where the wave energy (variance) spectr um is proportional to the inverse Fourth power of the frequency.