A simple formula for nonlinear wave-wave interaction

A simple expression is introduced as an approximation for the rate of change of the spectral energy density of surface gravity waves due to nonlinear wave-wave interaction. It has the Form of a second-order nonlinear diffusion operator, and conserves wave energy, momentum and wave action. It is independent of the details of the dispersion relation, so it can possibly be used for both deep and shallow water, although its application to shallow water is not explicitly considered. The directional dependence of the Formula is essential in permitting the wave momentum to be conserved, in addition to the wave energy and action. The formula may be useful in discussing the qualitative behavior of wave spectrum evolution without making elaborate calculations. It is consistent with the observed and modelled result that nonlinear effects tend to cause the wave energy to be transferred to lower wave frequencies. However, when applied to a JONSWAP wave spectrum it behaves rather 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 number is linked to the angular wave energy distribution. In general the directional spreading of the spectrum tends to increase as the scalar wavenumber increases. The limiting directionally-isotropic spectrum has the Kitaigorodskii equilibrium-range behavior, where the wave energy (variance) spectrum is proportional to the inverse Fourth power of the frequency.