A new expression is introduced as a simple approximation for the rate of change of the spectral energy density of surface gravity waves due to nonlinear wave-wave interaction. The proposed expression 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 general qualitative behavior of wave spectrum evolution without making elaborate calculations. It is consistent with the observed and modeled 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 may behave rather too diffusively, tending to directly reduce the amplitude of the spectral peak. In the absence of other wave energy source terms, the formula leads to a number of different time-independent wave spectra, whose dependence on scalar wave number is linked to the angular wave energy distribution. A special case is that of the Kitalgorodskii equilibrium-range form, where the wave energy (variance) spectrum is proportional to the inverse fourth power of the wave frequency.
Spectral wave forecasting models, based upon radiative transfer equations, have been used for a number of years to predict directional wave spectra worldwide, in the open ocean and in coastal/shelf seas. An important part of such models is the calculation of the flow of energy within the two-dimensional wavenumber space (or, alternatively, the space described by wave frequency and wave direction) due to nonlinear wave-wave interaction.
