Multi-wave interaction theory for wind-generated surface gravity waves

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Abstract
Consistently employing the assumption of localness of wave–wave interactions in the wavenumber space, the Kolmogorov treatment of the energy cascade is applied to the case of wind-generated surface gravity waves. The effective number v of resonantly interacting wave harmonics is not limited to four but is found as a solution of a coupled system of equations expressing: (i) the dependence of the spectrum shape on the degree of the wave nonlinearity, and (ii) the continuity of the wave action flux through the spectrum given a continuous positive input from wind. The latter is specified in a Miles-type fashion, and a simple scaling relationship based on the concept of the turnover time is derived in place of the kinetic equation. The mathematical problem is reduced to an ordinary differential equation of first order. The exponent in the ‘power law’ for the spectral density of the wave potential energy and the effective number of resonantly interacting wave harmonics are found as functions of the wave frequency and of external factors of wind—wave interaction. The solution is close to the Zakharov—Filonenko spectrum at low frequencies and low wind input while approaching the Phillips spectrum at high frequencies and sufficiently high wind.