On the nonlinear transfer of energy in the peak of a gravity-wave spectrum. II

Abstract

This paper presents the first accurate calculations of the nonlinear transfer of energy within a continuous spectrum of water waves. The spectrum is assumed to be narrow, that is, the wave energy is initially concentrated near one particular wavenumber, and use is made of the transfer equation derived previously in part I (Longuet-Higgins 1975) for this case. It is shown that when the spectrum is describable as a sum of normal distributions, then the sixfold multiple integral can be reduced to a single integration. Hence the accurate evaluation of the energy transfer (as a function of the two dimensional wavenumber) becomes practicable. For a symmetric normal spectrum it is found that the transfer function generally has the form of a clover-leaf, with four maxima lying in the characteristic directions dλ = ±√2 d μ , as seen from the peak. These are separated by troughs of negative transfer lying in the axial directions dλ = 0, d μ = 0. For a typically asymmetric spectrum, one of the negative troughs may be filled in, so that the transfer function more closely resembles a butterfly. An interpretation is given in terms of the balance of terms in the transfer equation. The (one dimensional) transfer function for the frequency-spectrum can be found by integration of the two dimensional transfer function. Typically it has a pronounced minimum near the peak frequency, indicat­ing strong negative transfer there, and two weaker maxima, one on each side. For asymmetric spectra, the maximum transfer is greater on the steeper face of the spectrum, usually on the low-frequency side. A com­parison with the rough calculations of Sell & Hasselmann (1972) for the JONSWAP R3C spectrum shows good agreement.