The Fully Developed Wind-Sea Spectrum as a Solution of the Energy Balance Equation

Abstract

We consider the energy transfer equation for well developed ocean waves under the influence of wind, and study the conditions for the existence of an equilibrium solution in which wind input, wave-wave interaction and dissipation balance each other. For the wind input we take the parametrization proposed by Snyder et al (1981), which was based on their measurements in the Bight of Abaco, and which agrees with Miles’ (1957, 1959) theory. The wave-wave interaction is computed with an algorithm given by Hasselmann et al (1984). The dissipation is less well-known, but we will make the general assumption that it is quasi-linear in the wave spectrum with a factor coefficient depending only on frequency and integral spectral parameters (cf. Hasselmann, 1974). Full details of this study are given elsewhere (Kamen, Hasselmann and Hasselmann, 1984). Here we summarize the main results. In the first part of our study we investigated whether the assumption that the equilibrium spectrum exists and is given by the Pierson-Moskowitz spectrum with a standard type of angular distribution leads to a reasonable dissipation function. We find that this is not the case. Even if one balances the total rate of change for each frequency (which is possible), a strong angular imbalance remains. This is illustrated in fig. 1, in which the assumed asymptotic spectrum and the corresponding source terms are given. The dissipation constant is chosen such that at any given frequency the total rate of change vanishes. As one can see there is no angular balance. Thus the assumed source terms are not consistent with this type of asymptotic spectrum. In the second part of the study we chose a different approach. We assumed that the dissipation was given and we performed numerical experiments simulating fetch limited growth, to see under which conditions a stationary solution can be reached. For the dissipation we took Hasselmann’s (1974) form with two unknown parameters. From our analysis it follows that for a certain range of values of these parameters a quasi-equilibrium solution results. We estimate the relation between dissipation parameters and asymptotic growth rates.