Wind forcing in the equilibrium range of wind-wave spectra

A new analytical model is developed for the equilibrium range of the spectrum of wind-forced ocean surface gravity waves. We first show that the existing model of Phillips (1985) does not satisfy overall momentum conservation at high winds. This constraint is satisfied by applying recent understanding of the wind forcing of waves. Waves exert a drag on the air flow so that they support a fraction of the applied wind stress, which thus leaves a smaller turbulent stress near the surface to force growth of shorter wavelength waves. Formulation of the momentum budget accounting for this sheltering constrains the overall conservation of momentum and leads to a local turbulent stress that reduces as the wavenumber increases. This local turbulent stress then forces wind-induced wave growth. Following Phillips (1985), the wind sea is taken to be a superposition of linear waves, and equilibrium is maintained by a balance between the three sources and sinks of wave action. These assumptions lead to analytical formulae for the local turbulent stress and the degree of saturation, B(k), of waves in the equilibrium range. We identify a sheltering wavenumber, k(s) over which the local turbulent stress is significantly reduced by longer waves. At low wavenumbers or at low winds, when k much less than k(s) the sheltering is weak and B(k) has a similar form to the model of Phillips (1985). At higher wavenumbers or at higher winds, k much greater than k(s) B(k) makes a transition to being proportional to k(0). The additional constraint of conservation of momentum also yields a formula for the coefficient that appears in the solution for B(k). The spectra for mature seas are calculated from the model and are shown to agree with field observations. In particular, our model predicts more realistic spectral levels toward the high wavenumber limit compared to the previous model of Phillips (1985). We suggest that the model may explain the overshoot phenomena observed in the spectral energy levels as the fetch increases.