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, ks, over which the local turbulent stress is significantly reduced by longer
waves. At low wavenumbers or at low winds, when k [Lt ] ks, the sheltering is weak
and B(k) has a similar form to the model of Phillips (1985). At higher wavenumbers
or at higher winds, ks, B(k) makes a transition to being proportional to k0. 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.