Wavenumber spectra of short gravity waves

The spectral balances involved in shaping the short gravity wave region of the ocean wave-height spectrum have been the subject of recent physical models. In terms of the wind friction velocity u*, gravitational acceleration g and local wavenumber k, these models predict a wavenumber dependence of $k^{-\frac{7}{2}}$, where k = |k|, and a linear dependence on u* for the equilibrium range of gravity waves above the spectral peak. In this paper we present the results of an experimental determination of the wavenumber spectrum for the wavelength range of 0.2−1.6 m, based on stereophotogrammetric determinations from an oil platform under open ocean conditions. From our observations, for this wavenumber range, the one-dimensional equilibrium wavenumber spectrum was determined as \[ \phi (k_i) \sim \left(\frac{u^2_*k}{g}\right)^{\gamma} k^{-3}_{i}\;\;\;\;\;\;\;(i=1,2 \;\;\; K = (k_1,k_2)) \] where γ = 0.09±0.09 at the 95% confidence level. These limits embrace wind-independent approximations to the observed one-dimensional and two-dimensional wavenumber spectra of the form \[ \phi (k_i) \sim B k^{-3}_i \;\;\; (i = 1,2), \] and \[ \psi(k_i) \sim A k^{-4}, \] respectively, with B ∼ 10−4 and A ∼ 0.3 × 10−4 for $(u^2_*k_i/g)=10^{-2}$ and k = |k| is expressed in cycles/metre. The present findings do not support the wavenumber dependence predicted by the recent models in this wavenumber range and are at variance with their predicted dependence on the friction velocity. However, our observations are generally consistent with the radar reflectivity dependence on wind direction and wind speed under Bragg scattering conditions within our wavenumber range. The experimental observations also point out the potentially important role of wave-breaking of longer wave components in influencing the spectral levels of short gravity wave components.