Wavenumber spectra of short gravity waves

Abstract

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.

Space–time measurements of oceanic sea states

“…In particular, directional spectra of short gravity waves were estimated from stereo-pairs by Shemdin et al (1988) and Banner et al (1989). Benetazzo (2006) has proposed and tested a Wave Acquisition Stereo System (WASS) for field measurements at the coast, and more recently Bechle and Wu (2011) have studied coastal waves over areas ~3 m 2 using a trinocular system (Wanek and Wu, 2006).…”

Directional wavenumber characteristics of short sea waves

“…In deep water (kh » 1) this simplifies to (2.2) where 9 is the gravitational constant (981 cm/sec 2 ), h is the water depth (em), T is the surface tension (~73 dynes/em), and p is the density of sea water (~1 g/cm 3 ).…”

“…The shape of the wavenumber spectrum for the short wave region in the dominant wave direction has generally taken the form (2.10) where m has taken on values from 3 to 5, depending on the experimental conditions [48,46,7,2]. The proportionality constant often includes parameters such as wind speed, wind stress, fetch, peak spectral frequency, and/or wave age.…”

“…2 For the purposes of consistency in the spectral comparisons between the CWWG data and the models, therefore, only the (symmetric) wave height spectrum will be used. The spreading models presented in Table 5.1 are not constrained to be symmetric, though, so a symmetrized or folded spreading function is used instead.…”

Wave modelling – The state of the art

“…Also, a dissipation function based on the breaking of short waves in the equilibrium interval does not account for dissipation due to dominant wave breaking, near the spectral peak, which may be more severe and can be quite frequent (Babanin et al, 2001;Young and Babanin, 2006). Finally, there is growing evidence that dominant waves and the breaking of dominant waves affect dissipation at smaller scales (Banner et al, 1989;Meza et al, 2000;Donelan, 2001;Young and Babanin, 2006). If that is true, dissipation in the saturation interval will not be a function local in wavenumber space.…”