Evolution of a narrow-band spectrum of random surface gravity waves

Abstract

Numerical simulations of the evolution of gravity wave spectra of fairly narrow bandwidth have been performed both for two and three dimensions. Simulations using the nonlinear Schrödinger (NLS) equation approximately verify the stability criteria of Alber (1978) in the two-dimensional but not in the three-dimensional case. Using a modified NLS equation (Trulsen et al. 2000) the spectra 'relax' towards a quasi-stationary state on a timescale ( 2 ω 0 ) −1 . In this state the low-frequency face is steepened and the spectral peak is downshifted. The three-dimensional simulations show a power-law behaviour ω −4 on the high-frequency side of the (angularly integrated) spectrum.

Imbalance of energy and momentum source terms of the sea wave transfer equation for fully developed seas

“…The computations of coefficients γ 1 and γ 2 depend not only on the functional forms of the source terms β w and β br , but also on the assumed shape of the sea wave spectrum in the (k, ϕ) space. The ratio R (not shown) computed when replacing Resio et al's (2011) spreading function by other spreading functions (Elfouhaily et al, 1997;Donelan and Pierson, 1987;Banner, 1990) at low wavenumbers did not allow reduction of the ratio R to a value close to 1.…”

“…(5a) and (5b), one needs to use a model for the sea wave spectrum based on observations. As concerns the omnidirectional spectrum, I used the empirical sea wave spectral model by Elfouhaily et al (1997) for different conditions of wind speed, assuming fully developed situations. An advantage of this model is that it describes the wave spectrum over the whole range of wavenumbers -from the spectral peak to capillary waves -on a purely empirical basis.…”

“…2.2.4, where the spectrum based on Elfouhaily et al (1997) and Resio et al (2011) is replaced by Kudryavtsev et al's (2003) spectrum at high wavenumbers. The results obtained with this composite spectral model are given in Table 2.…”

“…Following Benjamin and Feir's pioneering work, numerical simulations of the evolution of gravity wave spectra of fairly narrow bandwidth were performed (Alber, 1978;Dysthe et al, 2003), and a number of experimental investigations were conducted on the long-time evolution of nonlinear wave trains (Lake et al, 1977;Melville, 1982;Tulin and Waseda, 1999). The modulational instability is an interaction among three monochromatic wave trains: carrier (ω o ), upper (ω o + δω o ), and lower (ω o − δω o ) sideband waves.…”

“…The adjusted value of µ is seen to be of the order of 2 to 3, with only a weak dependence upon The wave breaking term βbr is from Komen et al (1984), and the wind input term βw is from Komen et al (1984), and the wind input term β w is from WAMDI (1988). The sea wave spectral model for k < 25k p is from Elfouhaily et al (1997), with directional spreading function by Resio et al (2011). Above k = 25k p , the sea wave spectral model is either from Elfouhaily et al (1997) wind speed over the 5-15 m s −1 range.…”