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The authors prove the existence and the linear stability of small amplitude time quasi-periodic standing wave solutions (i.e. periodic and even in the space variable x) of a 2-dimensional ocean with infinite depth under the action of gravity and surface tension. Such an existence result is obtained for all the values of the surface tension belonging to a Borel set of asymptotically full Lebesgue measure.

The observation-based source terms available in the third-generation wave model WAVEWATCH III (i.e., the ST6 package for parameterizations of wind input, wave breaking, and swell dissipation terms) are recalibrated and verified against a series of academic and realistic simulations, including the fetch/duration-limited test, a Lake Michigan hindcast, and a 1-yr global hindcast. The updated ST6 not only performs well in predicting commonly used bulk wave parameters (e.g., significant wave height and wave period) but also yields a clearly improved estimation of high-frequency energy level (in terms of saturation spectrum and mean square slope). In the duration-limited test, we investigate the modeled wave spectrum in a detailed way by introducing spectral metrics for the tail and the peak of the omnidirectional wave spectrum and for the directionality of the two-dimensional frequency–direction spectrum. The omnidirectional frequency spectrum E ( f ) from the recalibrated ST6 shows a clear transition behavior from a power law of approximately f −4 to a power law of about f −5 , comparable to previous field studies. Different solvers for nonlinear wave interactions are applied with ST6, including the Discrete Interaction Approximation (DIA), the more expensive Generalized Multiple DIA (GMD), and the very expensive exact solutions [using the Webb–Resio–Tracy method (WRT)]. The GMD-simulated E ( f ) is in excellent agreement with that from WRT. Nonetheless, we find the peak of E ( f ) modeled by the GMD and WRT appears too narrow. It is also shown that in the 1-yr global hindcast, the DIA-based model overestimates the low-frequency wave energy (wave period T > 16 s) by 90%. Such model errors are reduced significantly by the GMD to ~20%.

This volume describes the observation, analysis and prediction of wind-generated waves in the open ocean, shelf seas, and coastal regions. It introduces observation techniques for waves, both in-situ and through remote-sensing, and defines the parameters that characterise waves.

We analyze surface wave data taken in Currituck Sound, North Carolina, during a storm on 4 February 2002. Our focus is on the application of nonlinear Fourier analysis (NLFA) methods (Osborne 2010) to analyze the data set: The approach spectrally decomposes a nonlinear wave field into sine waves, Stokes waves, and phase-locked Stokes waves otherwise known as breather trains. Breathers are nonlinear beats, or packets which “breathe” up and down smoothly over cycle times of minutes to hours. The maximum amplitudes of the packets during the cycle have a largest central wave whose properties are often associated with the study of “rogue waves.” The mathematical physics of the nonlinear Schrödinger (NLS) equation is assumed and the methods of algebraic geometry are applied to give the nonlinear spectral representation. The distinguishing characteristic of the NLFA method is its ability to spectrally decompose a time series into its nonlinear coherent structures (Stokes waves and breathers) rather than just sine waves. This is done by the implementation of multidimensional, quasi-periodic Fourier series, rather than ordinary Fourier series. To determine preliminary estimates of nonlinearity, we use the significant wave height Hs, the peak period Tp, and the length of the time series T. The time series analyzed here have 8192 points and T =1677.72 s = 27.96 min. Near the peak of the storm, we find Hs ≈ 0.55 m, Tp ≈ 2.4 s so that for the wave steepness of a near Gaussian process, S=π5/2/gHs/Tp2\\({S} = \\left (\\pi ^{5/2}/g\\right )H_{s}/{T}_{p}^{2}\\), we find S ≈ 0.17, quite high for ocean waves. Likewise, we estimate the Benjamin-Feir (BF) parameter for a near Gaussian process, IBF=π5/2/gHsT/Tp3\\({I_{BF}} = \\left (\\pi ^{5/2}/g \\right ) H_{s} T/{T}_{p}^{3}\\), and we find IBF ≈ 119. Since the BF parameter describes the nonlinear behavior of the modulational instability, leading to the formation of breather packets in a measured wave train, we find the IBF for these storm waves to be a surprisingly high number. This is because IBF, as derived here, roughly estimates the number of breather trains in a near Gaussian time series. The BF parameter suggests that there are roughly 119 breather trains in a time series of length 28 min near the peak of the storm, meaning that we would have average breather packets of about 14 s each with about 5-6 waves in each packet. Can these surprising results, estimated from simple parameters, be true from the point of view of the complex nonlinear wave dynamics of the BF instability and the NLS equation? We analyze the data set with the NLFA to verify, from a nonlinear spectral point of view, the presence of large numbers of breather trains and we determine many of their properties, including the rise time for the breathers to grow to their maximum amplitudes from a quiescent initial state. Energetically, about 95% of the NLFA components are found to consist of breather trains; the remaining small amplitude components are sine and Stokes waves. The presence of a large number of densely packed breather trains suggests an interpretation of the data in terms of breather turbulence, highly nonlinear integrable turbulence theoretically predicted for the NLS equation, providing an interesting paradigm for the nonlinear wave motion, in contrast to the random phase Gaussian approximation often considered in the analysis of data.

We study nonlinear resonant wave–wave interactions which occur when ocean waves propagate into a thin floating ice sheet. Using multiple-scale perturbation analysis, we obtain theoretical predictions of the wave amplitude evolution as a function of distance travelled past the ice edge for a semi-infinite ice sheet. The theoretical predictions are supported by a high-order spectral (HOS) method capable of simulating nonlinear interactions in both open water and the ice sheet. Using the HOS method, the amplitude evolution predictions are extended to multiple (coupled) triad interactions and a single ice sheet of finite length. We relate the amplitude evolution to mechanisms with strong frequency dependence – ice bending strain, related to ice breakup, as well as wave reflection and transmission. We show that, due to sum-frequency interactions, the maximum strain in the ice sheet can be more than twice that predicted by linearised theory. For an ice sheet of finite length, we show that nonlinear wave reflection and transmission coefficients depend on a parameter in terms of wave steepness and ice length, and can have values significantly different than those from linear theory. In particular, we show that nonlinear sum-frequency interactions can appreciably decrease the total wave energy transmitted past the ice sheet. This work has implications for understanding the occurrence of ice breakup, wave attenuation due to scattering in the marginal ice zone and the resulting ice floe size distribution.

Presents an explaination of how sound-waves work.

We consider the full irrotational water waves system with surface tension and no gravity in dimension two (the capillary waves system), and prove global regularity and modified scattering for suitably small and localized perturbations of a flat interface. An important point of our analysis is to develop a sufficiently robust method, based on energy estimates and dispersive analysis, which allows us to deal simultaneously with strong singularities arising from time resonances in the applications of the normal form method and with nonlinear scattering. As a result, we are able to consider a suitable class of perturbations with finite energy, but no other momentum conditions. Part of our analysis relies on a new treatment of the Dirichlet-Neumann operator in dimension two which is of independent interest. As a consequence, the results in this paper are self-contained.