ALBERT

Description: A second-order nonlinear frequency-domain model extending the linear complementary mild-slope equation (CMSE) is presented. The nonlinear model uses the same streamfunction formulation as the CMSE. This allows the vertical profile assumption to accurately satisfy the kinematic bottom boundary condition in the case of nonlinear triad interactions as well as for the linear refractiondiffraction part. The result is a model with higher accuracy of wavebottom interactions including wavewave interaction. The model's validity is confirmed by comparison with accurate numerical models, laboratory experiments over submerged obstacles and analytical perturbation solutions for class III Bragg resonance. © 2009 Cambridge University Press.

Description: Nonlinear interactions between sea waves and the sea bottom are a major mechanism for energy transfer between the different wave frequencies in the near-shore region. Nevertheless, it is difficult to account for this phenomenon in stochastic wave forecasting models due to its mathematical complexity, which mostly consists of computing either the bispectral evolution or non-local shoaling coefficients. In this work, quasi-two-dimensional stochastic energy evolution equations are derived for dispersive water waves up to quadratic nonlinearity. The bispectral evolution equations are formulated using stochastic closure. They are solved analytically and substituted into the energy evolution equations to construct a stochastic model with non-local shoaling coefficients, which includes nonlinear dissipative effects and slow time evolution. The nonlinear shoaling mechanism is investigated and shown to present two different behaviour types. The first consists of a rapidly oscillating behaviour transferring energy back and forth between wave harmonics in deep water. Owing to the contribution of bottom components for closing the class III Bragg resonance conditions, this behaviour includes mean energy transfer when waves reach intermediate water depths. The second behaviour relates to one-dimensional shoaling effects in shallow water depths. In contrast to the behaviour in intermediate water depths, it is shown that the nonlinear shoaling coefficients refrain from their oscillatory nature while presenting an exponential energy transfer. This is explained through the one-dimensional satisfaction of the Bragg resonance conditions by wave triads due to the non-dispersive propagation in this region even without depth changes. The energy evolution model is localized using a matching approach to account for both these behaviour types. The model is evaluated with respect to deterministic ensembles, field measurements and laboratory experiments while performing well in modelling monochromatic superharmonic self-interactions and infra-gravity wave generation from bichromatic waves and a realistic wave spectrum evolution. This lays physical and mathematical grounds for the validity of unexplained simplifications in former works and the capability to construct a formulation that consistently accounts for nonlinear energy transfers from deep to shallow water. © 2016 Cambridge University Press.

Description: Abstract Common simplified models for surface gravity waves result in parabolic-type equations. These equations mostly assume a negligible reflection from the bottom variations but account for both refraction and diffraction effects. A common deficiency of these equations is an inherent assumption of normally incident waves, which cause an increasing error, as the incident waves propagate from an increasing attack angle. In the nonlinear formulation of this type of equation, previous research added the assumption of small crossing angles between interacting waves, which also limits the applicability of these nonlinear models to narrow directional spectra. The current paper presents a parabolic approximation for oblique incident waves that overcomes these limitations. This is done by introducing a perturbation solution for the wave's phase function, which in its lowest order corresponds to oblique incident waves on a bottom with no lateral changes. The resulting curved wave ray structure replaces the simplified straight one used in the derivation process of various former models. In the nonlinear model, the nonlinear interactions are calculated between the frequency modes while taking into account the different propagating directions. Numerical results of the oblique parabolic equation show great advantages compared to other formulations that use a straight wave ray structure and the small crossing angle assumption. The nonlinear model was used to investigate the nonlinear shoaling of two similar monochromatic waves approaching from different angles. This fundamental nonlinear interaction problem shows that there is an energy transfer to waves of the primary harmonic that approach at larger attack angles than the two incident waves. This is counter-intuitive. Unlike in the case of linear wave shoaling, where the waves reduce their attack angle in the refraction process, in the nonlinear shoaling process the attack angle can increase. Fundamental numerical simulations of the linear and nonlinear oblique parabolic models are shown to be in excellent agreement with the initial mild-slope equations. This confirms the benefits of applying these models to various shoaling scenarios for both linear and nonlinear waves. © Cambridge University Press 2013.

Description: Mild-slope (MS) type equations are depth-integrated models, which predict under appropriate conditions refraction and diffraction of linear time-harmonic water waves. Among these equations, the complementary mild-slope equation (CMSE) was shown to give better agreement with exact two-dimensional linear theory compared to other MS-type equations. Nevertheless, it has a disadvantage of being a vector equation, i.e. it requires solving a system of two coupled partial differential equations. In addition, for three-dimensional problems, there is a difficulty in constructing the additional boundary condition needed for the solution. In the present work, it is shown how the vector CMSE can be transformed into an equivalent scalar equation using a pseudo-potential formulation. The pseudo-potential mild-slope equation (PMSE) preserves the accuracy of the CMSE while avoiding the need of an additional boundary condition. Furthermore, the PMSE significantly reduces the computational effort relative to the CMSE, since it is a scalar equation. The accuracy of the new model was tested numerically by comparing it to laboratory data and analytical solutions. © 2010 Cambridge University Press.

Description: A new method to invert X-band radar images for linear shoaling conditions is proposed. The commonly used approach for this type of inverse problems is the Fourier transform. Unlike in deep water conditions, in the shoaling region, waves are modulated both in terms of wavelength and amplitude. However, Fourier analysis assumes spacial and temporal periodicity, and homogeneity limiting its applicability to this region. In order to overcome these limitations, a wavelet based technique is developed. The proposed technique treats every spatial radar image within the time sequence individually, so no information on the dispersion relation is required. For validation purposes, surface elevation range-time shoaling realizations based on the mild slope equation are prepared. A radar imaging model including tilt and shadowing modulations, speckle noise, and the radar equation is applied to these realizations to provide modeled grazing incidence radar images. The inversion process starts with the application of the continuous wavelet transform independently for each spacial image. The procedure continues with employing a successive range independent modulation transfer function to the wavelet spectra in the wavenumber domain. Then, after a phase shift correction, an inverse continuous wavelet transform is applied. The procedure is finalized by a calibration of the retrieved maps. After the calibration, a thorough comparison between the original and the reconstructed surface elevations is performed. It shows high efficiency of the proposed method in treating wave number and amplitude modulated signals, as well as in addressing local phase shifts due to tilt modulation and noise contamination. The new inversion method is proven to have high accuracy in inhomogeneous conditions. It shows high potential to be implemented for individual wave reconstruction using real aperture radars.