Spectral information of coastal waves and the associated statistical parameters (e.g., the significant wave height and mean wave period) over large spatial scales is essential for many applications (e.g., coastal safety assessments, coastal management and developments, etc.). This demand explains the necessity for accurate yet effective models. A well-known efficient modelling approach is the quadratic approach (often referred to as frequency-domain models, weakly nonlinear mild-slope models, amplitude models, etc.). The efficiency of this approach is achieved through modelling reduction of the original governing equations (e.g., Euler equations). Most significantly, wave nonlinearity is described solely by a single quadratic mode-coupling term. Therefore, doubts arise with regard to the predictive capabilities of the quadratic approach to reliably describe the nonlinear development of waves in the coastal environment where nonlinearity is typically significant. This study attempts to push the limit of the prediction capabilities of nonlinear coastal waves based on the quadratic approach. To this end, an optimization process is proposed, striving to extract the quadratic formulation which describes most adequately nonlinear wave developments over water depths and bathymetrical structures which characterize the coastal environment. The outcome is the model QuadWave1D: a fully dispersive quadratic model for coastal wave prediction in one-dimension. Based on a wide set of examples (including monochromatic, bichromatic and irregular wave conditions) and comparing to other representative quadratic formulations, it is found that QuadWave1D presents superior predictive capabilities of both the sea-swell components and the infragravity field.

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Coastal wave forecasting over large spatial scales is essential for many applications (e.g., coastal safety assessments, coastal management and developments, etc.). This demand explains the necessity for accurate yet effective models. A well-known efficient modelling approach is the quadratic approach (often referred to as frequency-domain models, nonlinear mild-slope models, amplitude models, etc.). The efficiency of this approach stems from a significant modelling reduction of the original governing equations (e.g., Euler equations). Most significantly, the description of wave nonlinearity essentially collapses into a single mode coupling term determined by the quadratic interaction coefficients. As a result, it is expected that the efficiency achieved by the quadratic approach is accompanied by a decrease in prediction accuracy. In order to gain further insight into the predictive capabilities of this modelling approach, this study examines six different quadratic formulations, three of which are of the Boussinesq type and the other three are referred to as fully dispersive. It is found that while the Boussinesq formulations reliably predict the evolution of coastal waves, the predictions by the fully dispersive formulations tend to be affected by false developments of modulational instability. Consequently, the predicted wave fields by the fully dispersive formulations are characterized by unexpectedly strong modulations of the sea-swell part and associated unexpected infragravity response. The impact of the modulational instability on wave prediction based on the quadratic approach is further demonstrated using existing laboratory results of bichromatic and irregular wave conditions.

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Weyl rule of association, proposed by Hermann Weyl for quantum mechanics applications (Weyl, 1931), can be used to associate between the dispersion relation of water waves and a non-local pseudo-differential operator. The central result of this study is that this operator correctly approximates the Dirichlet-to-Neumann operator derived for linear waves over a slowly varying bathymetry. This opens the door to a formal use of Weyl's operational calculus, and consequently, allowing straightforward derivations and generalizations of water waves’ models over mild slopes. Specifically, within the framework of linear wave theory, the formulation based on Weyl rule of association provides a generalized mild-slope model which does not impose a limit on the spectral width. Most significantly, the mild-slope formulation based on Weyl rule of association allows to derive a general linear kinetic equation for which the widely used energy balance equation (the central equation of forecasting models such as SWAN and WAVEWATCH) serves as a special case. This result not only provides a formal link between the deterministic description (i.e., Euler equations) and the stochastic description (i.e., the energy balance equation), but also establishes the theoretical foundations for the statistical description of bathymetry-induced wave interferences. Such a statistical description is especially important over coastal waters, where through the interaction with the bathymetry, waves are rapidly scattered and tend to form focal zones and associated interference patterns.

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Spectral wave models are widely used for wave prediction over large spatio-temporal scales. Over global scales, spectral models (e.g. WAM and WAVEWATCH III) are used regularly by environmental modelling centers, such as the European Centre for Medium-Range Weather Forecasts (ECMWF) and the American National Center for Environmental Prediction (NCEP), in order to support human activity at sea. Along the coasts, practitioners rely on spectral models which are designated to the coastal environment (e.g. SWAN and TOMAWAC) for applications such as coastal hazard assessment, future coastal development, planning of defense strategies for coastal safety, evacuation planning of coastal communities and so forth. An important property that characterizes the spectral approach and enables its applicability for large scales is efficiency. This property is achieved owing to the simple wave description that underlies its formulation. Specifically, the spectral approach represents ocean wave fields as quasi-Gaussian, quasi-homogeneous and quasi-stationary processes. These convenient statistical properties provide a full statistical description of wave fields based on the energy spectrum alone, and therefore, allow to describe the waves in the ocean in a complete statistical sense through the solution of a single transformation equation - the energy balance equation. The validity of this statistical modelling framework is based on the weak (in the mean) wave forcing and the dispersion effects. These two agents provide reasonable justifications that the deviation from the assumed statistical properties (i.e. Gaussianity, homogeneity and stationarity) is kept negligible in the course of wave evolution. While these arguments are reasonable in the open ocean (where dispersive effects are strong and wave processes are characterized by large scales), they become somewhat loose for the coastal environment (where wave dispersion weakens and wave processes develop rapidly). Evidently, processes like medium-induced wave interferences and energy exchanges due to shallow water nonlinearity are not properly represented under this statistical framework. This study is set forward with the aim of advancing the spectral modelling capabilities in coastal waters by allowing the development of inhomogeneous and non-Gaussian statistics. To this end, the effort of this work is directed to three different parts, concerning three principle issues. The first part considers the formal connection between the classical deterministic formulation (e.g. the Euler equations) and the statistical formulation given by the so-called Wigner-Weyl formulation (a statistical framework that includes the information of wave interferences and reduces to the energy balance equation when interference effects are negligible). The second parts aims to generalize the Wigner-Weyl formulation (which presently accounts for wave-bottom interactions) to allow for the interaction of waves and ambient currents. Finally, the third part is devoted to the investigation of the quadratic modelling approach which defines the starting point for the present phase-averaged formulation of shallow water nonlinearity. The objective of the first part of this study is achieved by showing the equivalence between a formal definition of the Dirichlet-to-Neumann operator of waves over variable bathymetry and the Weyl operator of the dispersion relation. This equivalence opens the door to a formal use of Weyl calculus, based on which the Wigner-Weyl formulation is formally derived. This result establishes the desired formal link between the deterministic formulation for water waves and the statistical formulation given by the Wigner-Weyl formulation, which includes the energy balance equation as a statistically well-defined limiting case. In the second part of this study, the Wigner-Weyl formulation for water waves is extended to account for wave-current interactions. The outcome is a generalized action balance model that is able to predict the evolution of the wave statistics over variable media, while preserving statistical contributions due to wave interferences. Comparisons with results of the SWAN model and the REF/DIF 1 model through several examples verify model performance and demonstrate that retention of interference contributions is essential for accurate prediction of wave statistics in shear-current-induced focal zones. Finally, the third part of this study explores the predictive capabilities of the quadratic approach. This is performed by analyzing the nonlinear properties of six different quadratic formulations, three of which are of the Boussinesq type and the other three are referred to as fully dispersive formulations. It is found that while the Boussinesq formulations predict reliably the nonlinear development of coastal waves, the predictions by the fully dispersive formulations tend to be affected by false developments of modulational instability. As a result, the predicted fields by the fully dispersive formulations are characterized by unexpectedly strong modulations of the sea-swell part and associated unexpected infragravity response. Additionally, this part of the study also presents an attempt to push the limits of the predictive capabilities of the quadratic approach. The outcome is the model QuadWave1D: a fully dispersive quadratic model for coastal wave prediction in one dimension. Based on a wide set of examples (including monochromatic, bichromatic and irregular wave conditions), it is found that the new formulation presents superior forecasting capabilities of both the sea-swell components and the infragravity field. In summary, the overall effort of this study provides an additional step toward the broader goal of efficient and accurate spectral modelling capabilities of coastal waves. This step includes strengthening the theoretical foundations of the spectral approach, improving the spectral description of wave transformation over spatial inhomogeneity and helping to minimize the errors associated with the spectral formulation of shallow water nonlinearity. Ultimately, this study also points on and prepares the background to additional required model developments. @en

Coastal safety assessments with wave-resolving storm impact models require a proper offshore description for the incoming infragravity (IG) waves. This boundary condition is generally obtained by assuming a local equilibrium between the directionally-spread incident sea-swell wave forcing and the bound IG waves. The contribution of the free incident IG waves is thus ignored. Here, in-situ observations of IG waves with wave periods between 100 s and 200 s at three measurement stations in the North Sea in water depths of O(30) m are analyzed to explore the potential contribution of the free and bound IG waves to the total IG wave height for the period from 2010 to 2018. The bound IG wave height is computed with the equilibrium theory of Hasselmann using the measured frequency-directional sea-swell spectra as input. The largest IG waves are observed in the open sea with a maximum significant IG wave height of O(0.3) m at 32 m water depth during storm Xaver (December 2013) with a concurrent significant sea-swell wave height in excess of 9 m. Along the northern part of the Dutch coast, this maximum has reduced to O(0.2) m at a water depth of 28 m with a significant sea-swell wave height of 7 m and to O(0.1) m at the most southern location at a water depth of 34 m with a significant sea-swell wave height of 5 m. These appreciable IG wave heights in O(30) m water depth represent a lower bound for the expected maximum IG wave heights given the fact that in the present analysis only a fraction of the full IG frequency range is considered. Comparisons with the predicted bound IG waves show that these can contribute substantially to the observed total IG wave height during storm conditions. The ratio between the predicted bound-and observed total IG variance ranges from 10% to 100% depending on the location of the observations and the timing during the storm. The ratio is typically high at the peak of the storm and is lower at both the onset and waning of the storm. There is significant spatial variability in this ratio between the stations. It is shown that differences in the directional spreading can play a significant role in this. Furthermore, the observed variability along the Dutch coast, with a substantially decreased contribution of the bound IG waves in the south compared to the northern part of the Dutch coast, are shown to be partly related to changes in the mean sea-swell wave period. For the southern part of the Dutch coast this corresponds to an increased difference with the typically assumed equilibrium boundary condition although it is not clear how much of the free IG-energy is onshore directed barring more sophisticated observations and/or modeling.

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Wave forecasting in ocean and coastal waters commonly relies on spectral models based on the spectral action balance equation. These models assume that different wave components are statistically independent and as a consequence cannot resolve wave interference due to statistical correlation between crossing waves, as may be found in, for instance, a focal zone. This study proposes a statistical model for the evolution of wave fields over non-uniform currents and bathymetry that retains the information on the correlation between different wave components. To this end, the quasi-coherent model (Smit & Janssen, J. Phys. Oceanogr., vol. 43, 2013, pp. 1741-1758) is extended to allow for wave-current interactions. The outcome is a generalized action balance model that predicts the evolution of the wave statistics over variable media, while preserving the effect of wave interferences. Two classical examples of wave-current interaction are considered to demonstrate the statistical contribution of wave interferences: (1) swell field propagation over a jet-like current and (2) the interaction of swell waves with a vortex ring. In both examples cross-correlation terms lead to development of prominent interference structures, which significantly change the wave statistics. Comparison with results of the SWAN model demonstrates that retention of cross-correlation terms is essential for accurate prediction of wave statistics in shear-current-induced focal zones.

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The present study investigates the role of hydroelasticity and nonlinearity in the fundamental problem of the interaction between non-breaking water waves and an elastic wall. To this end, two interaction scenarios are considered: the interaction of a rigid wall supported by springs and a pulse-type wave, and the interaction of an elastic deformable wall and an incident wave group. Both of these scenarios are numerically simulated in a computational domain representing a two-dimensional wave flume. The simplicity of the domain enables one to perform highly efficient simulations using the high-order spectral method (HOSM). Wave generation at the flume entrance and the wave-wall interaction at the flume end are simulated by means of the additional potential concept. In this way, the efficiency that characterizes the original HOSM is preserved for the present non-periodic problems. The investigation of the first scenario reveals the influence of the wall's dynamical response on the hydrodynamic values. The results show that the maximum wave run-up and wave force are prominently fluctuating around the values corresponding to a fixed wall as a function of the wall's eigenfrequency, revealing regions of relaxation and amplification. The second scenario studies the effect of the nonlinear evolution of the incident wave group. The high-order wave harmonics generated during the group evolution are found to be significant for predicting extreme hydrodynamic and structural values, and may result in resonant interactions in which hydroelasticity appears to play an important role.

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