Accurate prediction of Wave-Group-Forced (WGF) InfraGravity (IG) waves depends on resolving the corresponding phase shift, typically achieved through a coupled phase – amplitude equation. However, this approach requires a grid resolution that resolves the structure of the wave groups making it computationally expensive at regional scales. To address this limitation, an existing local expression for the phase shift of normally incident WGF-IG waves has been extended to account for directional seas. The extended formulation is verified against predictions from the coupled phase – amplitude model using bichromatic wave forcing over a uniformly sloping beach for a wide range of sea-swell conditions. Results show that the local approach performs well in the off-resonant region for obliquely incident waves. When applied outside this regime, however, its accuracy decreases, with performance varying depending on sea-swell and bathymetric conditions. The coupled and local phase shift approaches are also validated with observations obtained during the Coast3D field experiment. The total, incoming and outgoing IG waves are predicted with comparable skill and root mean square error for both methods. The good match using the local expression is attributed to the fact that the conditions during Coast3D correspond to directionally broad sea-swell spectra with relative short peak periods propagating over moderately sloping bathymetry for which the verification showed significant skill. Additional validation with field observations at other locations are necessary to firmly determine the limitations of the use of a local phase shift.

This study examines the importance of free infragravity (FIG) waves in the North Sea using a recent collection of wave measurements and a newly developed unstructured SWAN model. The measurements include new observations of infragravity waves at offshore (30–40 m water depth) and nearshore (10–20 m water depth) locations in the southern North Sea. These observations serve as the basis for model optimization and verification. Good agreement is obtained between model predictions and measurements during two recent storm periods, including severe storms with unusual wind directions and high wind speeds (e.g., “Storm Babet”). Model investigation along the coasts of Belgium and the Netherlands demonstrated a strong dependence between nearshore FIG conditions (i.e., energy intensity and sources) and storm characteristics (i.e., alongshore wind pattern and storm track). Specifically, several storms have demonstrated significant contributions of FIG energy originating from remote sources (e.g., the coasts of UK and Denmark). This suggests that nearshore FIG conditions in the North Sea cannot be determined based on the local sea-swell conditions alone and may be significantly underestimated if non-local contributions are ignored. Finally, modelled and measured results at nearshore locations along the Dutch coast revealed that under storm conditions FIG energy can be an order of magnitude higher than energy due to bound infragravity (BIG) waves. This result, augmented with estimated ratios of free and forced infragravity energy at the shoreline, emphasizes the necessity of considering the FIG waves as an integral part of coastal safety assessments along the coasts of the North Sea.

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.

Despite the increasing use of non-hydrostatic models in the study of wave processes in coastal regions, there is still limited understanding of the non-linear properties of the governing equations and how they improve with increased vertical resolution. In this study, the governing equations of the non-hydrostatic wave model SWASH are analysed and the linear and non-linear solutions up to third-order of all dependent variables are derived, considering one to four vertical layers. The analysis concludes that the model can achieve excellent non-linear properties with respect to the Stokes theory over a large range of water depths using only a few vertical layers. Furthermore, deriving solutions for all variables enables the formulation of improved wave generation and absorption boundary conditions for non-hydrostatic models. A well-known issue of non-linear wave models is related to the generation and propagation of spurious free waves, resulting to non-homogeneous wave fields. In this study, it is proven that by imposing the derived exact mathematical solutions of the governing equations at the model's boundaries, the target first- and second-order wave profiles can be generated with high accuracy, while the spurious waves can be entirely eliminated.

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.

Currents can affect the evolution of waves in nearshore regions through altering their wavenumber and amplitude. Including the effect of ambient currents (e.g., tidal and wind-driven) on waves in phase-resolving wave models is not straightforward as it requires appropriate boundary conditions in combination with a large domain size and long simulation duration. In this paper, we extended the non-hydrostatic wave-flow model SWASH with additional terms that account for the influence of a depth-uniform ambient current on the wave dynamics, in which the current field can be taken from an external source (e.g., from observations or a circulation model). We verified the model ability by comparing predictions to results from linear theory, laboratory experiments and a spectral wave model that accounts for wave interference effects. With this extension, the model was able to account for current-induced changes to the wave field (i.e., changes to the wave amplitude, length and direction) due to following and opposing currents, and two classical examples of sheared currents (a jet-like current and vortex ring). Furthermore, the model captured the wave dynamics in the presence of strong opposing currents. This includes reflections of relatively small amplitude waves at the theoretical blocking point, and transmission of breaking waves beyond the theoretical blocking point for larger wave amplitudes. The proposed model extension allows phase-resolving models to more accurately and efficiently simulate the wave dynamics in coastal regions with tidal and/or wind-driven flows.

Accurate modelling of waves in harbours and the response of moored ships to that type of forcing is of prime importance to determine the safety and workability of ships moored at berths exposed to local wave conditions. This study investigates the combination of a non-hydrostatic wave-flow model (SWASH) and a 3D boundary-integral diffraction model (Harberth) to compute wave forces acting on moored ships. A series of systematic numerical tests has been performed to develop the proposed methodology and gain insight on its limits of application. The approach is validated using physical scale model test data of waves and forces acting on a restrained ship. Results indicate a good performance even for extremely energetic wave conditions, setting the investigated modelling approach as a potential alternative for future applications.

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.

Predictions of the wave-induced response of floating structures that are moored in a harbour or coastal waters require an accurate description of the (nonlinear) evolution of waves over variable bottom topography, the interactions of the waves with the structure, and the dynamics of the mooring system. In this paper, we present a new advanced numerical model to simulate the wave-induced response of a floating structure that is moored in an arbitrary nearshore region. The model is based on the non-hydrostatic approach, and implemented in the open-source model SWASH, which provides an efficient numerical framework to simulate the nonlinear wave evolution over variable bottom topographies. The model is extended with a solution to the rigid body equations (governing the motions of the floating structure) that is tightly coupled to the hydrodynamic equations (governing the water motion). The model was validated for two test cases that consider different floating structures of increasing geometrical complexity: a cylindrical geometry that is representative of a wave-energy-converter, and a vessel with a more complex shaped hull. A range of wave conditions were considered, varying from monochromatic to short-crested sea states. Model predictions of the excitation forces, added mass, radiation damping, and the wave-induced response agreed well with benchmark solutions to the potential flow equations. Besides the response to the primary wave (sea-swell) components, the model was also able to capture the second-order difference-frequency forcing and response of the moored vessel. Importantly, the model captured the wave-induced response with a relatively coarse vertical resolution, allowing for applications at the scale of a realistic harbour or coastal region. The proposed model thereby provides a new tool to seamlessly simulate the nonlinear evolution of waves over complex bottom topography and the wave-induced response of a floating structure that is moored in coastal waters.

The authors regret that there was a typo in Equation (14b). 

The Simulating WAves Nearshore (SWAN) model has been extended with an infragravity module to predict the Wave-Group-Forced (WGF) infragravity response to a frequency-directional sea-swell spectrum at a mildly sloping alongshore uniform beach. To that end the SWAN model has been extended with an WGF-infragravity source term denoted S_sb where the subscript denotes surfbeat. The corresponding WGF infragravity energy model has been verified with a set of benchmark tests using the infragravity amplitude model of Reniers et al. (2002). Next the implementation of the energy balance in SWAN has been validated with both prototype-scale laboratory experiments and field observations, showing a good comparison with observations not affected by the nodal structure of the (partially) standing infragravity waves. This suggests that the model is capable of providing improved infragravity boundary conditions in relatively shallow water compared to the typical assumption of equilibrium forcing conditions using for instance Hasselmann's equilibrium theory (Hasselmann, 1962). These infragravity boundary conditions can subsequently can be used by other more sophisticated models to compute runup, overtopping and dune erosion.

The weakly reflective wave generation is a wave generation and absorption method in phase-resolving models, based on the assumption that the waves propagating towards the wave generation boundary are small amplitude shallow water waves with direction perpendicular to the boundary. This assumption makes the method weakly reflective for dispersive and directional waves. The internal wave generation method was proposed by Vasarmidis et al. (2019b) as an alternative, for the non-hydrostatic wave model, SWASH, to avoid reflections. In this study, a comparison is made between the performance of the new internal wave generation method and the weakly reflective wave generation method. It is shown that using the internal wave generation leads to a significantly more accurate prediction of the resulting wave field in case of waves reflected back to the numerical boundary. Additionally, the internal wave generation method is extended to short-crested waves and SWASH is validated for the first time with experimental data for the cases of wave propagation over a shoal and wave diffraction around a wall. The proposed extended internal wave generation method increases the capability of SWASH towards the study of wave propagation of highly dispersive short-crested waves in coastal environments with minimal reflection from the boundaries.

Infragravity waves are low-frequency surface waves that can impact a variety of nearshore and oceanic processes. Recent measurements in the North Sea showed that significant bursts of infragravity energy occurred during storm events. Using a spectral wave model, we show that a substantial part of this energy was radiated from distant shorelines where it was generated by the incident sea-swell waves. These radiated infragravity waves can cross the North Sea basin and reach distant shorelines. The origin of the infragravity wave energy varied between the different storms, and particularly depends on where largest sea-swell waves made landfall. Along the coastlines of the North Sea, shoreward directed infragravity waves that originate from a remote source were non-negligible during storm events. This suggests that radiated infragravity waves can potentially contribute to coastal dynamics and hazards away from their region of generation.

Analysis of the mean (wave-averaged) momentum balance is a common approach used to explain the physical forcing driving wave set-up and mean currents in the nearshore zone. Traditionally this approach has been applied to phase-averaged models but has more recently been applied to phase-resolving models using post-processing, whereby model output is used to calculate each of the momentum terms. While phase-resolving models have the advantage of capturing the nonlinear properties of waves propagating in the nearshore (making them advantageous to enhance understanding of nearshore processes), the post-processing calculation of the momentum terms does not guarantee that the momentum balance closes. We show that this is largely due to the difficulty (or impossibility) of being consistent with the numerical approach. If the residual is of a similar magnitude as any of the relevant momentum terms (which is common with post-processing methods as we show), the analysis is largely compromised. Here we present a new method to internally calculate and extract the depth-integrated, mean momentum terms in the phase-resolving non-hydrostatic wave-flow model SWASH in a manner that is consistent with the numerical implementation. Further, we demonstrate the utility of the new method with two existing physical model studies. By being consistent with the numerical framework, the internal method calculates the momentum terms with a much lower residual at computer precision, combined with greatly reduced calculation time and output storage requirements compared to post-processing techniques. The method developed here allows the accurate evaluation of the depth-integrated, mean momentum terms of wave-driven flows while taking advantage of the more complete representation of the wave dynamics offered by phase-resolving models. Furthermore, it provides an opportunity for advances in the understanding of nearshore processes particularly at more complex sites where wave nonlinearity and energy transfers are important.

Joint effects of the dynamic sea-level rise projected changes in the large-scale atmosphere/ocean circulation, and wave climate on hurricane-induced extreme water levels in the Caribbean region are assessed. We use the 2D-depth integrated ADCIRC + SWAN wave-ocean model, baroclinically coupled to an ocean-eddying version of the Community Earth System Model, to compare impacts of the September 2017 hurricanes with projected impacts of similar hypothetical tropical storms occurring in the future. The model predicts only minor changes in the hurricane-induced extreme water levels for those Caribbean islands which were severely devastated by the 2017 tropical storms (Irma and Maria). That is, provided that the hurricane intensity remains at the present-day level, the global mean sea-level rise is the main future coastal flood risk factor.

We are concerned with the numerical solution of a linear transport problem with nonzero divergence velocity field that originates from the spectral energy balance equation describing the evolution of wind waves and swells in coastal seas. The discretization error of the commonly used first-order upwind finite difference and first-order vertex-centered upwind finite volume schemes in one space dimension is analyzed. Smoothness of nondivergent velocity field plays a crucial role in this. No such analysis has been attempted to date for such problems. The two schemes studied differ in the manner in which they treat the scalar flux numerically. The finite difference variant is shock captured, whereas the vertex-centered finite volume approximation employs an arithmetic mean of the velocity and appears not to be flux conservative. The methods are subsequently extended to two dimensions on triangular meshes. Numerical experiments are provided to verify the convergence analysis. The main finding is that the finite difference scheme displays optimal rates of convergence and offers higher accuracy over the finite volume scheme, regardless the regularity of the velocity field. The latter scheme notably yields convergence rates of 0.5 and 0 in L²-norm and L^∞-norm, respectively, when the velocity field is not smooth. A test case illustrating wave shoaling and refraction over submerged shoals is also presented and demonstrates the practical importance of flux conservation.