In shallow water, the spectral evolution associated with energy transfers due to three-wave (or triad) interactions is important for the prediction of nearshore wave propagation and wave-driven dynamics. The numerical evaluation of these nonlinear interactions involves the evaluation of a weighted convolution integral in both frequency and directional space for each frequency-direction component in the wave field. For reasons of efficiency, operational wave models often rely on a so-called collinear approximation that assumes that energy is only exchanged between wave components travelling in the same direction (collinear propagation) to eliminate the directional convolution. In this work, we show that the collinear approximation as presently implemented in operational models is inconsistent. This causes energy transfers to become unbounded in the limit of unidirectional waves (narrow aperture), and results in the underestimation of energy transfers in short-crested wave conditions. We propose a modification to the collinear approximation to remove this inconsistency and to make it physically more realistic. Through comparison with laboratory observations and results from Monte Carlo simulations, we demonstrate that the proposed modified collinear model is consistent, remains bounded, smoothly converges to the unidirectional limit, and is numerically more robust. Our results show that the modifications proposed here result in a consistent collinear approximation, which remains bounded and can provide an efficient approximation to model nonlinear triad effects in operational wave models.

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Stochastic wave models play a central role in our present-day wave modelling capabilities. They are frequently used to compute wave statistics, to generate boundary conditions and to include wave effects in coupled model systems. Historically, such models were developed to predict the wave field evolution in deep water where the conditions of Gaussianity generally hold. However, in recent decades, such models have been applied to the shallower coastal environment where the stochastic representation of the dominant wave physics becomes questionable. This is primarily due to the increased influence of wave nonlinearity and the additional depth-induced wave processes that are dominant in this region. Unfortunately, the two most dominant wave processes in the surf zone: depth-induced wave breaking and nonlinear triad wave-wave interactions are also the least well represented and understood. This is due to both their complexity and the scarcity of analytical solutions for realistic wave fields. As such, they represent a significant obstacle in the accurate modelling of the wave dynamics in the coastal region. Providing accurate representations of these wave processes is essential to answering the questions demanded from stochastic wave models from coastal engineers for coastal management and design. Such advancements are necessary to improve our understanding of wave-induced processes, to reduce costs in managing the coastal environment and to tackle contemporary issues such as uncertainties with respect to increased sea level rise. Due to the complexity of depth-induced wave breaking, a complete representation of this wave process does not exist for both stochastic and deterministic modelling frameworks. Although there is extensive literature on the subject of parameterizing depth-induced wave breaking in a stochastic sense, these parameterizations are inconsistent with theory, observations and (deterministic) model predictions. In particular, present-day modelling defaults perform poorly over (near-)horizontal bathymetries with over-enhanced wave dissipation of locally-generated waves and insufficient dissipation of swell waves. Equally, nonlinear triad wave-wave interactions are poorly represented in stochastic wave models due to the problem of closure and the impractical computational expense of more accurate representations. In particular, the most commonly applied parameterization in the wave literature incorrectly predicts the evolution of the spectral shape, and the convergence to an equilibrium high-frequency tail deep in the surf zone. Correctly resolving these issues is essential for the management of many of the activities occurring at the coast; from the design of coastal defenses to feasibility studies for wave energy converters, from port operation and availability to vessel navigation, from understanding the ecology at the coast to the fisheries, and from managing leisure and tourism to safety at the coast. In this work, we investigate the process of depth-induced wave breaking through a comprehensive analysis of the literature and a comparison of modelling performance. Here, we use an extensive set of wave observations representing a large range of wave conditions and bathymetric profiles. The analysis demonstrates that no currently available depth-induced breaking source term is capable of sufficiently representing the process of depth-induced wave breaking. This is shown to be in agreement with the wave literature with parameterizations either over-predicting wave dissipation for locally generated waves or under-predicting wave dissipation for non-locally generated waves over (near-)horizontal bathymetries. To address this issue, a new joint scaling using both local wave and bathymetric conditions is proposed. Using both the normalized characteristic wave number and local bottom slope unifies two approaches prevalent in the wave literature. This is shown to improve the model performance for the dissipation of both locally and non-locally generated waves over (near-)horizontal bathymetries. Furthermore, the validity of the assumption that wave dissipation can be modelled as analogous to a 1D dissipative bore is explored. Subsequently, a heuristic directional modification is introduced for depth-induced wave breaking dissipation models. This directionally partitions the 2D spectrum into several directional partitions that are assumed to be unidirectional. Model results demonstrate that the effect of the directional partitioning is to reduce the dissipation of wave energy and to enhance the significant wave height; in agreement with field measurements. Not only is this modification shown to be applicable to the joint wave breaking parameterization proposed in this study, but also for well-established parameterizations. The effects of both the proposed scaling and directional modification are then reviewed from an operational context and are compared to state-of-the-art source terms, field observations and a hypothetical storm representative of Dutch design conditions. Such design conditions are expected to be representative of design conditions found globally. In an environment where storm intensities may be increasing, for example due to global warming, the results of wave breaking models near the coast under such extreme conditions become of greater relevance. The influence of wave breaking models in coupled model systems is anticipated to provide important new insights in understanding the various wave-driven processes along our coasts. Next, the representation of the nonlinear triad wave-wave interactions in stochastic wave models is reviewed. In particular, the collinear approximation used to transform 1D triad source terms for implementation in 2D stochastic wave models is revisited. These approximations are necessitated by considerations of computational efficiency. The conventional collinear approximation is shown to be inconsistent at the unidirectional limit and to be a primary source of modelling error. Instead of converging to the values predicted by the 1D triad source terms at the unidirectional limit, the energy transfers as computed by stochastic wave models are shown to become unbounded. This results in a dimensional calibration coefficient which is at least an order of magnitude smaller than that found in the wave literature. Consequently, for directional wave conditions, 1D triad source terms implemented with the conventional collinear approximation insufficiently capture the wave evolution. To address this problem, a new collinear approximation is presented which accounts for the wave energy contained within a finite directional bandwidth. This collinear approximation is shown to converge correctly at the unidirectional limit and to agree well with predictions from a second-order accurate deterministic wave model. In particular, better agreement is shown in the modelling prediction of the spectral shape and related integral parameters, e.g. wave period, under idealized wave conditions. Under certain conditions, these error reductions are shown to be more significant than differences between the underlying triad models. The contribution of this work demonstrates that while the underlying theory underpinning stochastic wave modelling in the coastal environment still remains questionable, the accurate determination of wave statistics in the coastal zone is tenable. With the advancements presented in this study, the new source terms correspond better with the current wave literature and are shown to provide significant steps forward over existing default source terms. The developments presented here are anticipated to form the foundation for future source term research, and to be used for the representation of the dominant wave physics in the coastal environment in operational wave models.@en