Spectral Modelling of Coastal Waves over Spatial Inhomogeneity

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.