Mild-slope Models Research Articles

QuadWave1D: An optimized quadratic formulation for spectral prediction of coastal waves

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.

The impact of modulational instability on coastal wave forecasting using quadratic models

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.

An optimized, parameter-free PML-FEM for wave scattering problems in the ocean and coastal environment

Water-wave propagation in nearshore regions and hydroacoustic scattering problems, in the presence of structures, are fundamental to ocean and coastal engineering. Efficient modelling of these phenomena can be achieved using the Helmholtz equation with spatially varying coefficients to which mild-slope models are reducible. Despite the relatively simple forms of these models, geometric and medium inhomogeneities and inclusions, yield complex wavefield solutions that can only be numerically approximated. However, the numerical treatment of such problems in infinite domains requires the truncation of the computational region. In this work, an optimal, parameter-free Perfectly Matched Layer (PML) model is implemented in a Finite Element scheme. The fundamental similarities between the governing equations for steady-state water wave and hydroacoustic scattering problems allow for a joint analysis of the proposed PML-FEM solution strategy. Excellent convergence characteristics are verified through comparisons against benchmark solutions. Water-wave propagation solutions for an uneven seabed featuring an elliptic shoal are compared with available experimental data. Also, wave diffraction by vertical cylinders in regions of variable bathymetry, and scattering by an elliptically shaped body in the ocean-acoustic waveguide, are studied. The proposed numerical scheme is found to be an efficient means to tackle challenging wave-seabed-body interaction problems in large spatial domains.

A Mild-Slope System for Bragg Scattering of Water Waves by Sinusoidal Bathymetry in the Presence of Vertically Sheared Currents

Extended mild-slope models (MMSs) are examined for predicting the characteristics of normally incident waves propagating over sinusoidal bottom topography in the presence of opposing shearing currents. It is shown that MMSs are able to provide quite good predictions in the case of Bragg scattering of waves over rippled bathymetry without a current, but fail to provide good predictions concerning the resonant frequency in the additional presence of a current. In order to resolve the above mismatch, a two-equation mild-slope system (CMS2) is derived from a variational principle based on the representation of the wave potential expressed as a superposition of the forward and backward components. The latter system is compared against experimentally measured data collected in a wave flume and is shown to provide more accurate predictions concerning both the resonant frequency and the amplitude of the reflection coefficient. Future work will be devoted to the examination of the derived model for a more general wave system over realistic seabed topography.

Elliptic Harbor Wave Model with Perfectly Matched Layer and Exterior Bathymetry Effects

AbstractStandard strategies for dealing with the Sommerfeld condition in elliptic mild-slope models require strong assumptions on the wave field in the region exterior to the computational domain. More precisely, constant bathymetry along (and beyond) the open boundary, and parabolic approximations–based boundary conditions are usually imposed. Generally, these restrictions require large computational domains, implying higher costs for the numerical solver. An alternative method for coastal/harbor applications is proposed here. This approach is based on a perfectly matched layer (PML) that incorporates the effects of the exterior bathymetry. The model only requires constant exterior depth in the alongshore direction, a common approach used for idealizing the exterior bathymetry in elliptic models. In opposition to standard open boundary conditions for mild-slope models, the features of the proposed PML approach include (1) completely noncollinear coastlines, (2) better representation of the real unbounded...

FINITE ELEMENT MODELING OF NONLINEAR WAVE TRANSFORMATION USING ELLIPTIC MILD SLOPE EQUATION

Accurate modeling of nonlinear wave transformation is important for studies related to harbor and nearshore design. While sophisticated finite-element models based on the elliptic mild-slope equation are often used for wave prediction, they do not include wave-wave interactions. These interactions, in general, involve transfer of energy and wave phase coupling among spectral components and are known to be quite significant especially in shoaling regions and inside harbors. To overcome this limitation, and to provide a basis for the eective modeling of nonlinear wave transformation in complex coastal and harbor environments, the development of a finite-element model based on the second-order nonlinear mild-slope equation of Kaihatu and Kirby (1995) is considered. The model uses an iterative procedure for solving the second-order boundary value problem. To ensure eective boundary treatment, a combination of frequently-used boundary conditions and the method of internal wave generation is used. Two cases involving nonlinear shoaling and harbor resonance are considered for model validation. Modeled results are compared with experimental data, and good agreement is observed in most cases. The methodology described in this paper can improve the applicability of existing finite-element based mild-slope models.

Three-dimensional wave diffraction in the vicinity of openings in coastal structures

Wave transformation through openings in coastal structures is dominated by 3D diffraction effects due to sudden changes of water depth, along with the finite width of the channel. In the present work, a novel coupled-mode model, based on eigenfunctions expansions of the Laplace equation, is developed and applied to the numerical solution and the detailed representation of the local 3D wave flow problem in the vicinity of the opening. The harmonic wave field is excited by plane incident wave propagating normally or at an angle with respect to the axis of the opening/channel. The numerical solution converges rapidly, permitting the series truncation at its first terms. The proposed method fully accounts for the 3D diffraction effects and produces necessary information to further couple with mild-slope models describing wave propagation and transformation in coastal regions in the presence of breakwaters and coastal structures containing openings. Calculated results are presented for waves propagating in regions with breakwaters with openings simulating flushing culverts and compared against experimental measurements obtained in a 3D wave basin.

Reflection in variational models for linear water waves

The reflection characteristics are analysed for a series of Hamiltonian water-wave models. These variational models have been derived by applying a Boussinesq-like approach to the vertical flow-structure. Both parabolic and hyperbolic-cosine approximations to the vertical structure are considered. Mild-slope approximations are made for the flow velocities, by neglecting horizontal derivatives of the mean water depth in the Hamiltonian density. In all cases, a positive-definite Hamiltonian is ensured, contributing to the good dynamical behaviour of the resulting flow equations. It is found that, in general, the mild-slope approximation results in less good predictions of the reflections, as compared to the steep-slope variants – i.e. without the mild-slope approximation – and the accurate model results of Porter and Porter [13]. However, by carefully choosing the normalisation for the mild-slope models, good reflection characteristics can be obtained while maintaining the simpler structure of the mild-slope model, as compared with the steep-slope variants.