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Spectral models

High-order spectral methods are undeniably the most computationally efficient methods for modelling nonlinear waves, being capable of simulating 3D random sea states at linear scales of tens of wavelengths during tens of wave periods (Ducrozet et al., 2007).

In an early application of a spectral method, Fenton and Rienecker (1982) represented the potential and surface elevation in a 2D periodic domain of constant depth via Fourier expansion by basic functions satisfying the Laplace equation and boundary conditions. If the initial values of the Fourier coefficients are known, the kinematic free-surface condition can be used to advance the surface elevation using a finite difference approximation of the time derivative. All spatial derivatives are computed in the Fourier space. Inverse Fourier transforms are then used to perform a time step in the physical space. This simple approach, however, can not be applied to advance the potential. Instead, the dynamic free surface condition is used to calculate the time derivatives of each Fourier coefficient , which are then used to find the values of the coefficients at the next time step. Calculating derivatives requires solving a large system of simultaneous equations, which is responsible for the low computational efficiency of the method.

This problem was solved in high-order spectral methods (Dommermuth and Yue, 1987a; West et ah, 1987). In this method, the potential is expressed as an asymptotic expansion by a small steepness parameter. In addition, the free surface potential is expanded in a Taylor series around the mean water level, and a double expansion is used to represent the surface potential. The known initial values of surface potential and surface elevation define a Dirichlet boundary value problem for each term of the expansion in the domain below the mean water level. The solution of these problems is sought in the form of a Fourier expansion by modal functions satisfying the Laplace equation and the boundary conditions at side boundaries and the bottom. This makes it possible to express the components of the vertical velocity at the free surface via modal coefficients, which themselves are defined by the surface elevation and the surface potential. This closes the evolution equations provided by the free surface conditions and allows to update the surface values. Fast Fourier Transforms are used to switch between spectral and physical spaces. The shape of the domain should be selected to define a simple spectral basis to expand the velocity potential. Therefore, either periodic domains in both horizontal dimensions or rectangular tanks are usually used.

An alternative approach uses the Dirichlet-Neumann (DN) operator, which expresses the normal surface velocity in terms of velocity potential at the surface. If such an operator is defined, the water wave problem is reduced to the integration over time of free-surface boundary conditions with unknown functions evaluated only at the free surface. The nonlinear DN operator is expanded in terms of a convergent Taylor expansion about the mean water level. This method was introduced by Craig and Sulem (1993) for 2D waves and extended by Bateman et al. (2001) to 3D cases. Schaffer (2008) demonstrated that different variants of HOS methods and methods that used DN operator are either identical or have only minor differences. The use of the additional potential allowed the modelling of a wavemaker (Ducrozet et ah, 2012b) and a variable bathymetry (Gouin et ah, 2016). This makes the HOS approach acceptable for numerical wave tanks. High efficiency and accuracy of spectral methods compare to other methods for wave propagation was demonstrated by Olrnez and Milgram (1995) and Ducrozet et ah (2012a).

One of the drawbacks of the spectral methods is that they can not model the overturning waves. However, this can not be considered as a serious disadvantage compared to the BEM and FEM models, if we consider their application to wave propagation. Neither of the models considered here is able to continue calculations after wave breaking. However, to model severe sea states and extreme waves, a model should continue calculations after waves break and provide a reasonable prediction of energy dissipation due to breaking. Seiffert and Ducrozet (2018) solved this problem by introducing eddy viscosity as a diffusive term to the free surface boundary conditions to simulate breaking waves in a HOS model. Breaking onset is determined by a breaking criterion. The model demonstrated an impressive comparison with experiments on the propagation of surging wave groups.

More details on formulation and application of HOS models can be found in Bonnefoy et ah (2010).

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