Choice of numerical parameters in HOS-ocean

This section describes the main elements driving the choice of the numerical parameters in HOS-ocean. The model is based on pseudo-spectral formalism, which decompose the different surface spatial quantities on the normal modes of the domain.

Spatial/Modal discretization

Horizontal: n1 and n2

It is possible to provide general recommendations in the choice of the discretization in x-direction n1. However, as any numerical parameter, a convergence study should be carried out to test the influence of changing n1. One should take into account the following points:

• With a pseudo-spectral approach, the discretization defines the largest wavenumber that can be solved k_max =(n1-1)*2π/xlen

• One wants to simulate a wavefield with a given peak wavenumber k_p (or eq. regular wavenumber k₀)

• In HOS-ocean one should use: k_max ≈ 8 k_p

For the discretization along y-direction n2, the choice should be done carefully after a convergence study. General recommendation cannot be provided since it is highly dependent on the directional spreading associated with the simulated wave field.

Orders of nonlinearity mHOS and M_l

The HOS method relies on an iterative procedure that is controlled by the so-called HOS order of nonlinearity mHOS (and M_l for varying bathymetry solved with i_method=2). These parameters can be seen as pure numerical parameters and consequently their effect can be characterized by a classical convergence study. However, regarding the HOS formulation, it is also possible to relate mHOS to physical processes at play during wave propagation. Indeed, this order of nonlinearity corresponds to the nonlinear wave interactions between wave components that you take into account.

• Flat bottom (original HOS method)

  • mHOS=3 corresponds to third-order of nonlinearity, or equivalently the so-called four-wave interactions. With this set-up, the HOS model is equivalent to Zakharov equation for water waves. This is the minimum value suggested, if one focuses on nonlinear wave properties since it includes the most important nonlinear features at play during wave propagation

  • mHOS=1 corresponds to a pure linear solution

  • For practical applications, this parameter is set in the range [3,8] and mHOS=5 is a standard value

• Varying bottom

  • i_method=1: the same order of nonlinearity mHOS is used for free surface nonlinearities and influence of bathymetry on wave propagation. Convergence study has to be carried out to ensure that the bottom variation is correctly taken into account.

  • i_method=2: the choice of mHOS is the same than for flat bottom configuration and a convergence study with respect to M_l needs to be carried out to ensure that the bottom variation is correctly taken into account.

Dealiasing p1 and p2

Pseudo-spectral methods face the possible issue of aliasing. A zero-padding procedure is set-up as dealiasing methodology in HOS-ocean, which is controlled thanks to the parameters p1 and p2. The following recommendations can be made

• Increased accuracy is achieved with total dealiasing p1=mHOS. This is at the expense of computational effort as well as numerical stability

• A typical optimal value for this parameter (robustness/accuracy) is p1=3

• For unidirectional configuration n2=1, one should use p2=1

• For multidirectional waves, one should use p2=p1

Temporal integration

The accuracy of the time integration scheme is controlled with the tolerance parameter toler. This is a numerical parameter, which convergence can be studied for a detailed study conducted with HOS-ocean. A value of toler=10^(-6) is suggested.

Note

In pseudo-spectral models, without dissipation terms or breaking models, the stability of the numerical solution is highly dependent on the choice of the numerical parameters. This has been studied in details in [Ducrozet et al., 2017]