2.1. Modelling Inputs
The model has been set-up with spherical coordinates and a resolution of
, corresponding to ≈2.5 km longitude (
) and ≈2 km latitude (
), also accounting for the Earth’s curvature. Coastline data have been obtained by Amante et al. [
24] and the latest Global Self-consistent, Hierarchical, High-resolution Geography Database (GSHHG) [
25]. Based on this information a bathymetry domain was constructed as input for the model, see
Figure 1. The Dutch coastlines are located at a continental shelf, neighbouring Denmark and the United Kingdom. As seen in
Figure 1, the depth is varying “smoothly” without the existence of very sharp depth gradients, for the domain of this database depth does not exceed 100 m.
As driver of wave generation the ERA-Interim wind dataset, by the European Centre for Medium-Range Weather Forecasts (ECMWF) was utilised [
26]. There is a high-correlation between wind resource as a driver, wind-wave generation/propagation and model performance [
27]. However, depending on the area, model, and tuning the higher temporal resolution wind fields are not always optimal, for the wider Atlantic [
28] and the North Sea [
29] both the ERA-Interim and a higher temporal wind field has been explored. The results indicated that “peaks” of high wave were captured better, but the overall performance was significantly lower with greater scattering and a over-bias in higher wave frequencies. Similar, behaviour between different datasets has been also reported in other studies, with ERA-Interim exhibiting good performance with reduced scattering [
27]. While, other datasets can have higher temporal resolution, this public domain re-analysis dataset is based on Regional Climate Models closely relatable for the region, and the wind speeds exhibit better performance with measured data for the European continent.
Considering that NSWD is developed for various usage spanning from wave energy to climate analysis, main focus is to reduce the scattering and maintain close agreement with higher wave values therefore minimising over-predicting, as this can lead to over-estimating return values [
30], and to higher capital expenditure estimates for infrastructure.
Spectral boundary conditions were re-constructed by the WAve Model (WAM) from ECMWF and applied at domain open boundaries. Most important boundary region is the open upper North side where swell waves from the Atlantic and Norwegian Sea propagate inwards. The model was set-up with a “warm-up” configuration to minimise initial ramp-up periods.
2.2. Calibration Parametrisation
SWAN is a third generation spectral phased-averaged wave model, that accounts multiple physical processes suitable for deep and shallow waters, although arguably it is more efficient for nearshore and Shelf Seas. The wave spectrum is described in time (
t) by the action density equation (
E), dependent upon angular frequency (
), direction (
), frequency (
f), energy propagation (
c) over latitude (
) and longitude (
). Sink source terms are used to estimate the wave parameters (see Equation (
1)), given a specific set of inputs and physical coefficients, with wind input (
), triads (
), quadruplet (
) interactions, whitecapping (
), bottom friction (
) and (
) depth breaking.
In wave models, generation, propagation and spectrum evolution is dependent on various parameters. Most important source terms are mechanisms of wind , and dissipation , as they are responsible for wave generation and dissipation. Waves are created by wind surface pressure on the ocean, in wave models this term is modelled by considering a wind drag coefficient () that contributes to the growth. Wind wave generation is a summation of energy density from the (over Spherical coordinates).Wind drag coefficients can differ and may enhance or reduce the wave generation capabilities in the model. With regards to dissipation mechanisms, the most obscure and least understood is the white-capping that is predominately based on a wave steepness coefficient (), depending on a term adjustable and quite different for each methodology. It is known that wave models tend to under-estimate at lower frequencies, with accuracy affected by wind components used.
Recently, SWAN 41.20 introduced an adjusted formulation for wind and whitecapping, similar but not the same to Wavewatch3 (WW3) ST6 [
31,
32]. The wind drag parametrisation requires fine tuning in the whitecapping coefficient. Interestingly with this new addition the solutions both for the wind drag formulation, stress re-computation, allows for bias wind corrections. In addition, the new formulation can also be configured to include swell dissipation mechanisms. For the models developed an exponential growth coefficient is assigned, and all models have a “hot” start configuration that ensures a fully developed wave field. The sink term of wind input that gives wave generation is given by Equation (
2)
where
A is the linear growth, and
is the exponential growth, both
A and
depend on wind parametrisations. This in turn affects the momentum flux that is the driver between atmosphere and the ocean surface for wave generation, as the model translates wind at 10 m (
) to a surface wind, see Equation (
3) with an estimation wind drag coefficient (
) that depends on
.
Wind drag estimations have limitations especially for higher wind speeds, where they are known to under-estimate and even limit wave growth, therefore, for every different configuration, the
should be adjusted. Kamranzad et al. [
33] indicated that even though wind drag parametrisations in models are good at generating waves, they are limited in their performance especially at higher wind values, where wave growth reduces, see
Figure 2. To alleviate this limitation, a modified formulation was used and since 41.20, a similar approach to that of Rogers et al. [
34] can be activated.
The performance of the wave model depends highly on the parametrisation of the wind sceme, therefore, four different wind schemes have been used and parameterized to obtain the optimal solution. The explanatory naming sequence of the models is based first on the wind configuration used, more specifically for the ST6 (wind 4) package the naming is
ST Wind4 x Opt x Scale x, resulting in a numeric name for the model i.e., STE121 meaning a model that utilises the WAM4 wind configuration, with option 2 (for local & cumulative dissipation) and Scale 1, see
Figure 3. For Wind 1 the configuration uses Komen et al. [
35] set-up, where the wind drag coefficient (
) is dependent on the friction velocity of wind speed (
) with adjustments
and
, see Equation (
4). For Wind 2 the adjustments are based on Janssen [
36], where critical height is iteratively estimated according to its non-dimensional value from
, see Equation (
5). For Wind 3 option drag is based on the alternative description of van der Westhuysen et al. [
37], that uses a re-formulation of whitecapping to weakly and strongly forced waves.
Wind 4 represents the newly added ST6 package, and evaluates a different parametrisation in wind drag (see Equation (
6)), wind stress and whitecaps [
38]. This newly adopted package is similar to WWIII but they are implemented differently. The package includes influence of swell dissipation in the estimations. Wind 4a the
is adjusted according to Hwang et al. [
39], in wind 4b according to Fan et al. [
40] and Wind 4c based on Janssen [
36]. Within all different wind configurations, the stress calculation is iteratively vectorally estimated. This means that higher wind speeds are better represented and higher magnitude waves are better resolved.
For whitecapping, Wind 1 and 4 use [
35] (WAM3 cycle), but a noticeable difference of the ST6 package, from WWIII and the other SWAN options for whitecaps is the use of a swell steepness dependent dissipation coefficient, is set at 1.2 according to Ardhuin et al. [
41]. Wind 2 uses the WAM4 cycle formulation [
42]. Bottom friction has been adjusted according to Zijlema et al. [
43] 0.038
, nearshore breaking, triad interactions, and diffraction are all enabled based on their respective suggested values in SWAN. Quadruplets interactions for deeper water are resolved with a fully explicit computation per sweep, which makes the computation a bit more “expensive”, but retains good agreement.
In the ST6, dissipation is described by local and cumulative terms, that can be accordingly scaled; based on previous works on derivation of these terms, the following “pairs” are utilised for dissipation (whitecapping effects) [
34,
38]. Option 1 has local dissipation (lds): 5.7
cumulative dissipation (cds): 8
, option 2 lds: 4.7
, cds: 6.6
, and option 3 lds: 2.8
, cds: 3.5
. The scaling option parametrisation aims to correct the mean square slope, in this new term, the suggestion is that the scale is over 28. Therefore, seeking to ensure a potential noticeable improvement, we opted for three different tuning parameters, scale 1:28, scale 2:32 and scale 3:35. Whilst more scaling can be attempted, it is expected that the difference from 28 to 35 will be adequate to display any impacts on the hindcast. Tuning this option has to do with how much energy (more or less) is allowed to migrate in higher frequencies. The higher the number, the lower the amounts that are allowed there, therefore, this can be beneficial to not under-estimate lower frequencies. All calibration models were tuned using the binned distribution of 36 directions and frequencies, with the latter using a
= 0.1. The calibrations were conducted with an Intel Xeon with 36 GB of RAM.
To assess model results, several indices are used, Pearson’s correlation coefficient (
R) indicates how well the hindcast performed (see Equation (
7)), the root-mean-square-error (
) underlines the differences between hindcast and buoy measurements (see Equation (
8)), the Scatter Index (
) give an indication on the relationship between observed and modelled data (see Equation (
9)). The goal of a good hindcast is to obtain high correlation values of significant wave height (
)
85–90%, with small
showing a close “positioning” with the mean values, a low
25–30% (or high inverse
85–90%) indicating that the trends are well followed. From experience we are aware that wave models have a tendency to under-estimate, therefore, we also compare the maximum values of significant wave height (
), to ensure that not only the mean bias is low (see Equation (
10)), but the bias of maxima of events is also reduced by the model. This is considered helpful as it will translate to improvements in statistically estimating extreme return wave periods, and making the final model more versatile.
where
is the simulated wave parameter,
recorded and
N measurements. Finally, the Model Performance Index (MPI) diagnosis performance, indicating the degree to which the model reproduces observed changes of the waves (
).
The primary focus of the calibration is to ensure a good re-production of past wave events in order to develop a wave power database. To examine the model, wave data from buoy measurements were gathered [
44], filtered by removing non-operational days, see
Table 1 and
Figure 4 for their locations in the domain.