1 Introduction

The dissipative effect of unresolved islands and small obstacles is a major source of error in wave models if neglected, both locally and on large scale. This is more evident in basins characterized by the presence of large number of islands, such as the Pacific Ocean or the Baltic Sea (Tolman 2003; Tuomi et al. 2014). The most established approach to tackle this problem is the attenuation of the wave energy traveling through the obstructed cells, as a function of transparency coefficients. Such approach is traditionally implemented in the numerical scheme modeling the spatial propagation of the waves (Booij et al. 1999; Tolman 2003; Chawla and Tolman 2008). The above studies typically consider 2 transparency coefficients, one for energy traveling along each Cartesian direction (x and y) of regular meshes. However, Hardy and Young (1996) and Hardy et al. (2000) showed the advantages of modulating the attenuation with wave direction, as small islands with irregular and elongated shape can lead to a non-isotropic dissipation of energy. The approach they proposed, which is based on the advection scheme and considers the wave direction, is currently implemented in the model ECWAM (ECMWF 2019).

More recently, Mentaschi et al. (2015) theorized a methodology for subscale modeling the dissipation due to unresolved obstacles based on source terms (the Unresolved Obstacles Source Term, hereinafter referred to as UOST). Given each cell of a mesh, UOST estimates, for each spectral component, the dissipative effect of the unresolved features located in the cell (local dissipation), and the shadow projected to the downstream cells (shadow effect, SE). The source term can be expressed as:

$$ {S}_{uo}={S}_{ld}+{S}_{se}, $$
(1)
$$ {S}_{ld}=-{\psi}_{ld}\frac{1-{\beta}_l}{\beta_l}\frac{c_g}{\Delta L}N, $$
(2)
$$ {S}_{se}=-{\psi}_{se}\left(\frac{\beta_u}{\alpha_u}-1\right)\frac{c_g}{\Delta L}N, $$
(3)

where Sld and Sse are the local dissipation and the shadow effect, N is the spectral density, cg is the group velocity, ΔL is the path length of the spectral component in the cell, and the ψ factors model the reduction of the dissipation in presence of local wave growth. The subscripts l and u of α and β indicate that these transparency coefficients can be referred, respectively, to the cell and to the upstream polygon. For a more detailed explanation on the theoretical framework of UOST, the reader is referred to Mentaschi et al. (2015, 2018). Appendix provides a discussion on the geometrical meaning of the coefficients α and β.

The viability of UOST in real-world applications on regular meshes was shown by Mentaschi et al. (2018), and an open-source software package for the automatic computation of the transparency coefficients needed by UOST was developed (alphaBetaLab, now supporting regular and triangular meshes, Mentaschi et al. 2019). Furthermore, UOST is now part of release 6.07 of the wave model WAVEWATCH III (hereinafter referred to as WW3, WW3DG 2019).

A significant benefit of using UOST with respect to other implementations is that it considers the swell direction and the layout of the obstacles in the obstructed cells. Furthermore, it can be applied with any type of mesh and numerical scheme, and this is useful as the number of solvers of the wave action equation is constantly increasing, the numerical schemes are becoming more and more sophisticated (e.g., Roland 2008; Zijlema 2010; Li 2011), and in some cases, parameterizing the effect of the unresolved obstacles within the scheme would not be straightforward.

In this contribution, we illustrated the abovementioned advantages by running and validating global models on both unstructured and regular meshes with UOST, and comparing against simulation results using the numerical scheme–based approach implemented in WW3 for regular grids (hereinafter referred to as GRIDGEN, Chawla and Tolman 2008), as well as without any parameterization of the unresolved obstacles.

Section 2 of the manuscript describes the model setup and the validation methodology employed in this study. Section 3 summarizes the results, which are discussed in more detail in Section 4. Final remarks are drawn in Section 5.

2 Model setup and validation

In this study, the wave model WW3 version 6.07 was implemented over regular and unstructured (triangular) global domains, forced by 6-h 10 m winds from the Climate Forecast System Reanalysis (CFSR Saha et al. 2010). The spectral grid is constant for all the simulations, with a directional resolution of 15° and 25 frequency bands ranging exponentially from 0.04 to 0.5 Hz, separated by a factor of 1.1. The parameterization of the physics of wave growth/dissipation as proposed by Ardhuin et al. (2010), hereinafter referred to as ST4, was employed. The wave growth tuning parameter BETAMAX was set to 1.33, a value suggested by WW3DG (2019) for CFSR wind data. The Discrete Interaction Approximation (DIA) was used to represent the non-linear interactions (Hasselmann and Hasselmann 1985). Although shallow water dynamics are not expected to play a significant role at the spatial scales examined in this manuscript, the model setup includes the JONSWAP parameterization for bottom friction (Hasselmann et al. 1973), and the approach of Battjes and Janssen (1979) for surf breaking. UOST was set up by compiling the executables with the corresponding WW3 switch, and post-processing the meshes with the alphaBetaLab utility (Mentaschi et al. 2019, see Appendix for details).

The two regular meshes employed in this study cover the globe (latitude 80° S–80° N) with resolutions of 1.5° and 0.4° respectively, and the ULTIMATE QUICKEST numerical scheme (Leonard 1991) was used for energy propagation. The global unstructured mesh has a resolution of about 150 km (~ 1.5°) offshore and 10 km (~ 0.1°) nearshore, and the implicit N-scheme was employed (Zijlema 2010; Roland 2012). The values of the global time step, and of the time steps of spatial and spectral propagation, were set to 900 s, 300 s and 300 s respectively for all the models. Such value of the global time step guarantees an accurate application of UOST in cells with a length > ~ 15 km in the direction of the wave propagation (Mentaschi et al. 2018). The simulations were carried out on a 10-year time frame (2000–2009) using different approaches for the parameterization of the unresolved obstacles: no parameterization (hereinafter no subscale modeling, NOSM), UOST, and, for regular domains, GRIDGEN (Table 1). Maps of bulk wave parameters (significant wave height Hs, mean period T0-1) were saved hourly for all the simulations, while the spectra were stored at selected locations.

Table 1 Simulations carried out, mesh type employed parameterization of the unresolved obstacles
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The accuracy of the runs was evaluated versus satellite altimeter data of Hs provided by the Globwave database (Queffeulou and Croizé-Fillon 2014). The dataset includes altimeter data from various satellites: ERS 1 and 2, ENVISAT, GEOSAT, Jason 1 and 2, Poseidon, TOPEX, and Cryosat 2. Since the resolution of the satellite data along the satellite tracks is much higher than that of the model runs, satellite data have been aggregated on 0.5°-long segments along the tracks and averaged. The model output was space-time interpolated to the corresponding points along the tracks and compared with the satellite observation. Wave heights lower than 0.5 m in either model or observations were excluded from the validation, due to the unreliability of the measurements for low values of Hs. The pairs simulation-observation were grouped in 1°-sided tiles, and these statistical indicators were used as proxies of performance:

$$ \mathrm{NBI}=\frac{\sum \left({S}_i-{O}_i\right)}{\sum {O}_i}, $$
(4)
$$ \mathrm{NRMSE}=\sqrt{\frac{\sum {\left({S}_i-{O}_i\right)}^2}{\sum {O}_i^2}}, $$
(5)

where S represents the simulated value of Hs; O represents the observation.

3 Results

On a regular mesh with a resolution of 1.5°, a model without subscale modeling has a mean positive bias of little less than 10% of Hs, exceeding 30% in vast areas characterized by the presence of many islands, like the tropical western part of the Pacific basin (Fig. 1a). Employing GRIDGEN considerably improves the bias, lowering it to 1.5%, while the NRMSE decreases from 17.5 to 14.2%. However, we notice that GRIDGEN-150 underestimates Hs in many areas, with a negative bias below − 10% in vast portions of the Pacific Ocean, and other areas filled with islands (Fig. 1b). The model employing UOST results in an overall bias similar to GRIDGEN, but with a narrower spatial distribution and fewer areas affected by relevant underestimation of Hs. The overall value of NRMSE is consequently reduced further to 13.7% (Fig. 1c).

Fig. 1
figure 1

Normalized bias (NBI) of Hs for simulations on the 1.5°-resolution domain. No subscale modeling (a), GRIDGEN (b), UOST (c). The red (blue) color indicates overestimation (underestimation) of Hs

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The results are similar for the simulations with the 0.4° regular mesh and on the unstructured mesh, though with biases of Hs reduced with respect to NOSM-150. For NOSM-040 (NOSM-UNST), the overall overestimation of Hs is 5.3% (2.6%), with local exceedance beyond 20% in many areas (Figs. 2a and 3a). Both GRIDGEN and UOST reduce such kind of overestimation, but at 0.4° as at 1.5° GRIDGEN comes with an underestimation of Hs in the western Pacific Basin and in other areas filled with islands (Fig. 2b). While in UOST-040 such negative bias is smaller, and in UOST-UNST is absent (Figs. 2c and 3b).

Fig. 2
figure 2

Same as in Fig. 1, for the 0.4°-resolution simulations

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Fig. 3
figure 3

Unstructured model bias, without subscale modeling (a) and with UOST (b). The red (blue) color indicates overestimation (underestimation) of Hs

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4 Discussion

The importance of taking into consideration the unresolved obstacles is shown by the better skill of models adopting any approach for subscale modeling, with respect to models neglecting them, for all the considered meshes. In particular, the skill of a 1.5° model on a regular mesh with GRIDGEN or UOST is better than that of a 0.4° model without. This fact is remarkable if one considers that a 0.4° simulation is about 14 times more expensive than a 1.5° one, from a computational point of view. Similar considerations hold comparing the 1.5° simulations with the ones of the unstructured mesh. Moreover, coarse resolutions as for example 1.5° are viable to provide boundary conditions to higher resolution local models, to perform climate projections, and in general in applications where the exact knowledge of the wave climate in specific locations and at specific times is not required.

The skill improvement in models adopting any approach for subscale modeling is also noticeable for the higher resolution domains (040 and UNST) and can be dominant in many locations, especially in the presence of intense swell events. For example, during a swell event in the shadow area of the Palliser Islands on 23/02/2008, NOSM-040 modeled values of Hs (T0-1) about 75% (35%) larger than UOST-040 and about twice larger than GRIDGEN-040. Accordingly, the peak of frequency spectrum modeled by NOSM-040 is 4.8 times larger than that of UOST-040, and ~ 10 times larger than that of GRIDGEN-040 (Fig. 4).

Fig. 4
figure 4

Swell event at the Palliser Islands (French Polynesia). Maps of Hs on 23/02/2008, for NOSM-040 (a), GRIDGEN-040 (b), and UOST-040 (c). Time series of Hs (d) T-10 (e), and frequency spectra at 23/02/2008 (f). The red dot in the planisphere inset in (a) corresponds to the location of the Palliser Islands

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The unstructured meshes allow the increase of the resolution in areas of interest for specific applications, potentially ensuring accurate results with respect to regular domains, for the same computational costs. In the unstructured mesh considered in this study, we reached higher resolutions along the coast than in our regular domains (about 10 km, or 0.1°, versus 0.4° and 1.5°). A lower (but still noticeable) global impact of the unresolved obstacles than in the regular domains is therefore expected (the global mean NBI of Hs is 2.6% for NOSM-UNST, versus 0.5% for UOST-UNST). Yet, employing UOST leads to major improvements in large basins characterized by the presence of many small islands, such as the whole Pacific Ocean, where the overestimation of Hs is reduced from approximately 4% to about 0% (Fig. 3). We remark that UOST is the only workable methodology for subscale modeling the unresolved obstacles on unstructured meshes. It comes with advantages with respect to the usual approach of increasing the resolution in proximity of (the majority of) the small islands, which would result in large meshes, expensive from the point of view of development, computation, and storage requirement for the output. The unstructured mesh employed in this study has computational costs and model skills comparable to the ones of the 0.4°-resolution regular mesh, but with higher resolution near the coast representing the nearshore dynamics in higher detail. In that respect, UOST can simplify the process of setting up and running large-scale unstructured meshes: a modeler can increase the resolution in the areas of interest for a specific application, letting UOST doing its job in areas where only an adjustment of the energy fluxes is needed.

Our results show that UOST can improve the model skill also on regular meshes: GRIDGEN comes with a significant underestimation of Hs at the obstructed cells, which is reduced in UOST (Figs. 1b, c and 2b, c). Such negative bias is most likely due to an overestimation of the dissipation in presence of “diagonal” swell, i.e., with direction angled with respect to the two main axes of the mesh (longitude and latitude). The difference in behavior between UOST and GRIDGEN is exemplified by the two swell events discussed above, the first with waves directed diagonally towards NE, at the Palliser Islands (French Polynesia, Fig. 4), the second with westward waves at the Hawaii islands (“straight” swell, Fig. 5). For the event at Hawaii, we notice that GRIDGEN and UOST behave closely, reproducing similar patterns and alike values of Hs, T-10, and frequency spectra. On the other hand, for the diagonal event at the Palliser Islands, the dissipation modeled by GRIDGEN is consistently stronger than that of UOST.

Fig. 5
figure 5

Same as in Fig. 4, for a swell event in Hawaii on 03/02/2008. A comparison with the measurements of Hs of the NDBC buoy 51201 is shown in panel d

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The over-dissipation of GRIDGEN with diagonal swells could be related to the fact that it modulates the attenuation only along the 2 main axes of the grid. To reproduce the propagation of diagonal swell, the numerical scheme ULTIMATE QUICKEST decomposes the advection of the wave action into its two components along the zonal and meridional axes, then applies the transparency coefficients estimated for these directions. Even in the case of isotropic circular unresolved islands, with a dissipative effect independent on the direction, this approach can lead to over-dissipation in regular grids, as the cross section of a circular island computed with respect to the cell side (i.e., for straight swell) is larger than that computed with respect to the cell diagonal, (i.e., for diagonal swell). In other words, UOST allows to better consider not only the geometry of the obstacles, but also the geometry of the cells, which can have different crosswise sizes for swells coming from different directions.

5 Final remarks

In this study, we investigate applications of UOST on global domains, showing how it can benefit the model skill on both structured and unstructured meshes. On regular grids, UOST provides a better parameterization of the unresolved obstacles with respect to approaches which only consider the 2 main axes of propagation, as it better modulates the energy dissipation with the swell direction. This is important not only to take into account the geometry of the small islands, which can dissipate different amounts of energy from different directions. It also allows to better consider the geometry of the cell, which can have different crosswise sizes for swells coming from different directions. Neglecting the geometry of the cells is believed to be the cause of the over-dissipation of GRIDGEN in many areas, which is consistently reduced in UOST.

For unstructured (triangular) meshes, UOST comes as the only possible way to subscale model the unresolved obstacles. To show its capability in this context, we employed a mesh with a relatively high resolution along the coasts (about 10 km) but neglecting the plethora of small islands present in the Pacific Ocean and in other basins. UOST consistently improved the model skill with respect to simulations not considering the effects of the unresolved obstacles, by eliminating the overestimation of Hs in wide portions of the domain. Therefore, UOST opens a new way of wave modeling on unstructured meshes, whereby the mesh resolution is not anymore required to fit the size of small features that might disrupt the flux of wave action: a modeler can refine the mesh where and up to the resolution required by the specific application, leaving to UOST the job on the coarser cells.