A Comparison of WAVEWATCH III Grid Models for a Typical Reef Lagoon Ze Sun1, Xiaolong Liu1, Zhiwen Cai1, Jian-Guo Li,3 Wenwei Chen2, Jun Ding1,Chao Tian1

1Department of Offshore Structures, China Ship Scientific Research Center Wuxi, Jiangsu Province, China

2Shanghai Branch, China Ship Scientific Research Center Shanghai, China

3Met Office Exeter, United Kingdom

ABSTRACT Three grid configurations in WAVEWATCH III wave model for a typical reef lagoon are compared, including a multi-resolution nested grid (Multi-grid), a spherical multiple-cell grid (SMC-grid) and a triangular unstructured grid (Tri-grid). Model simulations of ocean surface waves under typhoon conditions are validated against buoy observations. The comparison revealed that three grid models yield similar results at Significant Wave Height (SWH). A detailed spectra analysis indicates that the shapes of calculated spectra from outside the lagoon agree well with observation in three grid models. KEY WORDS: WW3; Multi-resolution nested grid; Spherical Multiple-Cell grid; Triangular unstructured grid; typhoon waves; reef lagoon. INTRODUCTION Wave predictions in deep water have experienced significant developments during the last few decades and the skill of the state of the art models has been shown to be generally good (Cavaleri et al., 2007; Holthuijsen et al., 2003; Komen, 1994; The SWAN Team, 2013; Tolman et al., 2002). In small and semi-enclosed seas like a reef lagoon, wave modeling becomes cumbersome due to a mix of swell from open sea and the local generated waves under complex bathymetry. To obtain accurate wave simulations in semi-enclosed seas, bathymetry grids with adequate temporal and spatial resolution are of prime interest. In the widely used WAVEWATCH III (WW3) wave model, the most traditional way (Multi-grid) is to nest one or several regional grids within a large coarse resolution grid with data exchange between grids at relevant model time step (Tolman, 2008). This multi-grid method calls for severe restriction on time steps of finite-difference schemes to keep model stable (advection and diffusion in particular). As an extension to multi-scale resolution modeling, the unstructured triangular cell grid (Tri-grid) has been employed in WW3 by using

numerical schemes based on contour residual distribution (Roland, 2008). This makes it possible to run the model on high resolution near coasts and islands with limited number of nodes. It is noted that CFL (Courant-Friedrich-Levy) criterion must be fulfilled in order to guarantee a stable integration in the space domain which requires careful mesh editing to limit CPU time (Roland, 2012). Another alternative method developed recently is the spherical multiple-cell grid (SMC-grid) which is originally developed to tackle the polar problems (Li, 2011). One remarkable feature of the SMC-grid is that it can handle several resolutions within the same model grid so that small semi-enclosed seas are resolved at high resolutions while the open sea is kept at an affordable resolution (Li, 2012). The main purpose of this article is to establish an efficient and accurate grid model in WW3 for a typical reef lagoon in the South China Sea (SCS). The typhoon Kalmaegi (2014) wind data covering the whole SCS over 8 days is applied for wave hindcast. Time series of model predicted SWH and mean wave period (Tm) are compared with observations at three buoy sites near the reef lagoon. The performances of different grids are quantified in terms of statistical parameters. Further, we assess the CPU cost of different grids using the same order of propagation scheme. Additionally, the calculated wave energy spectra are examined using observed data, which provides a detailed view of wave evolution around target reef lagoon during typhoon passage. DATA AND METHOD Study Zone The study zone in this paper is an isolated group of islands, reefs, banks which is located in the north east of SCS. This typical reef lagoon is enclosed by islands and reefs that form a semi-enclosed sea shown in the lower right corner of Fig. 1. The fringing reef becomes a barrier reef through biotic growth, while the inner part of the reef falls behind, becoming a lagoon (Kusky, 1978). Water inside the lagoon is

connected to the open sea by the channels between islands. Due to the barrier effect, the wave energy can be largely dissipated from open sea to the lagoon, which provides an ideal shelter for offshore engineering during storms. The average depth inside the lagoon is about 35m while the water depth varies abruptly to hundreds of meters outside the lagoon. Instrumented wave observations have been conducted at three locations labelled with red circles in Fig. 1.

Fig. 1 Map of the typical reef lagoon and layout of buoys. Wind Field Reliable and accurate high-resolution (both temporally and spatially) surface wind data are crucial for driving ocean wave models (Ardhuin et al., 2007; De León and Soares, 2008; Van Vledder and Akpınar, 2015). In this study, the initial wind data source is obtained from the Global Data Assimilation System (Derber et al., 1991; Kanamitsu, 1989). An improved NCAR-AWFA typhoon BOGUS scheme combined with 3DVAR is used to form typhoon initial field (Gao et al., 2010). Typhoon Kalmaegi is the 15th named storm in 2014. The path of Kalmaegi is shown in Fig. 1. Different center pressure levels are shown with different colors. On September 15, Kalmaegi entered the SCS as a tropical storm shown in pink color at the very first. Then it intensified and upgraded to a typhoon in purple color. The typhoon reached its peak strength while making its landfall over Hainan Island. Kalmaegi rapidly weakened to a large tropical storm as it continued to move in a westward direction. The forcing field of wind is defined on the computational domain of wave model (seeing Fig. 1) from 105°E to 125°E and 5° N to 25° N with a spatial resolution of 1°. Covering time ranges from UTC September 14, 2014 to UTC September 21, 2014, which is updated once an hour, with a total of 192 times. Buoy Data Two Datawell Directional Waverider buoys were respectively indexed as Buoy1 and Buoy3, while a Multi-parameter buoy was indexed as Buoy2. As shown in Fig.1, Buoy1 was moored inside the lagoon with a relatively shallow water depth of 25 m while Buoy3 was deployed at open sea with a depth of 160 m. Buoy2 was deployed at the entrance of

lagoon with a depth of 45 m where water depth varies abruptly. The anemometer above Buoy2 records wind speeds and directions every ten minutes which provides a calibration for the wind filed data. Fig. 2 and Fig. 3 show the comparison of modeled wind speed and wind direction with observations at Buoy2. As we can see, the modeled wind speed matches well with the measured wind speed. Although there are some small discrepancies between the modeled wind direction and the observations during September 14 and September 15, it has nearly no effect on our concerning area because this period is at the warm start time of computation.

Fig. 2 Comparison of modeled and observed wind speeds at Buoy2.

Fig. 3 Comparison of modeled and observed wind directions at Buoy2. Wave Model Description The Eulerian ocean surface wave model is based on a 2-D spectral energy balance equation (Tolman et al., 2002). The balance equation for the action density used in WW3 takes the following conservative form:

( ) ( )1g⋅

∂ ∂ ∂+∇ + + + =

∂ ∂ ∂C U N N kN θN S totxt k θ σ

(1)

( ) ( ) ( ) ( ) ( ) ( ) = + + + +S tot S in S nl S ds S bf S db (2)

The left-hand side of Eq. 1 represents the evolution of the wave action density spectrum, N, which considers linear wave propagation as a result of physical processes in the right-hand side (Young and Babanin, 2006). There are various choices for the source terms of in WW3.Different source terms may produce different wave results. In this paper, all grid models take the same source terms. The parameterization of these source terms used in this study are listed in table 1 and more details about these source terms can be found in WW3 manual V5.16 (WAVEWATCH III Development Group, 2016)

Table 1. Source terms parameterization used in this study.

Physical process Characteristics Switch Parameters

Wind input S(in) and wave energy dissipation due to whitecapping S(ds)

Tolman and Chalikov (1996) source term

ST2

Non-linear wave–wave interactions S(nl)

Discrete interaction approximation (DIA)

NL1

wave decay due to bottom friction in shallow water S(bf)

JONSWAP bottom friction formulation

BT1

wave energy dissipation due to depth-induced breaking S(db)

Battjes and Janssen (1978) formulation

DB1

Evaluation Methods Statistical tools are employed in this paper to evaluate the model predictive capacity. The similarity between model predictions (m) and observations (o) is quantified in terms of Pearson’s linear correlation coefficient (R) and root-mean-square deviation (RMSD) based on the following relations:

( )( )

( ) ( )1

2 21

− − ∑ == − −∑ =

N m m o oi iiRN m o o oi ii

(4)

( )1 2

1= −∑

=

NRMSD m oi iN i

(5)

The standard deviations of the test field (often representing model predictions) and the reference field (usually representing observations) are given as below:

( )1 221

= −∑=

Nσ m mm iN n

(6)

( )1 22

1= −∑

=

Nσ o oio N n

(7)

To provide a graphical representation of these results, we adopted Taylor diagrams (Taylor, 2001) to illustrate the performance of different grid methods. In general, the Taylor diagram characterizes the statistical relationship between test field and reference field. For a perfect model that reproduces the observations exactly, the value of R is one and that of RMSD is zero. These statistics for simulation results using different grid models are presented in this study for the assessment of model performance. In Taylor diagrams, the location of symbols on the diagram quantifies how closely the model results match observations. The radial distance from the origin is equal to the standard deviation of each symbol. The RMSD between the observations and the model results is proportional to their distance apart. The correlation between the observed value and the model result is given by the azimuthal position of the model result. COMPARISON OF THREE GRID MODELS Grid Configurations and Computing Cost

In this section, we introduce the configurations of three grids established for typhoon Kalmaegi waves hindcast. In Multi-grid, the computation domain consists of two scaled grids and part of the grid is shown in Fig. 4 for clarity. The bathymetric data of the open sea is obtained from the ETOPO1 (Divins and Metzger, 2008), covering the entire domain from 105°E to 125°E and 5° N to 25° N with 1/30°×1/30° spatial grid resolution. The latitude-longitude grid numbers are 600×600 respectively. The local-scale domain is obtained using a high resolution topographic map which is fine enough for target lagoon, consisting of 250×250 grids with 1/240°×1/240° spatial grid resolution. The total number of sea grids in Multi-grid is 327527 which is the sum of 266742 sea points (74.5%) for the outer grid and 60785 sea points (97.3%) for the inner grid.

Fig. 4 Illustration of Multi-grid. Fig. 5 is part of the unstructured SMC grid for target lagoon. In order to make the SMC-grid and Multi-grid comparable, the SMC-grid shares the same domain area and the intermediate resolution area as in the Multi-grid. The multi-resolution feature is illustrated with a 4-level multi-resolution grid which change from level 4(1/30°×1/30°) to level 1(1/240°×1/240°) in just a few rows or columns near the reef lagoon. There are totally 320012 cells in the SMC grid, including 257877 cells at level 4 resolution, 526 cells at level 3 resolution, 1003 cells at level 2 resolution and 60606 finest cells at level 1 resolution. The finest cells number is nearly equal to the number of inner grids of Multi-grid model. Unlike Fig. 4, the red and brown areas indicating water depth less than 3 m are painted in white in Fig. 5. This is because the SMC depth array should be non-zero and integer, here we removed the sea points under 3 m to keep the refraction rate not too large for the stability of model.

Fig. 5 Illustration of SMC-grid. Part of the Tri-grid is shown in Fig. 6. The resolution along target reef lagoon is the finest (up to 300 m) of the whole domain which is also the highest resolution among three grid methods. The total number of triangular grid elements in Tri-grid is 83577 which are composed of 42445 grid nodes. For comparison purpose of CPU cost, Tri-grid does not have much in common with the other two grids except the domain area. It has only about 1/8 number of grid points in Multi-grid and

SMC-grid. Here we still include it for the purpose of assessing its performance in typical reef lagoon.

Fig.6 Illustration of Tri-grid. The advection of SMC-grid and Multi-grid are both estimated with the upstream non-oscillatory 2nd order (UNO2) advection scheme (Li, 2008). From V5.16 onwards, the 2nd order UNO option becomes available to the nested grid while the UNO3 (similar to UQ) scheme option is also added to the SMC grid. The UNO2 scheme is about 30% faster than UNO3 (Li, 2012).Considering this 30% CPU time reduction, the small loss of accuracy by UNO2 is worthwhile. In the Tri-grid model, we apply the RD-FCT (Residual Distribution Flux Corrected Transport) scheme which is also 2nd order accuracy in space and time (Roland, 2008). A minimum propagation time step has been tested and specified for the maximum group speed over the smallest grid length for each grid. The minimum propagation time step is 20 s for Multi-grid, 20 s for Tri-grid and 60 s for SMC-grid. SMC-grid uses the same 60s minimum time step for propagation over the base-level (1/30°×1/30°) cells but reduced sub-time steps for the refined cells. Some key characters of the three grid models are listed in Table 2. The models are tested on a Shengwei supercomputer in an 8-day hindcast run using 240 cores (Fu et al., 2016). As shown in Table 2, the Tri-grid costs least CPU time among three grids. This is apparently attributable to the fewest grid point numbers. Table 2. Grid configurations and corresponding CPU cost.

Description Multi-grid SMC-grid1 Tri-grid Number of grid cells/elements 327527 320012 83577 Min advection time step(s) 20 60/30/15/7.5 20 Propagation scheme UNO2 UNO2 RDFCT Elapsed Time(hr) 25.02 7.91 6.69 It seems that SMC-grid can reduce the total CPU time by about 2/3 in comparison with Multi-grid using the same propagation scheme and under the same resolution in the same domain area with nearly the same sea points. This is partly due to that the wave energy spectra in SMC-grid are stored only at sea points, while in Multi-grid they have to be expanded to all grids (both sea and land points) before the transport calculation (Li, 2011). Hence, quite a lot of computing time can be saved in SMC-grid. Another advantage of irregularity in SMC makes it possible not to stick to rectangular domain in SMC-grid like Multi-grid. The CPU cost can be further reduced by removing the insignificant cells such as the southeast area beyond Phillips. Comparison with Observations As shown in Fig. 7, predicted SWH and Tm from the three grids are compared with buoy observations. Apparently, the results from three grids agree well with three buoy observations.

(a)Buoy1

(b)Buoy2

(c)Buoy3

Fig. 7 Time series of observed and calculated mean wave parameters at three sites of (a): inside lagoon, (b): at the channel of lagoon and (c): outside lagoon. The black dotted lines indicate the buoy observations. Multi-grid results are drawn in blue solid lines. SMC-grid results are in red solid lines and Tri-grid results are in green solid lines.

(a)

(b)

Fig. 8 Taylor diagrams displaying statistical comparisons with observations of three numerical experiments. The similarity between calculated results and observations is quantified in terms of R, RMSD and standard deviations. As shown in Fig. 8, Taylor diagrams for SWH and Tm are illustrated to evaluate model performances. In Fig. 8-(a), results from three grids are similar at each site. The values of R representing SWH in three sites are 0.967, 0.965, 0.984 for Multi-grid, 0.968, 0.975, 0.976 for SMC-grid and 0.979, 0.965, 0.978 for Tri-grid. In terms of the wave periods in Fig. 8-(b), the results are more scattered. The values of R representing Tm are 0.891, 0.683, 0.744 for Multi-grid, 0.823, 0.720, 0.790 for SMC-grid and 0.937, 0.531, 0.333 for Tri-grid. Wave Spectra Analysis In the previous sections, we have evaluated the performance of these three grid models in SWH and Tm. However, the two wave parameters only give limited description of wave conditions. For instance, a mixed sea state of wind waves (short, irregular, locally generated waves) and swell (long, smooth waves, generated in a distant storm) may have the same SWH and Tm as severe wind waves (Holthuijsen, 2010). For a complete description, the wave spectra S(f) calculated by different grids are compared with those measured by two Datawell waverider

buoys (Buoy1 and Buoy3). Fig. 9 and Fig. 10 show the 3-hourly evolution of wave spectrum in 12 hours at Bouy3 and Buoy1, respectively.

Fig. 9 Comparison of three grid models vs Buoy3 wave spectra. Observed data from Buoy3 is presented using black line while blue dots represent numerical results from Multi-grid model. SMC-grid and Tri-grid model results are indicated with red symbols and green dashes respectively. The covering time ranges from 17 o'clock on UTC September 15 to 2 o'clock on UTC September 16. During this period, waves of target reef lagoon are mostly affected by Typhoon Kalmaegi.

Fig. 10 Comparison of three grid models vs Buoy1 wave spectra (same note as Fig. 9). In the first graph of Fig. 9, a bimodal spectrum is found at open sea (Buoy3). The first peak is around 0.07 Hz while the second peak is around 0.16 Hz. If we take a look at the SWH distribution of three grids at that time, we can find the swell comes from the north–east area outside the reef lagoon (see Fig. 11). In Fig. 11, the SWH around our target lagoon is mainly affected by the other large reef lagoons from north and east. It seems that the color of Tri-grid SWH plane is slightly lighter than the other two grids, which also induces a lower peak than the other grids at low frequency in Fig. 9. At the second plot of Fig. 9, the swell peak decreases and the wind wave grows up. After that, the spectrum outside the reef lagoon is dominated by wind waves.

On the other hand, observed spectra at Buoy1 are unimodal. In the first two graphs of Fig. 10, the spectra of SMC-grid have a swell peak which deviates from the observation. This also results in a higher SWH curve in red than the other models during Sept. the descending period in Fig. 7 (a). From the directional spectrum of SMC-grid at 17hr, Sept. 15 (Fig. 12), we can find that the swell component is from east which should be totally blocked by the reef. It indicates that the dissipation of long waves in SMC is inadequate.

(a) Multi-grid

(b) SMC-grid

(c) Tri-grid

Fig. 11 SWH distribution of three grids at 17hr, Sept. 15.

Fig. 12 Directional wave spectrum of SMC-grid at 17hr, Sept. 15. CONCLUSIONS Discrepancies of grid methods in WW3 are rarely investigated for wave simulating in small and semi-enclosed seas. In this study, three grid models are constructed for a typical reef lagoon in South China Sea. In-situ measurement is carried out at three sites around the reef lagoon during the passage of Typhoon Kalmaegi in September 2014 and the observation data are used for evaluation of three grids. Among the three grids’ configurations, the Multi-grid and the SMC-grid use the same propagation scheme and share the same resolutions in the same domain area. This setup makes these two grids comparable in both computing cost and accuracy. Using an 8-day wind field of Typhoon Kalmaegi passage as wind input, three grids are tested. The calculated SWH of three grids agree quite well with Buoy data. The performance of Tri-grid in Tm is worse than the other two grids. On the other hand, SMC-grid can save about 2/3 CPU cost than Multi-grid while the Tri-grid takes about 85% CPU cost of SMC-grid. According to the wave spectra analysis at major times, a bimodal spectrum is found at open sea. The shapes of calculated spectra from at Buoy3 agree well with observation in three grids. However, it seems that SMC-grid tends to overestimate the swell peak at low frequency band inside the lagoon at Buoy1. We speculate that the block effect to long waves near reef is not adequate in SMC-grid which needs further validation. ACKNOWLEDGEMENTS This research was supported by the Ministry of Industry and Information Technology with the research project in the fields of high-tech ships under Grant number [2016] 22. REFERENCES Ardhuin, F, Bertotti, L, Bidlot, J-R, Cavaleri, L, Filipetto, V,

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