Abstract

The observation-based source terms available in the third-generation wave model WAVEWATCH III (i.e., the ST6 package for parameterizations of wind input, wave breaking, and swell dissipation terms) are recalibrated and verified against a series of academic and realistic simulations, including the fetch/duration-limited test, a Lake Michigan hindcast, and a 1-yr global hindcast. The updated ST6 not only performs well in predicting commonly used bulk wave parameters (e.g., significant wave height and wave period) but also yields a clearly improved estimation of high-frequency energy level (in terms of saturation spectrum and mean square slope). In the duration-limited test, we investigate the modeled wave spectrum in a detailed way by introducing spectral metrics for the tail and the peak of the omnidirectional wave spectrum and for the directionality of the two-dimensional frequency–direction spectrum. The omnidirectional frequency spectrum E(f) from the recalibrated ST6 shows a clear transition behavior from a power law of approximately f−4 to a power law of about f−5, comparable to previous field studies. Different solvers for nonlinear wave interactions are applied with ST6, including the Discrete Interaction Approximation (DIA), the more expensive Generalized Multiple DIA (GMD), and the very expensive exact solutions [using the Webb–Resio–Tracy method (WRT)]. The GMD-simulated E(f) is in excellent agreement with that from WRT. Nonetheless, we find the peak of E(f) modeled by the GMD and WRT appears too narrow. It is also shown that in the 1-yr global hindcast, the DIA-based model overestimates the low-frequency wave energy (wave period T > 16 s) by 90%. Such model errors are reduced significantly by the GMD to ~20%.

1. Introduction

In deep water, the evolution of wind-generated ocean waves can be described by the radiative transfer equation (e.g., Komen et al. 1994; Young 1999):

 
formula

where is the wave action density spectrum, is the two-dimensional wavenumber spectrum, σ is the intrinsic (radian) frequency, k is the wavenumber, and θ is the propagation direction of wave energy. For deep water the dispersion relation is

 
formula

and g is gravitational acceleration. The RHS of (1) represents different physical sources and fluxes of wave energy, including the wind input term Sin, wave breaking term Sds, nonlinear wave–wave interaction Snl, and swell decay Sswl, among others (Cavaleri et al. 2007; Holthuijsen 2007; Cavaleri et al. 2018).

The 3-yr field experiment carried out in Lake George, New South Wales, Australia, in 1997–2000 (Young et al. 2005; Donelan et al. 2005) revealed various novel features of wave dynamics. For the wind input Sin, the main novel features (Donelan et al. 2006) are the following: 1) Sin is a nonlinear function of the wave spectrum because the growth rate γ depends on wave steepness and hence on the spectrum itself; 2) the growth rate γ slows down in extreme conditions because of the flow separation (in relative terms; the growth still increases as the wind increases, but not as fast as one would expect by extrapolating the measurement in moderate wind-forcing conditions); and 3) wind input doubles over a breaking wave and hence can increase if the breaking rates are substantial (Babanin et al. 2007). For the whitecapping dissipation term Sds, the novel features are as follows: 1) the threshold for inherent wave breaking demonstrates its existence in terms of (significant) wave steepness (Banner et al. 2000; Babanin et al. 2001), and Babanin and Young (2005) established dimensionless value for such threshold across the entire spectrum; 2) the two-phase behavior of Sds is noteworthy: at any frequency the breaking can happen due to inherent reasons, but above the spectral peak the breaking is also enhanced due to the influence of longer waves on shorter ones (Babanin and Young 2005; Young and Babanin 2006); 3) the direct dependence of Sds on the wind speed when (the wind speed at 10 m above the sea surface) exceeds 14–15 m s−1 (Manasseh et al. 2006); and 4) the directional distribution of Sds is bimodal rather than isotropic (Young and Babanin 2006).

These Lake George observations resulted in a new set of source functions for wind input Sin (Donelan et al. 2006; Babanin et al. 2007) and whitecapping dissipation Sds (Babanin and Young 2005; Young and Babanin 2006), which were tested in academic models (Tsagareli et al. 2010; Babanin et al. 2010) and subsequently implemented in Simulating Waves Nearshore (SWAN; Booij et al. 1999) and WAVEWATCH III (WW3; Tolman 1991) by Rogers et al. (2012, hereafter RBW12) and Zieger et al. (2015, hereafter ZBRY15), respectively. Practical modeling also required introduction of further observation-based physics such as swell dissipation Sswl (Babanin 2006, 2011; Young et al. 2013) and negative wind input (ZBRY15; Aijaz et al. 2016; Liu et al. 2017). Once the waves stop breaking, the dissipation continues, but due to a different reason: turbulence production by wave orbital motion (i.e., the so-called swell decay Sswl) (Babanin 2006; Babanin and Haus 2009; Young et al. 2013). Note that other mechanisms responsible for Sswl based on the interaction of ocean waves and upper ocean turbulence or air turbulence are also available in the literature (e.g., Teixeira and Belcher 2002; Ardhuin and Jenkins 2006; Ardhuin et al. 2009). This complete set of new physics ready for practical forecast and hindcast received the name of ST61 in 2014 and 2016 public releases of WW3 (WAVEWATCH III Development Group 2016, hereafter T16). Besides, ST6 is now formally part of the SWAN model as well (SWAN Team 2018, version 41.20A). Further academic developments related to ST6 included a new nonlinear interaction term based on the general kinetic equation (Gramstad and Babanin 2016), modules for wave–current interactions (Rapizo et al. 2017), infragravity waves (Nose et al. 2017), and wave–ice interactions (to be released in the 2019 version of WW3).

Since its implementation in SWAN and WW3, this unique source term package, ST6, of has been proven skillful for different spatial scales and under different weather conditions (e.g., ZBRY15; Aijaz et al. 2016; van Vledder et al. 2016; Liu et al. 2017). ZBRY15 (their Fig. 5) and Stopa et al. (2016, hereafter SABZ16, their Fig. 7), however, also suggested that ST6 was inclined to overestimate the energy level of the high-frequency tail of the spectrum, indicating an inaccurate balance of different source terms in this specific frequency range.

As will be shown in this paper, this shortcoming can be solved by increasing the wind input Sin slightly and then recalibrating tunable parameters of Sds [i.e., and in (9) and (10)]. From a practical point of view, a relatively stronger input allows a higher dissipation, which in turn pulls the overestimated spectral tail down to the correct energy level. Through detailed analyses of academic and realistic simulations performed with WW3 (version 5.162; T16), we demonstrate that the recalibrated ST6 package not only performs well in predicting commonly used bulk wave parameters (e.g., significant wave height and wave periods;  appendix A), but also yields a clearly improved estimation of the high-frequency energy level [or specifically, the saturation spectrum in (21) and mean square slope ;  appendix A]. Besides, the updated ST6 is also able to produce a realistic transition behavior from to , where is the omnidirectional frequency spectrum.3

The Discrete Interaction Approximation of Snl (DIA; Hasselmann et al. 1985) is the crucial component permitting routine application of third-generation wave models (e.g., Hasselmann et al. 1988; Tolman 1991). It however also has some well-known shortcomings as an approximation (see an extended discussion about this issue in section 2b). To investigate the errors in spectral wave models attributable to the DIA, we first specifically optimize another more accurate nonlinear solver, that is, the Generalized Multiple DIA (GMD; Tolman 2013) for ST6, and then conduct a thorough comparison of model simulations with these two different nonlinear solvers. The most prominent advantage of the GMD-based model over the DIA-based model, as later illustrated in this paper, is that the former shows a much higher accuracy in simulating the energy of long-period waves ( s). The computational expense of the GMD approach used here, however, is about 5 times larger than that of the DIA.

This paper is organized as follows. Section 2 provides a brief overview of ST6 source functions (Sin + Sds + Sswl) and the four-wave resonant interactions Snl. Section 3 describes the updates of ST6 over its predecessor, particularly focusing on the retuning procedure. Section 4 presents a detailed analysis of modeled wave spectra from duration-limited simulations, followed by a thorough validation of model performance with a 1-yr global hindcast in section 5. Conclusions in section 6 finalize this paper.

2. Parameterizations

a. ST6 source term package

A brief overview of the ST6 source terms is given here for completeness. The reader is referred to RBW12 and ZBRY15 and references therein for more details.

1) Wind input

The wind input parameterization Sin, formulated by Donelan et al. (2006), is given as follows:

 
formula
 
formula
 
formula

where and are air and water densities, is the wind forcing parameter, is the scaling wind speed, is the phase velocity, and is a spectral measure of wave steepness—the saturation spectrum (Phillips 1984) normalized by the spreading function A [(1); Babanin and Soloviev 1998b]. Also, represents the degree of flow separation (whether the full separation occurs or not) and is the wind growth parameter:

 
formula

Here is a tuning parameter controlling the strength of negative wind input (ZBRY15; Liu et al. 2017, and references therein). The specific values of the four parameters in (4) vary with the scaling wind speed adopted in . For , Donelan et al. (2006) suggested

 
formula

The wave model community, however, prefers scaling in order to assure a consistent fetch law across different wind speeds (Komen et al. 1994, p. 253), where is the friction velocity. Therefore, RBW12 advocated using an approximation

 
formula

by following Komen et al. (1984).

2) Wave breaking

The wave breaking parameterization Sds of the ST6 package incorporates two different mechanisms: 1) the inherent wave breaking occurring at each frequency once the steepness of that wave component exceeds a threshold value (Banner et al. 2000; Babanin et al. 2001) and 2) the induced breaking of relatively short waves due to the modulation of longer waves (Donelan 2001; Young and Babanin 2006). The source term reads (RBW12; ZBRY15)

 
formula
 
formula
 
formula

where , , , and are tunable parameters, fmin is the lowest discrete frequency defined in the spectral grid, is the spectral threshold, is the dimensionless saturation-threshold value, and is the exceedance level (Babanin et al. 2010). RBW12 found that highly nonlinear and are required to balance the strong wind input [(3)] beyond the spectral peak. This was achieved by setting .

3) Swell dissipation

The swell dissipation term Sswl of ST6 characterizes the loss of wave energy as a result of the turbulence production by nonbreaking surface waves (Babanin 2006; Babanin and Haus 2009). According to Babanin (2011) and Young et al. (2013), ZBRY15 implemented Sswl as

 
formula

where the dimensionless proportionality coefficient is hypothesized to be steepness-dependent in the following way:

 
formula

Here is the peak wavenumber and is a tunable scaling coefficient. Note that Sswl is applied to both wind sea and swell. But in the context of wind sea, it does not contribute significantly to the source term balance (e.g., ZBRY15).

b. Nonlinear wave–wave interactions

Hasselmann (1962) pointed out that four wave components satisfying the resonance condition

 
formula

could exchange energy and momentum, where is the wavenumber vector. The set of these four waves is also known as a quadruplet. The nonlinear term describes the redistribution of wave energy over the spectrum resulted from such resonant wave–wave interactions. The important role of Snl in the evolution of wave spectrum is well founded (e.g., Hasselmann et al. 1973; Young and van Vledder 1993). The computation of its exact solutions, such as by the Webb–Resio–Tracy (WRT) method (Webb 1978; Tracy and Resio 1982; Resio and Perrie 1991; van Vledder 2006), however, is extremely time-consuming, and this prohibits its applicability to large-scale wave hindcasting and forecasting. Hasselmann et al. (1985) proposed the DIA approach to overcome this difficulty by accounting for interaction contributions for a single representative quadruplet only, defined by (13) and

 
formula

where λ is a free shape parameter. The DIA is not only several orders of magnitude more efficient (e.g., Tolman 2013, his Table 4), but also retains the dominant features of the exact solutions (Hasselmann et al. 1985; Young et al. 1987). Accordingly, it has been widely used for decades in third-generation spectral wave models (Hasselmann et al. 1988; Tolman 1991; Booij et al. 1999).

Nonetheless, the shortcomings of DIA have also been extensively discussed. First, the two lobes of beyond the peak frequency —one negative lobe close to and one positive lobe at higher frequencies—computed by the DIA are clearly different from the exact solutions (Hasselmann et al. 1985; Cavaleri et al. 2007). As a result, the DIA may not be able to yield a correct form of the equilibrium range () of the wind wave spectra (Resio et al. 2016). Second, wave spectra from the DIA are too broad in both frequency and directional space (e.g., Young et al. 1987; Cavaleri et al. 2007; Rogers and van Vledder 2013). Third, the DIA could fail to reproduce the directional bimodality of short waves because of its tendency to misplace the two lobes of Snl beyond (van der Westhuysen et al. 2007; Cavaleri et al. 2007). Finally, it is also noteworthy that under hurricane conditions the DIA may give rise to ~20% errors in the simulated , as shown in Tolman (2013) and Liu et al. (2017). Liu et al. (2017, their Fig. 13) also demonstrated that for cross swell the low-frequency wave energy simulated by DIA is clearly higher than that by WRT.

Tolman (2013) developed a more accurate parameterization of Snl (i.e., the GMD) by expanding the DIA in several ways. The definition of the representative quadruplet [(14)] is extended as follows:

 
formula

where μ is the second shape parameter and is the angle between and . Clearly, (14) is a simplified form of (15) for and . Another important feature of the GMD is its capacity of incorporating interaction contributions for multiple representative quadruplets, rather than one quadruplet only by the DIA. For brevity, other advantageous features of the GMD over the DIA are not described here. The reader is referred to Tolman (2010), Tolman (2013), and Tolman and Grumbine (2013) for further details. These three studies showed that the accuracy of the GMD increases with increasing numbers of free parameters used in (15) [i.e., one-parameter , two-parameter , and three-parameter quadruplet layout] and increasing numbers of quadruplets . For the three-parameter quadruplet definition, the improvement of GMD accuracy will saturate at or 6. Considering this, we chose the GMD configuration with five quadruplets and a three-parameter quadruplet definition4 in our following analysis. For convenience, we will also simply refer to this specific GMD configuration as GMD. Figure 1, as an example, clearly illustrates the differences in quadruplet layouts used by the DIA (Fig. 1a) and the GMD (Fig. 1b). Note that only the deep scaling function of the GMD is used in our manuscript, which represents “weak” four-wave nonlinear interactions in deep water and intermediate depths for (d is water depth; Tolman 2013). This is comparable to the applicable range of the DIA and its -dependent scaling relation for (e.g., Hasselmann and Hasselmann 1985; Hasselmann et al. 1988). The application of ST6 with other nonlinear solvers such as the simplified Research Institute for Applied Mechanics (RIAM; Komatsu and Masuda 1996; Tamura et al. 2008) and two-scale approximation (TSA; Resio and Perrie 2008; Perrie et al. 2013) is beyond the scope of this paper, and therefore may be pursued in the future.

Fig. 1.

Interaction diagram for [after Hasselmann (1963)] with solid lines for contours of and vectors for quadruplet layouts: (a) the DIA quadruplet and (b) the five GMD quadruplets as specified in Table 1. For the given values of , only one solution of (15) is shown for the clarity of this figure.

Fig. 1.

Interaction diagram for [after Hasselmann (1963)] with solid lines for contours of and vectors for quadruplet layouts: (a) the DIA quadruplet and (b) the five GMD quadruplets as specified in Table 1. For the given values of , only one solution of (15) is shown for the clarity of this figure.

3. Model calibrations

Herein ST6 is applied with different parameterizations of Snl. To distinguish these model configurations, we will refer to ST6 + DIA as ST6D, ST6 + GMD as ST6G, and ST6 + WRT as ST6W. When necessary, the combination of the ST4 source terms (Ardhuin et al. 2010) and DIA (hereafter ST4D), which is used for the operational forecasting in NOAA’s National Centers for Environmental Prediction (NCEP; Alves et al. 2015), is also included for comparisons. The wind growth parameter of ST4D used is 1.33 (Ardhuin et al. 2010; Rascle and Ardhuin 2013), unless otherwise specified.

a. Calibration of ST6D

As mentioned in section 1, our (re)calibration of ST6D is conducted by increasing the (positive) wind input term slightly, and then finding the new tunable parameters existing in the wave breaking and swell decay terms (i.e., ). The amplification of Sin is achieved by setting , as compared with in (7). It was found by coauthor W. Erick Rogers (2014, unpublished work) that using could improve model skills in estimating tail level in the ST6 implementation in SWAN [see also Rogers (2017)]. RBW12 calibrated and [see (9) and (10)] by using a single-grid-point, duration-limited simulation. Because of the scarcity of observations in such idealized cases, the authors had to adopt growth curves simulated by other widely used packages (Komen et al. 1984; Rogers et al. 2003) as references. Unlike RBW12, here we decided to tune the model with fetch-limited simulations, particularly considering the extensive field studies of fetch-limited wind wave spectra (e.g., Hasselmann et al. 1973; Kahma and Calkoen 1992; Babanin and Soloviev 1998a; Romero and Melville 2010a, hereafter RM10).

For tuning purposes, we utilized a fetch-limited test under a homogeneous wind forcing with m s−1, blowing perpendicularly to a straight shoreline. Similar to Tolman and Chalikov (1996), we employed three 40-point grids with different spatial resolutions ( 2.5, 25, and 250 km, respectively) to guarantee a wide range of fetches. The spectral grid was discretized as and , with Hz, . The fetch law for stable stratification, as suggested by Kahma and Calkoen (1992, hereafter KC92) (see also Komen et al. 1994, p. 181), was selected as the tuning reference:

 
formula

where , X is the fetch, and is the peak frequency. It is interesting to note that the fetch law from Babanin and Soloviev (1998a) is remarkably consistent with (16), except for at short fetches (Fig. 2). The power law suggested by RM10 is also in excellent agreement with the former two studies, particularly for the dimensionless energy .

Fig. 2.

(a) Dimensionless energy and (b) peak frequency as a function of dimensionless fetch . The solid black lines represent the KC92 growth curves [(16)] and the Pierson–Moskowitz asymptotic limits [(17)]. The gray dash-dotted lines and black dotted lines are growth curves from Babanin and Soloviev (1998a) and Romero and Melville (2010a), respectively. Results from different model configurations (1D runs; see section 3c) after a 72-h model run are color-coded by blue for ST6D, red for ST6G, yellow for ST6W, and purple for ST4D. The corresponding overall error metrics (normalized bias and RMSE ; see  appendix A) are also printed. Only model points within the valid range of KC92 growth curves (highlighted by the thick gray horizontal lines) were included in error analyses. Besides, the first five points of each grid were abandoned to reduce the effects of numerical errors (Tolman 1992). The dashed, dash-dotted, and solid colored lines denote points from grids with the resolution of 2.5, 25, and 250 km, respectively. The black arrows indicate the point where km, which was chosen to compute for fully developed waves. Note that the negative Sin and Sswl are activated here.

Fig. 2.

(a) Dimensionless energy and (b) peak frequency as a function of dimensionless fetch . The solid black lines represent the KC92 growth curves [(16)] and the Pierson–Moskowitz asymptotic limits [(17)]. The gray dash-dotted lines and black dotted lines are growth curves from Babanin and Soloviev (1998a) and Romero and Melville (2010a), respectively. Results from different model configurations (1D runs; see section 3c) after a 72-h model run are color-coded by blue for ST6D, red for ST6G, yellow for ST6W, and purple for ST4D. The corresponding overall error metrics (normalized bias and RMSE ; see  appendix A) are also printed. Only model points within the valid range of KC92 growth curves (highlighted by the thick gray horizontal lines) were included in error analyses. Besides, the first five points of each grid were abandoned to reduce the effects of numerical errors (Tolman 1992). The dashed, dash-dotted, and solid colored lines denote points from grids with the resolution of 2.5, 25, and 250 km, respectively. The black arrows indicate the point where km, which was chosen to compute for fully developed waves. Note that the negative Sin and Sswl are activated here.

We first made an attempt to determine () for ST6D by the following subjective, loose rules:

  1. ST6D-simulated dimensionless energy and peak frequency should match the KC92 growth curves in (16) reasonably well. Specifically, , , and , where and are the normalized bias and root-mean-square error (RMSE;  appendix A). The positiveness of is imposed due to the exclusion of the negative and terms at this stage (i.e., ).5

  2. For fully developed waves, for example, waves at the 30th point of the coarsest grid where km (black arrows in Fig. 2), the two-phase wave breaking term in (8) should satisfy (RBW12).

Through our tuning exercises, we found that the two restrictions described above yield a narrow corridor in the parameter space [see section 1 of the supplemental online material (SOM)]. To further refine these two parameters, we added another important supplementary constraint (rule 3, below).

  • 3) For a realistic 75-day wave hindcast in Lake Michigan,6 ST6D should predict both and mean square slope quite accurately (e.g., the RMSE of and are less than 0.2 m and 10−3, respectively), as compared with measurements from a single buoy 45007 (see section 2 of the SOM).

Waves prevailing in Lake Michigan are generally young and free of wind-swell interactions (Rogers and van Vledder 2013). Accordingly, the deactivation of Sswl and negative Sin in this hindcast experiment is still physically sound.

After the optimal for ST6D was established by the above-mentioned approach (; Table 1), we continued the calibration of the negative wind input parameter in (6) and the swell decay coefficient in (12). Following ZBRY15, these two parameters were determined through a global hindcast of the year 2013, using a coarse longitude–latitude grid (1.25° × 1.0°) forced by winds from the NCEP Climate Forecast System version 2 (hereafter CFSv2; Saha et al. 2014).7 By comparing with measurements from the cross-calibrated altimeter dataset produced by Young et al. (2017), we found that ST6D with and (Table 1) provided acceptable skills in specification of in global basins (section 3 of the SOM; bias m and RMSE m). Thus far, this finalizes the entire calibration procedure of ST6D.

Table 1.

Parameter setting for ST6D, ST6G, and ST6W, including five parameters from the ST6 source terms (Sin, Sds, Sswl), and four parameters () pertaining to the DIA-like (i.e., DIA and GMD) parameterizations of Snl, where is a proportionality constant [C in Hasselmann et al. (1985) and in Tolman (2013)].

Parameter setting for ST6D, ST6G, and ST6W, including five parameters  from the ST6 source terms (Sin, Sds, Sswl), and four parameters () pertaining to the DIA-like (i.e., DIA and GMD) parameterizations of Snl, where  is a proportionality constant [C in Hasselmann et al. (1985) and  in Tolman (2013)].
Parameter setting for ST6D, ST6G, and ST6W, including five parameters  from the ST6 source terms (Sin, Sds, Sswl), and four parameters () pertaining to the DIA-like (i.e., DIA and GMD) parameterizations of Snl, where  is a proportionality constant [C in Hasselmann et al. (1985) and  in Tolman (2013)].

Figure 2 shows the ST6D-simulated dimensionless energy and peak frequency as a function of dimensionless fetch in the fetch-limited test under m s−1. In general, wave parameters given by ST6D are in reasonable agreement with KC92 curves in (16), as forced in our tuning process. Within the valid range of the fetch-limited observations from KC92 (; see Komen et al. 1994, p. 180, RM10),8 the overall of and are 19% and 5%, respectively. At short and intermediate fetches (e.g., ), ST6D overestimates moderately with an overall of 21%. At extremely long fetches, ST6D agrees quite well with the Pierson–Moskowitz (PM) asymptotic limits, given by

 
formula

where and are the -scaled PM limits from Alves et al. [2003, their (17)] and is the wind-dependent drag coefficient. Here we took for m s−1 by following the drag law described in Hwang (2011), which is also the drag law utilized by ST6 to estimate (RBW12). In this case ST4D presents an improved accuracy for () and a slightly degraded accuracy for (). The fully developed sea states reached by ST4D are very close to those for ST6D ().

b. Calibration of ST6G

For the GMD parameterization of with five quadruplets and a three-parameter quadruplet definition [(15)], there are a total of 20 free parameters to be determined.9 Tolman and Grumbine (2013) designed a holistic genetic optimization (GO) technique to efficiently optimize these parameters altogether. The five quadruplet layouts,10 yielded by the GO algorithm specifically for the ST6 package, are presented in Table 1 and Fig. 1. For idealized test cases selected in Tolman and Grumbine (2013) and relative to the WRT, such a GMD configuration reduces errors of the DIA by more than 50% (section 4 of the SOM), consistent with the findings in Tolman (2013).

Using the same from ST6D, ST6G shows approximately 20% lower and 1% higher with slightly improved (13%) and unchanged (5%) in the fetch-limited test (Fig. 2). The overestimation of at intermediate fetches by ST6D is noticeably improved by ST6G. Considering that all three tuning rules described in the previous section are still well satisfied, we decided not to adjust for ST6G.11 By contrast, the global hindcast experiment for 2013 suggested an update of to in ST6G (Table 1) is useful to further enhance model accuracy of (section 3 of the SOM).

Since it is extremely expensive, if not impossible, to run ST6W in realistic large-scale applications, we did not try a further calibration of ST6W. For simplicity, ST6W directly inherits all the parameters (i.e., ) from ST6G, as shown in Table 1. Examination of Fig. 2 demonstrates that ST6W conforms very well to ST6G, except at short fetches (). Among the three ST6 model configurations, ST6W produces the lowest errors with and . The asymptotic values for from ST6G and ST6W are slightly higher than the value suggested by Alves et al. (2003) (6.00 × 10−3 vs 5.64 × 10−3).

c. Fetch geometry

The last topic we would like to discuss in this section is the effect of fetch geometry on wave growth (Young 1999, p. 109). Field studies (Pettersson and Kahma 2005; Ataktürk and Katsaros 1999) suggested that for the same dimensionless fetch χ, the dimensionless energy ε values of mature waves were remarkably lower for the narrow fetch than for the broad fetch. The dimensionless frequency ν was also affected to some extent, but was clearly less sensitive than ε. It was believed that the narrow geometry constrained the development of waves propagating along directions oblique to the long axis of narrow fetches (bays or lakes). As demonstrated in Rogers and Wang (2006, their Fig. 15) and Ataktürk and Katsaros (1999, p. 643), the third-generation wave models are able to provide qualitatively consistent behavior. An interesting detail, which we think is worth mentioning and which we found from our simulations, is that the DIA-based results are more sensitive to the fetch geometry than those from the GMD and WRT.

Figure 3 presents the fetch-limited results from the one-dimensional (1D) run (i.e., the propagation of wave energy along the y direction was switched off, and the model domain became essentially infinitely wide) and the two-dimensional (2D) run (i.e., wave energy was allowed to propagate along y direction) with an aspect ratio , where x and y are the length and width of the model domain, respectively. Clearly, the fully developed asymptotic energy (frequency ) is lower (higher) in the 2D run (gray dotted lines in Fig. 3) due to the narrow geometry of the fetch. The differences between these asymptotic values can be quantified by

 
formula

where and are the asymptotic values from the 1D (broad fetch) and 2D (narrow fetch) cases. For , the DIA-based models (ST6D and ST4D) show and , whereas these metrics for ST6G and ST6W are markedly lower ( and ) (Table 2). Results for 2D runs with higher aspect ratios (; i.e., narrower fetches) present similar features (Table 2). The wave spectrum from the DIA is generally too broad (e.g., Hasselmann et al. 1985; Young et al. 1987) and thus becomes more constrained by the narrowness of the fetch, explaining the results we obtained here. An intuitive indication of these results is the DIA-based models may have problems simulating mature waves in narrow bays/lakes.

Fig. 3.

Fetch-limited simulations from the third grid only ( km km). The black solid lines with markers show the 1D (broad fetch) model results (i.e., the propagation of the wave energy along the y direction is turned off), the gray dashed lines with markers represent 2D (narrow fetch) model results (i.e., wave energy is allowed to propagate along the y direction) where the aspect ratio of the model domain is . The black asterisks highlight the PM asymptotic limits [(17)].

Fig. 3.

Fetch-limited simulations from the third grid only ( km km). The black solid lines with markers show the 1D (broad fetch) model results (i.e., the propagation of the wave energy along the y direction is turned off), the gray dashed lines with markers represent 2D (narrow fetch) model results (i.e., wave energy is allowed to propagate along the y direction) where the aspect ratio of the model domain is . The black asterisks highlight the PM asymptotic limits [(17)].

Table 2.

Relative differences between the asymptotic values of the dimensionless energy and frequency ( and , respectively) from the 1D (broad fetch) model run and the 2D (narrow fetch) model runs with different aspect ratios (i.e., ).

Relative differences between the asymptotic values of the dimensionless energy and frequency ( and , respectively) from the 1D (broad fetch) model run and the 2D (narrow fetch) model runs with different aspect ratios (i.e., ).
Relative differences between the asymptotic values of the dimensionless energy and frequency ( and , respectively) from the 1D (broad fetch) model run and the 2D (narrow fetch) model runs with different aspect ratios (i.e., ).

4. Duration-limited wave growth

The previous section focuses on integrated wave parameters (e.g., significant wave height and peak frequency ) only. However, as demonstrated by many previous studies (e.g., Banner and Young 1994; Alves and Banner 2003; Romero and Melville 2010b; Resio et al. 2016), high skills in predicting those bulk parameters do not necessarily guarantee high accuracies in modeled spectral shape. In this section, we will analyze simulated wave spectra in a much detailed way, mainly using spectral metrics suggested by RM10 and Resio et al. (2016). Most attention is dedicated to studying the following:

  1. to what extent modeled spectra reflect measured properties of ocean waves, and

  2. how well the GMD configuration presented in the previous section represents the exact solutions of (i.e., WRT).

A single-grid-point, duration-limited wave growth experiment is selected here, mainly due to its computational efficiency and its reduced sensitivity to numerical errors (e.g., RBW12). The model setup is the same as the one used in fetch-limited simulations, except that the directional grid is refined from to .12 It is, however, particularly noteworthy that in the duration-limited simulations, the high-frequency spectral tail evolves freely without any prescribed slope.

a. Equilibrium and saturation ranges

Based on a dimensional analysis, Phillips (1958) proposed that the high-frequency range of the wave spectrum should follow a form

 
formula

if the wave breaking term is dominant in this specific frequency range, where is the so-called Phillips constant. Toba (1973) argued the role of wind stress is also essential for small scale waves, and thus should be alternatively parameterized as

 
formula

where is known as Toba’s “constant.” Later, assuming all the three physical processes (Sin, Sds, Snl) are important and comparable, Phillips (1985) reached the same form as (20). With the theoretical and observational progress in ocean waves over the past several decades (Hasselmann et al. 1973; Forristall 1981; Donelan et al. 1985; Ewans and Kibblewhite 1990; KC92; Hwang and Wang 2001; Resio et al. 2004; Babanin 2010; Lenain and Melville 2017; Zakharov 2018, among others), it has been gradually recognized that the equilibrium range () and saturation range () could coexist in wave spectra, with the equilibrium range typically located between (Donelan et al. 1985; Resio et al. 2004) and the saturation range being applicable at higher frequencies.

Figure 4 shows the evolution in time of the modeled omnidirectional frequency spectrum and saturation spectrum over 48 h of duration-limited simulations, where

 
formula

Inspection of Fig. 4 suggests that all the three ST6 models (i.e., ST6D, ST6G, and ST6W in Figs. 4a,c) yield a clear transition from a power law of approximately to the power law of about . At intermediate frequencies, wave spectra from ST4D (Fig. 4e) also follow the slope of very well, whereas they fail to present the saturation range at higher frequencies. Consistent with ZBRY15 (their Fig. 5), here we find ST4D gives a high-frequency tail roughly proportional to . Another important result is that wave spectra from ST6G are in excellent agreement with those from ST6W (Fig. 4c), indicating the high accuracy of the GMD approach in reproducing exact solutions of Snl. As expected, from ST6W and ST6G are narrower in frequency space than those from DIA-based models (Fig. 4c vs Figs. 4a,e; see also our Figs. 7 and 8).

Fig. 4.

The evolution of (left) omnidirectional frequency spectrum and (right) saturation spectrum from (a),(b) ST6D, (c),(d) ST6G [and ST6W; dashed lines in (c)], and (e),(f) ST4D. Only spectra at h are displayed and color-coded according to the wave age . In (a), (c), and (e), reference slopes proportional to and are shown as solid black and gray lines, respectively. The black lines in (b), (d), and (f) correspond to the converging constant saturation level of field measurements from Lenain and Melville (2017), and the gray line is a reference spectral slope proportional to f.

Fig. 4.

The evolution of (left) omnidirectional frequency spectrum and (right) saturation spectrum from (a),(b) ST6D, (c),(d) ST6G [and ST6W; dashed lines in (c)], and (e),(f) ST4D. Only spectra at h are displayed and color-coded according to the wave age . In (a), (c), and (e), reference slopes proportional to and are shown as solid black and gray lines, respectively. The black lines in (b), (d), and (f) correspond to the converging constant saturation level of field measurements from Lenain and Melville (2017), and the gray line is a reference spectral slope proportional to f.

The saturation spectra from ST6D (Fig. 4b), for different wave ages (where is the peak phase velocity), converge at frequencies between 0.3 and 0.6 Hz to a constant level (hereafter Bc), indicating that, for m s−1, at this frequency range is practically independent of the stage of development of ocean waves. It is worth mentioning that for lower winds (e.g., m s−1), the dependence of at saturation ranges on wave age or wind speed is clearly visible (Ewans and Kibblewhite 1990, their Fig. 3). RM10 found that the average of over the interval of rad m−1 (vertical dashed lines in Figs. 4b,d,f) are (8 ± 2) × 10−3 for m s−1. The value of given by ST6D over this wavenumber range is about 7 × 10−3, falling within the measured range of RM10. It is also noteworthy that the ST6D-favored constant level of 7 × 10−3 is in excellent agreement with field measurements from Babanin and Soloviev (1998a; ) and Lenain and Melville (2017; ). ST6G (Fig. 4d) and ST6W show similar except is slightly lower (~6 × 10−3).13 For Hz, from ST6 models starts to increase again, which may not be impossible as pointed out by Lenain and Melville (2017). As seen from Fig. 4f, ST4D overestimates at spectral tails as a result of the deviation of from the form.

To scrutinize the equilibrium range, we present Toba’s parameter in Fig. 5 as estimated from modeled spectra using

 
formula

where and are the lower and upper limits of integration. For consistency with RM10, we adopted and Hz.14 The values of from all three ST6 models (ST6D, ST6G, and ST6W) are quite close, and higher than those from RM10. Nonetheless, they are consistent with values suggested by Resio et al. (2004) for most wave ages and by Hwang et al. (2000a) for . For very young waves (), ST6-computed is noticeably lower because in such cases, equilibrium ranges are fairly narrow or might not exist (see our Fig. 6 and Babanin 2010). ST4D overpredicts compared with ST6 models (ST6D, ST6G, and ST6W) and the three just-mentioned field studies. But for , ST4D-modeled is still within the range of [0.06, 0.11] as summarized in Phillips (1985).

Fig. 5.

Toba’s parameter vs the wave age as simulated by ST6D (circles), ST6G (triangles), ST6W (diamonds), and ST4D (squares). The solid red and blue lines with error bars show measurements from Resio et al. (2004) and Romero and Melville (2010a). The thick is the measurement from Hwang et al. (2000a), collected for a mature wind sea under quasi-steady winds m s−1. The gray shaded region indicates the range of historical measurements summarized in Phillips (1985). Figure adapted from Romero and Melville (2010a).

Fig. 5.

Toba’s parameter vs the wave age as simulated by ST6D (circles), ST6G (triangles), ST6W (diamonds), and ST4D (squares). The solid red and blue lines with error bars show measurements from Resio et al. (2004) and Romero and Melville (2010a). The thick is the measurement from Hwang et al. (2000a), collected for a mature wind sea under quasi-steady winds m s−1. The gray shaded region indicates the range of historical measurements summarized in Phillips (1985). Figure adapted from Romero and Melville (2010a).

Fig. 6.

As in Fig. 4, but for against the dimensionless frequency according to (a) ST6D, (b) ST6G, and (c) ST6W. The color-coded markers “|”, “Y”, “+”, and “×” highlight transition frequencies for corresponding wave ages as estimated by Forristall (1981), KC92, RM10, and Babanin (2010), respectively. Note that only horizontal coordinates of these markers are meaningful. The gray solid (dashed) line highlights the reference spectral slope proportional to f ().

Fig. 6.

As in Fig. 4, but for against the dimensionless frequency according to (a) ST6D, (b) ST6G, and (c) ST6W. The color-coded markers “|”, “Y”, “+”, and “×” highlight transition frequencies for corresponding wave ages as estimated by Forristall (1981), KC92, RM10, and Babanin (2010), respectively. Note that only horizontal coordinates of these markers are meaningful. The gray solid (dashed) line highlights the reference spectral slope proportional to f ().

b. Transition frequency

Since ST6 models are able to produce a transition behavior from to (Fig. 4), it is interesting to check how well the transition frequency given by ST6 compares with previous studies. Forristall (1981) analyzed over 4000 wave spectra measured in situ and found that could be determined by

 
formula

Based on an analysis of wave spectra collected from various wave-growth experiments, KC92 suggested that the transition from a to tail occurred at

 
formula

Similar to Forristall (1981), RM10 derived through finding the intersection of (19) and (20):

 
formula

where the authors adopted and (blue line in Fig. 5). Using an alternative scaling for the equilibrium range as suggested by Resio et al. (2004),

 
formula

Babanin (2010) quantified by

 
formula

where and m s−1 according to Resio et al. (2004) and is a wave age-dependent spectrum-tail level as proposed by Babanin and Soloviev (1998a, gray solid line in our Fig. 7a).

Fig. 7.

The (left) high-frequency energy level α and (right) peak enhancement factor γ vs the dimensionless frequency for wave spectra of (a),(b) the JONSWAP form () (Hasselmann et al. 1973) and (c),(d) the Donelan et al. (1985) form (). The relationships in (a) and (b) are Hasselmann et al. (1976; black solid lines), Janssen [2004; black dotted line for his (2.109)], Babanin and Soloviev (1998a; gray solid lines), and Lewis and Allos (1990; gray dotted line). The gray filled circles in (b) are the JONSWAP data sourced from Young (1999, his Fig. 5.17). The dependences in (c) and (d), shown as the solid lines, are defined in Donelan et al. (1985).

Fig. 7.

The (left) high-frequency energy level α and (right) peak enhancement factor γ vs the dimensionless frequency for wave spectra of (a),(b) the JONSWAP form () (Hasselmann et al. 1973) and (c),(d) the Donelan et al. (1985) form (). The relationships in (a) and (b) are Hasselmann et al. (1976; black solid lines), Janssen [2004; black dotted line for his (2.109)], Babanin and Soloviev (1998a; gray solid lines), and Lewis and Allos (1990; gray dotted line). The gray filled circles in (b) are the JONSWAP data sourced from Young (1999, his Fig. 5.17). The dependences in (c) and (d), shown as the solid lines, are defined in Donelan et al. (1985).

In Fig. 6 we replotted from ST6 models against the dimensionless frequency , together with estimated from (23)(27). As anticipated, ST6G and ST6W match the slope better than ST6D for , particularly when . ST6D presents a slope close to at such frequency regions for those wave ages15 (dashed line in Fig. 6a). Visual inspection of this figure suggests from the ST6 models is generally compatible with Forristall’s estimation. The increases from below for very young waves to above for old wind seas; from KC92 is on average higher. As already reported by RM10, their is also higher than Forristall’s values; but the difference gradually reduces as waves develop. Unlike the former three studies, Babanin (2010) favors an first increases with wave age and then decreases for . This counterintuitive behavior is mainly attributed to the use of the wave age-dependent (Babanin and Soloviev 1998a) in (27). During the 1990s, the form of the high-frequency wave spectrum was in dispute (e.g., Young 1999, p. 123). Babanin and Soloviev (1998a) adopted the Joint North Sea Wave Project (JONSWAP) form (Hasselmann et al. 1973) for obtaining from their field data. Unavoidably, their calculation of must have included contributions from the equilibrium range, particularly for high wave ages. This is clearly demonstrated in our Fig. 7a: the modeled spectra with a nearly constant saturation level correspond to the JONSWAP spectra with an value agreeing well with that from Babanin and Soloviev (1998a) (see the next subsection for more details).

c. Spectral peakedness

The spectral peakedness is a critical metric linked to the modulational instability of spectral waves and thus is frequently utilized in the freak wave literature (e.g., Janssen 2003; Onorato et al. 2006; Ribal et al. 2013). To examine modeled peakedness, we made an attempt to fit the generalized JONSWAP spectral form to each model spectrum. Following Young (2006), the generalized JONSWAP expression reads

 
formula

where α is the high-frequency energy level, γ is the peak enhancement factor, and σ is the peak width parameter. For , (28) corresponds to the JONSWAP form established by Hasselmann et al. (1973) and for to the form proposed by Donelan et al. (1985).

Both of these forms of (28) with and were applied in the curve fitting process, and spectral parameters α and γ obtained from each of the best-fit spectra are illustrated in Fig. 7. The fit was attempted over the full-frequency range (i.e., Hz) with a nonlinear least squares method,16 which minimizes the cost function , where and denote the fitted and simulated discrete wave spectra, respectively (see also Battjes et al. 1987). For the JONSWAP form (Figs. 7a,b), the DIA-based models (ST6D and ST4D) yield an α marginally higher than that from ST6G and ST6W; α from all these four models is generally consistent with the power law proposed by Hasselmann et al. (1976) and slightly lower than power laws described in Babanin and Soloviev (1998a) and Janssen (2004). The most striking result in Fig. 7b is that γ from ST6G and ST6W is remarkably greater than that predicted by Hasselmann et al. (1976), Lewis and Allos (1990), and Babanin and Soloviev (1998a), although it is still within the range of values observed in JONSWAP experiment (Hasselmann et al. 1973; Young 1999); whereas ST6D and ST4D conform to the power law of Hasselmann et al. (1976) much better. For the form of Donelan et al. (1985) (Figs. 7c,d), α from the four models is fairly consistent and follows the dependence on wave age found by Donelan et al. (1985) reasonably well. In contrast, all models display a γ generally higher than that from Donelan et al. (1985). In spite of these discrepancies, the consistent features of Fig. 7 are as follows: 1) γ from ST4D is in general the lowest, then ST6D-simulated γ is marginally higher, and γ from ST6G and ST6W is the highest; and 2) except for very young waves (), both modeled α and γ decrease as the wave develops (), which is analogous to previous field studies (e.g., Hasselmann et al. 1976; Donelan et al. 1985; Babanin and Soloviev 1998a; Janssen 2004).

Another three metrics connected to spectral peakedness are illustrated in Fig. 8, including the spectral width [ν in Babanin and Soloviev (1998a)], spectral narrowness [ in Rogers and van Vledder (2013)], and the Benjamin–Feir index (BFI; Janssen 2003; Onorato et al. 2006; Xiao et al. 2013):

 
formula
 
formula
 
formula

where is the nth-order moment ( appendix A) and is the half-width at the half-maximum of . Remarkably, similar to what we have seen in Fig. 7, the spectral narrowness (Fig. 8b) from ST6G and ST6W are clearly higher than that for DIA-based models (ST6D and ST4D), and all the models underestimate spectral width from Krivinskii (1991) for most wave ages (; Fig. 8a). A close inspection of Fig. 8c suggests that ST6W- and ST6G-simulated BFI appears unrealistically high (BFI > 1).

Fig. 8.

The evolution of (a) spectral width (Babanin and Soloviev 1998a), (b) spectral narrowness (Rogers and van Vledder 2013), and (c) BFI (Onorato et al. 2006) against dimensionless frequency . The solid black line highlights the dependence of on wave development suggested by Krivinskii (1991).

Fig. 8.

The evolution of (a) spectral width (Babanin and Soloviev 1998a), (b) spectral narrowness (Rogers and van Vledder 2013), and (c) BFI (Onorato et al. 2006) against dimensionless frequency . The solid black line highlights the dependence of on wave development suggested by Krivinskii (1991).

Therefore, a consistent finding from our Figs. 7 and 8 is that the WRT- and GMD-based models (i.e., ST6W and ST6G), contrary to our expectation, noticeably overestimate the peakedness/narrowness (or alternatively, underestimate the width) of wave spectra. Being an approximation to the WRT, the DIA provides a slightly improved, but still problematic in general, estimation of the spectral peakedness due to its inherent tendency to unrealistically broaden the exact solutions in frequency space. Although counterintuitive, this finding is remarkably supported by a recent numerical study by Annenkov and Shrira (2018). The authors showed (their Figs. 6 and 11) that relative to the WRT results based on the Hasselmann kinetic equation (Hasselmann 1962), their direct numerical simulations based on the Zakharov integrodifferential equation (Zakharov 1968) predict “considerably wider frequency spectra with much less pronounced peaks.” Considering that the Zakharov equation is “the primitive equation for a weakly nonlinear wave field” and “does not employ any statistical assumptions,” and considering that the Hasselmann kinetic equation can be derived from the Zakharov equation “by applying standard closure hypothesis,” Annenkov and Shrira (2018) argued these systematic mismatches “call for revision of the fundamentals” of the Hasselmann kinetic equation. Besides, the poor performance of wave models in simulating the spectral peak implies a difficulty in predicting the occurrence of freak waves.

d. Directional properties

The last two metrics selected in this section are associated with directional properties of wave spectra. The first metric is the directional spreading. For comparison purpose, here we adopt the definition described in RM10:

 
formula

where ϑ is the angle relative to the dominant wave direction (; see, e.g., Fig. 16 of RM10).

Figure 9 illustrates from different models as a function of . Three directional distribution parameterizations are also shown as references, including 1) the unimodal form proposed by Donelan et al. (1985) with the extension at high frequencies suggested by Banner (1990) (hereafter the Donelan–Banner form), where the parameter β determines the spectral spread; 2) the bimodal form as described in Ewans (1998); and 3) the same form of Donelan et al. (1985) but with β suggested in Babanin and Soloviev (1998b) (hereafter the Babanin–Soloviev form). Note that unlike the former two forms, the latter form depends on the stage of wave development. Nonetheless, all these forms correspond to a that is a minimum in the neighborhood of the spectral peak and then increasing toward lower and higher frequencies. Among the four wave models, ST4D (Fig. 9d) compares the best with the two wave age-independent parametric forms (black lines in Fig. 9). The directional spreading estimated by ST4D agrees well with Ewans’ values below the peak, and then starts to match Donelan–Banner’s prediction for higher frequencies. At , ST4D-computed wave spectra are moderately broader than measurements from Donelan et al. (1985) and Ewans (1998), but are still comparable to the spatial measurements from RM10 (Fig. 9e). On the other hand, wave spectra from ST6D (Fig. 9a) are too broad at and too narrow beyond . The overestimation of spreading near the spectral peak by ST6D is partially reduced by ST6G and ST6W (Figs. 9b,c) due to their more accurate computations of Snl. However, for , from ST6G and ST6W is fairly low, particularly when compared against measurements from Donelan et al. (1985), Ewans (1998), and RM10. The dependence of the peak spreading on deserves special attention. Wave models and observations from RM10 show an upward trend in at as waves develop. This is, however, in contradiction to measurements from Babanin and Soloviev (1998b) and Donelan (2017), which suggest the peak of wave spectrum becomes narrower as increases (e.g., see gray lines in Fig. 9). It should be remarked that there is relatively large uncertainty associated with the peak spreading measurements ( at ) by RM10. Their measurement error of peak spreading generally increases with wave age, and for , the error can be as large as 10° (Fig. 9e). Similarly, the directional spreading estimated from measured by in situ wave buoys is rather sensitive to instrument or analysis noise, in particular for the peak spreading, which is generally low (Kuik et al. 1988, their Fig. 3).

Fig. 9.

Directional spreading according to (a) ST6D, (b) ST6G, (c) ST6W, (d) ST4D, and (e) spatial measurements from RM10. The solid and dashed black lines illustrate calculated from the directional distribution models of Donelan–Banner and Ewans (1998), respectively. The solid and dashed gray lines show based on the form of Babanin and Soloviev (1998b) for and 25, respectively. Also shown in (e) are measurement errors of peak spreading (i.e., at ) from RM10 (vertical colored lines), starting from 3° for the lowest wave age group () to 10° for the largest wave age group ().

Fig. 9.

Directional spreading according to (a) ST6D, (b) ST6G, (c) ST6W, (d) ST4D, and (e) spatial measurements from RM10. The solid and dashed black lines illustrate calculated from the directional distribution models of Donelan–Banner and Ewans (1998), respectively. The solid and dashed gray lines show based on the form of Babanin and Soloviev (1998b) for and 25, respectively. Also shown in (e) are measurement errors of peak spreading (i.e., at ) from RM10 (vertical colored lines), starting from 3° for the lowest wave age group () to 10° for the largest wave age group ().

The second directional metric to be evaluated here is the bimodality of wave components above the spectral peak, as reported by a number of previous studies (Young et al. 1995; Ewans 1998; Hwang et al. 2000b; Lenain and Melville 2017; RM10, among others). Following Wang and Hwang (2001), for a given the angular separation between the two local maxima of is quantified by the metric :

 
formula

where and are the locations of the maximum on each side of the mean wave direction. With directional wave spectra measured with a heave-pitch-roll buoy, Ewans (1998) found that weakly depends on wave age, and its average over all stages of wave development17 can be fitted by

 
formula

for . Similarly, RM10 also found a weak dependence of on wave age from their spatial measurements, showing that for

 
formula

Figure 10 presents estimated from the four wave models, together with measurements from Ewans (1998) and RM10. For low wave ages, ST6D (Fig. 10a) is able to exhibit a bimodal structure, although is around 7° lower than measurements. For high wave ages, however, wave spectra from ST6D generally lose the bimodality, as represented by . ST6G and ST6W (Figs. 10b,c) show enhanced ability to reproduce the bimodal behavior, except for fairly old waves (). ST4D (Fig. 10d) exhibits a superior performance over ST6D in terms of its capacity to reproduce a bimodal spectral structure. Nonetheless, it appears that the dependence of from ST4D on wave development is relatively stronger than those from ST6G, ST6W, and field measurements (Ewans 1998; RM10). The lobe ratios, defined by , from the four wave models are lower than values from RM10 for most of wave ages and thus are not reproduced here (section 5 of the SOM).

Fig. 10.

The , scaled with , against according to (a) ST6D, (b) ST6G, (c) ST6W, and (d) ST4D. The black line with error bars show measurements collected by RM10, from which (35) was fit. The dashed line is estimated from Ewans’ formula (34) with the mean wave age of his measurements.

Fig. 10.

The , scaled with , against according to (a) ST6D, (b) ST6G, (c) ST6W, and (d) ST4D. The black line with error bars show measurements collected by RM10, from which (35) was fit. The dashed line is estimated from Ewans’ formula (34) with the mean wave age of his measurements.

5. Global hindcast

After a thorough analysis of model skills in specifying the wave spectrum in academic tests, it is necessary to verify performances of ST6D and ST6G in realistic global hindcasts. ST6W is excluded from the analysis here because of its computational infeasibility at the global scale.

SABZ16 conducted a comprehensive study on the comparison and assessment of different source term packages (Tolman and Chalikov 1996; Janssen 2004; Ardhuin et al. 2010; ZBRY15) available in WW3 with a global hindcast of the year 2011. For easy intercomparison with other packages evaluated in SABZ16, we selected the same year (2011) in our simulations. The global model domain is bounded within 78°S–78°N, with a resolution of 1/2° × 1/2°. The resolution of the spectral grid is and with frequencies ranging from 0.037 to 0.953 Hz. As for SABZ16, we used winds from CFSv2 (Saha et al. 2014) as the external forcing (~0.2°; 3 hourly), and for ST4D. It was shown that the CFSv2 winds of 2011 agreed well with altimeter observations (SABZ16, their Fig. 2), with and . (The corresponding spatial distribution of the errors of CFSv2 winds is presented in Fig. B1 in  appendix B.)

a. Comparison against altimeters

Figure 11 presents comparisons of the simulated wave height by different wave models and altimeter measurements from Young et al. (2017). Records of from four altimeters, including Jason-1/2, Envisat, and CryoSat-2, were sourced, leading to a large set of model–altimeter collocations []. Relative to altimeter observations, the overall bias b and RMSE in ST6D-simulated (Fig. 11a) are 0.04 m and 0.39 m, respectively. The scatter index (SI) is low (0.14) and the correlation coefficient ρ is as high as 0.96. The performance of ST6G (Fig. 11b) is slightly improved, showing a b of 0 m, reduced (0.35 m) and SI (0.13), and an elevated ρ of 0.97. ST4D (Fig. 11c) provides a remarkably similar skill to that of ST6G. Note that error metrics shown here are in good agreement with those from ZBRY15 (their Fig. 13), where the authors also found that ST4D was marginally superior to ST6D in modeling for the global basin.

Fig. 11.

Comparison of for the 2011 global hindcast between altimeters and wave model (a) ST6D, (b) ST6G, and (c) ST4D. The dashed line is the 1:1 line. Error metrics printed in the inset of each panel includes the total number of model–altimeter collocations N, bias b, RMSE , correlation coefficient ρ, and scatter index SI (see  appendix A).

Fig. 11.

Comparison of for the 2011 global hindcast between altimeters and wave model (a) ST6D, (b) ST6G, and (c) ST4D. The dashed line is the 1:1 line. Error metrics printed in the inset of each panel includes the total number of model–altimeter collocations N, bias b, RMSE , correlation coefficient ρ, and scatter index SI (see  appendix A).

The spatial distributions of the normalized bias and RMSE of simulated by each of the three models are illustrated in Fig. 12. Consistent with previous studies (e.g., Ardhuin et al. 2010; ZBRY15), values from all the three models (Figs. 12a,c,e) are biased low in the western Pacific Ocean, the majority of the Atlantic Ocean, and the tropical Indian Ocean, with a about −5%. The underestimation of becomes slightly more noticeable in the western tropical Pacific Ocean and the tropical Atlantic Ocean, particularly in the proximity of the Indonesia archipelago (). This could result from the bias in CFSv2 winds in equatorial zones (SABZ16; see our Fig. B1), the neglect of shoreline reflection (Ardhuin et al. 2010), and the overblocking effect of obstruction grids utilized to represent unresolved islands (Tolman 2003). In contrast, the eastern Pacific Ocean and the Southern Ocean are dominated by positive biases in . In the Southern Ocean, is roughly zonally distributed and increases toward the Antarctica from 5% to 25%. Ardhuin et al. (2011) demonstrated that including the blocking effect of small-size icebergs could reduce positive biases of south of 50°S. All the three wave models also share a similar pattern of the normalized RMSE (Figs. 12b,d,f): in most of oceans is below 15%, whereas regions such as the equatorial western Pacific Ocean and the midlatitudes in the North Atlantic Ocean correspond to a moderately higher (~20%); adjacent to Antarctica is the highest, and could be above 25%. Once again, this is comparable to results from Ardhuin et al. (2010) and ZBRY15. A close inspection of Figs. 12b, 12d, and 12f suggests that ST6G and ST4D perform slightly better than ST6D, analogous to what we have seen in Fig. 11.

Fig. 12.

Error metrics of gridded in 2° × 2° bins for the 2011 global hindcast according to (a),(b) ST6D, (c),(d) ST6G, and (e),(f) ST4D, showing (left) normalized bias and (right) normalized RMSE .

Fig. 12.

Error metrics of gridded in 2° × 2° bins for the 2011 global hindcast according to (a),(b) ST6D, (c),(d) ST6G, and (e),(f) ST4D, showing (left) normalized bias and (right) normalized RMSE .

The geophysical mismatches presented in Fig. 12 can be partially, but not fully, explained by the errors of wind forcing (Fig. B1). For example, the overestimation of in the eastern Bay of Bengal, the western Arabian Sea, the Hudson Bay, and the Argentine Sea can be attributed to the biased-high CFSv2 wind forcing. The overall negative bias of dominated in the central Pacific, Atlantic, and Indian Oceans is in line with the overall underestimation of by CFSv2 as well. Besides, the relatively high of the simulated present in the western midlatitude Pacific and Atlantic Oceans and the northern Indian Ocean correlates well with the comparatively high of . Nonetheless, the signs of the biases of and in the eastern Pacific Ocean and the entire Southern Ocean are opposite, suggesting that other physical factors are operative. A close inspection of Fig. 12 indicates that wave energy was overestimated in the zone of the southern westerlies and then these overestimated wave energy propagates northeastward as swell to the eastern coast of the South Pacific Ocean. A detailed analysis of the wave model biases in the Southern Ocean, as done by Ardhuin et al. (2011) and Rapizo et al. (2018), is left for future studies. The reader is also referred to Ardhuin et al. (2010, p. 1930) for further discussions about the spatial distribution of wave model errors.

b. Comparison against NDBC buoys

The validation of model simulations against in situ buoys was also conducted using observations from a total of 21 stations (Fig. 13) maintained by the U.S. National Data Buoy Center (NDBC; http://www.ndbc.noaa.gov). Closely following SABZ16, we only selected wave buoys that provide two-dimensional wave spectra and that are located in deep water (depth m).

Fig. 13.

A total of 21 NDBC buoys (filled circles) used in the model–buoy comparison.

Fig. 13.

A total of 21 NDBC buoys (filled circles) used in the model–buoy comparison.

Figure 14 displays Taylor diagrams (Taylor 2001) summarizing the statistical comparison between wave models and buoys. Apart from wave parameters scrutinized by SABZ16 (e.g., wave height , wave period , mean square slope , directional spreading ; see  appendix A for definitions), we include another three complementary wave quantities, namely the wave period appendix A), the narrowness of one-dimensional spectrum [(30)], and the partial significant wave height (Rogers and Wang 2007; Li and Holt 2009), given by

 
formula

where and are lower and upper integral limits. Similar to Li and Holt (2009), we divided the overlapping frequency range of buoys and models (i.e., [0.037, 0.485] Hz) into four bands with band boundaries locating at . Based on a 10-day duration hindcast for Lake Michigan, Rogers and van Vledder (2013) clearly demonstrated that wave models using DIA for Snl tend to overestimate energy below the spectral peak and underspecify spectral narrowness and that using WRT for Snl instead of DIA could significantly alleviate such model biases. Therefore, examinations of and could be beneficial to identify possible advantages of GMD over DIA. Similarly, the wave period is especially relevant for the low-frequency part of the wave spectrum, and thus an evaluation of this specific period may also be informative. The Taylor diagram is not applicable to vector quantities. To give a rough idea of model skills in computing wave direction, we also plotted zonal and meridional wave heights , defined in a similar fashion as zonal and meridional winds , where and are wind direction and mean wave direction ( appendix A), respectively. It may be also useful to mention that the mean square slope used here is actually a low-pass mean square slope as we are just looking at the waves in the buoy frequency range ( Hz).

Fig. 14.

Taylor diagram summarizing the statistical comparison between NDBC buoys and wave models: (a) ST6D, (b) ST6G, and (c) ST4D. The wave parameters represented by different colored-markers are interpreted in figure legends. Values in parentheses identify the normalized bias of each wave parameter. The last two variables and in (a), shown as white and gray stars, illustrate comparisons of zonal and meridional winds between CFSv2 and NDBC buoys.

Fig. 14.

Taylor diagram summarizing the statistical comparison between NDBC buoys and wave models: (a) ST6D, (b) ST6G, and (c) ST4D. The wave parameters represented by different colored-markers are interpreted in figure legends. Values in parentheses identify the normalized bias of each wave parameter. The last two variables and in (a), shown as white and gray stars, illustrate comparisons of zonal and meridional winds between CFSv2 and NDBC buoys.

Our Fig. 14 suggests that wave height , wave periods and , mean square slope , and zonal wave height are generally well estimated by all three wave models, corresponding to a correlation coefficient ρ greater than 0.9, a normalized standard deviation between 0.85 and 1.1, and a normalized centered RMSE less than 0.5. Overall, the simulated is around 6% biased low at these selected buoy locations, consistent with the model–altimeter comparison shown in our Fig. 12. Wave energy at frequencies beyond 0.06 Hz (i.e., in Fig. 14) is simulated with a remarkably high accuracy as well. Surprisingly, model skills in estimating meridional wave height is appreciably inferior to those for zonal wave height , as indicated by a lower ρ and higher . By contrast, error metrics for zonal and meridional winds (, ; see white and gray stars in Fig. 14a) are not that noticeably different. Interestingly, ST6G indeed provides more accurate () as compared with ST6D () and ST4D ().

The directional spreading and spectral narrowness are the two most poorly simulated quantities. Both of these wave parameters show a within [0.8, 0.9]. The ρ of simulated is about 0.7–0.75, and the ρ for is even lower (). The variability of is overestimated by around 20%; in contrast, models underestimate the variability of more than 35%. In terms of correlation coefficient ρ, ST6G yields a slightly improved estimation of ( for ST6G vs for ST6D and ST4D, respectively) and ( for ST6G vs for ST6D and ST4D). The DIA is known to overpredict directional spreading of wave spectrum (see, e.g., Fig. 9a vs Figs. 9b,c). When compared with buoy observations, model spectra however turn out to be slightly too narrow, indicated by the negative (−7% and −5% for ST6D and ST4D). As anticipated, from ST6G is more biased (−12%). The underestimation of spreading might be attributed to inherent errors in directional distribution of different source functions and the neglect of effects of currents on waves, as already pointed out by SABZ16. Numerical simulations (e.g., Fan et al. 2009a,b; Komen et al. 1994, p. 359), laboratory experiments (Rapizo et al. 2016), and field observations (Romero et al. 2017) have shown that ocean currents could broaden wave spectrum due to their refraction and scattering of wave rays. A detailed analysis of the impact of ocean currents on directionality of ocean waves will be pursued in the future. In line with Rogers and van Vledder (2013), a more accurate parameterization of Snl is helpful for reducing the negative bias in spectral narrowness given by DIA ( for ST6D vs for ST6G). The improvement in modeling shown here, however, is considerably less than that reported in Rogers and van Vledder (2013).

The most noticeable advantage of GMD over DIA is seen in comparisons of long-period wave energy ( Hz or equivalently wave period s). Clearly from Fig. 14, the accuracy of modeled is not as high as that for the three other counterparts (). ST6D (Fig. 14a) exhibits a normalized bias of 90% for , indicating a serious overestimation of energy at frequencies below 0.06 Hz. ST4D (Fig. 14c) provides a somewhat improved performance in specifying (). This large positive bias is significantly reduced in the ST6G simulation, as corroborated by a much lower (19%). The variability of is underestimated by ST6G by around 20%.

To highlight improvements in simulating high-frequency energy brought about by the recalibrated ST6D over its predecessor (ZBRY15), and improvements in modeling low-frequency energy brought about by use of the GMD over the DIA, we replotted the mean square slope and partial wave height in Figs. 15 and 16, respectively. Similar to Ardhuin et al. (2010) and SABZ16, Fig. 15 illustrates the averaged mean square slope over each 1 m s−1 bin of and each 0.5 m bin of