1 INTRODUCTION

Rogue waves are wind-generated sea waves that are considerably higher than the surrounding waves. A traditional definition of a rogue wave is H_max/H_s > 2 (Dysthe et al., 2008; Kharif et al., 2009), where H_max is the maximum wave height in a certain wave field and H_s is the significant wave height. Various theories have been proposed to explain the formation of rogue waves (Kharif and Pelinovsky, 2003; Kharif et al., 2009) including ⅰ) nonlinear focusing due to third-order quasi-resonant wave-wave interaction (Janssen, 2003) and ⅱ) purely dispersive focusing due to second-order nonresonant or bound harmonic waves, which does not satisfy the linear dispersion relation (Fedele, 2008; Fedele and Tayfun, 2009). Both theories include complex nonlinear effects but their physical mechanisms operate in different sea state conditions.

Third-order quasi-resonant interactions and associated modulational instabilities are observed more easily in a sea state where wave energy is concentrated in direction and frequency (Janssen and Bidlot, 2009; Mori et al., 2011; Fedele, 2015) and the water depth influences the modulational instabilities (Janssen and Onorato, 2007; Janssen and Bidlot, 2009; Janssen, 2018), too. Some rogue waves do indeed occur in sea states with concentrated energy distributions (Waseda et al., 2012, 2014) but some occur under completely different conditions (Fedele et al., 2016, 2017). Purely dispersive focusing is more likely to occur in a sea state dominated by second-order nonlinearities (Tayfun and Fedele, 2007; Fedele, 2008). However, the questions of whether second-order nonlinearities dominate the dynamical process of rogue waves or whether third-order nonlinearities also play an important role remain under discussion (Fedele et al., 2016, 2017).

It is evident that the sea state condition is important in the formation of rogue waves. With current wave modeling technology, large-scale sea state simulation still depends on third-generation wave models. Largescale spatiotemporal simulation can elucidate wave field evolution including the evolution of wave systems; however, the directional spectra simulated by such models tend to smooth the information on both wave phase and surface elevation, which are crucial for rogue waves. Although directional spectra remain the basis of rogue wave research, it is inadequate to simply consider only bulk parameters like significant wave height and mean wave period integrated from the spectra.

This study investigated the sea state conditions pertaining at the time three rogue waves were observed during Typhoon Trami in the East China Sea in 2013. The third-generation WAVEWATCH Ⅲ model (The WAVEWATCH Ⅲ Development Group (WW3DG), 2016) was adopted to simulate the sea state conditions and spectra. The evolution of wave systems in the region of the observations was analyzed using an algorithm for spatial and temporal wave system tracking (Hanson and Phillips, 2001; Devaliere et al., 2009). Furthermore, taking advantage of the High-Order Spectral (HOS) method (Ducrozet et al., 2016), wave surface elevations were simulated and used to determine the relative importance of nonlinearities. Certain characteristic parameters of the two theories mentioned above were selected to assess whether such physical mechanisms were in operation during the studied events.

The remainder of this paper is organized as follows. In Section 2, the records of the observed rogue waves and the meteorological conditions pertaining at the time of their occurrence are introduced, and the methodology used in the analysis is described. The simulated results and analyses are presented in Section 3. Finally, our conclusions are offered in Section 4.

2 MATERIAL AND METHOD 2.1 Observational records and meteorological conditions

In August 2013, three rogue wave events H_max/H_s > 2 were recorded by a wave buoy located at 29.068 7°N, 124.921 5°E, with a local water depth of about 88 m. The TRIAXYSTM Directional Wave Buoy (AXYS Technologies Inc., Canada) recorded horizontal and vertical displacement of the water surface hourly with a sampling interval of 0.78 s (i.e., 1.28 Hz). Details of the recorded rogue waves are shown in Table 1.

The time series of normalized profiles of the three rogue waves are shown in Fig. 1. Much sharper crests are evident accompanied by asymmetric troughs at the time of the occurrence of the maximum elevations (locating at 0 of the x-axes of Fig. 1), denoting significant nonlinear effects. Moreover, the influence of external forces such as fishing-boat dragging or a signal fault of the instrument can be excluded based on Fig. 1.

According to the tropical cyclone database of the China Meteorological Administration (tcdata. typhoon.org.cn) (Ying et al., 2014), all three rogue waves might have been affected primarily by Typhoon Trami (Table 2 and Fig. 2). When event RW2106 occurred, Trami was generating large waves near the observation site, but when the final two events (i.e., RW2215 and RW2222) occurred, Trami's center had made landfall to the southwest of the observation site. Several other typhoons mentioned in Table 2 had become tropical depressions before the studied events occurred. In August, the prevailing wind in the South China Sea is from the southwest, while that in the East China Sea is from the south and southwest.

2.2 Methodology 2.2.1 Wave model settings

The third-generation WAVEWATCH Ⅲ model (Tolman, 1991) version 5.16 (The WAVEWATCH Ⅲ Development Group (WW3DG), 2016) was used to simulate the wave field and to track the wave systems (Section 2.2.2). The spectra used as initial conditions of the HOS method simulation (Section 2.2.3) and in the assessment of certain characteristic parameters (Section 2.2.4) were also derived from the wave model. Thus, all of the results based on the same numerical modeling were consistent.

The spectral space of WAVEWATCH Ⅲ was set as 36 directions with intervals of 10 degrees and 35 frequencies logarithmically spaced from the minimum frequency 0.042 Hz up to 1.05 Hz with intervals of f_i+1/f_i=1.1. In the current version of WAVEWATCH Ⅲ, several input (S_in) and dissipation (S_ds) source term packages have been provided to model the wave fields. In this study, the ST4 source package (Ardhuin et al., 2010) was adopted, which had performed better than other S_in+S_ds combinations in our previous experiments with current settings of the model. And the nonlinear wave-wave interactions were parameterized by the DIA method (Hasselman et al., 1985) in the model.

With consideration of the swells generated by non-local wind systems and because the prevailing winds were from south and southwest, the southern border of our computational grid extended to 5°N and the western border extended to 100°E. The parent computational grid (5°–45°N, 100°–150°E) also covered the tracks of the typhoons listed in Table 2. In anticipation of high resolution for the wave system tracking, we designed a set of three nested grids (see Fig. 2): nst0, ns1, and nst2 with resolutions of 1/4°, 1/8°, and 1/16°, respectively. The bathymetric data were obtained from the ETOPO1 1 Arc-Minute Global Relief Model (DOI: 10.7289/V5c8276m) of the National Centers for Environmental Information of the National Oceanic and Atmospheric Administration, and the depth read from the ETOPO1 at the observation site was 89 m, which was considered sufficiently precise for the numerical simulation. Finally, no wave level change was considered because the effect of wave level changing was negligible for the local water depth.

The CFSv2 Selected Hourly Time-Series Products (Saha et al., 2011) were used to drive the wave model. With consideration of the typhoon tracks and the non-local wind systems, we decided to allow sufficient time for waves to develop before the occurrence of the first rogue wave and to see how the sea state might be changed after the occurrence of the final event. Thus, the numerical simulation was executed from 20130810 00:00:00 to 20130824 00:00:00 (all dates and times used in this paper are referred to UTC).

2.2.2 Spatial and temporal tracking of wave systems

A wave field might comprise one wind sea system generated by local winds and several swell systems generated by distant storms. The composition of a wave system could play an important role in the formation of a rogue wave. Since version 4.08, wave system tracking postprocessing has been available in the WAVEWATCH Ⅲ source code package. Based on the spectral partitioning procedure (Vincent and Soille, 1991; Hanson and Jensen, 2004; Hanson et al., 2009), some spatial and temporal correlation steps have been introduced (Voorrips et al., 1997; Hanson and Phillips 2001; Devaliere et al., 2009) to track the spatial and temporal changes of a certain spectral partition (wave system). Taking advantage of this tracking can provide visualization of the process of generation, coexistence, and diminishment of wave systems for a single point or within a certain area. Although the accuracy might be influenced by the computational resolution or by the parameters of the algorithm, the trend of variation of the wave systems can still be elucidated.

2.2.3 High-Order Spectral method

Nonlinear effects are important in the formation of rogue waves. The two most popular theories on how rogue waves occur have a very close relationship with the order of nonlinearity. Modulational instability is induced by third-order quasi-resonant wave-wave interactions (Janssen, 2003), while purely dispersive focusing is caused by second-order nonresonant or bound harmonic waves (Fedele, 2008; Fedele and Tayfun, 2009). Thus, the order of nonlinearity that dominates the wave field might indicate the possible physical mechanism behind the occurrence of a rogue wave.

High-order statistical characteristics of wave surface elevations such as skewness and kurtosis denote the nonlinear effects of a given wave field. However, it is widely known that the directional spectra of such wave fields simulated by third-generation wave models have smoothed signals of wave phase and surface elevation. To represent those signals again, highly and fully nonlinear potential flow solvers must be applied. The HOS method pseudo-spectral numerical algorithm (Dommermuth and Yue, 1987; West et al., 1987) represents a potential flow solver with acceptable efficiency and accuracy. Based on a perturbation expansion of the wave potential function up to a prescribed M order of nonlinearities in terms of a small parameter, the HOS method can be used to solve for nonlinear wave-wave interactions up to the specified order M.

As the potential flow solver, this study adopted the HOS-ocean software version 1.5 (Ducrozet et al., 2016). The suitability of HOS-ocean has been verified in studies of modulational instability (Toffoli et al., 2010; Fernandez et al., 2014) and rogue waves (Ducrozet et al., 2007; Sergeeva and Slunyaev, 2013; Xiao et al., 2013). Potential flow solvers are limited to nonbreaking waves, but if the wave field presents marginal breaking that is highly localized in space and time, the simulation might be able to continue. Considering this limitation (Ducrozet et al., 2017), the wave field was resolved with the spatial discretization of N_x×N_y=192×192 points (total number of points: 384×384) on a 2 km×2 km area and temporal duration of 1 800 s. With consideration of that, the simulated peak wavelengths of the sea states where three events occurred are 240, 200, 160 m respectively, such settings allowed accurate estimation of the possible presence of wave breaking and the energy dissipation due to wave breaking could be neglected. In consideration of the spatial-temporal scales in this study, the growth of waves due to wind input is practically insignificant (Dysthe et al., 2003; Janssen, 2003), thus the wind effect was ignored too. Furthermore, only constant water depth was considered in the simulations.

Initial conditions for the wave potential and surface elevation were specified from the directional spectra output by the WAVEWATCH Ⅲ simulation at the moments and locations when and where the rogue waves occurred. The HOS-ocean nonlinear order was set separately as M=3 and M=2. If no discrepancies were observed, second-order nonlinearities were dominant; otherwise, third-order nonlinearities had a reasonably important role. For each order M, the HOS-ocean simulation was performed 50 times, and the mean value and 95% confidence interval were considered to ensure coherence of the HOS-ocean simulation runs.

2.2.4 Characteristic parameters of sea state

Certain characteristic parameters based on the geometry of a directional spectrum denote the status of a sea state condition that is closely related to the formation and occurrence probability of rogue waves. Based on the spectra output by WAVEWATCH Ⅲ, this study estimated four sets of such parameters. Single values of these parameters might be meaningless in relation to the entire physical process; however, their relative differences and trends of variation at each stage of the process illustrate further details of the rogue waves. Discussion on the physical significance of those parameters and on how best they could be estimated is presented in Section 3.4.

3 RESULT 3.1 Numerical simulation of the wave field

Illustrations of the acceptable performance of the simulation of bulk parameters, e.g., significant wave height, mean cross-zero wave period and mean wave direction, are shown in Fig. 3a with vertical dashed lines identifying the times at which the rogue waves occurred. And the mean absolute errors (MAE) between observed and simulated parameters are listed in Table 3 and calculated as

where N is the number of total samples, OBS_i and SIM_i are the i^th observed and simulated parameters in their respective time series which are synchronous. The corresponding CSFv2 wind speed and direction at the observation site are shown in Fig. 3b.

The wave and wind fields at the times of occurrence of the rogue waves are shown in Fig. 4a–f. At 06:00 on 21 August 2013, Typhoon Trami was centered near the northeastern corner of the Taiwan, China, with its area of maximum wind speed of > 20 m/s located at approximately 27.5°N, 122.5°E (Fig. 4d). Similarly, the maximum significant wave height of > 7 m can also be seen at the same location, which was very close to the observation site (Fig. 4a). The maximum simulated significant wave height at the observation site was 6.13 m, the mean direction of the waves was toward NW and the corresponding wind direction was toward WNW. Thus, it can be inferred that event RW2106 occurred in a wave field that was influenced intensely by the typhoon.

The wind and wave fields at the times when events RW2215 and RW2222 occurred are shown in Fig. 4b and e and Fig. 4c and f, respectively. Typhoon Trami had made landfall before event RW2215 occurred. At this time, the wind direction at the observation site was toward NNW, mean wave direction was toward N and simulated significant wave heights were 3.77 and 3.50 m. The area of maximum significant wave height, which had reduced to approximately 3–4 m, was located at 28°–29°N, 122.5°E at the end of the wind fetch and not far from the observation site. In comparison with event RW2106, the influence of Trami in the region of the observation site had weakened considerably during the final two events.

The PM spectrum (Pierson and Moskowitz, 1964) which presents fully developed sea states was adopted to assess the degree of development of the sea states where the rogue waves occurred. The peak period of the PM spectrum is

where U_19.5 is the wind speed at 19.5 m above the sea surface and g is the gravitational acceleration (=9.81 m/s²). Because the u-v wind data used in the CFSv2 dataset is at 10 m, a simple transformation needed to be applied

The T₀s of the three events and related parameters are shown in Table 4. It can be seen that the simulated peak period (T_p) of events RW2215 and RW2222 are much larger than PM peaks under the wind speed at the time when the two events occurred, denoting those two events occurred in fully developed sea states. However, T_p of event RW2106 is a little smaller than the T₀ assessed under the corresponding wind speed, implying that the sea state might have not completely developed at that time. With consideration of that the simulated H_s and T_p were attaining the peak of the whole process at the time of the occurrence of event RW2106 (see Fig. 3a) and the conclusion we have drawn that the sea state was intensely influenced by the typhoon at that time, it is inferred that the sea state was quasi-fully developed where event RW2106 occurred.

3.2 Tracking of wave systems

The wave system tracking in WAVEWATCH Ⅲ was used to track the evolution of the wave systems at the observation site and in the area covered by nst2 (20.0°–33.0°N, 120.0°–130.0°E). Detected wave systems are shown in the upper subplot with different colors and the wind-sea and swell systems in coexistence can be easily identified by their peak periods which are usually larger for swell systems. The magnitude of each vector represents the significant wave height of each system and the direction of that represents which direction the system is propagating to. Similar representation for local wind is illustrated in the lower part of Fig. 5.

As shown in Fig. 5, system sys-1 (black arrows) was present for all three events. Based on the wind vector shown in the lower part of Fig. 5 and the previous analysis of the wind and wave fields, sys-1 represents the dominant wind sea system of Typhoon Trami. The system sys-2 (blue arrows), which was in existence before sys-1, i.e., during August 16–18, might represent the influence of the "unnamed" tropical depression (see Fig. 2 and Table 2).

Only one wind sea system was evident at the observation point when event RW2215 occurred, although the mean wave direction deviated slightly from the local wind direction (Fig. 5). At the time of event RW2222, two systems were in coexistence: one toward NW (in the same direction as the wind) and the other toward NE. After the occurrence of the final event, wave systems in the region of the observation site became much more complex, and the wind sea system no longer reflected a single wave system.

The outputs of the tracking algorithm could help elucidate the evolution of the wave systems; however, inaccuracies can occur when the significant wave height or peak period of different systems are similar. This situation was in existence when events RW2215 and RW2222 occurred because the wind and wave fields were both in a transitional period as the influence of Typhoon Trami diminished. As shown in Fig. 6, two systems were present in the region of the observation site when event RW2215 occurred, both with a significant wave height of 4–5 m in peak period of 10–12 s. The primary difference between the two systems was the wave direction (sys-1 and sys-3). The system with wave direction toward NW and the system with wave direction toward NE both influenced the observation site at the moment event RW2215 occurred. A similar situation can be observed in Fig. 7 at the time when event RW2222 occurred (sys-1 and sys-4).

Based on the above analysis, we conclude that Typhoon Trami affected all three events but to differing degrees. For event RW2106, the storm was powerful and the wave field was dominated totally by its wind sea system. As Trami made landfall, the storm diminished to a tropical depression and weakening of its influence was inevitable. Events RW2215 and RW2222 occurred near the end of the storm when another system with a wave direction different from the local wind developed and gradually became increasingly stable. According to the wave direction, this new system was probably generated by the southwest wind in the southern East China Sea. Furthermore, the spectrum of event RW2106 might have had a narrower directional distribution than events RW2215 and RW2222, which implies a greater probability of triggering modulational instability.

3.3 HOS simulation

High-order statistics of water surface elevation η such as skewness λ₃ and excess kurtosis λ₄₀ denote the importance of the nonlinearity of sea state:

where < > means the statistical average and σ is the standard deviation of surface wave elevation. Skewness is contributed totally by second-order bound waves (Tayfun, 1980; Tayfun and Fedele, 2007; Fedele and Tayfun, 2009), whereas excess kurtosis includes a dynamical part λ₄₀^d, which represents contributions from third-order nonlinear quasi-resonant wave-wave interactions (Janssen, 2003; Janssen and Bidlot, 2009) and a bound part λ₄₀^b that represents contributions from crest-trough asymmetry caused by both second- and third-order nonlinearities (Tayfun, 1980; Tayfun and Lo, 1990; Tayfun and Fedele, 2007; Fedele, 2008; Fedele and Tayfun, 2009; Janssen, 2009; Janssen and Bidlot, 2009):

Numerical simulations of HOS-ocean were performed with settings of M=2 (considering only second-order nonlinearities) and M=3 (both second- and third-order nonlinearities) using the WAVEWATCH Ⅲ simulated directional spectrum as initial conditions. The time evolutions of statistical skewness and excess kurtosis of the 50 runs for each value of M (M=2: black, M=3: red) and for each event are shown in Fig. 8 (RW2106: Fig. 8a & d, RW2215: Fig. 8b & e, and RW2222: Fig. 8c & f). Solid lines in Fig. 8 are the mean values and dashed lines are the 95% confidence bands; blue lines are theoretical values of skewness and kurtosis in a narrow band condition, which is discussed in Section 3.4.

An artificial transient period of about 100 s can be seen in each subplot of Fig. 8, which is introduced for the relaxation scheme (Dommermuth, 2000) to stabilize the transition from linear initial conditions to a fully nonlinear calculation. The value of λ₃ should be contributed totally by second-order bound waves; therefore, no discrepancies between the black and red lines are evident in Fig. 8a–c, i.e., the third-order nonlinearity had nothing to do with skewness. As mentioned at the end of Section 3.2, event RW2106 might have had a greater likelihood of being affected by the modulational instability. However, there is no obvious departure of λ₄₀ based on M=2 and 3 in Fig. 8d, and similar situations can be seen in Fig. 8e & f. Thus, third-order quasi-resonant wave-wave interactions did not play an important role in the formation of any of the three rogue wave events, and the pertaining sea states were all dominated by second-order nonlinearities. As for the relative values of λ₃ and λ₄₀, the statistical parameters of event RW2106 are slightly larger than events RW2215 and RW2222, which means more significant deviation from the Gaussian structure of linear seas and an increased occurrence of large waves.

3.4 Characteristic parameters

In this section, characteristic parameters based on the geometry of the directional spectra simulated using the WAVEWATCH Ⅲ model are presented. Further details of the sea states and nonlinear effects are discussed and the reason why the modulational instability was not the primary factor in the formation of the observed rogue waves is illustrated. The characteristic parameters are introduced as:

3.4.1 Freakish sea index

The freakish sea index maps the trajectory of the directional spreading σ_θ (Kuik et al., 1988) and frequency bandwidth Q_p (Goda, 1970), and it has been used successfully in studies of the Suwa-Maru incident (Tamura et al., 2009) and the Onomichi-Maru incident (In et al., 2009):

where S(ω, θ) is the directional spectrum with , and . The freakish sea index indicates the temporal change of the geometry of directional spectra. In most of the incidents investigated (Waseda et al., 2012), the freakish sea index changed rapidly with time. Moreover, in some cases, the incidents occurred exactly when the index indicated the narrowest directional spectrum. The freakish sea index can also be introduced into a third-generation wave model to predict "freakish" or "dangerous" sea states (Waseda et al., 2014).

3.4.2 The degree of short crestedness and irregularity parameters

The mean wavelength L_x and mean wave crest length L_y can be expressed as follows (Baxevani and Rychlik, 2006; Fedele, 2012):

where m_ijl is the moment of a spectrum S(ω, θ) and m_ijl=∫k_xⁱk_y^jω^lS(ω, θ)dωdθ where the x direction is the direction of wave propagation and the y direction is vertical to the x direction. The degree of short crestedness can be expressed as:

which in sea states with long crestedness tends toward 0 and in sea states with short crestedness tends toward 1 (Longuet-Higgins, 1957). The irregularity parameters (Baxevani and Rychlik, 2006; Fedele, 2012):

express the correlation between gradients of sea surface elevation along spatial and temporal coordinates. The irregularity parameters account for the correlation between space and time (α_xt and α_yt) or space and space (α_xy) of sea surface gradients. They can assume absolute values within the range 0–1 that correspond to "confused" and "organized" sea state conditions, respectively (Barbariol et al., 2015).

3.4.3 Theoretical skewness and kurtosis in a narrowband condition

When discussing the consequences of the canonical transformation in the Hamiltonian theory of water waves (Zakharov, 1968) using Krasitskii's canonical transformation, general expressions for the skewness and the kurtosis of the sea surface have been proposed (Janssen, 2009; 2018). The general expressions are complex but they can be simplified in narrowband conditions (Tayfun, 2006; Janssen, 2009) and fitted for moderate finite depth:

with

and

where d is the local depth, k₀ is the theoretically dominant wave number and represents the wave steepness of the sea state. In this paper, we use the mean wave number k_m instead of k₀, which is introduced from the dispersion relation , where m_n=∫ωⁿS(ω, θ)dωdθ, n=0, 1.

3.4.4 Envelope skewness, kurtosis, and associated parameters

To improve the performance of the ECMWF freak wave warning system (ECMWF, 2016), new parametrizations of skewness and kurtosis of the envelope wave height (Janssen, 2014; Janssen and Bidlot, 2009) have been presented (Janssen, 2018). Envelope skewness and kurtosis are important factors used to depict the maximum envelope wave height distribution for weakly nonlinear systems, and these two parameters and other associated variables can reveal further details of sea state (Janssen, 2009, 2018; Janssen and Bidlot, 2009) though they are not the real skewness and kurtosis of a certain sea state. The parametrizations were validated by numerical experiments on the Draupner wave and the Andrea Storm. In this paper, we use exactly the same expressions and free parameter choices as in ECMWF Technical Memorandum 813 (Janssen, 2018) except the dynamical kurtosis expression which followed Fedele's work (Fedele, 2015).

The envelope wave height theory (Janssen, 2009; Janssen and Bidlot, 2009; Janssen, 2014, 2018) also includes a skewness parameter C₃ that represents the contributions solely from second-order bound waves and a kurtosis parameter C₄, which has a dynamical part C₄^dyn and a bound part C₄^bound, which represent the contributions of third-order quasi-resonant wavewave interactions and the contributions of both second- and third-order bound waves, respectively:

The definitions of α, γ and Δ in Eqs.15–16 are similar to Eq.13 but with an extra term that expresses the effect of shallow water. The term represents wave steepness with a characteristic wave number k. The reader is referred to ECMWF Technical Memorandum 813 for further details regarding those expressions.

The dynamical part of envelope kurtosis C₄^dyn is associated with the modulational instability due to third-order quasi-resonant wave-wave nonlinear interactions (Janssen, 2003; Mori and Janssen, 2006; Janssen and Onorato, 2007; Janssen and Bidlot, 2009; Mori et al., 2011; Fedele, 2015) and influenced by the relative directional width and local water depth. With a Gaussian type directional spectrum, Fedele (2015) derived the dynamical kurtosis analytically under the narrowband assumption. The results show that, in the focusing regime (0 < R < 1, where R is an important parameter discussed later), the C₄^dyn of an initially homogeneous Gaussian wave field grows, attaining a maximum at an intrinsic time scale. Thus, the sea state initially deviates from being Gaussian, but eventually, the dynamical kurtosis tends monotonically to zero as energy spreads directionally. The maximum of dynamical kurtosis is well approximated by

where the Benjamin-Feir Index (BFI) denotes the degree of modulational instability and will be discussed later, and the function with and b=2.48 is associated with R, an important parameter that expresses the importance of directional width δ_θ with respect to frequency width δ_ω:

When R tends toward 0, the wave field will be unidirectional. This could lead to nonlinear focusing because of modulational instability, which raises the possibility of the formation of extreme waves. As R becomes larger (0 < R≤1), the sea state becomes multidirectional and the degree of focusing weakens. When R becomes larger than 1, the sea state will become defocused.

Another important parameter in the expression of C₄^dyn is the BFI. The BFI is the ratio of wave steepness to the spectral bandwidth in a unidirectional sea state, and it indicates the probability of occurrence of nonlinear focusing due to modulational instability. In deep water:

where is the steepness of waves, as in Eqs.12 and 13, and δ_ω' is another relative frequency width, which is proportional to the reciprocal of frequency bandwidth Q_p (Goda, 1970).

In the expression of C₄^dyn, the BFI² should be considered with shallow water effects:

In deep water α_S=1, and as the depth limit k₀d decreases, where k₀ is the dominant wavenumber and d is the local depth, α_S decreases slowly until k₀d=1.363 and becomes negative, which means defocusing of nonlinear energy for k₀d < 1.363 (Janssen and Onorato, 2007; Janssen and Bidlot, 2009). In this paper, we use the characteristic wave number k instead of k₀.

3.4.5 Freakish sea index

As indicated by the red stars in Fig. 9, none of the three rogue waves occurred at the "narrowest" moment. Affected by two wave systems with different wave directions, a slightly wider directional spreading is evidenced when events RW2215 and RW2222 occurred. In comparison with the other two events, it can be seen that event RW2106 has narrower directional spreading but the frequency bandwidth, indicated by Q_p is not so narrow. The wide frequency bandwidth of all three events might be related to the stage of wave development. As discussed in Section 3.1, the wave field is quasi-fully developed when event RW2106 occurred, and the observation site was near the end of the wind fetch, which also had a fully developed sea state at the time the other two events occurred.

3.4.6 The degree of short crestedness and irregularity parameters

The time evolutions of the degree of short crestedness and the irregularity parameters are shown in Fig. 10, in which the dashed lines indicate the times at which the rogue waves occurred. According to our simulated spectra at the observation site, the values of γ_s are 0.82, 0.85 and 0.81, indicating a higher degree of short crestedness of sea state at the times at which the rogue waves occurred. The irregularity parameter α_xt is > 0.75 and the values of α_yt and α_xy are between 0.1–0.3, which indicate that the sea states were more organized along the direction of propagation and more confused along the y direction. For event RW2106, most wave energy was directed along the mean wave direction (toward WNW), which caused the confused wave field in the y direction. In events RW2215 and RW2222, the mean wave directions were toward N but the coexisting wave systems were toward NW and NE. The pair of opposite vector components toward W and E led to the confusion indicated by the low values of α_yt and α_xy.

3.4.7 Theoretical skewness and kurtosis in narrowband condition

From Eqs.11–14, we find λ_{3, NB} and λ_{40, NB} influenced primarily by wave steepness , dominant wave number k₀ (k_m) and the local depth d. As the local depth was a constant in our simulation, the time evolutions of the two parameters show similar trends in Fig. 11. Consistent with the previous HOS simulation, the values of theoretical skewness and kurtosis of event RW2106 are slightly larger than the other two events, whereas the parameters of the latter two events are similar. Figure 8 shows that the narrow band predictions (blue lines) are larger than the HOS simulated results. This is simply because narrow band approximations do not consider the finite directional and frequency bandwidth (Fedele et al., 2016). Characteristic parameters like Q_p and σ_θ of the Freakish-sea Index and δ_ω and δ_θ of the parameter R can represent both frequency and directional width of the spectrum, the values of those parameters at the times when rogue events occurred are shown in Table 5, the evolution of those parameters during the whole typhoon process can be found in Sections 3.4.1 and 3.4.4. The influence of wave steepness is discussed again later.

3.4.8 Envelope skewness, kurtosis, and associated

Figure 12 shows the time evolutions of envelope skewness and kurtosis. It can be seen that the values of C₃² (black line) and C₄ (green line) for event RW2106 are slightly larger than for events RW2215 and RW2222, which is consistent with our previous conclusions. It can also be found that the envelope kurtosis (green line) is dominated by the bound part (blue line), which contributes up to approximately 83%, 77% and 80% of the total C₄ at the times at which the rogue wave events occurred. In comparison with the bound part, the dynamical envelope kurtosis (red line) is much smaller and the dynamical part in all three events is similar, denoting that even the narrower directional distribution in event RW2106 did not trigger active nonlinear focusing due to modulational instability. We show later that C₃^bound and C₄^bound are dominated primarily by wave steepness and that damping of wave steepness is caused by the degree of development of the waves. Many other factors contribute to C₄^dyn; however, despite the three peaks of C₄^dyn shown in Fig. 12, because no rogue wave was observed at the buoy at those moments, will do not discuss this further in this paper.

In Janssen (2018), R is shown affected by water depth. The time evolution of R with and without the effect of shallow water, and the relative directional and frequency width are all shown in Fig. 13. It can be seen that R decreased after the first event occurred (Fig. 13a), which led to a more unidirectional wave state (Fig. 13b). That decrease was caused by the widening of δ_ω because of the increasingly developed sea state. However, parameter R is a ratio determined by both δ_ω and δ_θ. Therefore, as another wave system was added, as concluded in Section 3.2, the relative directional width δ_θ became wider just before event RW2215 occurred. Thus, we observe an increase of R during the final two events, which means the sea state, became less unidirectional again.

As shown in Fig. 14a, a damping trend of BFI_d can be seen from the time before event RW2106 occurred, except for a temporary "peak" due to a temporarily narrower δ_ω', as shown in the top subplot in Fig. 14b. Several hours before event RW2215, the BFI_d started to increase again and it kept increasing gradually throughout the period during which events RW2215 and RW2222 occurred. Similar trends were noted in the evolutions of λ_{3, NB} and λ_{40, NB} in Fig. 11 and C₃^bound and C₄^bound in Fig. 12. Apparently, those parameters are all associated with wave steepness or. As shown in Fig. 15, both types of wave steepness exhibit the same trend as the parameters in Figs. 11 and 12. As discussed in Sections 3.1, the sea state was quasi-fully developed when event RW2106 occurred. Then, as the center of Typhoon Trami moved westward and made landfall, the local wind speed showed an inevitable decrease. However, the local wind field pattern also changed. The observation site was located at the end of the local wind fetch when events RW2215 and RW2222 occurred. With a fully developed sea state and decreasing wind speed, damping of wave steepness occurred. This was the main reason for the decrease of skewness and kurtosis, and it was also the main cause of the damping of BFI_d, which means the modulational instability was suppressed. The final increase of steepness was attributable totally to an increase of local wind speed at the observation site, as shown in Fig. 3b.

Greater damping of can be observed in Fig. 14a compared with BFI_d, and the same damping of α_S and depth limit kd can be seen in Fig. 14b. Apparently, α_S is determined by the depth limit, and with constant water depth, the damping is dominated totally by the wavenumber. Because of the nonlinear interactions among four resonant waves, the wave energy transfers from the peak frequency of the spectrum to lower and higher frequencies, which makes the peak move to lower frequencies (Hasselmann, 1962, 1963a, b ); thus, a sea state that is more developed obtains larger wave period and a smaller wavenumber. Considering the development of the sea state discussed previously, the damping of BFI_S is reasonable. In conclusion, further suppression of the modulational instability was manifest by the shallow water effects due to the fully developed waves.

4 CONCLUSION

During August 2013, three rogue waves were recorded with an observation buoy deployed in the East China Sea at 29.068 7°N, 124.921 5°E. From the perspectives of wind and wave fields, wave system tracking, HOS method simulation, and characteristic parameters of sea state, we undertook a detailed analysis on three events using simulated results from the WAVEWATCH Ⅲ model. The principal conclusions are as follows.

1) All three rogue waves were influenced by Typhoon Trami (TY201312 CMA) but in different degrees. The sea state was quasi-fully developed and highly influenced by Trami when event RW2106 occurred. Then, as Trami made landfall and the storm became a tropical depression, weakening of Trami's influence was inevitable. After the storm made landfall and during the period when events RW2215 and RW2222 occurred, the local wind pattern changed at the observation site, which was located at the end of the local wind fetch.

2) The wave field was dominated completely by Typhoon Trami when event RW2106 occurred; thus, there was only one system in the wave field. Events RW2215 and RW2222 occurred as the storm diminished, which is when another system with a wave direction different to the local wind was added and became increasingly stable. According to the wave direction, the new system was probably generated by the southwest wind over the southern part of the East China Sea. Consequently, the directional distribution of event RW2106 was narrower than for the other two events.

3) All three events occurred in sea states dominated by second-order nonlinearities, and third-order quasi-resonant interactions were considered negligible in the formation of the rogue waves. Furthermore, the degree of nonlinearity was influenced primarily by wave steepness, which was closely associated with sea state development.

4) Modulational instability was considered negligible in the extreme events for three reasons. First, even with a narrower directional distribution, the fully or quasi-fully developed wind-generated waves broadened the frequency bandwidth in three events, i.e., the directional spectra had no chance to be "narrowest." Second, the damping of wave steepness reduced the BFI, which represents suppression of modulational instability; with a fully developed sea state and decreasing wind speed, such damping of wave steepness was unavoidable. Finally, shallow water effects dominated by the smaller wavenumber of a more developed sea state further stabilized the modulational instability. To conclude, the fully developed sea state suppressed the third-order quasi-resonant interactions and associated modulational instabilities.

The nonlinearities of sea state, which play an important role in the formation of rogue waves, are closely related to the degree of development of the waves. With the presence of a moving typhoon center, the development of the sea state becomes complex. The method adopted in this study can provide comprehensive and full-scale analysis of rogue waves in the real world. The case studied in this paper is not considered unique, and rules could be found and confirmed in relation to other typhoon sea states through the application of our proposed method.

5 DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available on request from the corresponding author: JIANG Xingjie (jiangxj@fio.org.cn).