Abstract
We present a study on the prediction of rogue waves during the 1hour sea state of Hurricane Joaquin when the Merchant Vessel El Faro sank east of the Bahamas on October 1, 2015. Highresolution hindcast of hurricanegenerated sea states and wave simulations are combined with novel probabilistic models to quantify the likelihood of rogue wave conditions. The data suggests that the El Faro vessel was drifting at an average speed of approximately 2.5 m/s prior to its sinking. As a result, we estimated that the probability that El Faro encounters a rogue wave whose crest height exceeds 14 meters while drifting over a time interval of 10 (50) minutes is ~1/400 (1/130). The largest simulated wave is generated by the constructive interference of elementary spectral components (linear dispersive focusing) enhanced by bound nonlinearities. Not surprisingly then, its characteristics are quite similar to those displayed by the Andrea, Draupner and Killard rogue waves.
Introduction
The tragic sinking of the SS El Faro vessel occurred while it was traveling from Florida to Puerto Rico^{1}. The vessel with a crew of 33 sank about 1140 Hrs UTC on Oct. 1, 2015. As part of their official investigation into the sinking of the El Faro, the National Transportation Safety Board (NTSB) has requested us to carry out an analysis on the occurrence of rogue waves during Hurricane Joaquin around the time and location of the El Faro’s sinking^{2}. Here, we provide a plain presentation of the main results of our analysis avoiding interpretations, considerations or claims that can be drawn from our studies.
The data suggests that the El Faro vessel was drifting at an average speed of approximately 2.5 m/s prior to its sinking^{2}. As a result, El Faro has a higher probability to encounter a rogue wave while drifting over a period of time than that associated with an observer located at a fixed point on the ocean surface. Indeed, the encounter of a rogue wave by a moving vessel is analogous to that of a big wave that a surfer is in search of the surfer’s likelihood to encounter a big wave increases if he moves over a large area instead of staying still. Indeed, if he spans a large area the chances to encounter a large wave increase^{3, 4}. This is a spacetime effect very important for ship navigation and it cannot be overlooked. Such an effect is considered in our rogue wave analysis by way of a new probabilistic model for the exceedance probability, or occurrence frequency of a rogue wave encountered by a vessel along its navigation path^{3, 5}. The proposed spacetime model provides the basis for the next generation of wave forecast models for a predictive capability of wave extremes and early warnings for shipping companies and others to avoid dangerous areas at risk of rogue waves.
Results
Our rogue wave analysis is focused on the 1hour sea state of Hurricane Joaquin during which the El Faro vessel sank. This will hereafter be referred to as the El Faro sea state. The wave parameters and statistical models relevant to and required for our analysis are presented in the Methods section.
Metocean parameters of Hurricane Joaquin in the region of the sinking of El Faro
We use the hindcast directional spectra predicted by WAVEWATCH III and describe the wave characteristics of the sea states generated by Hurricane Joaquin at and around the time and location where the El Faro vessel sank^{6}. The top panel on the left of Fig. 1 shows the hourly variation of the significant wave height H _{ s } during the event. The topright panel displays the time history of the dominant wave period T _{ p }, and the dominant wave direction, the neutral stability 10m wind speed U _{10} and direction are shown in the bottompanels, respectively. The red vertical lines delimit the 1–hour interval during which the El Faro vessel sank.
The 1hour sea state experienced by El Faro at and around the time and location of sinking had a significant wave height of H _{ s } ≈ 9 m and the maximum wind speed was U _{10,max } = 51 m/s. Waves were multidirectional (shortcrested) as indicated by the large values of both the spectral bandwidth ν and angular spreading σ _{ θ } as shown in Fig. 2.
In Table 1 we report the metocean parameters of the El Faro sea state in comparison to those of the Draupner, Andrea and Killard rogue sea states^{7}. Note that the four sea states have similar metocean characteristics. However, El Faro is a steeper sea state as the mean wavelengh L _{0} is shorter than those observed in the other three cases.
Statistical properties of Hurricane Joaquingenerated seas
The relative importance of ocean nonlinearities can be measured by integral statistics such as the coefficients of skewness λ _{3} and excess kurtosis λ _{40} of the zeromean surface elevation η(t). The skewness is a measure of asymmetry, and it describes the effects of secondorder bound nonlinearities on the geometry and statistics of the sea surface with higher sharper crests and shallower more rounded troughs^{8,9,10}. The excess kurtosis indicates whether the tails of the distribution of surface elevations is heavy or lighttailed relative to a Gaussian distribution. It comprises a dynamic component \({\lambda }_{40}^{d}\) measuring thirdorder quasiresonant wavewave interactions and a bound contribution \({\lambda }_{40}^{b}\) induced by both second and thirdorder bound nonlinearities^{8,9,10,11,12,13}.
In deep waters, the dynamic kurtosis^{14} depends on the BenjaminFeir index BFI and the parameter R, a dimensionless measure of the multidirectionality of dominant waves^{11, 14, 15}. For unidirectional (1D) waves R = 0. The bottom panel of Fig. 2 displays the hourly variations of the directional factor R during Hurricane Joaquin near the location where El Faro sank. Around the peak of the hurricane, the generated sea states are quite multidirectional (shortcrested) as R > 1. As wave energy also spreads directionally, nonlinear focusing due to modulational instability effects diminishes^{14, 16,17,18} and becomes essentially insignificant under such realistic oceanic conditions^{7, 14, 19, 20}.
The top panel of Fig. 3 displays the hourly variation of the Tayfun steepness μ (solid line) with associated bounds (dashed lines). The coefficient of excess kurtosis λ _{40} mostly due to bound nonlinearities is shown in the center panel and the associated Λ parameter at the bottom. The red vertical lines delimit the 1hour interval during which the El Faro vessel sank.
In Table 1 we compare the statistical parameters of the El Faro sea state and the Draupner, Andrea and Killard rogue sea states (from ref. 7). Note that the El Faro sea state has the largest directional spreading. Moreover, for all the four sea states the associated BFI are less than unity and the maximum dynamic excess kurtosis is of O(10^{−3}) and thus negligible in comparison to the associated bound component. Thus, thirdorder quasiresonant interactions, including NLStype modulational instabilities play an insignificant role in the formation of large waves^{7, 14} especially as the wave spectrum broadens^{21} in agreement with oceanic observations available so far^{9, 22, 23}. On the contrary, NLS instabilities have been proven to be effective in the generation of optical rogue waves^{24}.
Higher Order Spectral (HOS) simulations of the El Faro sea state
We have performed HigherOrder pseudoSpectral (HOS) simulations^{25, 26} of the El Faro sea state over an area of 4 km × 4 km for a duration of 1 hour (see Methods section for a description of the numerical method). The initial wave field conditions are defined by the WAVEWATCH III hindcast directional spectrum S(f, θ) around the time and region of the El Faro sinking as shown in Fig. 4. This is the result of a balance of the energy fluxes due to wind input (S _{ in }), exact fourwave resonance nonlinearities (S _{ nl }) and dissipation due to wave breaking (S _{ ds }). Wind gustiness and currents are not modeled. Our WW3 hindcast indicates that the flux S _{ in } is balanced out by S _{ ds }. In particular, around the spectral peak 60% of wind input is lost to dissipation. This offset increases away from the peak. Any wave growth associated with S _{ in } + S _{ ds } and S _{ nl } is accounted for in the WW3 model. It is the wave growth associated with quasiresonant and bound harmonics nonlinear effects that is not modeled. In our study, we exploit the HOS wave solver to simulate the ElFaro sea state by accounting for quasiresonant and bound nonlinearities up to fourth order in wave steepness. An estimate of the most likely rogue wave amplitude is then provided as discussed below. Note that both wind input and wave breaking are somewhat modeled in our HOS simulations as these are initialized with the WW3 spectrum. Clearly, our analysis suggests future studies on the relative importance of possible effects such wind gustiness^{27} and wave breaking^{28, 29} on the HOS model results.
The wavenumberfrequency spectrum S(k, ω) estimated from the HOS simulations is shown in Fig. 5. Here, dashed lines indicate the theoretical dispersion curves related to the firstorder (1^{st}) free waves as well as the second (2^{nd}) and thirdorder (3^{rd}) bound harmonic waves. The HOS predictions indicate that secondorder nonlinearities are dominant with a weak effect of thirdorder nonlinear bound interactions, in agreement with recent studies of rogue sea states^{7}. It appears that fourthorder effects are insignificant.
The wave skewness and kurtosis rapidly reach a steady state after a few (mean) wave periods as an indication that thirdorder quasiresonant wavewave interactions are negligible in agreement with theoretical predictions^{14} and simulations^{7}. Note that the theoretical narrowband (NB) predictions slightly overestimate the simulated values for skewness and excess kurtosis (see Table 1). The same trend is also observed in recent studies on rogue waves^{7}. This is simply because NB approximations do not account for the directionality and finite spectral bandwidth of the El Faro wave spectrum.
Occurrence frequency of a rogue wave by a fixed observer: the return period of a wave whose crest height exceeds a given threshold
To describe the statistics of rogue waves encountered by an observer at a fixed point of the ocean surface, we consider the conditional return period N _{ h }(ξ) of a wave whose crest height exceeds the threshold h = ξH _{ s }, namely
where P(ξ) is the probability or occurrence frequency of a wave crest height exceeding ξH _{ s } as encountered by a fixed observer. In other words, P(ξ) is the probability to randomly pick from a time series observed at a fixed point of the ocean a wave crest that exceeds the threshold ξH _{ s }. Equation (1) also implies that the threshold ξH _{ s }, with H _{ s } = 4σ, is exceeded on average once every N _{ h }(ξ) waves. For weakly nonlinear random seas, the probability P is hereafter described by the thirdorder TayfunFedele^{9} (TF), secondorder Tayfun^{8} (T), secondorder Forristall^{30} (F) and the linear Rayleigh (R) distributions (see Methods section).
Our statistical analysis of HOS wave data suggests that secondorder effects are the dominant factors in shaping the probability structure of the El Faro sea state with a minor contribution of excess kurtosis effects. Such dominance is seen in Fig. 6, where the HOS numerical predictions of the conditional return period N _{ h }(ξ) of a crest exceeding the threshold ξH _{ s } are compared against the theoretical predictions based on the linear Rayleigh (R), secondorder Tayfun (T) and thirdorder (TF) models from Eq. (17). It is noted that the HOS predictions are based on a sample population of 10^{6} crests. In particular, N _{ h }(ξ) follows from Eq. (1) as the inverse 1/P(ξ) of the empirical probabilities of a crest height exceeding the threshold ξH _{ s }. An excellent agreement is observed between simulations and the thirdorder TF model up to crest amplitudes h/H _{ s } ~ 1.5. For larger amplitudes, the associated confidence bands of the estimated empirical probabilities widen, but TF is still within the bands. Donelan and Magnusson^{31} suggest that the TF model agrees with the Andrea rogue wave measurements up to h/H _{ s } ~ 1.1, concluding that TF is not suitable to predict larger rogue crest extremes (see their Fig. 7 in ref. 31). Unfortunately, their analysis is based on a much smaller sampled population of ~10^{4} crest heights and they do not report the confidence bands associated with their probability estimates, nor they provide any parameter values to validate their data analysis. The deviation of their data from the TF model is most likely due to the relatively smaller population of crests observed. Note also that TF slightly exceeds both the T and F models as an indication that secondorder effects are dominant, whereas the linear R model underestimates the return periods.
For both third and fourthorder nonlinearities, the return period N _{ r } of a wave whose crest height exceeds the rogue threshold 1.25H _{ s } ≈ 11 m^{32} is nearly N _{ r } ~ 10^{4} for the El Faro sea state and for the simulated Andrea, Draupner and Killard rogue sea states^{7}. This is in agreement with oceanic rogue wave measurements^{23}, which yield roughly the same return period. Similarly, recent measurements off the west coast of Ireland^{33} yield N _{ r } ~ 6 · 10^{4}. In contrast, N _{ r } ~ 3 · 10^{5} in a Gaussian sea.
Note that the largest simulated wave crest height exceeds the threshold 1.6H _{ s } ≈ 14 m (see Table 1). This is exceeded on average once every 10^{6} waves in a time series extracted at a point in third and fourthorder seas and extremely rarely in Gaussian seas, i.e. on average once every 10^{9} waves. This implies that rogue waves observed at a fixed point of the ocean are likely to be rare occurrences of weakly random seas, or Tayfun sea states^{34}. Our results clearly confirm that rogue wave generation is the result of the constructive interference (focusing) of elementary waves enhanced by bound nonlinearities in agreement with the theory of stochastic wave groups developed by Fedele and Tayfun (2009)^{10} as an extension of Boccotti’s (2000) theory of quasideterminism^{35}. Our conclusions are also in agreement with observations^{9, 10, 12, 22}, recent rogue wave analyses^{7, 31, 36,37,38,39,40,41} and studies on optical rogue waves caustics analogues^{42}.
Time profile of the simulated rogue waves
The wave profile η with the largest wave crest height (>1.6H _{ s } ≈ 14 m) observed in the time series of the surface fluctuations extracted at points randomly sparse over the simulated El Faro domain is shown in the left panel of Fig. 7. For comparison, the Draupner, Andrea and Killard rogue wave profiles are also shown^{7}. In the same figure, the mean sea level (MSL) below the crests is also shown. The estimation of the MSL follows by lowpass filtering the measured time series of the wave surface with frequency cutoff f _{ c } ~ f _{ p }/2, where f _{ p } is the frequency of the spectral peak^{43}. An analysis of the kinematics^{44, 45} of the simulated rogue waves indicate that such waves were nearly incipient breaking^{28, 29, 44} suggesting that larger rogue events are less likely to occur^{21, 44}. The saturation of the crest height is mainly due to the nonlinear dispersion and it is an energy limiter for rogue waves.
The four wave profiles are very similar suggesting a common generation mechanism of the rogue events. The manner waves are generated by Hurricane Joaquin or the northerly storm of the Draupner, Andrea and Killard sea states, all four waves and their statistics cannot differ in a fundamental way from each other as the spectral shape of the four sea states is similar showing only some variations in terms of directionality or frequency characteristics.
Further, we observe a setup below the simulated El Faro rogue wave, most likely due to the multidirectionality of the sea state. A setup is also observed for the actual Draupner rogue wave. Indeed, recent studies showed that Draupner occurred in a crossing sea consisting of swell waves propagating at approximately 80 degrees to the wind sea^{46, 47}. This would explain the setup observed under the large wave^{43} instead of the secondorder setdown normally expected^{48}.
Spacetime statistics of the sea state encountered by El Faro before sinking
The largest crest height of a wave observed in time at a given point of the ocean represents a maximum observed at that point. Clearly, the maximum wave surface height observed over a given area during a time interval, i.e. spacetime extreme, is much larger than that observed at a given point. Indeed, in relatively shortcrested directional seas such as those generated by hurricanes, it is very unlikely that an observed large crest at a given point in time actually coincides with the largest crest of a group of waves propagating in spacetime. In contrast, in accord with Boccotti’s (2000) QD theory^{35}, it is most likely that the sea surface was in fact much higher somewhere near the measurement point.
Spacetime wave extremes can be modeled stochastically^{3, 4} drawing on the theory of Euler Characteristics of random fields^{49,50,51} and nonlinear wave statistics^{14}. In the following, we present the Fedele’s SpaceTime (FST) stochastic model for the prediction of spacetime extremes^{3} that accounts for both second and thirdorder nonlinearities^{5}. Fedele’s work^{3, 5} considers a 3D nonGaussian field η(x, y, t) in spacetime over an area A for a time period of D (see Fig. 8). The area cannot be too large since the wave field may not be homogeneous. The duration should be short so that spectral changes occurring in time are not significant and the sea state can be assumed as stationary. Then, the thirdorder nonlinear probability \({P}_{{\rm{FST}}}^{(nl)}(\xi ;A,D)\) that the maximum surface elevation \({\eta }_{\max }\) over the area A and during the time interval D exceeds the generic threshold ξH _{ s } is described by^{5}
where
denotes the Gaussian probability of exceedance, and P _{R}(ξ) is the Rayleigh exceedance probability of Eq. (19).
Here, M _{1} and M _{2} are the average number of 1D and 2D waves that can occur on the edges and boundaries of the volume Ω, and M _{3} is the average number of 3D waves that can occur within the volume^{3}. These all depend on the directional wave spectrum and its spectral moments m _{ ijk } defined in the Methods section.
The amplitude ξ relates to ξ _{0} via the Tayfun (1980) quadratic equation^{8}
Given the probability structure of the wave surface defined by Eq. (2), the nonlinear mean maximum surface or crest height \({\overline{h}}_{{\rm{FST}}}={\xi }_{{\rm{FST}}}{H}_{s}\) attained over the area A during a time interval D is given, according to Gumbel (1958), by^{4, 5}
where the most probable surface elevation value ξ _{m} satisfies P _{ST}(ξ _{m}; A, D) = 1 (see Eq. (2)) and the EulerMascheroni constant γ _{ e } ≈ 0.577.
The nonlinear mean maximum surface or crest height h _{T} expected at a point during the time interval D follows from Eq. (5) by setting M _{2} = M _{3} = 0 and M _{1} = N _{D}, where \({N}_{{\rm{D}}}=D/\bar{T}\) denotes the number of waves occurring during D, and \(\bar{T}\) is the mean upcrossing period (see Methods section). The secondorder form of the FST model (Λ = 0) has been implemented in WAVEWATCH III^{52}. The linear limit follows from Eq. (5) by setting μ = 0 and Λ = 0.
The statistical interpretations of the probability \({P}_{{\rm{FST}}}^{(nl)}(\xi ;A,D)\) and associated spacetime average maximum \({\overline{h}}_{{\rm{ST}}}\) are as follows. Consider an ensemble of N realizations of a stationary and homogeneous sea state of duration D, each of which has similar statistical structure to the El Faro wave field. On this basis, there would be N samples, say \(({\eta }_{\max }^{\mathrm{(1)}},\ldots ,{\eta }_{\max }^{(N)})\) of the maximum surface height \({\eta }_{\max }\) observed within the area A during the time interval D. Then, all the maximum surface heights in the ensemble will exceed the threshold \({\overline{h}}_{{\rm{FST}}}\). Clearly, the maximum surface height exceeds by far such average. Indeed, only in a few number of realizations \(N\cdot {P}_{{\rm{FST}}}^{(nl)}(\xi ;A,D)\) out of the ensemble of N sea states, the maximum surface height exceeds a threshold \(\xi {H}_{s}\gg {\overline{h}}_{{\rm{FST}}}\) much larger than the expected value. To characterize such rare occurrences in thirdorder nonlinear random seas one can consider the threshold h _{ q } = ξ _{ q } H _{ s } exceeded with probability q by the maximum surface height \({\eta }_{{\rm{\max }}}\) over an area A during a sea state of duration D. This satisfies
The statistical interpretation of h _{ q } is as follows: the maximum surface height \({\eta }_{\max }\) observed within the area A during D exceeds the threshold h _{ q } only in qN realizations of the above mentioned ensemble of N sea states.
Note that for large areas, i.e. \(\ell \gg {L}_{0}\), our FST model as any other similar models available in literature^{47, 53,54,55,56} will overestimate the maximum surface height over an area and time interval because they all rely on Gaussianity. This implies that there are no physical limits on the values that the surface height can attain as the Gaussian model does not account for the saturation induced by the nonlinear dispersion^{21} of ocean waves or wave breaking. Thus, the larger the area A or the time interval D, the greater the number of waves sampled in spacetime, and unrealistically large amplitudes are likely to be sampled in a Gaussian or weakly nonlinear Gaussian sea.
This point is elaborated further and demonstrated explicitly by way of the results displayed in Fig. 9. Here, the theoretical (FST) ratio \({\overline{h}}_{{\rm{FST}}}/{\overline{h}}_{{\rm{T}}}\) as a function of the area width \(\ell /{L}_{0}\) is shown for the El Faro, Draupner and Andrea sea states respectively. The FST ratios for Draupner and Andrea are estimated using the European Reanalysis (ERA)interim data^{5}. For comparisons, the empirical ST ratio from the El Faro HOS simulations together with the experimental observations at the Acqua Alta tower^{4} are also shown. Recall that \({\overline{h}}_{{\rm{FST}}}\) is the mean maximum surface height expected over the area \({\ell }^{2}\) during a sea state of duration D = 1 hour and \({\overline{h}}_{{\rm{T}}}\) is the mean maximum surface height expected at a point. Clearly, the theoretical FST ratio for El Faro fairly agrees with the HOS simulations for small areas (\(\ell \le {L}_{0}\)), whereas it yields overestimation over larger areas. We argue that the saturation of the HOS FST ratio over larger areas is an effect of the nonlinear dispersion which is effective in limiting the wave growth as a precursor to breaking^{21, 44}.
Note that the FST ratios for all the three sea states are nearly the same for \(\ell \le {L}_{0}\). These results are very encouraging as they suggest possible statistical similarities and universal laws for spacetime extremes in wind sea states^{5}. Moreover, for \(\ell \sim {L}_{0}\) the mean wave surface maximum expected over the area is 1.35 times larger than that expected at a point in agreement with Acqua Alta sea observations^{4}.
The occurrence frequency of a rogue wave by the El Faro vessel
The data suggests that the El Faro vessel was drifting at an average speed of approximately 2.5 m/s prior to its sinking. This is considered in our analysis as follows. First, define the two events R = “El Faro encounters a rogue wave along its navigation route” and S = “El Faro sinks”. We know that the event S happened. As a result, one should consider the conditional probability
Here, Pr[S] is the unconditional probability of the event that El Faro sinks. This could be estimated from worldwide statistics of sunk vessels with characteristics similar to El Faro. Pr[SR] is the conditional probability that El Faro sinks given that the vessel encountered a rogue wave. This probability can be estimated by Monte Carlo simulations of the nonlinear interaction of the vessel with the rogue wave field.
Our rogue wave analysis provides an estimate of the unconditional probability Pr[R] that El Faro encounters a rogue wave along its navigation or drifting route by means of the exceedance probability, or occurrence frequency P _{ e }(h). This is the probability that a vessel along its navigation path encounters a rogue wave whose crest height exceeds a given threshold h. The encounter of a rogue wave by a moving vessel is analogous to that of a big wave that a surfer is in search of. His likelihood to encounter a big wave increases if he moves around a large area instead of staying still. This is a spacetime effect which is very important for ship navigation and must be accounted for^{3, 57,58,59,60}.
The exceedance probability P _{ e }(h) is formulated as follows. Consider a random wave field whose surface elevation at a given point (x, y) in a fixed frame at time t is η(x, y, t). Consider a vessel of area A that navigates through the wave field at a constant speed V along a straight path at an angle β with respect to the x axis. Define also (x _{ e }, y _{ e }) as a cartesian frame moving with the ship. Then, the line trajectories of any point (x _{ e }, y _{ e }) of the vessel in the fixed frame are given by
where for simplicity we assume that at time t = 0 the center of gravity of the vessel is at the origin of the fixed frame.
The surface height η _{ c }(t) encountered by the moving vessel, or equivalently the surface fluctuations measured by a wave probe installed on the ship, is
If η is a Gaussian wave field homogeneous in space and stationary in time, then so is η _{ c } with respect to the moving frame (x _{ e }, y _{ e }, t). The associated spacetime covariance is given by
where \({k}_{x}=k\,\cos (\theta )\), \({k}_{y}=k\,\sin (\theta )\) and k is the wavenumber associated with the frequency f by way of the wave dispersion relation. As a result of the Doppler effect, the encountered, or apparent frequency is given by^{57,58,59,60}
and S(f, θ) is the directional wave spectrum of the sea state. Note that when the vessel moves faster than waves coming from a direction θ, the apparent frequency f _{ e } < 0 and for an observer on the ship waves appear to move away from him/her. In this case, the direction of those waves should be reversed^{57}, i.e. θ = θ + π, and f _{ e } set as positive.
The spectral moments \({m}_{ijk}^{(e)}\) of the encountered random field readily follow from the coefficients of the Taylor series expansion of Ψ(X, Y, T) around (X = 0, Y = 0, T = 0). In particular,
The nonlinear spacetime statistics can then easily processed by using the encountered spectral moments \({m}_{ijk}^{(e)}\) using the FST model^{3, 5}, which is based on Eq. (2) as described above. Note that for generic navigation routes the encountered wave field η _{ c } is a nonstationary random process of time. Thus, the associated spectral moments will vary in time. The spacetime statistics can be still computed by first approximating the route by a polygonal made of piecewise straight segments along which the random process η _{ c } is assumed stationary.
Figure 10 illustrates the HOS and theoretical predictions for the normalized nonlinear threshold h _{ n }/H _{ s } exceeded with probability 1/n, where n is the number of waves. In particular, consider an observer on the vessel moving along the straight path Γ spanned by El Faro drifting against the dominant sea direction over a time interval of 10 minutes. In spacetime the observer spans the solid red line shown in Fig. 8. In this case, he has a probability P _{ e } ~ 3 · 10^{−4} to encounter a wave whose crest height exceeds the threshold 1.6H _{ s } ≈ 14 m (blue lines), and the expected spatial shape is shown in Fig. 11. If we also account for the vessel size (base area A = 241 × 30 m ^{2}), in spacetime El Faro spans the volume of the slanted parallelepiped V _{ a } shown in Fig. 8. In this case, the exceedance probability P _{ e }(V _{ a }) further increases to 1/400 (black lines in Fig. 10). Note that if the vessel would be anchored at a location for the same duration, in spacetime it would span instead the volume of the vertical parallelepiped V _{ c } shown in the same Figure. Note that the two parallelepipeds cover the same spacetime volume A × D, with the base area A and height D = 10 min. For the case of the anchored vessel, the associated exceedance probability P _{ e }(V _{ c }) is roughly the same as P _{ e }(V _{ a }) since El Faro was drifting at a slow speed. Larger drift speeds yield larger P _{ e }(V _{ a }) since the vessel encounters waves more frequently than if it was anchored, because of the Doppler effect^{58, 59}. Moreover, the drifting vessel covers the strip area (1500 × 30 m ^{2}) in the 10minute interval and the associated spacetime volume is that of the parallelepiped V _{ b } shown in Fig. 8, which has a larger volume than that of V _{ a }. As a result, the occurrence frequency P _{ e }(V _{ b }) of a rogue wave associated with V _{ b } is larger and it increases to ~1/100 (see red lines in Fig. 10). However, El Faro does not visit the entire volume V _{ b }, but it only spans the smaller volume V _{ a }. Thus, the conditional probability P _{ e }(V _{ a }V _{ b }) that the drifting El Faro encounters a rogue wave given that a rogue wave occurred over the larger spacetime volume V _{ b } is P _{ e }(V _{ a })/P _{ e }(V _{ b }) ~ 1/4. Furthermore, a fixed observer has a much lower probability P _{ e } ~ 10^{−6} to pick randomly from a time series extracted at a point a wave whose crest height exceeds 1.6H _{ s } (see Fig. 6, TF model, black solid line). Finally, we observed that the exceedance probability P _{ e }(V _{ a }) for the drifting El Faro does not scale linearly with time because of nonlinearities that reduce the natural dispersion of waves. Indeed, assuming that El Faro drifts over a time interval 5 times longer (50 minutes), P _{ e }(V _{ a }) just increases roughly by 3 times, ~1/130.
Discussions
Our present studies open a new research direction on the prediction of rogue waves during hurricanes using the WW3 wave model combined with HOS simulations and the new stochastic FST model^{5} for the prediction of spacetime wave extremes^{3, 4}. Any wave growth associated with wind stresses, dissipation due to wave breaking and exact nonlinear resonant fourwave interactions is accounted for in the WW3 model. It is the wave growth associated with quasiresonant and bound harmonics nonlinearities that is not modeled. Such nonlinear effects can locally increase the wave amplitude over the expected values of the WW3 simulations. In our analysis, quasiresonant and bound nonlinearities are modeled by way of a HOS wave solver that simulated the sea state around the time and location of the accident. The HOS simulations provided an estimate of the most likely rogue wave amplitude over a given area and time interval indicating that bound nonlinearities are dominant, in agreement with recent roguewave studies^{7}. Our analysis also suggests new studies on the possible effects of factors such as wind gustiness^{27} and wave breaking^{28, 29} on generating rogue waves and associated statistics.
Methods
Wave parameters
The significant wave height H _{ s } is defined as the mean value H _{1/3} of the highest onethird of wave heights. It can be estimated either from a zerocrossing analysis or more easily but approximately from the wave omnidirectional spectrum \({S}_{o}(f)={\int }_{0}^{2\pi }S(f,\theta ){\rm{d}}\theta \) as H _{ s } ≈ 4σ, where \(\sigma =\sqrt{{m}_{0}}\) is the standard deviation of surface elevations, m _{ j } = ∫S _{ o }(f)f ^{j}df are spectral moments. Further, S(f, θ) is the directional wave spectrum with θ as the direction of waves at frequency f, and the cyclic frequency is ω = 2πf.
The dominant wave period T _{ p } = 2π/ω _{ p } refers to the cyclic frequency ω _{ p } of the spectral peak. The mean zerocrossing wave period T _{0} is equal to 2π/ω _{0}, with \({\omega }_{0}=\sqrt{{m}_{2}/{m}_{0}}\). The associated wavelength L _{0} = 2π/k _{0} follows from the linear dispersion relation \({\omega }_{0}=\sqrt{g{k}_{0}\,\tanh ({k}_{0}d)}\), with d the water depth. The mean spectral frequency is defined as ω _{ m } = m _{1}/m _{0} ^{8} and the associated mean period T _{ m } is equal to 2π/ω _{ m }. A characteristic wave steepness is defined as μ _{ m } = k _{ m } σ, where k _{ m } is the wavenumber corresponding to the mean spectral frequency ω _{ m } ^{8}. The following quantitites are also introduced: \({q}_{m}={k}_{m}d,{Q}_{m}=\,\tanh \,{q}_{m}\), the phase velocity c _{ m } = ω _{ m }/k _{ m }, the group velocity c _{ g } = c _{ m }[1 + 2q _{ m }/sinh(2q _{ m })]/2.
The spectral bandwidth \(\nu ={({m}_{0}{m}_{2}/{m}_{1}^{2}1)}^{1/2}\) gives a measure of the frequency spreading. The angular spreading \({\sigma }_{\theta }=\sqrt{{\int }_{0}^{2\pi }D(\theta ){(\theta {\theta }_{m})}^{2}{\rm{d}}\theta }\), where \(D(\theta )={\int }_{0}^{\infty }S(\omega ,\theta ){\rm{d}}\omega /{\sigma }^{2}\) and \({\theta }_{m}={\int }_{0}^{2\pi }D(\theta )\theta {\rm{d}}\theta \) is the mean direction. Note that \({\omega }_{0}={\omega }_{m}\sqrt{1+{\nu }^{2}}\).
The wave skewness λ _{3} and the excess kurtosis λ _{40} of the zeromean surface elevation η(t) are given by
Here, overbars imply statistical averages and σ is the standard deviation of surface wave elevations.
For secondorder waves in deep water^{10}
and the following bounds hold^{61}
Here, ν is the spectral bandwidth defined above and the characteristic wave steepness μ _{ m } = k _{ m } σ, where k _{ m } is the wavenumber corresponding to the mean spectral frequency ω _{ m } ^{8}. For narrowband (NB) waves, ν tends to zero and the associated skewness λ _{3,NB } = 3μ _{ m } ^{8,9,10}.
For thirdorder nonlinear random seas the excess kurtosis
comprises a dynamic component \({\lambda }_{40}^{d}\) due to nonlinear quasiresonant wavewave interactions^{11, 62} and a Stokes bound harmonic contribution \({\lambda }_{40}^{b}\) ^{63}. In deep water it reduces to the simple form \({\lambda }_{40,NB}^{b}=18{\mu }_{m}^{2}=2{\lambda }_{3,NB}^{2}\) ^{11, 63, 64} where λ _{3,NB } is the skewness of narrowband waves^{8}.
As for the dynamic component, Fedele^{14} recently revisited Janssen’s^{62} weakly nonlinear formulation for \({\lambda }_{40}^{d}\). In deep water, this is given in terms of a sixfold integral that depends on the BenjaminFeir index \(BFI=\sqrt{2}{m}_{m}/v\) and the parameter \(R={\sigma }_{\theta }^{2}/2{\nu }^{2}\), which is a dimensionless measure of the multidirectionality of dominant waves^{11, 15}. As waves become unidirectional (1D) waves R tends to zero and a random narrowband wave train becomes unstable if BFI > 1^{65}.
The TayfunFedele model
We define P(ξ) as the probability that a wave crest observed at a fixed point of the ocean in time exceeds the threshold ξH _{ s }. For weakly nonlinear nonlinear seas, this probability can be described by the thirdorder TayfunFedele model^{9},
where ξ _{0} follows from the quadratic equation \(\xi ={\xi }_{0}+2\mu {\xi }_{0}^{2}\) ^{8}. Here, the Tayfun wave steepness μ = λ _{3}/3 is of O(μ _{ m }) and it is a measure of secondorder bound nonlinearities as it relates to the skewness λ _{3} of surface elevations^{10}. The parameter Λ = λ _{40} + 2λ _{22} + λ _{04} is a measure of thirdorder nonlinearities and is a function of the fourth order cumulants λ _{ nm } of the wave surface η and its Hilbert transform \(\hat{\eta }\) ^{9}. In particular, \({\lambda }_{22}=\overline{{\eta }^{2}{\hat{\eta }}^{2}}/{\sigma }^{4}1\) and \({\lambda }_{04}=\overline{{\hat{\eta }}^{4}}/{\sigma }^{4}3\). In our studies Λ is approximated solely in terms of the excess kurtosis as Λ _{appr} = 8λ _{40}/3 by assuming the relations between cumulants^{66} λ _{22} = λ _{40}/3 and λ _{04} = λ _{40}. These, to date, have been proven to hold for linear and secondorder narrowband waves only^{12}. For thirdorder nonlinear seas, our numerical studies indicate that Λ ≈ Λ _{appr} within a 3% relative error in agreement with observations^{67, 68}.
For secondorder seas, referred to as Tayfun sea states^{34}, Λ = 0 only and P _{ TF } in Eq. (17) yields the Tayfun (T) distribution^{8}
For Gaussian seas, μ = 0 and Λ = 0 and P _{ TF } reduces to the Rayleigh (R) distribution
Note that the Tayfun distribution represents an exact result for large second order wave crest heights and it depends solely on the steepness parameter defined as μ = λ _{3}/3^{10}.
The Forristall model
The exceedance probability is given by^{30}
where α = 0.3536 + 0.2561S _{1} + 0.0800U _{ r }, \(\beta =21.7912{S}_{1}0.5302{U}_{r}+0.284{U}_{r}^{2}\) for multidirectional (shortcrested) seas. Here, \({S}_{1}=2\pi {H}_{s}/(g{T}_{m}^{2})\) is a characteristic wave steepness and the Ursell number \({U}_{r}={H}_{s}/({k}_{m}^{2}{d}^{3})\), where k _{ m } is the wavenumber associated with the mean period T _{ m } = m _{0}/m _{1} and d is the water depth.
SpaceTime Statistical Parameters
For spacetime extremes, the coefficients in Eq. (3) are given by^{3, 69}
where
are the average number of waves occurring during the time interval D and along the x and y sides of length \({\ell }_{x}\) and \({\ell }_{y}\) respectively. They all depend on the mean period \(\overline{T}\), mean wavelengths \(\overline{{L}_{x}}\) and \(\overline{{L}_{y}}\) in x and y directions:
and
Here,
are the moments of the directional spectrum S(f, θ) and
The Higher Order Spectral (HOS) method
The HOS, developed independently by Dommermuth & Yue^{25} and West et al.^{26} is a numerical pseudospectral method, based on a perturbation expansion of the wave potential function up to a prescribed order of nonlinearities M in terms of a small parameter, the characteristic wave steepness. The method solves for nonlinear wavewave interactions up to the specified order M of a number N of free waves (Fourier modes). The associated boundary value problem is solved by way of a pseudospectral technique, ensuring a computational cost which scales linearly with M ^{2} Nlog(N)^{70, 71}. As a result, high computational efficiency is guaranteed for simulations over large spatial domains. In our study we used the West formulation^{26}, which accounts for all the nonlinear terms at a given order of the perturbation expansion. The details of the specific algorithm are given in Fucile^{70} and Fedele et al.^{2}. The wave field is resolved using 1024 × 1024 Fourier modes on a spatial area of 4000 m × 4000 m. Initial conditions for the wave potential and surface elevation are specified from the directional spectrum as an output of WAVEWATCH III^{72}.
Data Availability
All the publicly available data and information about the El Faro accident are posted on the National Transportation Safety Board (NTSB) website^{1}.
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Acknowledgements
This manuscript is based on a study on the prediction of rogue waves during Hurricane Joaquin provided as a supplement to the National Transportation Safety Board (NTSB) to assist them in their investigation of the sinking of the Merchant Vessel El Faro, which occurred east of the Bahamas on October 1, 2015. The authors thank Emilio F. Campana for his support and incisive intellectual discussions and Fabio Fucile for helping with the HOS simulations. C. Lugni was supported by the Research Council of Norway through the Centres of Excellence funding scheme AMOS, project number 223254 and by the Flagship Project RITMARE  The Italian Research for the Sea  coordinated by the Italian National Research Council.
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The concept and design was provided by F. Fedele, who coordinated the scientific effort together with C. Lugni. C. Lugni performed numerical simulations and developed specific codes for the analysis. The wave statistical analysis was performed by F. Fedele together with C. Lugni. The overall supervision was provided by F. Fedele; A. Chawla performed the WAVEWATCH simulations and made ongoing incisive intellectual contributions. All authors participated in the analysis and interpretation of results and the writing of the manuscript.
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Fedele, F., Lugni, C. & Chawla, A. The sinking of the El Faro: predicting real world rogue waves during Hurricane Joaquin. Sci Rep 7, 11188 (2017). https://doi.org/10.1038/s41598017115055
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DOI: https://doi.org/10.1038/s41598017115055
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