Occurrence of rogue sea states and consequences for marine structures

Ocean Dynamics (2014) 64:1457 1468 DOI 10.1007/s10236-014-0753-2 Occurrence of rogue sea states and consequences for marine structures Elzbieta M. Bitner-Gregersen & Alessandro Toffoli Received: 5 February 2014 /Accepted: 14 July 2014 /Published online: 22 August 2014 # Springer-Verlag Berlin Heidelberg 2014 Abstract The frequency of occurrence of combined wave systems like wind sea and swell may increase in some ocean areas due to the observed change of storm tracks. These combined sea states, when crossing at a particular angle, may lead to more frequent occurrence of rogue events. The present study addresses these rogue-wave-prone sea states and their probabilities of occurrence. The analysis is based on hindcast data from the North Atlantic, the North Sea, the Norwegian Sea, Nigeria and Australia and supported by numerical simulations performed by the Higher Order Spectral Method (HOSM, West et al. J Geophys Res 92:11803 11824, 1987). The hindcast data have been generated by the wave model WAM. Long-term probabilistic description of significant wave height and spectral peak period is established for the selected locations and probability of occurrence of crossing rogue-wave-prone sea states is indicated. Further, the occurrence of individual rogue waves in low, intermediate and high sea states is also evaluated. The results are discussed from the perspective of design and operations of ships and offshore structures. Responsible Editor: Alexander V. Babanin This article is part of the Topical Collection on the 13th International Workshop on Wave Hindcasting and Forecasting in Banff, Alberta, Canada October 27 - November 1, 2013 E. M. Bitner-Gregersen (*) DNV GL AS, DNV GL Strategic Research and Innovation, Veritasveien 1, 1322 Høvik, Norway e-mail: Elzbieta.Bitner-Gregersen@dnvgl.com A. Toffoli Centre for Ocean Engineering, Science and Technology, Swinburne University of Technology, P.O. Box 218, Hawthorn 3122, VIC, Australia e-mail: toffoli.alessandro.@gmail.com Keywords Crossing seas. Rogue waves. Probability of occurrence. Design and operations of marine structures 1 Introduction Current knowledge of ocean waves has significantly advanced in the last decade owing to many research efforts (see, e.g., Dysthe et al. 2008; Kharif et al. 2009, Osborne 2010, for a general overview). The occurrence of rogue waves, their mechanism and detailed dynamic properties are now becoming clear and consistency between numerical models and experimental data has been documented by several studies (e.g., Toffoli et al. 2010, 2013; Shemeret al. 2010). Despite these recent achievements, however, a full consensus on probability of occurrence of rogue waves has not been achieved yet, although some progress on the topic has been made recently. Such consensus, nonetheless, is essential for the evaluation of possible revision of offshore standards and classification society rules, which currently do not include rogue waves explicitly. This is because design practice is moving towards a more consistent probabilistic approach, where extremes are determined for a given return period (e.g., expected lifetime of a structure). Probability of occurrence of rogue waves is related to mechanisms generating them such as linear Fourier superposition (frequency or angular linear focussing), wave current interactions, crossing seas, quasi-resonance nonlinear interactions (modulational instability) and shallow water effects (see, for example, Onorato et al. 2006a, b, 2010, 2013; Toffoli et al. 2010, 2011, 2013; Didenkulova 2010; Didenkulova and Pelinovsky 2011; Slunyaev2010; Sergeeva et al. 2011, among many others). In the last decade, most of the attention was given to the formation of rogue waves due to quasi-resonance nonlinear interactions referred to as modulational instability. It has been shown that the sea states responsible for occurrence of

1458 Ocean Dynamics (2014) 64:1457 1468 modulational instability in deep water are characterized by high steepness and a narrow wave spectrum, both in frequency and direction, and can be identified by the Benjamin Feir Index (BFI) (Onorato et al. 2001; Janssen 2003); such sea states can be addressed as Rogue Sea States (M. Onorato, personal communication). The BFI is a measure of the relative importance of nonlinearity and dispersion. It can be defined as BFI=(k p H s /2)/(Δω/ω p ), where k p H s /2 is the wave steepness (k p is the wave number at the spectral peak) and Δω/ω p is the frequency spectral bandwidth (Δω is the halfwidth at halfmaximum of the spectrum and ω p is the spectral peak frequency; cf. Onorato et al. 2006a). It should be noted that the above definition of BFI is valid for stationary conditions (A. Slunyaev, personal communication). Provided the wave field is sufficiently steep, narrow banded, and unidirectional, random waves are expected to become unstable when BFI=O(1). This results in an increase of probability of occurrence of rogue waves associated with an enhancement of non- Gaussian properties of the surface elevation (e.g., Onorato et al. 2006a; Socquet-Juglard et al. 2005; Toffoli et al. 2010; Xiao et al. 2013). To investigate the frequency of occurrence of seas states which may trigger the modulational instability in deep water in the North Atlantic, Bitner-Gregersen and Toffoli (2012) have studied hindcast data from a few North Atlantic locations generated by Oceanweather Inc. and European Centre for Medium-Range Weather Forecast (ECMWF). The Oceanweather Inc. hindcast wind and wave covered the period 1988 1998 and were sampled every 3 h. Data have been post-processed by Shell using the program APL Waves for the partitioning of 3D spectra (i.e. directional wave spectra) into separate peaks, Hanson and Phillips (2001). The ECMWF wind and wave data covered the period 2001 2009 and were archived at a sampling frequency of 6 h. Results revealed that such rogue-wave-prone sea states can actually occur in the North Atlantic ( the North Atlantic wave climate is used for design of ships) more often than once in the 20/25-year period with the latter being the current design return period for ship structures. Also the highest sea state within the 10-yr time period analysed (H s > 15 m) is characterised by k p H s /2=0.13, the conditions which may trigger the modulational instability. The findings of Bascheck and Imai (2011) support the above conclusions. Note that this value of steepness is also consistent with the steepness recorded during a large number of ship accidents reported as being due to bad weather conditions (Toffoli et al. 2005). It is established that the directional spreading of wave energy weakens effects related to modulational instability. This forces a transition from strongly to weakly non- Gaussian properties as the sea state become more directionally spread (Onorato et al. 2009; Waseda et al. 2009; Socquet- Juglard et al. 2005; Toffoli et al. 2010; Xiao et al.2013). Interestingly enough, however, Onorato et al. (2006, 2010) have shown that the modulational instability and rogue waves can be triggered by a peculiar form of directional sea state, where two identical, crossing, narrow-banded random wave systems interact between each other. Such results have been confirmed through recent numerical simulations of the Euler equations and experimental work carried out at the MARINTEK Laboratories (Toffoli et al. 2011). Results also indicated a robust dependence over the angle between the mean direction of propagation of the two crossing systems, with a maximization of rogue wave probability of occurrence for angles of approximately 40. It is worth mentioning that such an unusual sea state condition of two almost identical wave spectra with high steepness, and different direction was observed during the accident to the cruise ship Louis Majesty (et al. 2012). The present study discusses such rogue-wave-prone crossing seas and their probabilities of occurrence in the ocean, using hindcast data from different locations in the North Atlantic, North Sea and Norwegian Sea, Equatorial Atlantic off coast of Nigeria and NWS Australia. The analysis is supported by numerical simulations performed by the Higher Order Spectral Method (HOSM as derived by West et al. 1987). Implications for design and operations of ship and offshore structures are discussed. The paper is organised as follows. Section 2 is dedicated to rogue waves in crossing seas. In Section 3 hindcast data used in a study is presented while in Section 4 occurrence of rogue waves in crossing seas is evaluated. Implication of crossing seas for design and operations of marine structures is discussed in Section 5. The paper closes with conclusions, recommendations and references. 2 Rogue waves in crossing seas 2.1 Theoretical background To the leading order in dispersion and nonlinearity, the evolution of the free surface during crossing sea states can be described mathematically by a set of two coupled Nonlinear Schrödinger (CNLS) equations (Roskes 1976; Onorato et al. 2006b; Shukla et al. 2006): A t A iα 2 x 2 þ i ξjaj2 þ 2ςjBj 2 A ¼ 0 B t B iα 2 x 2 þ i ξjbj2 þ 2ςjAj 2 B ¼ 0 ð1þ ð2þ

Ocean Dynamics (2014) 64:1457 1468 1459 where A and B denote complex wave envelopes. α, ξ,and ζ are coefficients, which can be defined as follows (see Onorato et al. 2006b, 2010 for details): α ¼ ωκ ð Þ 8κ 4 2l k 2 ð3þ ξ ¼ 1 2 ωκ ð Þκ2 ς ¼ ωκ ð Þ k 5 k 3 l 2 3kl 4 2k 4 κ þ 2k 2 l 2 κ þ 2l 4 κ 2κ 2k 2 2l 2 þ kκ ð4þ ð5þ where (k, l)and(k, l) are the coordinates in Fourier space of the two carrier waves; ω=(gκ) 1/2 with κ=(k 2 +l 2 ) 1/2. The surface elevation η(x, y, t) is related to the envelops A and B through the expression: η ¼ 1 2 kxþly ωt Aei ð Þ þ Be ikx ly ωt ð Þ þ c:c: ð6þ where c.c. stands for complex conjugate and g is the acceleration due to gravity. The angle between the two wave systems is defined as β=2arctan(l/k). A linear stability analysis of plane wave solutions of Eqs. (1) and(2) (Ioualalenand Kharif 1994; Badulin et al. 1995; Onorato et al. 2006b; Shukla et al. 2006) indicates that the growth rates of perturbations moving along the main direction of propagation depend not only on the length of the perturbation but also on the angle between the two wave systems. Growth rates different from zero have been found for 0<β<arctan(2/2) 1/2 70.53. As β approaches β c 70.53, the nonlinear terms in the coupled system become increasingly more important. Consequently, the ratio between nonlinearity and dispersion, a measure for the presence of extreme waves in single NLS (Onorato et al. 2001; Janssen 2003), increases substantially (see Onorato et al. 2010). For β>β c, however, the ratio changes sign and the coupled NLS change from focusing to defocusing. For random waves, we can expect that larger deviations from Gaussianity already begin for β 40 (Onorato et al. 2010). As the growth rate decreases and eventually becomes zero for β approaching β c, deviations from Gaussian statistics should decrease for angle β close to 70.53. 2.2 Occurrence of extremes Experiments in a large directional basin and numerical simulations based on the Euler equations were used to confirm the validity of theoretical results presented in Section 2.1 (see Toffoli et al. 2011, for details). By monitoring the nonlinear evolution of the surface elevation associated to two crossing wave systems with identical energy content (i.e. H s ), peak period (T p ) and with a very narrow directional distribution (i.e. each individual system is a unidirectional wave field), it was demonstrated that the kurtosis, a measure of the probability of occurrence of extreme waves, notably increases with the increase of the angle β between the crossing systems. More specifically, it was observed that extreme events are more likely to occur when 40 <β<60. It is interesting to note that Waseda et al. (2011) provided evidence of an increased likelihood of rogue waves when directional spreading was relatively low (approximately 7.6 narrower compared to observations without rogue waves present), and they showed this corresponded to a wave spreading of 30 at the Kvitebjørn platform in the North Sea. On the other hand, the analyses of measured data from many locations in the North Sea, including the Kvitebjørn platform, by Christou and Ewans (2011) did not find a link between low spreading and the occurrence of rogue waves in the measured data. Christou and Ewans (2011) found that the most probable directional spreading was ca. 30 for wind sea and. 13 for swell, irrespective of whether a rogue wave was observed or not. For a simple unimodal wave system (one peak only), it is well known that the increase of wave directional spreading reduces the kurtosis (Onorato et al. 2009; Waseda et al. 2009; Socquet-Juglard et al. 2005; Toffoli et al. 2010; Xiao et al. 2013). In order to verify what effect the directional spread produces in crossing sea states, direct numerical simulations were carried out by solving the Euler equations with HOSM as derived in West et al. (1987). It is important to note that the HOSM is a pseudo-spectral method, which uses a series expansion in the wave slope of the vertical velocity about the free surface and hence the method does not allow the solution of the fully nonlinear problem. For the purpose of the present investigation, nonetheless, a third-order expansion was applied so that four-wave interactions are included (see Tanaka 2001, 2007). The selected order of the expansion makes HOSM consistent with the modified nonlinear Schrödinger equation as both models describe third-order (in nonlinearity) effects (see, e.g., Toffoli et al. 2010 for a comparison). Aliasing errors generated in the nonlinear terms are removed (West et al. 1987; Tanaka 2001). The time integration is performed by means of a fourth-order Runge Kutta method by applying a small time step, Δt=T p /50 (see additional detail in e.g. Toffoli et al. 2010). The model was initialized by providing an input linear surface η(x,y,t=0), which was computed by means of an inverse Fast Fourier Transform with random amplitudes and random phases approximation from an initial spectrum. Spectral conditions were defined by adding together two JONSWAP-like spectral peak with identical T p =1 s, H s = 0.065 m (and hence, steepness k p H s /2=0.13), and peak enhancement factor γ=6. Directional spreading was modeled by

1460 Ocean Dynamics (2014) 64:1457 1468 a cos N (ϑ) function with identical spreading coefficient N for each peak. For the present study, several spreading coefficients were used, ranging from fairly unidirectional to fairly directional cases, i.e. N=100, N=50, N=25and N=6, respectively. Note that, whereas N=100 may represent an unrealistically narrow directional spectrum, N=6 describes the directional spreading of common swells and hence more likely to be observed in realistic oceanic conditions. Crossing seas were eventually obtained by imposing different mean wave directions to each peak. In this regard, the following angles (β) between the mean wave directions of the two systems were applied: β=20, 40 and 60. As a benchmark, tests were also carried out to simulate the evolution of the wave surface in the presence of only one peak. A series of Monte Carlo simulations was undertaken to simulate the temporal evolution of the initial surface within a temporal window equivalent to 10 peak periods. The physical domain was discretized by 256 256 grid point with Δx=Δy=0.0758 m. In order to have enough records to perform a statistical analysis, 25 realizations of the surface elevation with different random amplitudes and phases were carried out for each spectral configuration (cf. Toffoli et al. 2010). The probability density function of the surface elevation and the concurrent fourth order moment (i.e. the kurtosis, a measure of the percentage of extremes in a record) were calculated at temporal steps of 4T p. The temporal evolution of the kurtosis and the associated input spectrum for different directional spreading coefficients are presented in Figs. 1, 2, 3 and 4. Overall, the kurtosis increases monotonically as the wave field evolves. Furthermore, the kurtosis enhances as a function of the angle β and maximizes for approximately β=40. It is important to mention, though, that the increase of kurtosis is rather weak, especially if compared to the one observed as a result of the interaction between two crossing long crested systems (cf. Fig. 3 in Toffoli et al. 2011), albeit with slightly higher steepness k p H s /2=0.14. As expected, kurtosis is observed to drop for β=60, because of the defocussing effect. Interestingly enough, the same trends are observed also when the spectral peaks broaden, i. e. for N=50, 25 and 6 (see Figs. 2, 3 and 4), corroborating that maximum kurtosis occurs for the angle β=40, independently on the wave directional spreading. 3 Hindcast data Hindcast data were provided by the Norwegian Meteorological Institute (met.no), ECMWF (European Centre for Medium-Range Weather Forecasts) and Oceanweather Inc. The data set includes the wind speed, significant wave height and spectral peak period (apart from ECMWF which includes only zero-crossing wave period) and mean wave direction for the total sea, wind sea and swell. The wind sea component of wave spectral energy is detected as the part of spectrum where the wind input source term is positive, thus the components of the wind sea are those which are still under the influence of the local wind forcing. The remaining part of the spectrum is considered as swell (see, for example, Hauser et al. 2005, for details). Time series from the Norwegian database NORA10, generated by the 3G WAM model and developed at MET NORWAY with major support from a consortium of oil companies (The Fig 1 Input spectra (upper panels) and temporal evolution of the kurtosis (lower panels). Directional spreading of each peak is N=100

Ocean Dynamics (2014) 64:1457 1468 1461 Fig. 2 Input spectra (upper panels) and temporal evolution of the kurtosis (lower panels). Directional spreading of each peak is N=50 Norwegian Deep Water Programme), have been used in the study. The deep water North Atlantic location 59 N, 12.0 W (north-west of UK) is used in the analysis. The NORA10 data cover the period 1958 2009 and are sampled every 3 h. The NORA10 database validates better towards satellite and buoy observations than the ECMWF s ERA40 database. The ECMWF ERA-Interim data used herein have been generated also by the 3G WAM model for the same North Atlantic location as the NORA10 data (59 N, 12.0 W) and cover the period 1989 2008. They are sampled every 6 h. The data have been received from MET NORWAY. In the considered North Atlantic location wind sea and swell is mostly always present. Additionally, three locations, one in the Northern North Sea and two in the Norwegian Sea, are considered in the study: Statfjord (61.09 N 1.40 E), Halten (64.94 N 7.98 E) and Vøring (67.02 N 6.93 E) with the water depth ca. 150, 250 and 1,000 m, respectively. The data covering the period (1955 2000) were generated by the 2G WAM model by MET NORWAY and are sampled every 6 h. Simultaneous presence of wind sea and swell is also common for these three locations. The Oceanweather Inc. hindcast data have been provided by Shell. The data were generated by the Oceanweather wave model for three locations characterized by very different wave climate: NWS Australia, off coast of Nigeria and the Southern North Sea (SNS). At the Australia location both wind sea and swell are present. The SNS location is strongly dominated by wind-sea with very limited swells present, while in off coast of Nigeria primarily swells are present with a few significant wind-sea events. The NWS Australia data were generated for the period 1994 2005 (water depth 250 m), the Nigeria Fig. 3 Input spectra (upper panels) and temporal evolution of the kurtosis (lower panels). Directional spreading of each peak is N=25

1462 Ocean Dynamics (2014) 64:1457 1468 Fig. 4 Input spectra (upper panels) and temporal evolution of the kurtosis (lower panels). Directional spreading of each peak is N=6 data for the period 1985 1999 (water depth 1,000 m), while Southern North Sea (SNS) for the period 1964 1995 (water depth33m).theyweresampledevery3hforthesouthern North Sea and Nigeria locations, while every hour for the NWS Australia one. Note that the Southern North Sea data include shallow water effects. The hindcast data applied in the study were available to the authors. Note that they cover different time periods, from 10 to 50 years. They are, therefore, not consistent, but the present study is aiming at investigating in which locations such rogueprone crossing sea states can occur and not at comparing probability of occurrence of rogue-prone sea states between different locations. On the other hand, the data represent sufficiently long time history to provide satisfactory statistics of wave parameters and to indicate probability of occurrence of potential rogue-prone crossing sea. 4 Occurrence of rogue-prone crossing seas In the following section, we discuss the frequency of occurrence of potential rouge-prone crossing seas, assuming that the hindcast data used in the analysis are good representation of the nature. The scatter diagrams of significant wave height H m0 and spectral peak period T p for total sea, wind sea and swell have been established for the considered locations. Examples of empirical scatter plots are shown in Figs. 5 and 6. The locations considered are characterized by different wave climate. At the North Atlantic, Northern North Sea, Norwegian Sea and Australia locations wind sea and swell are present most of the time. However, the North Atlantic, Northern North and Norwegian Sea scatter diagrams of significant wave height and spectral peak period for total sea have one pronounced peak (see Fig. 5), while in the Australia data, due presence of long swell show two pronounced well separated peaks in the scatter diagram, one for wind sea and one for swell (see Fig. 5). The Southern North Sea is dominated strongly by wind-sea while off coast of Nigeria by swell with few wind sea components present. These location specific features of wave climate will influence joint environmental modelling and occurrence of rogue-prone crossing wave systems. Wind sea and swell wave systems registered during the Louis Majesty accident Cavaleri et al. (2012) had approximately the same wave energy and frequency but a typical wave directional spreading. Swell had only a slightly narrower spectrum, both in frequency and direction and the crossing angle between wind sea and swell was approximately 50. The results presented in Section 2.2 show that although directionality has an effect on occurrence of extreme waves in crossing seas, extreme waves can occur also when wave directional spreading is broader. In this case the increase of kurtosis is observed also for β=40. It seems that the most critical condition for occurrence of rogue waves in crossing seas is associated with energy and frequency of two wave systems while the angle between the wave systems and wave directional spreading will decide how large extreme waves will grow. Conditions with the 40 angle and narrow directional distributions are expected to generate the largest waves. In the present study, the occurrence of wind sea and swell having almost the same spectral period (T pw~ T ps ) and significant wave height (H sw~ H ss ) and crossing at the angle 40 <β< 60 o is investigated. Figures 7 and 8 show results for the North Atlantic, Statfjord, Halten and Vøring. The study does not show detail analysis of wave spectra assuming that they are

Ocean Dynamics (2014) 64:1457 1468 1463 Fig. 5 Contour plots of long term significant wave height H m0 and spectral peak period T p for total sea: a NORA10 data (1958 2009) and b data from the SNS location (1964 1995) a) b) characterized by a typical directional wave spreading and that energy and frequency criteria are the driving ones for occurrence of rogue waves in crossing sea states. Wind sea and swell crossing at the angle 40 <β<60 o and having almost the same spectral period (T pw ~T ps ) and significant wave height (H sw ~H ss ) are shown in Fig. 7 as a function of H sw (H ss ) for the North Atlantic location and NORA10 data (1958 2009). Such seas have been observed only for low and intermediate sea states with the total significant wave height (2(H 2 sw +H 2 ss )) 1/2 in the range 1.0 7.2 m and more of them occur for significant wave height lower than 2.5 m. Totally 27 crossing seas satisfying the significant wave height, spectral period and the angle β criterion have been identified (151,840 observations are in the period 1958 2009). If the angle range is extended to 30 <β<60 o than wave systems with H sw ~H ss =6.85 m (total H m0 =9.69 m) and the same spectral period T pw ~T ps =14.9 s have been found but none sea states with H sw >7 m.

1464 Ocean Dynamics (2014) 64:1457 1468 Fig. 6 Contour plots of longterm significant wave height H m0 and spectral peak period T p : a total sea data from NWS Australia (1994 2005) and b Nigeria swell data (1985 1999) a) b) The ERA-Iterim data available to the authors do not include the spectral period T p, only the zero-crossing wave period T m02. Within this data set, several wind sea and swell systems with the same significant wave height and crossing at the angle 40 <β<60 have been identified, but again only for low and intermediate sea states. Wind sea and swell systems satisfying the angle (40 <β< 60 ), H sw /H ss and T p criterion for Statfjord, Halten and Vøring are plotted in Fig. 8. They include 56 crossing seas for Statfjord, 74 for Halten and 81 for Vøring (65,700 observations are in the period 1955 2000); again majority of them are found in low sea states and some in the intermediate sea states with the total significant wave height (2(H 2 sw +H 2 ss )) 1/2 in the range 0.4 7.2 m. As shown in Fig. 6, at the Australia location, wind sea and swell are well separated and the swell component has significantly longer periods than the wind sea component. Therefore none crossing rogue-prone wave

Ocean Dynamics (2014) 64:1457 1468 1465 Fig. 7 The angel between wind sea and swell having almost the same spectral period and significant wave height as a function of H sw (H ss ). NORA10 data, the North Atlantic location β (degree) 100 80 60 40 20-40 -60-80 -100 Angle β between crossing wind sea and swell systems Tp and Hs for wind sea and swell equal 0-20 0 2 4 6 8 10 Significant wave height of wind sea and swell (m) NORA10: angle β systems have been found in this location. Note, however, that the spectral partitioning procedure used in the analysis of hindcast data may miss identifying swell in large wind seas, and this may be responsible for not observing any crossing seas in the Australian location. However, the large wind sea components in this hindcast data base will likely be associated with tropical cyclones and it might be expected that some tropical cyclone sea states would have wind sea and swell components that meet the search criteria. Similarly as at the Australia location, in off coast of Nigeria the swell has significantly different spectral periods than the wind sea and none crossing rogue-prone wave systems have been identified at this location. Such conditions could be present for different swell components; however, in the considered data set all swell components have been recombined in one swell component not allowing investigating different swell components separately. As the highest swell in off coast of Nigeria has significant wave height in the range 3.0<H ss < 3.5 m rogue-prone crossing seas could be expected only in low and intermediate sea states. The SNS location is strongly wind sea dominated and none rogue-prone crossing wave systems have been found in this location either. 5 Implication for marine structural design and operations According to current design practice marine structural strength is evaluated for a given return period (i.e. a time period during which a hazard that can endanger the structure integrity appears not more than once). Ship structural strength and ship stability are calculated, following international standards, in extreme events with the return period of 20/25 years (Ultimate Limit State, ULS, in the structural reliability methodology). Offshore structures (including FPSOs: Floating Production Storage and Offloading units) follow a different approach to design of ship structures and are designed for the 100-year return period (ULS). The Norwegian offshore standards (NORSOK 2007) require that there must be enough space for the wave crest to pass beneath the deck to ensure Fig. 8 The angle between wind sea and swell having almost the same spectral period and significant wave height as a function of H sw (H ss ). The Statfjord, Halten and Vøring locations β (degree) 100 80 60 40 20 0-20 -40-60 -80-100 Angle β between crossing wind sea and swell systems Tp and Hs for wind sea and swell equal 0 2 4 6 8 10 Statfjord: angle β Halten: angle β Vøring: angle β Significant wave height of wind sea and swell (m)

1466 Ocean Dynamics (2014) 64:1457 1468 that a 10000-yr wave load does not endanger the structure integrity (Accidental Limit State, ALS). Knowledge about probability of occurrence of rogue waves is necessary for providing a consistent risk-based approach for design of ship and offshore structures which combines new information about extreme and rogue waves. Visual observations of waves collected from ships in normal service (BMT 1986) are currently used in the design of ship structures sailing world-wide. Four ocean areas in the North Atlantic, regarded as having the most severe wave climate, represent the wave base for ship design. The last significant wave height class in the design North Atlantic scatter diagram is in the range 16 17 m, IACS (2000). Classification rules, in fact, permit the design of ships for restricted service (in terms of geographical zones and the maximum distance the ship will operate from a safe anchorage); in which case reduced design loads apply. Many aspects of the design, approval and operation require a detailed knowledge of local weather conditions. While in principle open to all ship types, the use of such restricted service is in practice mainly confined to high speed vessels. Unlike ship structures, offshore structures normally operate at fixed locations and often represent a unique design. As a result, platform design and operational conditions need to be based on location specific met-ocean climate. Note that Floating Production Storage and Offloading systems are designed for the North Atlantic wave climate if location specific wave environment cannot be proved more appropriate. The investigations carried out by Bitner-Gregersen and Toffoli (2012) based on the hindcast data from the few North Atlantic locations show that rogue-prone seas due to quasi-resonant interactions (modulational instability) can occur in low, intermediate and high sea states. Therefore this type of seas can have impact not only on design loads and responses of marine structures but also on marine operations, but their importance will depend on their frequency of occurrence in different ocean locations. The present results indicate that rogue-prone crossing seas can be found only in low and intermediate sea states. Their occurrence depends on wave climate features typical for a specific location. These rogue-prone sea states can be expected to impact operational conditions of marine structures but may also influence weather restricted design as well as design of local loads. The accident that took place to the Louis Majesty ship in the Mediterranean Sea on March 3, 2010, is an example of rogue-prone crossing seas, Cavaleri et al. (2012). The ship was hit by a large wave that destroyed some windows at deck number five and caused two fatalities. Using the WAM wave model, driven by the COSMO-ME winds, a detailed hindcast of the local wave conditions has been performed. The results have revealed the presence of two comparable wave systems characterized by almost the same frequency (around 0.1Hz) and significant wave heights of approximately 3.5 m. The total significant wave height, H m0, at the time of the accident was estimated around 5 m. These sea state conditions are discussed by Cavaleri et al. (2012) in the framework of a system of two coupled Nonlinear Schrödinger (CNLS) equations, each of which describe the dynamics of a single spectral peak. Even though, due to the lack of measurements, it is impossible to establish the nature of the wave that caused the accident, it has been shown that the angle between the two wave systems during the accident is close to the condition for which the maximum amplitude of the breather solution is observed (40 <β<60 ). 6 Conclusions The study investigates, based on hindcast data, whether rogueprone crossing sea states can occur in the ocean. The analysis is supported by numerical simulations carried out by the HOSM method. The numerical simulations show that although directionality has an effect on occurrence of extreme waves in crossing seas, extreme waves can occur not only for narrow-banded wave directional spreading but also when it is broader. It seems that the most critical condition for occurrence of rogue waves in crossing seas is associated with energy and frequency of two wave systems while the angle between the wave systems and directional spreading will decide how large extreme waves will grow. The 40 angle and narrow-banded directional spreading is generating the largest waves. It is indicated that rogue-prone crossing wave systems that are responsible for the generation of abnormal waves can occur primarily in low and intermediate sea states. Their occurrence is location specific, depending strongly on local features of wave climate. This type sea states are not expected to be present in locations where wind sea and swell components, or several swell components, are well separated, characterized by significantly different spectral periods. The rogue-prone crossing wave systems can be dangerous for marine structures but their impact will depend on a type and size of a structure. They are expected to affect most operations of ships and offshore structures, but may also influence weather restricted design as well as design of local loads. The investigations indicate that ships may experience such crossing wave systems several times during their life time. It should be noted that the present study is limited to the few locations only and do not include detail investigations of the found crossing seas. A systematic analysis of wave spectra associated with the identified rogue-prone sea states, supported by numerical simulations, needs to be carried out in the future to reach firm conclusions regarding occurrence of rogue-prone crossing sea states in the ocean.

Ocean Dynamics (2014) 64:1457 1468 1467 Further, uncertainties of the data used in the study affect the presented results. They are due to the assumptions of the wave models applied for generation of the hindcast data, validation of the wave models as well as due to an approach adopted for estimation of wind sea and swell components. Particularly, the latter may impact the presented results. Furthermore, it is assumed that the hindcast data approximate satisfactory the nature. The investigations of these uncertainties have been outside the scope of the analysis. However, it is not expected that accounting for them will change the present conclusions significantly; rogue-prone crossing wave systems will probably be observed primarily in low and intermediate sea states and they will not appear in locations when wind sea and swell (or several swell components) are well separated. Developments of warning criteria for rogue-prone crossing seas for marine structures are also called for. The research on the topic has already been initiated by ECMWF, with which the EC EXTREME SEAS project has collaborated, and Meteorological Offices are focusing on it too. This work needs to continue to enhance safety at sea. Acknowledgments This work has been supported by the Norwegian Research Council project ExWaCli (the Project No. 226239) and the E.U. project EXTREME SEAS (SCP8-GA-2009-234175). The authors are indebted to the Norwegian Meteorological Institute and Shell International Exploration and Production B.V. for providing the wave data. The authors thank Miguel Onorato and Alexey Slunyaev for fruitful discussions. References Badulin SI, Shrira VI, Kharif C, Ioualalen M (1995) On two approaches to the problem of instability of short crested water waves. J Fluid Mech 303:297 326 Bascheck B, Imai J (2011) Rogue wave observations off the US West Coast. 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