JUNE 2018 G R A M S T A D E T A L . 1317

Modulational Instability and Rogue Waves in Crossing States

ODIN GRAMSTAD AND ELZBIETA BITNER-GREGERSEN Group Technology and Research, DNV GL, Høvik, Norway

KARSTEN TRULSEN Department of Mathematics, University of Oslo, Oslo, Norway

JOSÉ CARLOS NIETO BORGE Department of Physics and Mathematics, University of Alcalá, Alcalá de Henares, Spain

(Manuscript received 19 January 2018, in final form 9 April 2018)

ABSTRACT

Wave statistical properties and occurrence of extreme and rogue waves in crossing sea states are in- vestigated. Compared to previous studies a more extensive set of crossing sea states are investigated, both with respect to spectral shape of the individual wave systems and with respect to the crossing angle and separation in peak frequency of the two wave systems. It is shown that, because of the effects described by Piterbarg, for a linear the expected maximum crest elevation over a given surface area depends on the crossing angle so that the expected maximum crest elevation is largest when two wave systems propagate with a crossing angle close to 908. It is further shown by nonlinear phase-resolving numerical simulations that nonlinear effects have an opposite effect, such that maximum sea surface kurtosis is expected for relatively large and small crossing angles, with a minimum around 908, and that the expected maximum crest height is almost independent of the crossing angle. The numerical results are accompanied by analysis of the modu- lational instability of two crossing Stokes waves, which is studied using the Zakharov equation so that, dif- ferent from previous studies, results are valid for arbitrary-bandwidth perturbations. It is shown that there is a positive correlation between the value of kurtosis in the numerical simulations and the maximum unstable growth rate of two crossing Stokes waves, even for realistic broadband crossing sea states.

1. Introduction wave-height Hs. In the last decades there has been a considerable interest in rogue waves, with a particular In many regions sea states often consist of more focus on understanding the mechanisms behind the than one wave system, so that the wave spectrum has formation of such waves as well as understanding in multiple peaks with different directions of propaga- what types of sea conditions such waves may occur more tion and different peak wave periods. Such sea states frequently. Recent reviews on oceanic rogue waves can are often referred to as crossing and are typical in be found in Adcock and Taylor (2014) and, with a par- the case that a local wind sea coexists with a system ticular focus on applications to the marine industry, in of . Bitner-Gregersen and Gramstad (2016). From a sea captain’s point of view, a crossing sea may In the absence of inhomogeneities such as currents be a more challenging situation, since waves may hit the and variable bottom topography, the main mecha- ship from different directions simultaneously. Crossing nism that has been suggested to explain the occur- sea states have also attracted some attention in the rence of rogue waves beyond what is expected from context of rogue and extreme waves. Rogue waves standard linear or second-order wave theories is the (also called freak or abnormal waves) are waves that effect of modulational instability. In its basic form the are unlikely to occur in a given sea state, given the modulational instability is the phenomenon that a averaged properties of a sea state, typically the significant uniform wave train (a ) is unstable to side-band perturbations, as first shown by Benjamin Corresponding author: Odin Gramstad, [email protected] and Feir (1967). This instability, which may result in

DOI: 10.1175/JPO-D-18-0006.1 Ó 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses). Unauthenticated | Downloaded 09/24/21 08:13 PM UTC 1318 JOURNAL OF PHYSICAL VOLUME 48 the formation of large individual waves, may also occur in modulational instability, just as in the case of unimodal the more realistic case of irregular wave fields described seas (Alber 1978; Crawford et al. 1980). by a finite-width wave spectrum. It is, however, shown that The results from these existing studies indicate that a the modulational instability is suppressed for spectra with coupled system may experience increased instability broad bandwidth or wide directional spreading (Alber growth rates compared to a single Stokes wave, in par- 1978; Crawford et al. 1980). This is in agreement with both ticular when the difference in propagation directions of numerical (Onorato et al. 2001; Gramstad and Trulsen the two waves is small. However, the instability results 2007; Xiao et al. 2013) and experimental (Onorato et al. based on NLS-type equations suffer from nonphysical 2009, 2004) studies, which have shown that the occurrence extensions of the instability regions beyond the band- probability of rogue waves is reduced when the wave width constraints of the equations, and it may be difficult spectrum becomes broader and has a wider directional to distinguish the physical instability regions with non- spreading. Based on such results it has recently been an physical finite-bandwidth effects (see, e.g., Gramstad increasing concern whether effects of modulational insta- and Trulsen 2011). bility are still relevant in realistic ocean conditions In this paper we derive the modulational instability (Christou and Ewans 2014; Fedele et al. 2016; Benetazzo of a system of two Stokes waves using the Zakharov et al. 2015, 2017). equation (Zakharov 1968). The Zakharov equation has In this context, crossing seas have been suggested no bandwidth constraints, and the instability results are as a situation that may be associated with increased therefore valid for a wider range of configurations than probability of rogue waves, even in realistic ocean existing results derived using NLS-type equations. The conditions, and there are a number of previous studies instability analysis of a single wave train using the addressing this topic, both in terms of modulational Zakharov equation was first presented in Crawford et al. instability of systems of two crossing wave trains and in (1981) as well as in the following review by Yuen and terms of numerical simulations of crossing random Lake (1982). The modulational instability of standing wave fields. waves (i.e., two counterpropagating waves with the The modulational instability of a system of two Stokes same ) was previously considered within the waves with different directions of propagation has pre- framework of the Zakharov equation by Okamura viously been considered within the framework of cou- (1984). The derivation in this paper roughly follows the pled nonlinear Schrödinger (NLS) equations. Roskes procedure of Crawford et al. (1981) and Okamura (1976) studied the modulational instability of a system (1984) but with the necessary modifications to take the of two uniform waves with the same frequency but dif- presence of two wave trains with general ferent directions of propagation using a set of coupled ka and kb into account. NLS equations. Assuming perturbations along the mean Numerical simulations of two nearly unidirectional propagation direction of the two Stokes waves, he crossing wave systems, using a higher-order spectral showed that such a system is unstable for crossing method (HOSM), was presented in Onorato et al. angles 08,u , 70.58 and 136.18,u # 1808. Based on (2010), who showed that the numerical simulations were this result it was suggested by Onorato et al. (2006) that in qualitative agreement with the results from the more rogue waves may be expected in crossing sea states analysis of modulational instability, in the sense that with crossing angles smaller than about 708. Similar more rogue waves were observed for crossing angles in analysis was presented also in Onorato et al. (2010), the range of 408–608. Later, laboratory experiments of who, based on the growth rate and amplification factor crossing unidirectional wave fields (Toffoli et al. 2011) of an Akhmediev breather solution, suggested that a also indicated an increased sea surface kurtosis for maximum amount of rogue waves could be expected crossing angles of about 408. for crossing angles between 208 and 608.Themore While the works mentioned above considered the general case of two-dimensional perturbations was crossing of two unidirectional or nearly unidirectional considered by Shukla et al. (2006), and stability analysis wave systems, numerical simulations of crossing sea based on a more accurate fourth-order coupled NLS states with more realistic directional spread wave equation was derived in Gramstad and Trulsen (2011), spectra were performed by Bitner-Gregersen and who also considered the more general case where the Toffoli (2014) using HOSM. They showed that similar, two Stokes waves both have different directions of although less pronounced, dependence on crossing propagation and different frequencies. The effect of fi- angle could be identified also when the two wave sys- nite spectral width on the modulational instability of tems were represented by more realistic spectra. Con- crossing systems was investigated by Shukla et al. (2007), sistent with the theoretical and numerical results who showed that finite bandwidth suppresses the discussed above, it was shown by Cavaleri et al. (2012)

Unauthenticated | Downloaded 09/24/21 08:13 PM UTC JUNE 2018 G R A M S T A D E T A L . 1319 that the accident of the cruise ship Louis Majesty, 2. Modulational instability of two crossing Stokes which was hit by a that destroyed windows waves on deck 5 and caused two fatalities, which took place As mentioned in the introduction, it has previously been in a crossing sea state with crossing angle of about 408– suggested that there is a link between the enhancement in 608, and where the two wave systems had similar peak modulational instability experienced by a system of two frequencies. A similar study on the sea state during the waves and the increased occurrence of rogue waves in Prestige accident (Trulsen et al. 2015) showed that this crossing sea states. These suggestions have been based on accident also happened in a crossing sea state, with a modulational instability results obtained from coupled crossing angle of about 908. They were however not able NLS equations that suffer from bandwidth constraints. To to identify any increased probability of rogue waves for obtain more accurate results of the modulational instabil- this sea state, also somewhat consistent with previous ity of two Stokes waves, we here derive the modulational results mentioned above. instability of two crossing Stokes waves using the frame- Note that the investigations that have indicated in- work of the Zakharov equation. This represents a gener- creased occurrence of rogue waves in crossing sea states alization of the modulational instability of a single Stokes have studied two wave systems with the same peak wave, derived from the Zakharov equation by Crawford frequency and the same energy (same H ). The opposite s et al. (1981), and an extension of the bandwidth constraint case that two crossing wave systems are well separated results obtained from coupled nonlinear Schrödinger in frequency, typical in crossing seas consisting of wind equations by Onorato et al. (2006), Shukla et al. (2006), sea and swell, was for example considered in Gramstad and Gramstad and Trulsen (2011). and Trulsen (2010), who in numerical simulations of an The analysis of modulational instability of two cross- NLS-type equation, found no evidence of increased ing Stokes waves using the Zakharov equation was rogue wave occurrence due to the swell. previously carried out by Okamura (1984). Compared to Since most of the studies on crossing seas have consid- the present analysis, the less general case of standing ered either the case that two wave systems have the same waves was considered; that is, k 52k , where k and peak frequency and same energy or the case that two wave a b a k are the wavenumbers of the two Stokes waves. The systems are very well separated in frequency, it is not clear b analysis is a generalization of the results Okamura to what extent existing results are robust also for more (1984) for general choices of k and k . realistic cases, both with respect to spectral shape of the a b two crossing systems and with respect to the differences in a. Derivation of equations for instability peak frequency and energy of the two systems. The main objective of this paper is to investigate a somewhat wider For a system of Nm discrete wave modes defined in 5 range of crossing sea states, with different types of spectra terms of the complex amplitude spectrum B(kj), j ... and by letting two wave systems have different peak fre- 1, , Nm, the Zakharov equation (Zakharov 1968) can be written in the form quencies and different significant wave-height Hs.Thisis investigated both through theoretical analysis of the dB modulational instability of two crossing Stokes waves and i n 5 å å å T eiDvtd B*B B . (1) dt n,p,q,r n1p2q2r i q r by performing a large number of numerical simulations p q r using the numerical wave model HOSM to simulate gen- Here, * denotes complex conjugate, v is the angual eral broadband nonlinear sea states. In addition, linear frequency, and Dv 5 v 1 v 2 v 2 v . The indices effects on the statistical properties of crossing sea states are p i j m n, p, q, and r run from 1 to N . Subscripts denote also considered. m functional dependency of wavenumbers; for example, The paper is organized as follows. First, in section 2 the B 5 B(k ), T 5 T(k , k , k , k ), and d is the analysis of the modulational instability of two crossing j j n,p,q,r n p q r n,p,q,r Kronecker delta Stokes waves is presented. Then, in section 3 the more realistic case of random crossing sea states are consid- 1 when k 1 k 5 k 1 k d 5 n p q r . (2) ered, both in the context of linear theory, taking the ef- n1p2q2r 0 otherwise fects described by Piterbarg (Piterbarg 1996; Krogstad et al. 2004) into account, and by numerical simulations of The exact expression for the Zakharov kernel function nonlinear phase-resolving numerical simulations. The T can be found in, for example, Krasitskii (1994) or relationship between the numerical results and the Janssen and Onorato (2007). theoretical results from the instability analysis is Extending the analysis of Crawford et al. (1981),we discussed in section 4. Final discussion and conclusions now consider a wave system consisting of two Stokes are given in section 5. waves with wavenumbers ka and kb with corresponding

Unauthenticated | Downloaded 09/24/21 08:13 PM UTC 1320 JOURNAL OF VOLUME 48 complex amplitude functions A(t) and B(t), as well as a into the Zakharov equation [(1)] and keeping only linear set of perturbations ka 6 K and kb 6 K with associated terms in the perturbation amplitudes gives the following amplitudes A6(t) and B6(t), respectively. Inserting this system of six equations:

dA i 5 jAj2AT 1 2AjBj2T , (3a) dt a ab

dB i 5 jBj2BT 1 2BjAj2T , (3b) dt b ab

6 7 7 7 dA6 2 2 2i D 1D t 2i D 1D t i 5 2 T jAj 1 T jBj A 1 T A2A* e ðÞa a 1 2T ABB* e ðÞa b dt a6,a a6,b 6 a6,a7,a,a 7 a6,b7,a,b 7 2 D61D6 1 iðÞa b t 2Ta6,b,a,b6AB*B6e , and (3c)

6 7 7 7 dB6 2 2 2i D 1D t 2i D 1D t i 5 2 T jBj 1 T jAj B 1 T B2B* e ðÞb b 1 2T BAA* e ðÞb a dt b6,b b6,a 6 b6,b7,b,b 7 b6,a7,b,a 7 2 D61D6 1 iðÞb a t 2Tb6,a,b,a6BA*A6e , (3d)

where subscripts a6 denote ka 6 K and b6 denote kb 6 One can show that (3) admits solutions in the fol- K, and where the following short-hand notation is in- lowing form: troduced, for example, 2iTjaj212T jbj2 t 2iTjbj212T jaj2 t 6 A 5 ae ðÞa ab , B 5 be ðÞb ab D 5 v 2 v 5 a a a6, Ta Ta,a,a,a, 2 21 21D61V 5 iTðÞajaj 2Tabjbj a t 5 5 A6 a6e , Tab Ta,b,a,b, and Ta6,a Ta6,a,a6,a, (4) 2 21 21D61V 5 iTðÞbjbj 2Tabjaj b t (5) B6 b6e , In the derivation of (3) we have made use of the symmetry properties of the Zakharov kernel func- where V is an angular frequency associated with the tion; that is, Ta,b,c,d 5 Tb,a,c,d, Ta,b,c,d 5 Tb,a,d,c,and perturbations. Inserting (5) into (3) yields a set of four Ta,b,c,d 5 Tc,d,a,b. Note that, as expected, the system algebraic equations for the amplitudes a6 and b6. One of equations in (3) is invariant to the interchange of can show that this system of equations can be written as T A and B. Mx 5 0, where x 5 [a1, a*2, b1, b*2] and where

2 3 1 1V 2 2 2 Ma aaTa1 a2 a a 2ab*Ta1 b a b1 2abTa1 b2 a b 6 2 , , , , , , , , , 7 6 2a*a*T M 2V 22a*b*T 22a*bT 7 M 5 6 a1,a2,a,a a a2,b1,a,b a2,b,a,b2 7 4 2 2 1 1V 2 5, (6) 2a*bTa1,b,a,b1 2abTa2,b1,a,b Mb bbTb1,b2,b,b 2 2 2 2 2V 2a*b*Ta1,b2,a,b 2ab*Ta2,b,a,b2 b*b*Tb1,b2,b,b Mb where gives a fourth-order equation for V, and the corre- sponding unstable growth rate for the perturbations is 6 5 2 1 2 2 2 2 2 1D6 V 5 5 5 Ma Tajaj 2Tabjbj 2Ta6,bjbj 2Ta6,ajaj a , given by Im . We note that by setting b b1 b2 0 in the above equations the results of Crawford et al. and (1981) for a single Stokes wave are recovered. (7a) 6 5 2 1 2 2 2 2 2 1D6 b. Results of the instability analysis Mb Tbjbj 2Tabjaj 2Tb6,bjbj 2Tb6,ajaj b . (7b) In the following we consider two Stokes waves propagat- ing with an angle u relative to each other and with a possible For this system of equations to have nontrivial solutions difference in amplitude and . To limit the the determinant must be equal to zero, det(M) 5 0. This number of free parameters, we restrict ourselves to the case

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FIG. 1. Instability diagram for the case u 5 458, s 5 1. Shown are (left),(center) the instability of waves A and B separately and (right) the instability of the combined system. The red stars show the points of maximum growth rate. that the two Stokes waves have the same wave steepness. (1981) for a single wave train and in Okamura (1984) for

That is, letting ka and aa and kb and ab be the wavenumber standing waves. Generally, increased steepness leads to vectors and amplitudes of the two Stokes waves, we let enhanced growth rates and enlargement of the instability regions. However, the general features of the instability 5 ka ka[cos(u/2), sin(u/2)], seem to remain, and in the following we consider only one k 5 sk [cos(u/2), 2sin(u/2)], value of the wave steepness. b pffiffiffia pffiffiffi 5 5 5 Examples of instability regions in the K–plane are a 2aa, b 2ab, ab aa/s, (8) shown in Figs. 1–4. Generally, the instability regions of for u 2 [0, p]ands 2 (0, 1]. For the sake of comparison with the combined system consist of regions that resemble the numerical results presented in the nextp section,ffiffiffi we let the instability regions of the two waves separately (see 5 (a) 2 5 21 5 (a) 5 ka [2p/Tp ] /g 0.04 m and aa Hs / 8 1.414 m, also figures in Crawford et al. 1981), but in addition new (a) (a) where Tp and Hs are the peak period and significant regions of instability are also present, and in several of wave system A chosen in the numerical cases these new regions have higher growth rates than simulations in the next section. The corresponding steep- the original instability regions. Note also that the in- nesses of the two crossing Stokes waves are consequently stability regions are qualitatively similar to the ones aajkaj 5 abjkbj 5 0.057. The effects of the wave steepness derived from the coupled NLS-type equations in Shukla on the instability arediscussedbothinCrawford et al. et al. (2006) and Gramstad and Trulsen (2011). Different

FIG.2.AsinFig. 1, but for u 5 908, s 5 1.

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FIG.3.AsinFig. 1, but for u 5 1358, s 5 1. from the coupled NLS results, however, results also 2015) as well as with the numerical results in this study. outside the narrow-bandwidth regions can be trusted. For larger u, a new figure-eight-like instability region For the case of standing waves (u 5 1808 and s 5 1), we with increased growth rates emerges. It is, however, reproduce the results of Okamura (1984). As shown by, worth mentioning that in this case, the growth rates of for example, Fig. 3, the additional figure-eight-like in- the ‘‘original’’ instability regions are actually slightly stability region that was found by Okamura (1984) for decreased, as shown by the dashed lines in Fig. 5. standing waves is present also for not purely counter- It is also interesting to note from Fig. 5 that in the case propagating waves. the of the two waves differ (s , 1), there is a Figure 5 shows the normalized maximum growth rates weaker increase in the unstable growth rate. However, as functions of the angle u between the two waves for somewhat surprisingly, the enhanced modulational in- different values of the wavelength- and amplitude-ratio stability persists even for s 5 0.5. For s 5 0.25, however, s. It is interesting to see that for relatively small angles the interaction between the two wave systems seems to there is a significant increase in the modulational in- be minor, and the growth rate is similar to that of one stability compared to each of the wave systems alone. wave only, independent of the crossing angle. These For angles close to 908, however, there is almost no results suggest that even typical wind-sea and swell change in the instability compared to the unimodal case. systems may interact to increase the modulational in- This is consistent with previous studies on crossing sea stability, provided that the difference in wavelength of states with crossing angle close to 908 (Trulsen et al. the two systems is not too large.

FIG.4.AsinFig. 1, but for u 5 1808, s 5 1.

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corresponding domain in the physical space covers an area (max) 3 (max) 5 of nxp/kx nyp/ky . For waves with Tp 10 s on infinite water depth this corresponds to a computational domain of 5 km 3 5 km. Note that these values are for the fully dealiased grid; the corresponding values before

dealiasing when M 5 3arenx 3 ny 5 1024 3 1024 and (max) 5 (max) 5 (a) (b) kx ky 16 max[kp , kp ]. The wave fields are simulated in time for a total du-

ration of tmax 5 1800 s using a fixed-step ODE solver with integration time step Dt 5 0.3125 s, corresponding

to Dt 5 Tp/32 for Tp 5 10 s. A weak dissipation of high wavenumbers is included to model the energy dissipa- tion due to wave breaking, using the dissipation model suggested in Xiao et al. (2013). Some testing has been carried out to ensure that neither the choice of time step D FIG. 5. The maximum unstable growth rate max(ImVab)ofthe t nor the dissipation model influence the statistical coupled system, normalized by the maximum growth rate of wave A results significantly. alone, as function of angle u for different values of the wavelength In all simulations the initial condition was chosen as a and amplitude ratio s. The dashed lines show the maximum growth system of two JONSWAP spectra with cosN(f 2 f ) rates when instability regions far from the ‘‘original’’ instability re- p gions are excluded (e.g., the figure-eight region in Figs. 3 and 4). directional distribution with different peak directions of (a) (b) (a) propagation fp and fp and different peak periods Tp (b) and Tp . Random phases and amplitudes were assigned 3. Statistical properties of random bimodal wave to the initial spectrum in all cases. Hence, each of the fields wave systems were chosen according to a spectrum of the type E(k) 5 F(k)D(f), where k 5 (k , k ) 5 k(cosf, While the analysis of modulational instability of two x y sinf) and where crossing Stokes waves presented above indicates that a system of two waves may experience enhanced unstable hiÀÁ   pffiffiffiffiffiffiffi 2 growth rates, it is not clear to what extent such theoretical a 5 22 exp 2 k/k 21 =ðÞ2s2 F(k) 5 exp 2 k/k g p A , results are relevant also for the more realistic case of 2k3 4 p random, short-crested bimodal wave fields in nature. In (9a) the following, linear and nonlinear numerical simulations of short-crested bimodal wave fields are performed, with and the objective of investigating statistical properties and occurrence of rogue waves in crossing seas. D(f) 8 > 1 G(N/2 1 1) p a. Simulation setup < pffiffiffiffi cosN f 2 f ,ifjf 2 f j # G 1 p p 5 k p (N/2 1/2) 2 . The numerical simulations in this study have been :> carried out using a numerical solver based on the HOSM, 0, otherwise first proposed by Dommermuth and Yue (1987) and West (9b) et al. (1987). Short-crested bimodal wave fields have been simulated in a quadratic spatial domain with periodic Here, G is the gamma function and the parameter sA has boundary conditions. The nonlinear order M in the HOSM the standard values 0.07 for k # kp and 0.09 for k . kp. simulations was in this study set to M 5 3, which includes The other spectral parameters a, g, kp, fp, and N were the leading-order nonlinear dynamical effects, including chosen to give the desired spectral shape, significant wave- the effect of modulational instability. height Hs, peak period Tp, and peak direction fp.Provided The horizontal plane is discretized using nx 3 ny 5 that the two wave systems are uncorrelated, the spectrum 512 3 512 grid points, and the maximum resolved wave- of the total wave field is E(k) 5 Ea(k) 1 Eb(k), where Ea(k) (max) 5 (max) 5 (a) (b) numbers were chosen as kx ky 8max[kp , kp ], and Eb(k) are the individual spectra for the two crossing (a) (b) where kp and kp are the peak wavenumbers of the two wave systems. The total significant wavepÐffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi height of the wave systems in the crossing sea state. Thus, the shortest wave field can be defined as Hs 5 4 E(k) dk, while wave that is resolved by the simulations, corresponding to the significant wavepÐffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi heights of the individualpÐffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sys- (a) 5 (b) 5 the Nyquist frequency, spans two grid points, and the tems are Hs 4 Ea(k) dk and Hs 4 Eb(k) dk.

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TABLE 1. Parameters of the JONSWAP spectra employed in the and the maximum observed crest height hmax.That numerical simulations. is, for a surface snapshot from the numerical simula-

5 5 (a) (a) (b) (b) tions hi,j 5 h(xi, xj) where i 5 1, ..., nx and j 5 1, ..., ny Case ga gb Na Nb Tp Hs Tp Hs ffiffi 5 5 (a) p (a) (here nx ny 512), we can define A 1.0 4 10 s 4 m Tp / ffiffis Hs /s (a) p (a) B 3.3 16 10 s 4 m Tp / s Hs /s n pffiffi nx y (a) (a) 1 4 C 6.0 100 10 s 4 m Tp / s Hs /s å å 2 hi,j h n n i51 j51 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 "#x y 5 k4 2, hmax max hi,j, (10) 2 2 n n i,j 5 (a) 1 (b) 1 x y 2 If the wave fields are uncorrelated, Hs Hs Hs . å å 2 hi,j h Note that the energy-based definition of the significant nxny i51 j51 wave height used in this paper (often denoted Hm0) 5 nx ny differs from the definition based on the average of the where h 1/(nxny)åi51åj51hi,j is the mean of hi,j. Note 1/3 largest zero-crossing waves in a wave record (often that by construction of the numerical simulations, h 5 0. denoted H1/3). For linear and narrowband wave fields, In the presentation of results below, the values for k4 and these two definitions are almost equivalent. However, hmax are calculated as averages over all random realiza- for broadband wave fields like crossing seas, Hm0 and tions of the same sea state, and the maximum observed H1/3 will be different. However, in numerical simula- values over the duration of the simulations are plotted. tions as carried out in this paper, H basically defines s b. Linear results the energy of the initial spectrum, and an energy- based definition is most natural. As a background for nonlinear numerical simulations The spectral parameters for the different cases con- presented in the following it is useful to briefly discuss sidered in this study are summarized in Table 1.For the statistical parameters k4 and hmax in the case of a each case, different values of the angle u between the purely linear random sea surface, in which case the two wave systems are considered, as well as different surface is Gaussian distributed and can be viewed as a (b) choices for the peak period Tp and significant wave- Gaussian random field. (b) height Hs of the second wave system. However, as for It is well known that the kurtosis of a Gaussian sea the analysis of modulational instability in section 2b,in surface is equal to three. Deviation from three is used (b) (b) all cases Tp and Hs are chosen so that the mean as a measure of the degree of non-Gaussianity of the sea 5 5 2 2 wave-steepness « kpHs/2 2p Hs/(Tp g)isthesame surface, in the sense that a kurtosis larger than three for both wave systems. Hence, for a given 0 , s # 1, the indicates more extreme waves than expected according period and significant wave heights are chosen so that to linear wave theory. However, one should keep in 5 5 (a) (b) 5 (a) (b) 2 s kb/ka Hs /Hs [Tp /Tp ] .Forthecaseoftwo mind that k4, as defined in (10), is not the ‘‘real’’ pop- identical wave systems (s 5 1), 17 equally spaced values ulation kurtosis, but the sample kurtosis of a realization of u between 08 and 1808 are considered. For the cases of of a Gaussian random field, and the sample kurtosis may two wave systems with different peak periods and dif- be a biased estimator for the real population kurtosis. ferent significant wave heights (s 5 0.75, 0.5, and 0.25), In fact, even for independent and identically distributed the five angles u 2f08,458,908, 1358, 1808g are consid- (IID) normal samples the expected value of the stan- (a) (b) ered. The peak directions fp and fp are chosen so that dard estimator for the sample kurtosis is biased so that (a) 5 (b) 52 8 8 5 2 1 fp u/2 and fp u/2, for u 2 [0 , 180 ]. E[k4] 3(n 1)/(n 1), where n is the number of samples The numerical simulations provide the full surface (here n 5 nxny 5 262 144). However, as further dis- elevation h(x, y) at every integration time step (i.e., cussed in the appendix, samples from a spatial snapshot about every 0.3 s), and the solution was stored every 60 s, of the surface elevation are not IID but contain values providing 31 snapshots of the surface in the spatial do- that are statistically dependent according to the relevant main of 5 km 3 5 km. All sea states were simulated with cross correlation of the sea surface, which in turn de- 20–50 repetitions with different initial random phases pends on the wave spectrum through the Wiener– and amplitudes each time. Hence, in total this corre- Khinchin theorem. This dependency has the effect that sponds to an area of about 500–1250 km2 of wave data the bias of the kurtosis estimator is significantly en- for each output time. hanced (Bai and Ng 2005), and deviation from the the- The focus of the present study is the occurrence oretical value of three must be expected, even for a of rogue waves, and in the following we discuss the Gaussian field. Note that the alternative formula for following two statistical parameters that provide in- sample kurtosis that is unbiased for IID samples also formation about the occurrence of extreme and rogue suffers from this bias. Moreover, the magnitude of the waves in a wave field: the sea surface kurtosis k4 bias depends on the cross correlation of the field so

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FIG. 6. Expected value of hmax/Hs according to Piterbarg theory as a function of crossing angle u for different values of the wavelength and significant wave-height ratio s. Solid lines with symbols show the corresponding results from linear numerical simulations, for the case s 5 1. The transparent solid lines show the 95% confidence intervals for the linear numerical results: (a) g 5 1, N 5 4; (b) g 5 3.3, N 5 16; and (c) g 5 6, N 5 100. that a more narrow wave spectrum will give a stronger value distributions of homogeneous Gaussian fields. bias. However, as shown in the appendix, this effect is Hence, the results based on the Piterbarg theory must be most pronounced for small simulation domains, so for viewed as an approximation, which is approximately the present numerical results, which applies to a rela- valid for linear wave fields. tively large computational area, this effect is of limited For crossing sea states, it is clear that lwave and lcrest importance, but it highlights the importance of having a depend on the crossing angle of the two wave systems, sufficiently large computational domain. which will affect the expected value for hmax through the The theoretical treatment of the maximum wave el- relation in (12). In particular, it is obvious that the typ- evation hmax is more complicated, even in the linear ical crest length in the wave field will be shorter for case. When considering a finite surface area, as in the crossing angles close to 908. This effect is shown in Fig. 6, present numerical simulations, the expected value for which shows the expected value E[hmax/Hs], according hmax naturally depends on the characteristics of the sea to the Piterbarg theory, for the different types of cross- state, in particular on the typical number of waves in ing sea states considered in this study. Thus, from purely the domain. This, in turn, depends on the average linear considerations a higher maximum wave elevation values for the wave and crest length of the wave field. is expected for crossing sea states with crossing angles

These effects are discussed in the context of Gaussian close to 908, while the lowest values for hmax are ex- random fields in general by Piterbarg (1996),andinthe pected for co- and counterpropagating wave systems. It context of ocean surface waves by, for example, is also clear from the figure that the expected value of

Krogstad et al. (2004). It can be shown that the typical hmax/Hs is reduced when the two wave systems differ in number of waves Nw in a rectangular domain of size V peak frequency and energy (s , 1). The theoretical re- can be estimated by sults are qualitatively confirmed by linear numerical simulations, shown by solid lines with symbols in Fig. 6. pffiffiffiffiffiffi 5 V The quantitative difference between the Piterbarg the- Nw 2p , (11) lwavelcrest ory and the linear simulations is due to the fact that the Piterbarg theory is, as mentioned above, based on ap- where lwave and lcrest are the mean wave and crest lengths, proximations as well as due to sampling variability in the respectively, which can be derived from the wave vector numerical results obtained from a limited number of spectrum E(k) as explained in Krogstad et al. (2004).It simulations. can further be shown that the expected value for hmax can be estimated as c. Results from nonlinear simulations ÂÃqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀÁ ÂÃÀÁ Keeping in mind the results from the theoretical in- 5 1 E hmax/Hs log Nw log 2 log Nw . (12) stability analysis of section 2, as well as the linear results of the previous section, we now turn to the nonlinear Note that the Piterbarg theory, on which the above ex- case and present results from the numerical simulations pressions are based, considers the asymptotic extreme using the HOSM model. The main focus is how k4 and

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FIG. 7. Results from the nonlinear HOSM simulations, as a function of the angle u between two identical wave systems (s 5 1). All results are averaged over many independent runs using different random phases and amplitudes of the initial spectra. Symbols show the actual simulated cases, while solid lines show the line fitted by GP regression. The transparent lines show the corresponding 95% confidence intervals for the estimated means. (a) Sea surface kurtosis. (b) Maximum crest elevation hmax normalized by the significant wave-height (p) Hs. (c) Maximum crest elevation hmax normalized by the expected value according to Piterbarg theory hmax (as shown in Fig. 6).

hmax depend on both the angle u between two crossing approximately u 5 458 for the case g 5 6, N 5 100. This 5 5 (a) (b) wave systems and the ratio s kb/ka Hs /Hs of the is remarkably consistent with previous works (Onorato wavelengths and significant wave heights of the two et al. 2010; Toffoli et al. 2011) that have suggested that a systems, as well as on other properties of crossing maximum probability of rogue waves is obtained for sea states. 408,u , 608. It was suggested by Onorato et al. (2010) To visualize how the statistical properties depend si- that this feature could be explained from the modula- multaneously on u and s, Gaussian process (GP) re- tional instability of crossing waves and that this range of gression has been used to fit surfaces in the u–s plane angles represents a compromise between large growth representing the best fit to the simulated values of k4 and rates and large amplification factors of the instability. hmax. This approach is useful both with respect to visu- Similar results have also been obtained previously from alization of the results and also for estimating the un- numerical simulations of two nearly unidirectional certainty related to lack of observations in regions of the crossing wave systems. The present results indicate that u–s plane where few simulations are available. The GP this feature is relevant only for relatively narrowbanded regression is performed so that the resulting fitted sur- crossing systems. However, one should note that it is faces goes exactly through the actual values obtained likely that this conclusion may also depend on the wave from the numerical simulations. Similarly, correspond- steepness of the two crossing wave systems, which is not ing surfaces are fitted to the 95% confidence intervals, investigated here. which are calculated using the standard deviation over For the maximum crest height hmax shown in Fig. 7b, runs with different random seeds. no clear dependence of the crossing angle or differences

Figure 7 shows how k4 and hmax depend on the angle between the different types of spectra are observed. between the two wave systems, for the case that the two When interpreting this result in light of the linear results systems have the same peak frequency (i.e., s 5 1), for discussed above, however, it is clear that the nonlinear the three different types of crossing sea states listed in contribution has an opposite effect compared to the Table 1. linear effects shown in Fig. 6. This is confirmed by (p) (p) It is evident from Fig. 7a that the kurtosis is larger for Fig. 7c, which shows hmax/hmax, where hmax is the ex- relatively small and relatively large angles and attains a pected value of hmax according to the linear Piterbarg minimum around u 5 908. Note also that, consistent with theory [(12)], shown in Fig. 6. Thus, nonlinearities seem previous works on crossing seas (Bitner-Gregersen and to have a similar effect on hmax as observed for the Toffoli 2014) as well as well-known results for unimodal kurtosis, but since the linear results due to Piterbarg wave systems (e.g., Gramstad and Trulsen 2007; Janssen theory are to a large extent opposite to the effects of

2003; Mori and Janssen 2006; Mori et al. 2011), kurtosis nonlinearity, the actual observed hmax in the nonlinear is higher for the most narrow spectrum (g 5 6, N 5 100) simulations is almost independent of the crossing angle. than for the more broad and directionally spread cases. For all three types of crossing seas shown in Table 1, One should also notice the local peak in kurtosis at several cases were run for which the two crossing systems

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(p) FIG. 8. (top) Kurtosis, (middle) hmax/Hs, and (bottom) hmax/hmax as a function of wavenumber and significant wave-height separation ratio s for different angles u between the wave systems. Results are shown for g 5 1, N 5 4 (blue line with crosses), g 5 3.3, N 5 16 (red line with circles), g 5 6, N 5 100 (green line with asterisks). Symbols show actual simulated cases. Solid lines show the line fitted by GP regression. Transparent lines show the corresponding 95% confidence intervals of the estimated means. have different peak frequency and significant wave height namely, that increasing the separation frequency tends to in addition to different propagation angles. The results decrease the observed value of hmax. from these simulations are shown in Fig. 8, which shows For the most broadbanded case (g 5 1, N 5 4), the (p) how k4 and hmax depend on the wavenumber and signifi- picture is less clear, and both the kurtosis and hmax/hmax 5 5 (a) (b) cant wave-height separation ratio s kb/ka Hs /Hs show a flat, and in some cases even increasing, trend for for different crossing angles. decreasing values of s. However, hmax/Hs is still de- The general feature of Fig. 8 is that for the two most creasing with decreasing s, likely because of the fact that narrow spectra (cases B and C in Table 1) both the the linear effects dominate in this case. kurtosis and hmax show a decreasing trend when the two wave systems become more separated in frequency 4. Comparison of instability analysis and results space (corresponding to decreasing values for s). How- from numerical simulations ever, for several crossing angles the magnitude of both kurtosis and hmax show almost no change for s $ 0.5. For Given the results of the previous sections, it is now of the largest separation (s 5 0.25), however, lower values interest to discuss the statistical results from the nu- for kurtosis and hmax are consistent for most angles. This merical simulations in light of the theoretical stability indicates that the nonlinear effects shown in Fig. 7, analysis in section 2. In fact, it is interesting to observe which tend to increase the occurrence of rogue waves, is that the curves of unstable growth rate in Fig. 5 share relevant also for wave systems that are moderately some common characteristics with Figs. 7a and 7c, separated in frequency space. Note also that, as shown in showing the kurtosis and normalized maximum crest (p) Fig. 6b, both the effect of nonlinearity and the linear ef- elevation hmax/hmax, in the sense that these quantities are fects due to Piterbarg theory have the same consequence, largest for small and large separation angles and have a

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FIG. 9. Relation between the statistical parameters from the numerical simulations presented in section 3 and the unstable growth rate from the instability analysis. The growth rate is found by using the same value of u and s as in the numerical simulations. Straight lines show linear regression fits. minimum for angles around 908. Although this similarity For the most broadband case (g 5 1, N 5 4) there is no seems to be most pronounced for the most narrow statistically significant correlation between the growth rate 5 5 (p) spectrum (g 6, N 100), some resemblance is seen and the kurtosis or hmax/hmax, indicating that, as one might also for the two broader cases. expect, as the spectrum becomes broader, the effect of the This is confirmed in Fig. 9, which shows the relation modulational instability is suppressed. This feature is well between the unstable growth rate from the instability known in unimodal sea states, and it is not surprising that analysis and the statistical results from the numerical this also applies to crossing seas, which is also consistent simulations, where the growth rate is taken from the with the theoretical work of Shukla et al. (2007). case of two Stokes waves with the same crossing angle One should also note that for the two broadest cases and the same amplitude ratio as used in the corre- (cases A and B in Table 1), there is a positive correlation sponding numerical simulations (i.e., same values for between the unstable growth rate and hmax/Hs. This is u and s). It is clear from the figure that for the two most due to the fact that the Piterbarg theory predicts de- narrow cases (cases B and C in Table 1) there is a pos- creasing hmax with decreasing s (see Fig. 6), and simi- itive correlation between the unstable growth rate and larly, the modulational instability predicts decreasing both the kurtosis and the maximum surface elevation growth rate for decreasing s (see Fig. 5). However, since normalized by the expected value from the linear theory the Piterbarg effect is a purely linear effect while the (p) hmax/hmax. However, because of the effects discussed in modulational instability is a nonlinear effect, these two section 3b, no clear correlation is observed between the effects are not related, and this correlation does not maximum growth rate and the actual observed maxi- represent a causal relationship. mum crest height hmax/Hs. These conclusions are also confirmed by Fig. 10, which shows the corresponding 5. Conclusions correlation coefficients together with the p value for the statistical test that the correlation is different The motivation for this work is several previous in- from zero. vestigations suggesting that crossing sea states may lead

FIG. 10. Calculated correlation coefficients corresponding to the plots in Fig. 9. White cells correspond to cases where the correlation is not statistically significant different from 0 (p . 0.05).

Unauthenticated | Downloaded 09/24/21 08:13 PM UTC JUNE 2018 G R A M S T A D E T A L . 1329 to increased probability of rogue waves. In particular, it states. Namely, the fact that the typical crest and wave has been suggested that more rogue waves may be ex- lengths in a crossing sea state strongly depend on the pected if two wave systems with similar peak frequency crossing angle and frequency difference of the two wave and similar energy propagate with crossing angle close systems, and hence influence the expected value of, for to 508. example, hmax even in the linear case. Interestingly, this In this paper, several types of crossing sea states have effect acts in opposite direction to the nonlinear effects, been simulated numerically, investigating the effects of so that hmax in a nonlinear crossing sea state seems to be crossing angle, differences in peak frequency and en- almost independent of both the crossing angle and the ergy, and the influence of the spectral shape of two spectral bandwidth of the crossing systems. crossing systems. For narrowband and almost unidirec- It should be mentioned that the nonlinear effects in tional crossing wave systems with the same peak fre- crossing sea states depend on the average wave steep- quency and the same energy, our numerical results nesses of the two wave systems. In this paper only one confirm an increased sea surface kurtosis for crossing value for the wave steepness of the two crossing wave angles close to 508, although even higher values of kur- systems is considered, with the restriction that the wave tosis are observed for very small crossing angles, where steepnesses of the two systems are equal. Hence, for an the resulting total spectrum essentially is a unimodal even more complete picture of the effect of crossing sea narrow spectrum. Minimum kurtosis is observed for states on wave statistical properties, a larger set of crossing angles close to 908, with larger values for both steepnesses should be investigated, including the effect smaller and larger separation angles. of varying the steepnesses of the two wave systems Similar results are also obtained for more realistic independently. broadband and directionally spread crossing wave sys- tems, which also show a minimum kurtosis for crossing Acknowledgments. This work is funded by the Research angles close to 908 and largest values for small and large Council of Norway Grant 256466. crossing angles. However, in these more-broadband cases, the local peak in kurtosis for crossing angles of APPENDIX about 508 does not appear. Further, the nonlinear simulations show that kurtosis Effect of Domain Size on Estimation of Sea is generally decreasing when two wave systems become Surface Kurtosis more separated in frequency space. However, for rela- tively weak separation in frequency space the kurtosis is An effect that, to our knowledge, has to a large extent only slightly reduced. been overlooked in studies on statistical properties of Interestingly, the dependence of kurtosis on crossing waves is the fact that a realization of a sea surface in a angle and frequency difference in the numerical sim- limited spatial area (or a realization of a time series of a ulations is in qualitative agreement with the results limited duration) contains a large number of dependent from the analysis of the modulational instability of two statistical samples compared to overall number of sam- crossing Stokes waves, in the sense that the largest growth ples. In such a case, the expected values of the estimates rate is observed for large and small crossing angles, with a of, for example, the kurtosis and skewness differ from minimum around 908, and that increasing difference in their theoretical values for a linear sea surface. In fact, the frequency of the two waves results in smaller growth the effect of statistical dependence in a realization of a rates. This correlation between the maximum growth rate linear sea surface will tend to underestimate the kurtosis of two interacting Stokes waves and the simulated kur- (Bai and Ng 2005; Bao 2013) because the estimators for tosis is observed also for the relatively broadband cross- kurtosis and skewness are biased for dependent data of ing system with g 5 3.3, N 5 16, but not for the broadest limited sample sizes. spectrum with g 5 1, N 5 4. This suggests that for broader Figure A1a shows the estimated kurtosis of linear spectra the effect of the modulational instability is realizations of a sea state with peak period Tp 5 10 s, weakened; however, the results suggest that nonlinear calculated as a spatial average over two different do- modulational-instability effects due to the interaction of main sizes: a 5 km 3 5 km area discretized with nx 3 ny 5 the two wave systems play a role also in relatively 512 3 512 points and a 1.25 km 3 1.25 km area dis- broadband, realistic types of crossing sea states. cretized with nx 3 ny 5 128 3 128 points. For neither of For wave-statistical properties that depend on the the two domain sizes does the kurtosis converge ex- average size of the individual waves in a wave field, such actly to the true value of three with increasing number as the maximum crest height hmax in a given area, there of realizations, and for the smallest domain size, the is another effect that also plays a role in crossing sea kurtosis is underestimated by about 3%–4%. This is also

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FIG. A1. (a) Kurtosis calculated as a spatial average over a linear sea surface as a function of the number of realizations used to estimate the kurtosis. (b) Histogram of the individual kurtosis estimates of the 1000 realizations used in the average in (a). (c) Estimated kurtosis calculated from time series of linear surface elevation corresponding to two different JONSWAP sea states, averaged over 100 000 random realizations for each sea state. illustrated by the plot in Fig. A1b, which shows that the Bai, J., and S. Ng, 2005: Tests for skewness, kurtosis, and normality distributions of the sample kurtosis are skewed with for time series data. J. Bus. Econ. Stat., 23, 49–60, https:// mean below three. doi.org/10.1198/073500104000000271. Bao, Y., 2013: On sample skewness and kurtosis. Econometric Rev., The same effect is also relevant for kurtosis calculated 32, 415–448, https://doi.org/10.1080/07474938.2012.690665. from time series of linear surface elevation, as illustrated Benetazzo,A.,F.Barbariol,F.Bergamasco,A.Torsello,S.Carniel, in Fig. A1c, which shows the averaged kurtosis obtained and M. Sclavo, 2015: Observation of extreme sea waves in a from 100 000 random time series realizations of two space–time ensemble. J. Phys. Oceanogr., 45, 2261–2275, https:// different sea states. Consistent with the results in Bai doi.org/10.1175/JPO-D-15-0017.1. ——, and Coauthors, 2017: On the shape and likelihood of and Ng (2005), the kurtosis is underestimated for time oceanic rogue waves. Sci. Rep., 7, 8276, https://doi.org/10.1038/ series of short duration and only slowly approaches s41598-017-07704-9. three with increasing duration. Note also that, as shown Benjamin, T. B., and J. E. Feir, 1967: The disintegration of wave in Bai and Ng (2005), the bias of the sample kurtosis will trains on deep water: Part 1. Theory. J. Fluid Mech., 27, 417– naturally depend on the serial dependence in the wave 430, https://doi.org/10.1017/S002211206700045X. Bitner-Gregersen, E., and A. Toffoli, 2014: Occurrence of rogue field. Consequently, a narrowband wave field with a long sea states and consequences for marine structures. Ocean Dyn., correlation length will give a stronger underestimation 64, 1457–1468, https://doi.org/10.1007/s10236-014-0753-2. of the kurtosis compared to a more broadband wave ——, and O. Gramstad, 2016: Rogue waves: Impact on ships and field with a shorter correlation length. This effect is offshore structures. Det Norske Veritas Germanischer shown in Fig. A1c by the fact that the JONSWAP Lloyd Strategic Research and Innovation Position Paper 5 5-2015, 60 pp. spectrum with g 7 has lower kurtosis than the more Cavaleri, L., L. Bertotti, L. Torrisi, E. Bitner-Gregersen, M. Serio, broadband spectrum with g 5 1. and M. Onorato, 2012: Rogue waves in crossing seas: The Hence, one should be aware that the kurtosis, which is Louis Majesty accident. J. Geophys. Res., 117, C00J10, https:// often used as an indicator for effect of nonlinearity in a doi.org/10.1029/2012JC007923. sea state, may be underestimated in numerical simula- Christou, M., and K. Ewans, 2014: Field measurements of rogue water waves. J. Phys. Oceanogr., 44, 2317–2335, https:// tion or field measurements when using a small compu- doi.org/10.1175/JPO-D-13-0199.1. tational domain or a short time series, purely because of Crawford, D. R., P. G. Saffman, and H. C. Yuen, 1980: Evolution the effect of bias in the estimators for kurtosis for de- of a random inhomogeneous field of nonlinear deep-water pendent samples. gravity waves. Wave Motion, 2, 1–16, https://doi.org/10.1016/ 0165-2125(80)90029-3. ——, B. M. Lake, P. G. Saffman, and H. C. Yuen, 1981: Stability REFERENCES of weakly nonlinear deep-water waves in two and three dimensions. J. Fluid Mech., 105, 177–191, https://doi.org/ Adcock, T. A. A., and P. H. Taylor, 2014: The physics of anomalous 10.1017/S0022112081003169. (‘‘rogue’’) ocean waves. Rep. Prog. Phys., 77, 105901, https:// Dommermuth, D. G., and D. K. P. Yue, 1987: A high-order spectral doi.org/10.1088/0034-4885/77/10/105901. method for the study of nonlinear gravity waves. J. Fluid Mech., Alber, I. E., 1978: The effects of randomness on the stability of 184, 267–288, https://doi.org/10.1017/S002211208700288X. two-dimensional surface wavetrains. Proc. Roy. Soc. London, Fedele, F., J. Brennan, S. Ponce de León, J. Dudley, and F. Dias, 363A, 525–546, https://doi.org/10.1098/rspa.1978.0181. 2016: Real world ocean rogue waves explained without the

Unauthenticated | Downloaded 09/24/21 08:13 PM UTC JUNE 2018 G R A M S T A D E T A L . 1331

modulational instability. Sci. Rep., 6, 27715, https://doi.org/ ——, ——, and ——, 2006: Modulational instability in crossing 10.1038/srep27715. sea states: A possible mechanism for the formation of freak Gramstad, O., and K. Trulsen, 2007: Influence of crest and group waves. Phys.Rev.Lett., 96, 014503, https://doi.org/10.1103/ length on the occurrence of freak waves. J. Fluid Mech., 582, PhysRevLett.96.014503. 463–472, https://doi.org/10.1017/S0022112007006507. ——, and Coauthors, 2009: Statistical properties of mechanically ——, and ——, 2010: Can swell increase the number of freak waves generated surface gravity waves: A laboratory experiment in a in a wind sea? J. Fluid Mech., 650, 57–79, https://doi.org/ three-dimensional wave basin. J. Fluid Mech., 627, 235–257, 10.1017/S0022112009993491. https://doi.org/10.1017/S002211200900603X. ——, and ——, 2011: Fourth-order coupled nonlinear Schrödinger ——, D. Proment, and A. Toffoli, 2010: Freak waves in crossing equations for gravity waves on deep water. Phys. Fluids, 23, seas. Eur.Phys.J.Spec.Top., 185, 45–55, https://doi.org/ 062102, https://doi.org/10.1063/1.3598316. 10.1140/epjst/e2010-01237-8. Janssen, P. A. E. M., 2003: Nonlinear four-wave interactions and Piterbarg, V. I., 1996: Asymptotic Methods in the Theory of Gaussian freak waves. J. Phys. Oceanogr., 33, 863–884, https://doi.org/ Processes and Fields. American Mathematical Society, 206 pp. 10.1175/1520-0485(2003)33,863:NFIAFW.2.0.CO;2. Roskes, G. J., 1976: Nonlinear multiphase deep-water wavetrains. ——, and M. Onorato, 2007: The intermediate water depth limit of Phys. Fluids, 19, 1253–1254, https://doi.org/10.1063/1.861609. the Zakharov equation and consequences for wave predic- Shukla, P. K., I. Kourakis, B. Eliasson, M. Marklund, and tion. J. Phys. Oceanogr., 37, 2389–2400, https://doi.org/10.1175/ L. Stenflo, 2006: Instability and evolution of nonlinearly in- JPO3128.1. teracting water waves. Phys.Rev.Lett., 97, 094501, https:// Krasitskii, V. P., 1994: On reduced equations in the Hamiltonian doi.org/10.1103/PhysRevLett.97.094501. theory of weakly nonlinear surface waves. J. Fluid Mech., 272, ——, M. Marklund, and L. Stenflo, 2007: Modulational instability 1–20, https://doi.org/10.1017/S0022112094004350. of nonlinearly interacting incoherent sea states. JETP Lett., Krogstad, H. E., J. Liu, H. Socquet-Juglard, K. B. Dysthe, and 84, 645–649, https://doi.org/10.1134/S0021364006240039. K. Trulsen, 2004: Spatial extreme value analysis of nonlinear Toffoli, A., E. M. Bitner-Gregersen, A. R. Osborne, M. Serio, simulations of random surface waves. Proc. 23rd Int. Conf. J. Monbaliu, and M. Onorato, 2011: Extreme waves in random on Offshore Mechanics and Arctic Engineering, Vancouver, crossing seas: Laboratory experiments and numerical simula- BC, Canada, American Society of Mechanical Engineers, tions. Geophys. Res. Lett., 38, L06605, https://doi.org/10.1029/ OMAE2004-51336, https://doi.org/10.1115/OMAE2004-51336. 2011GL046827. Mori, N., and P. A. E. M. Janssen, 2006: On kurtosis and occur- Trulsen, K., J. C. Nieto Borge, O. Gramstad, L. Aouf, and J. M. rence probability of freak waves. J. Phys. Oceanogr., 36, 1471– Lefèvre, 2015: Crossing sea state and rogue wave probability 1483, https://doi.org/10.1175/JPO2922.1. during the Prestige accident. J. Geophys. Res. , 120, ——, M. Onorato, and P. A. E. M. Janssen, 2011: On the estimation 7113–7136, https://doi.org/10.1002/2015JC011161. of the kurtosis in directional sea states for freak wave fore- West, B. J., K. A. Brueckner, R. S. Janda, D. M. Milder, and R. L. casting. J. Phys. Oceanogr., 41, 1484–1497, https://doi.org/ Milton, 1987: A new numerical method for surface hydrody- 10.1175/2011JPO4542.1. namics. J. Geophys. Res., 92, 11 803–11 824, https://doi.org/ Okamura, M., 1984: Instabilities of weakly nonlinear standing 10.1029/JC092iC11p11803. gravity waves. J. Phys. Soc. Japan, 53, 3788–3796, https://doi.org/ Xiao, W., Y. Liu, G. Wu, and D. K. P. Yue, 2013: Rogue wave 10.1143/JPSJ.53.3788. occurrence and dynamics by direct simulations of nonlinear Onorato, M., A. R. Osborne, M. Serio, and S. Bertone, 2001: Freak wave-field evolution. J. Fluid Mech., 720, 357–392, https:// waves in random oceanic sea states. Phys. Rev. Lett., 86, 5831– doi.org/10.1017/jfm.2013.37. 5834, https://doi.org/10.1103/PhysRevLett.86.5831. Yuen, H. C., and B. M. Lake, 1982: Nonlinear dynamics of deep- ——, ——, ——, L. Cavaleri, C. Brandini, and C. T. Stansberg, water gravity waves. Adv. Appl. Mech., 22, 67–229, https:// 2004: Observation of strongly non-Gaussian statistics for doi.org/10.1016/S0065-2156(08)70066-8. random sea surface gravity waves in wave flume experi- Zakharov, V. E., 1968: Stability of periodic waves of finite ampli- ments. Phys. Rev., 70E, 067302, https://doi.org/10.1103/ tude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys., PhysRevE.70.067302. 9, 190–194, https://doi.org/10.1007/BF00913182.

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