JUNE 2018 G R A M S T A D E T A L . 1317
Modulational Instability and Rogue Waves in Crossing Sea 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 sea state 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 ocean 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 seas 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 swell. 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 Stokes wave) 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 OCEANOGRAPHY 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 wavelength) 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 wavenumbers 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 rogue wave 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 PHYSICAL OCEANOGRAPHY 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