Wave amplification in the framework of forced nonlinear Schrödinger equation: The rogue wave context
Introduction
In this paper we consider the evolution of ensembles of irregular nonlinear waves, and also the dynamics of envelope solitons subjected to external forcing. This problem appears in different realms and has a general physical significance. In the present study we focus on the most simplified setting, when narrow-banded waves propagate along one coordinate; the evolution equation includes terms of weak linear dispersion and nonlinear four-wave interactions. The evolution of the conservative system is governed by the nonlinear Schrödinger (NLS) equation of focusing type, which supports existence of localized stable wave groups — ‘bright’ envelope solitons; it is known to be integrable by means of the Inverse Scattering Technique after Zakharov and Shabat [1]. The external action on the wave system is taken into account through an additional term of linear forcing/damping, which in dimensionless form is of the order of magnitude similar to the squared wave steepness. This modification violates integrability of the NLS equation, thus approximate or numerical approaches become preferable.
So-called rogue waves are a particular case when the nonlinear Schrödinger theory sheds light on the physical mechanisms, responsible for a sudden occurrence of unexpectedly high waves. Rogue waves form a specific class of extreme waves which occur much more frequent than it is prescribed by the statistics of the random Gaussian process. The role of the modulational (Benjamin–Feir) instability in generation of rogue waves on the sea surface is generally accepted at present (see reviews Kharif and Pelinovsky [2], Dysthe et al. [3], Kharif et al. [4], Slunyaev et al. [5]). A recent survey of rogue wave phenomena in various media supporting nonlinear wave self-modulation may be found in Onorato et al. [6]. The theory for modulational instability of wind driven plane waves which is described by the forced NLS equation was developed by Leblanc [7], though the opposite problem of stability of weakly damped waves was discussed in Segur et al. [8]. The Benjamin–Feir instability of uniform waves may become explosive in the case of external pumping. In the context of water waves the NLS equation with forcing term was discussed in paper by Onorato and Proment [9] (see also references therein). More general extensions of the nonlinear Schrödinger equation in application to the wind driven surface waves and possible mechanisms of wave growth saturation were considered by Fabrikant [10]. The theory under assumption of stronger wind, which is of the same order as wave steepness, was developed by Brunetti et al. [11]. New terms which appear in that modification of the NLS equation may be eliminated after appropriate change of variables, and then the equation turns to the form considered in Leblanc [7], Kharif et al. [12], Onorato and Proment [9].
Numerical simulations of individual wave groups affected by wind were performed within different frameworks in Yan and Ma [13], Adcock and Taylor [14]. Laboratory and numerical simulations of transient and modulated wave trains in the presence of wind were conducted by Touboul et al. [15], Kharif et al. [16], [12], Chabchoub et al. [17].
It is important that most of statistical theories are based on assumptions of stationarity and proximity to gaussianity. Meanwhile, strongly non-stationary situations seem prone to higher probability of extreme waves, such as: the transition from non-equilibrium condition to the quasi-stationary one [18], [19], [20], [21], [22], rapidly changing winds [21], [23], change of the waveguide properties [24], [25], [26], [27], [28], [29]. The problem of statistical description of these situations is still challenging.
Higher probability of extreme waves in random fields relates to large values of the fourth statistical cumulant, kurtosis, which may be split into two parts: the dynamical kurtosis due to resonant and near-resonant interactions and the kurtosis due to non-resonant interactions (bound waves) [30], [21]. The transition regimes of intense irregular waves observed e.g. by Annenkov and Shrira [21], Shemer et al. [22] were accompanied by an increase of the dynamical kurtosis, which constituted the main part of total kurtosis. In contrast, in typical for sea states quasi-stationary conditions the kurtosis due to bound waves seems to prevail [21]. In Slunyaev [31], Slunyaev and Sergeeva [32] the increase of dynamical kurtosis was associated with formation of soliton-like wave patterns due to coherence in wave harmonics, which obviously breaks the assumption of random wave phases.
In this paper we consider how time-limited pumping changes ensembles of irregular nonlinear waves and in particular their statistics. Since the concerned problem is complicated, the role of constrained by nonlinearity wave groups (envelope solitons) is studied in parallel with the help of supplementary numerical simulations. We consider situations of different ratios between the two characteristic time scales which may be singled out: the effective time of nonlinearity and the characteristic time of energy input. In Section 2 the problem statement is given, and the key controlling parameters are formulated. In Section 3 the results of simulations of irregular waves are reported and discussed from the general viewpoint. They are interpreted later with the help of auxiliary numerical experiments described in Section 4, where deterministic envelope solitons are simulated. The solution of the Cauchy problem for the integrable NLS equation in terms of the Inverse Scattering Technique, and the asymptotic solution of the perturbed NLS equation yield qualitative understanding of the observed nonlinear phenomena; they provide with approximate formulas. We discuss the main conclusions including the relation to rogue waves and their forecasting at the end.
Section snippets
Problem statement and controlling parameters
The framework adopted in this study is the modified nonlinear Schrödinger equation, which may be also considered as a particular case of the complex Ginzburg–Landau equation or more generalized equations for wind driven waves [10], [33], [11]. For the situation of gravity waves over deep water the equation reads where the last term is responsible for damping/pumping. In what follows we assume that the system is conservative or experiences energy input
No pumping (conservative system)
There are numerous studies both in numerical and laboratory frameworks dealing with investigations of irregular wave evolution in conservative (or with negligible dissipation) systems (among others, Onorato et al. [35], [36], [20], Janssen [18], Socquet-Juglard et al. [19], Gramstad and Trulsen [37], Shemer and Sergeeva [38], Shemer et al. [34], [22], Mori et al. [39]). The general picture of this problem when waves propagate in one direction was depicted in Slunyaev and Sergeeva [32]. The
Fast and adiabatic growth of solitary waves
It follows from Section 3 that the adiabatic pumping results in effective formation of intense and short soliton-like wave groups, which are readily seen in Fig. 5. The following numerical experiments (series S in Table 1) aim at consideration of a purified situation, when only one envelope soliton, the exact solution of the NLS equation, experiences external time-limited pumping. This problem was considered by Adcock and Taylor [14] in case of an adiabatically slow change of an envelope
Discussion
This research is aimed at investigation of the evolution of irregular nonlinear waves affected by external forcing. As a particular case, the target setting should model long-crested deep water waves driven by wind due to Miles’ mechanism. We restrict our interest to only one very simplified model, the unidirectional nonlinear Schrödinger equation with linear forcing term. The ultimate benefit of this equation is the possibility of approximate transformation of its solutions to the framework of
Acknowledgements
The authors were supported by RFBR grants 14-02-00983 and 14-05-00092, and also State Contract 2014/133 (project 2839). The support from Volkswagen Foundation is gratefully acknowledged.
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