Extreme wave events at the initial stage of the modulational instability in a forced and damped wave medium
EExtreme wave events at the initial stage of themodulational instability in a forced and dampedwave medium
E V Belkin, A V Kirichok , V M Kuklin Kharkov National University, Institute for High Technologies, 4 Svobody Sq.,Kharkov 61077, UkraineE-mail: [email protected]
Abstract.
The modulational instability of waves in a medium under the action of anexternal monochromatic force and dissipation is considered. The model which describesthe nonlinear stage of the modulation instability was constructed with using methodsof so-called S -theory. Following this theory, we take into account only interactionsbetween modes with slowly varying relative phases. Within framework of this model,it is shown that the waves satisfying the criterion of abnormality appears as a ruleat the initial stage of the instability. With further development of the process, theabnormal waves can be also observed but their amplitudes decrease gradually withdecreasing of the average wave height. Keywords : modulational instability, extreme waves, S-theory
PACS numbers: 41.60.-m, 52.35.-g
Submitted to:
Phys. Scr. a r X i v : . [ n li n . PS ] N ov xtreme wave events at the initial stage of the modulational instability...
1. Introduction
In recent years, the attention of researchers was attracted to the phenomenon of large-amplitude short-lived waves observed in various nonlinear dispersive wave media. Thesewaves was named as freaks or rogue waves. Of particular interest are the experimentalobservations of extreme ocean waves. A complete review on the various phenomenayielding to rogue waves in ocean can be found in the book [1]. Later, it has beenfound experimentally that freak waves can be generated in optical systems [2], [3] andin space plasma [4]. A possibility of generation of ion-acoustic freak waves in plasmawith negative ions and generation of Alfv´en freak waves in astrophysical plasma wasconsidered in [5].Discussion about the possible mechanisms for the formation of freak waves haspersisted for all last years. Now, the modulation instability is considered as mainpretender on this role [6]. The phenomenon of modulational instability of finiteamplitude wave was first investigated in [7], [8], [9]. In the conservative media, thatis in the absence of external energy sources and mechanisms of energy absorption,the final stage in the development of modulational instability is the formation of atrain of solitary waves or regimes with peaking (collapse) [10], [12]. That’s why thereare persistent attempts to describe the development of modulational instability usingmodified in various ways solutions relating to conservative systems (see, for example[13]). However, this approach may not necessarily be constructive, since it does nottake into account the rather active and long transition process of the formation ofstable wave structures. What is more, the stability of these structures can be achievedonly in narrow ranges of parameters.Recently, Segur et al [14] and Wu et al [15] shown that dissipation stabilizes themodulational instability. In the wavenumber space, the region of instability shrinksas time increases. This means that any initially unstable mode of perturbation doesnot grow for ever. Damping can stop the growth of the side-bands before nonlinearinteractions become important. Hence, when the perturbations are small initially, theycannot grow large enough for nonlinear resonant interaction between the carrier and theside-bands to become important. The amplitude of the side-bands can grow for a whileand then oscillate in time. Kharif et al [16] considered the modulational instability ofStokes wavetrains suffering both effects of external forcing and dissipation. They foundthat the modulational instability depends on both frequency of the carrier wave andstrength of the external force.This work shows that the presence of energy flow through such an open systemresults in complete or partial suppression or delay the process of forming a train ofsolitary waves and creates a new metastable state. This state, at least at the initialstage of the instability (i.e. during only a few times greater than the inverse linearincrement of instability), characterized by the appearance of short-lived splashes ofmodulation of the fundamental wave with abnormally large amplitude. The averageamplitude of the fundamental wave therewith remains large enough. In the case of a xtreme wave events at the initial stage of the modulational instability... k → ω = ω ( k ) + ω ( − k ) and 2 k = 0 = k − k . Growingside-band spectrum acts back on the pump that leads to ”freezing” of its amplitude atthe threshold level and restriction of the instability.Further improvement of the theory [19] takes into account the weaker interactionbetween excited low-amplitude modes, and the main contribution to this interactionis provided by pairs of waves symmetric with respect to the pump ω ( k ) + ω ( − k ) = ω ( k (cid:48) )+ ω ( − k (cid:48) ), which ensure that mentioned above conditions of space-time synchronizmare satisfied for all interacting modes. Later, Zaharov and co-authors (see detailed review[20] and book [11]) have formulated the theory of the nonlinear stage of modulationalinstability based on this approach, which subsequently became known as S -theory.Within framework of this theory they come to description in the language of correlationfunctions (cid:104) A K A ∗ k (cid:48) (cid:105) = n K ∆( k − k (cid:48) ) and (cid:104) A K A k (cid:48) (cid:105) = σ K ∆( k + k (cid:48) ), taking into accountthe effect of total coupling (correlation) of phases ϕ K ϕ − k for modes synchronouslyinteracting with the uniform pump field and representing σ K = n K exp {− iψ K } .Among important results of this extremely constructive S -theory are the discovereddominant mechanism of instability saturation, which consist in back action of the excitedspectrum on the pump wave under condition of low above-threshold level, and growthof influence of the phase mismatch on the saturation of the instability with increasingabove-threshold level [21].The model, considered below, is a continuation of the study [22], where we haveanalyzed the evolution of individual modes in the spectrum of plasma oscillations in theprocess of modulational instability.In present work we study more carefully not only the dynamics of individualmodes but also the spatial-temporal evolution of wave packets (e.g. waves and theirenvelope). In addition, we analyze some specific futures of the instability, in particular,the symmetry breaking of the excited spectrum during the progress of the modulationalinstability in a medium with strong dispersion. This model, as shown below, allows todetect the formation of abnormally high peaks of field modulation and the appearanceof extremely high waves. xtreme wave events at the initial stage of the modulational instability...
2. Mathematical model
Consider, as an example, the modulational instability of externally driven wave in amedium with sufficiently strong dispersion and weak dissipation, which can be observedin plasma wave-guides, as well as on the water surface and other physical situations.Weak dissipation, as noted in [14], provides necessary stabilizing effect on the modulationof fundamental waves, as an external source provides the permanent energy transfer tothe wave motion.It is significant that here, such as in the above discussed cases, the amplitude ofexternally forcing wave retains large during the instability development and greatlyexceeds the amplitudes of the excited spatial spectrum. Therefore, in conditions offinite level of dissipation the side-band modes excited by the modulational instabilitywill interact with one other only if this interaction is supported by the fundamentalwave . In other words, the dominant regime of interaction, at least in the early stage ofthe nonlinear regime of the instability is the interaction of modes symmetrically locatedin (cid:126)k -space of the spectrum relative to the central mode of large-amplitude fundamentalwave.We use below the following dispersion relation that characteristic, as an example,of gravity waves on deep water [23]: ω = (cid:112) gk (cid:26) A k ... (cid:27) , (1)where g is some dimensional coefficient (for the gravity waves on deep water it is theacceleration of gravity).The equations for the field amplitude which satisfies the dispersion equation (1)can be represented as follows: ∂A K ∂t = − δA K − i (cid:112) g ( k + K ) A K − i (cid:112) g ( k + K ) ( k + K ) {| A | A } K == − δA K − i (cid:112) g ( k + K ) A K − i (cid:112) g ( k + K ) ( k + K ) ×× (cid:32) A K (cid:34) | A | + 2 (cid:88) κ (cid:54) = K, | A κ | + | A K | (cid:35) + A ∗− K (cid:34) A + (cid:88) κ (cid:54) = K, A κ A − κ (cid:35)(cid:33) , (2)where λ = 2 π/k is the wave-length of the fundamental wave, δ is the linear dissipationcoefficient.Excluding the fundamental frequency ω = √ gk : A K ∝ exp {− iω t + i ( k + K ) x + iϕ K } → A K ∝ exp { + i ( k + K ) x + iϕ K } , (3)we can rewrite Eq.(2): ∂A K ∂t = − δA K − i [ (cid:112) g ( k + K ) − (cid:112) gk ] A K − i (cid:112) g ( k + K ) ( k + K ) ×× (cid:32) A K (cid:34) | A | + 2 (cid:88) K (cid:48) (cid:54) = K, | A K (cid:48) | + | A K | (cid:35) + A ∗− K (cid:34) A + (cid:88) κ (cid:54) = K, A κ A − κ (cid:35)(cid:33) . xtreme wave events at the initial stage of the modulational instability... A K = | u K | exp( iϕ K ) we have obtained thesystem of equations which describes the modulational instability in a medium withstrong dispersion: ∂u K ∂τ = − δu K + (1 + K ) . (cid:34) u − K u sin Φ K + u − K (cid:88) κ (cid:54) = K, u κ u − κ sin(Φ κ − Φ K ) (cid:35) , (4)where Φ K = 2 ϕ − ϕ K − ϕ − K is the total phase (or the phase of the instability channel).A distinction needs to be drawn between modes with wave numbers K and κ and phasesΦ K and Φ κ . ∂ϕ K ∂τ = − α ( (cid:112) (1 + K ) − − (1 + K ) . (cid:20) u + u K + 2 (cid:88) K (cid:48) (cid:54) = K, u K (cid:48) ++ u − K u K u cos Φ K + u − K u K (cid:88) κ (cid:54) = K, u κ u − κ cos(Φ κ − Φ K ) (cid:35) , (5)where the mode frequencies are ω ( K ) − ω = ω [ k (1 + K )] − ω ( k ) = (cid:112) gk (1 + K ) −√ gk . We use the following notations k ξ = ζ , ω t/ τ /α , α = k | A | , τ = t √ gk ( k | A | ) / K = ( k − k ) /k , ω tK / τ / α ) K , A K /A ( τ = 0) = a K = u K exp { iϕ K } , and also∆ K = 2 (cid:16)(cid:112) (1 + K ) −
1] + (cid:112) (1 − K ) − (cid:17) /α, P K = 2(1 + K ) . + 2(1 − K ) . − . Note that radicands may be not expanded in further calculations.Equations for amplitude and phase of the fundamental wave take the form: ∂u ∂τ = − δu − u (cid:88) K (cid:54) =0 u K u − K sin Φ K + G, (6) ∂ϕ ∂τ = − u − (cid:88) K (cid:54) =0 u K − (cid:88) K (cid:54) =0 u K u − K cos Φ K . (7)Here G is an external source, supporting the monochromatic fundamental wave. Thesumming over K can be replaced by summing over K m = m ∆ K/k , where ∆ K =2 K max /N and m = ± (1 , , ..., N ) with K max /k = √ k | A | = √ α , N is the number ofmodes. Note that we consider direct external forcing rather than parametric as in [16].Equations (4)-(7) form a closed system that should be solved numerically.The wave profile in ζ -space looks as follows: E = exp { + iζ + iφ }{ u + (cid:88) i> (cid:20) u i exp (cid:18) − i τα ( √ K − − K ) + iKζ + i ( φ i − φ ) (cid:19) ++ u − i exp (cid:18) − i τα ( √ − K − K ) − iKζ + i ( φ − i − φ ) (cid:19)(cid:21) . (8) xtreme wave events at the initial stage of the modulational instability... Wave height U M A X U S W H U C P
T i m e
Figure 1.
Time evolution of wave height parameters: U CP - the average height of allwaves in the observation domain, U SW H - the average height of a highest third, U MAX - the maximum wave height [25].
Here, the fundamental wave is at rest and modulation of moves.Since the dispersion is sufficiently strong in the vicinity of the fundamental wavefrequency, one can readily see from above equations the symmetry breaking in excitationof the Stokes and anti-Stokes part of the spectrum, which can significantly change thecharacter of the modulation.
3. Numerical results
In order to analyze the wave height distribution (e.g. the distribution of verticaldistances between the wave crest and the deepest trough preceding or following thecrest ), we take a third of highest waves. Then we find the average height of allwaves U CP , average height of a highest third U SW H and the maximum wave height U MAX in consideration domain ( ζ ⊂ L = 2 π/ (∆ K/k ) = πN/K m = πN/ √ α , where∆ K = 2 K max /N , ζ = k x ). Calculations were performed for 600 modes in the spectrum.The ratio of dissipation level δ to the maximum growth rate was chosen as 0.1 (e.g. δ = 0 . G = δ = 0 . U AG > U SW H (9) xtreme wave events at the initial stage of the modulational instability... E x (a) - 1 5 0 - 1 0 0 - 5 0 0 5 0 1 0 0 1 5 0- 3 , 0- 2 , 5- 2 , 0- 1 , 5- 1 , 0- 0 , 50 , 00 , 51 , 01 , 52 , 02 , 53 , 0 E x (b) Figure 2.
Wave profile (see (8)) near extremely high waves for various realization ofthe process: (a) τ ∝
10; (b) τ ∝ or something like this, should be used with caution because this criterion as usual isapplied to statistics obtained on sufficiently large observation periods, but the highestamplitudes are observed at the initial stage of the instability.Nevertheless, much smaller wave amplitudes or heights also satisfy this criterionin the regime of developed instability, since a decrease in both medium and large waveamplitudes is observed with time. Let give two examples of realizations where somewaves satisfy the criterion (9) (see Fig.2).Analysis of observations and numerical simulation show [1] that extreme waveevents often occur in wave groups having the form of soliton-like structures.With decreasing dissipation level δ the processes of energy exchange between theside spectrum and the fundamental wave is amplified, which is clear seen in Figure 3.It can be seen that the maximum of instability growth rate shifts to lower valuesof K with decreasing amplitude of the fundamental wave at the initial stage ofinstability. Since the maximum growth rate corresponds to a value of the total phaseΦ K = 2 ϕ − ϕ K − ϕ − K equal to π/
2, the majority of mode pairs with different values of K are synchronized with the phase of the fundamental wave [26] during instability progress.The fact that the total phase Φ K is focused near the π/ ϕ K remain randomly distributed (inparticular, no symmetry of the form ϕ K (cid:54) = ϕ − K , ϕ K (cid:54) = − ϕ − K ), the instability spectrumsynchronized with the fundamental wave forms a different interference structure foreach realization. Nevertheless, the degree of coherence of excited modes achieves xtreme wave events at the initial stage of the modulational instability... d = 0.7d = 0.6d = 0.5d = 0.4d = 0.3 Intensity
T i m e
Figure 3.
The total intensity of side-band spectrum for different values of δ ( N = 100) the maximum at the initial stage of the instability. Incidentally, the phenomenonof synchronization of total phases in the spectrum of modulational instability in thepresence of the symmetry ϕ K = ϕ − K , usually leads to regimes with peaking [27]. Theabsence of the phase symmetry in pairs of interacting waves could weaken the intensityof interference peaks and reduce the time of their existence.Considering the dynamics of instability spectrum, it is possible to detect thephenomenon of its noticeable shift in relation to the spectral domain of the linearinstability. This shift can be explained by reduced amplitude of the fundamental wave.In addition, note that the amplitudes of individual modes of the spectrum remain muchsmaller than the amplitude of the fundamental wave. It should be noted also theasymmetry of the instability spectrum relative to the fundamental wave due to strongdispersion and large enough modulational instability growth rate for fundamental waves.Analyzing the instability spectra one can see that the modulation length increasesalmost 3.5 times with development of the instability. Evolution of the relativecharacteristic modulation length is shown in Figure 5.Dynamics of two-dimensional wave processes turns out to be similar. Thus, thenumber of waves on the modulation length at the initial stage of the instability is muchless than at the later stages of the process.Analyzing the probability of extreme wave events for different realizations of theprocess, we have found that one such wave appears among the 10000 waves, which is xtreme wave events at the initial stage of the modulational instability... Figure 4.
The spectrum of instability at τ ∝
10 (a), τ ∝
35 (b), τ ∝
140 (c) qualitatively consistent with the known observations of ocean waves.
4. Conclusion
The considered model shows that the appearance of waves with abnormally largeamplitudes is characteristic of the initial stage of the modulational instability. Theaverage and maximum wave heights noticeably decrease with the development of theinstability. However, even in the later case ( τ ∝
30) abnormally high waves can bedetected according to the criterion (9), although their amplitude is already one and ahalf - two times less than in the most interesting case ( τ ∝ xtreme wave events at the initial stage of the modulational instability... The number of waves over the modulation length
T i m e
Figure 5.
Evolution of the relative characteristic modulation length with developmentof the instability.
Figure 6.
Distribution of wave amplitudes in different realizations at the moment of τ ∝ under consideration (both in time and space) and the condition that the amplitudes ofindividual side-band modes remain much smaller of the fundamental wave amplitudeis satisfied during the entire simulation time. The latter allows us to assume such adescription of the modulational instability sufficiently correct.The authors are grateful to Prof. E.A. Kuznetsov for his interest and helpfulcomments. xtreme wave events at the initial stage of the modulational instability... References [1] C. Kharif, E. Pelinovsky, A. Slunyaev, Rogue Waves in the Ocean, Springer-Verlag, Berlin,Heidelberg, 2009.[2] D.R. Solli, C. Ropers, P. Koonath, B. Jalali, Optical rogue waves, Nature 450 (2007) 1054.[3] D.-I. Yeom, B.J. Eggleton, Photonics: rogue waves surface in light, Nature 450 (2007) 953.[4] L.F. Burlaga, N.F. Ness, M.H. Acuna, Linear magnetic holes in a unipolar region of the heliosheathobserved by Voyager 1, J. Geophys. Res., 112 (2007) A07106.[5] M.S. Ruderman, Freak waves in laboratory and space plasmas, Eur. Phys. J. Special Topics 185(2010) 57-66.[6] Kharif, C. and Pelinovsky, E.: Physical mechanisms of the rogue wave phenomenon, Eur. J. Mech.B-Fluid., 22(6) (2003) 603-633.[7] M.J. Lighthill, Contribution to the theory of waves in nonlinear dispersive system, J. Inst. Math.Appl. 1 (1965) 269-306.[8] T. B. Benjamin and J. E. Feir, The disintegration of wave trains on deep water, J. Fluid Mech. 27(1967) 417-430.[9] V.E. Zakharov, The instability of waves in nonlinear dispersive media, Sov. Phys. JETP 24 (1967)740.[10] V.E. Zakharov, A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensionalself-modulation of waves in nonlinear media, Sov. Phys. JETP, 34(1) (1972) 62.[11] V.S. L’vov, Wave Turbulence under Parametric Excitations. Applications to Magnetics, Springer-Verlag, 1994.[12] V.E. Zakharov, Collapse of Langmuir Waves, Sov. Phys. JETP 35(5) (1972) 908-914.[13] E.V. Zemlyanaya and N.V. Alexeeva, Oscillation solitons of the driven, damped nonlinearSchr¨odinger equation, Theoretical and Mathematical Physics 159 (2009), 870-876.[14] H. Segur, D. Henderson, J. Carter et al., Stabilizing the Benjamin-Feir instability, J. Fluid Mech.539 (2005) 229-271.[15] Wu, G., Liu, Y., and Yue, D. K. P.: A note on stabilizing the Benjamin-Feir instability, J. FluidMech., 556 (2006) 45-54.[16] C. Kharif and J. Touboul, Under which conditions the Benjamin-Feir instability may spawn anextreme wave event: A fully nonlinear approach, Eur. Phys. J. Special Topics 185 (2010) 159-168[17] V. M. Kuklin. Effect of Induced Interference and the Formation of Spatial Perturbation FineStructure in Nonequlibrium Open-Ended System, Problems Of Atomic Science And Technology(Ukrainian) 5 (2006), 63-68.[18] H. Suhl, Effective Nuclear Spin Interactions in Ferromagnets, Phys. Rev. 109(2) (1958) 606.[19] E.L. Schlomann, J.H. Saunder, M.H. Sirvets, Band Ferromagnetic Resonance Experiments at HighPeak Power Levels, J. Appl. Phys. 31 (1960) 386S-395S.[20] V.E. Zakharov, V.S. L’vov, S.S. Starobinets, Spin-wave turbulence beyond the parametricexcitation threshold, Sov. Phys. Usp. (1975) 896-919.[21] V.E. Zakharov, V.S. L’vov, S.S. Starobinets, A new mechanism of limitation of the amplitude ofspin waves in parallel pumping, Soviet Physics - Solid State 11 (1969) 2047- 2055.[22] E.V. Belkin, A.V. Kirichok, V.M. Kuklin, On interference effects in multi-mode regimes ofmodulational instability, Problems Of Atomic Science And Technology (Ukrainian), 4 (2008)222-227.[23] L. Debnath, Nonlinear Water Waves, Academic Press, Boston, 1994[24] L.W. Schwartz, J.D. Fenton, Strongly nonlinear waves, Ann. Rev. Fluid. Mech., 14 (1982) 39-60.[25] E.V. Belkin, V.M. Kuklin, A.S. Petrenko, About quasi-linear character of multi-mode instabilitiesin nonlinear media,Thesis of Int. Conf. MSS-09 ”Mode Convertion, Coherent Structures andTurbulence” Moscow, 23-25 Nov. 2009.[26] V.S. Vorob’ev and V.M. Kuklin, On Mechanism of Space Structures Generation in DissipativeInequilibrium Media 13(22) Sov. Tech. Phys. Lett. (1987) 1354-1360. xtreme wave events at the initial stage of the modulational instability...12