PHYSICAL REVIEW E 67, 046305 ~2003!

Landau damping and coherent structures in narrow-banded 1¿1 deep water gravity waves

Miguel Onorato,1 Alfred Osborne,1 Renato Fedele,2 and Marina Serio11Dipartimento di Fisica Generale, Universita` di Torino, Via Pietro Giuria 1, 10125 Torino, Italy

2Dipartimento di Scienze Fisiche, Universita` Federico II di Napoli and INFN Sezione di Napoli, Complesso Universitario di MonteS. Angelo, Via Cintia, I-80126 Napoli, Italy

~Received 19 February 2002; revised manuscript received 24 January 2003; published 21 April 2003!

We study the modulational instability in surface gravity waves with random phase spectra. Starting from thenonlinear Schro¨dinger equation and using the Wigner-Moyal transform, we study the stability of the narrow-banded approximation of a typical wind-wave spectrum, i.e., the JONSWAP spectrum. By performing numeri-cal simulations of the nonlinear Schro¨dinger equation we show that in the unstable regime, the nonlinear stageof the modulational instability is responsible for the formation of coherent structures. Furthermore, a Landau-type damping, due to the incoherence of the waves, whose role is to provide a stabilizing effect against themodulational instability, is both analytically and numerically discussed.

DOI: 10.1103/PhysRevE.67.046305 PACS number~s!: 47.20.2k, 47.35.1i, 92.10.Hm

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I. INTRODUCTION

In many different fields of nonlinear physics, local nolinear effects such as the modulational instability~MI ! haveplayed a very important role@1#. In plasmas, in the largeamplitude regime a nonlinear coupling between higfrequency Langmuir and low-frequency ion-acoustic wavtakes place@2#. Under suitable physical conditions, the dnamics can be described by a nonlinear Schro¨dinger ~NLS!equation and the modulational instability~MI ! can be ana-lyzed directly with this equation@3#. In nonlinear optics, thepropagation of large amplitude electromagnetic waves pduces a modification of the refractive index which, in tuaffects the propagation itself and makes possible the fortion of wave envelopes. In the slowly varying amplitude aproximation, this propagation is governed again by suitaNLS equations@4# and the MI plays a very important rol@5#. The charged-particle beam dynamics in high-energycular accelerating machines has been suitably describeterms of NLS for a complex wave function whose squamodulus is proportional to the beam density@6#. In this con-text, the so-called ‘‘coherent instability’’ due to the collectivinteraction of the beam with the surroundings@7#, has beenrecently interpreted as the MI of the NLS equation@6#.

For ocean gravity waves, the subject of this paper, the~also known as the Benjamin-Feir instability! has been dis-covered independently by Benjamin and Feir@8# and by Za-kharov@9# in the 1960s. The instability predicts that in dewater a monochromatic wave is unstable under suitasmall perturbations. This instability is well described by tNLS equation and has been recently addressed as responfor the formation of freak waves@10–13#.

While the role of the modulational instability for a monochromatic wave has been widely studied, its role in a ctinuous wave spectrum characterized by random phasesdeserved less attention. In order to approach statistically,nonlinear energy transfer processes involved, one is inested in finding a suitablekinetic equation. As far as oceanwaves are concerned, this kinetic equation has been derindependently by Hasselmann@14,15# and by Zakharov@9,16#. Besides the quasi-Gaussian approximation@17#, one

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of the major hypothesis required to derive the kinetic eqtion is that of homogeneity, i.e.,A(k)A* (k8)&5n(k)d(k2k8), whereA is a complex wave amplitude describing thenvelope of the wave train,k andk8 are wave numbers,n(k)is the spectral density function, and brackets indicatesemble averages. According to the kinetic equation, theergy is transferred in an irreversible manner only when fowaves interact resonantly. Unfortunately, the HasselmaZakharov theory is not able to predict the modulationalstability because the latter results from a correlation betwthe carrier wave and the sideband perturbations and, mover, it is not the result of an exact resonance. Neverthelif the hypothesis of homogeneity of the system is relax~correlation between different wave numbers is allowed!, animproved kinetic equation can be derived which is ableshow a random version of the Benjamin-Feir instability. Fsurface gravity waves, this improvement is contained inpioneering work by Alber@18#, followed by the works ofCrawford et al. @19# and Janssen@20–22#. Independent ofthese works@18–20#, a similar approach has been developfor the large-amplitude electromagnetic wave-envelopropagation in nonlinear media@24#, for the quantumlike de-scription of the longitudinal charged-particle beam dynamin high-energy accelerating machines@25# and for the reso-nant interaction between an instantaneously produced disbance and a partially incoherent Langmuir wave@26#. In allthe above approaches, the basic idea is to transit fromconfiguration space, where the NLS equation governswave-envelope propagation, to the phase space, wherappropriate kinetic equation is able to show a random vsion of the MI. This has been done by using the mathemcal tool provided by the Wigner-Moyal transform@27#. Con-senquently, the governing kinetic equation is nothing buvon Neumann-Weyl-like equation@28#.

In this paper, we outline the approach formulated in Re@18–20# and discuss the modulational instability for randowave spectra. In particular, we identify the values of tparameters of the JONSWAP spectrum~see, for example,Ref. @14#! for which the spectrum itself is unstable. Furthemore, within the framework of the theory developed in Re@18–20#, we show that a phenomenon similar to the Land

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damping @29# can be predicted for ocean gravity wavwhose role is to provide a stabilizing mechanism againstMI. In particular, we show, both analytically and numecally, that this phenomenon is due to the incoherence ofbackground solution~random wave spectra!. By performingnumerical simulations of the NLS equation, we show thathe unstable regime coherent structures naturally appeathe space-time plane. We stress that our focus herein is nattempt to model ocean waves but instead to study leaorder effects using the simplest weakly nonlinear and dispsive model in deep water, i.e., the NLS equation. The resobtained are very similar to the ones recently given inliterature that have shown the existence of a Landau-tdamping in the dynamics of small perturbations on a ptially incoherent background, consisting of a constant amtude and a stochastically varying phase, in nonlinear op@24#, in charged-particle beam dynamics@25# and plasmaphysics@26#.

II. STATISTICAL DESCRIPTION OF THE NLS EQUATIONFOR WATER WAVES

The NLS equation for water waves in infinite depth wderived under the hypothesis of small steepness and narbanded spectra for the first time by Zakharov in 1968@9#.While this equation is not appropriate for describing the dnamics in the tail of the spectrum or in the inertial ran~exact four-wave resonant interactions are forbidden!, itshould describe with satisfactory accuracy the dynamaround the spectral peak. In dimensional form, in a framereference moving with the group velocity, the equation re

]A

]t1 im

]2A

]x21 inuAu2A50, ~1!

where in deep waterm5v0/8k02 andn5v0k0

2/2 with v0 thecarrier angular frequency andk0 the respective wave number. The free surface elevationz(x,t) is related to the com-plex envelopeA(x,t) in the following way:

z~x,t !5Re@A~x,t !ei (k0x2v0t)#. ~2!

Equation~1! is the starting point for deriving the requirekinetic equation. Following Alber@18#, the Wigner-Moyaltransform@27# can be applied directly to the NLS equatioThis transform allows one to give a representation of a fution A(x,t) both in configurationx and in wave numberkspace:

n~x,k,t !51

2pE2`

1`

^A* ~x1y/2,t !A~x2y/2,t !&e2 ikydy.

~3!

n(x,k,t) is a second-order correlator. In order to deriveevolution equation forn(x,k,t), we take the time-derivativeof Eq. ~3! and use the NLS equation to remove the timderivative of the complex envelopeA in the right-hand sideof Eq. ~3!. The nonlinear term in NLS will generate a fourthorder correlator@a term of the form^A1A1* A1A2* & where

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A15A(x1y/2) andA25A(x2y/2)], therefore, a new vari-able is introduced. In order to proceed in the calculationclosure that relates fourth- and second-order correlamust be introduced. This closure is achieved by intducing the quasi-Gaussian approximation^A1A1* A1A2* &52^A1A2* &^A1A1* &. The fourth-order correlator can be spas the product of the sum of second-order correlators,carding the fourth-order cumulants. This procedure is wknown for the statistical description of water waves@16# andof many other fields such as plasma physics@30#. The result-ing kinetic equation is the following von Neumann-Weylike equation:

]n~x,k,t !

]t12mk

]n~x,k,t !

]x14n (

m50

`~21!m

~2m11!!22m11

3]2m11^uA~x,t !u2&

]x2m11

]2m11n~x,k,t !

]k2m1150, ~4!

with

^uA~x,t !u2&5E2`

1`

n~x,k,t !dk. ~5!

If only the first term in the infinite sum is considered, thequation reduces to the Vlasov-Poisson equation in plaphysics that is well known to describe the Landau dampphenomenon@29#. This damping is due to the interactionsresonant electrons of the system; the theory predicts tharate of decay of the wave energy is proportional to the fiderivative of the equilibrium distribution function of thelectrons.

A. Stability of wave spectra

In order to study the stability of wave spectra, a standlinear stability analysis of the von Neumann-Weyl-like eqution is performed: we let the distribution functionn(x,k,t)be expressed in terms of an equilibrium distributionn0(k)plus a small perturbation,

n~x,k,t !5n0~k!1n1~x,k,t ! ~6!

with n1(x,k,t)!n0(k). After substituting Eq.~6! into Eq.~4!and linearizing we obtain the following equation for the peturbation:

]n1~x,k,t !

]t12mk

]n1~x,k,t !

]x14n (

m50

`~21!m

~2m11!!22m11

3]2m11^uA~x,t !u2&

]x2m11

]2m11n0~k!

]k2m1150. ~7!

We then look for solutions of the form

n1~x,k,t !5n1~k!ei (Kx2Vt). ~8!

After standard algebra, the following implicit form of thdispersion relation is obtained~see also, Refs.@24–26#!:

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LANDAU DAMPING AND COHERENT STRUCTURES IN . . . PHYSICAL REVIEW E67, 046305 ~2003!

11n

mE n0~k1K/2!2n0~k2K/2!

K~k2V/~2mK !!dk50, ~9!

wheren0(k) is the homogeneous envelope spectrum. Othe equilibrium solutionn0 is given, Eq.~9! represents thedispersion for the perturbation.

We now look for an equilibrium solutionn0 for sea sur-face gravity waves. According to experimental works coducted more than 25 years ago~see, for example, Ref.@14#!,it has been found that the spectrum for the free surfaceevationz is well approximated by the JONSWAP spectru~we give its form in wave-number space using the dispersrelation in infinite water depthv5Agh):

P~k!5a

2k3e2(3/2)[k0 /k] 2

gexp[2(Ak2Ak0)2/2d2k0] , ~10!

with a, g, andd constants (d is usually set to 0.07, whileaand g depend on the state of the ocean!. As a and g areincreased, the wave amplitude and, therefore, the nonlinity of the wave train increases.g rules also the spectrawidth: for large values ofg the spectrum becomes monarrow banded. Since we are interested in the dynamaround the peak of the spectrum, as is done in Ref.@21#, weconsider a second-order Taylor expansion ofP(k) around thepeakk0. It turns out that the spectrum in Ref.~10! reduces tothe following Lorentzian one:

P~k!5Hs

2

16p

p

p21~k2k0!2, ~11!

where

p5A 8k02d2

24d21 ln~g!and Hs54Ap

agp

2E3/2k03.

~12!

Hs is the significant wave height calculated as four timesstandard deviation of the wave field andp corresponds ex-actly to the half-width at half-maximum of the spectrumNote that the spectrum in Eq.~11! or in Eq. ~10! is thespectrum for the free surfacez and not for the envelopeA.Nevertheless, it can be shown@19# that for a symmetric spectrum P(k) of the surface elevation, the spectrum for tcomplex envelope is given byn0(k)54P(k1k0), therefore,a factor of 4 must be taken into account. Solving Eq.~9! withrespect toV, we obtain the following dispersion relation:

V5K~AK2m22Hs2nm22imp!. ~13!

Positive complex roots of Eq.~13! result in a growing insta-bility of the perturbation: ifK2,Hs

2n/m, the first term on theright-hand side is responsible for the MI~note that in thelimit as p→0, the dispersion relation~13! gives theBenjamin-Feir instability!. The last term on the right-hanside has a stabilizing effect and plays the same role plaby the Landau damping in plasma physics@29#, i.e., a damp-ing of the perturbation. There is a competition between

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ponential growth and damping of the perturbation thatpends on the parametersa and g of the Lorenzian~orJONSWAP! spectrum. Ifp.Im@A1K2/42Hs

2n/(4m)# thedamping dominates the MI, the opposite will occur ifp,Im@AK2/42Hs

2n/(4m)#. In Fig. 1 we show the marginastability curve in thea-g plane in the limit ofK→0. Spectrawith higher values ofa and g are more likely to show theMI. In the following section, we will perform numericasimulations of the NLS equation in order verify the resfrom the dispersion relation~13! and study the effect of theinstability in physical space.

B. Analogy with plasma physics

Let us now present some remarks about the physicalgin of the Landau-type damping predicted above. In plasphysics, this damping, for instance, is caused by resoninteractions between a plasma wave and the electronsdenoting withv the single-particle velocity and withv andkthe frequency and wave number of the electron plasma wrespectively, the Landau theory, based on the Vlasov kinequation, clearly shows that the decay rate of the waveergy is proportional to the first derivative of the equilibriudistribution functionr0(v) of the electrons. Typically, theshape ofr0(v) is such thatdr0(v5v/k)/dv,0, which im-plies that there are more particles withv,v/k ~which gainenergy from the wave! than withv.v/k ~which give energyto the wave!. This statistical circumstance leads to a ndamping of the plasma wave. This is usually referred as‘‘weak Landau damping.’’ Additionally, as the thermal dispersion of the electrons of the plasma becomes neglig~for example, the equilibrium distribution becomes more amore sharp!, the Landau damping becomes more and mweak. In principle, a cold plasma, whose thermal distributcorresponds to ad-function, does not exhibit the phenomenon of Landau damping.

On the other hand, we observe that, in the limiting cafor K!k dispersion relation~9! becomes

11n

mE dn0 /dk

k2V/~2mK !dk50, ~14!

FIG. 1. Instability diagram in thea-g plane.a andg are non-dimensional variables.

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ONORATO et al. PHYSICAL REVIEW E 67, 046305 ~2003!

where we have used the approximation

n0~k1K/2!2n0~k2K/2!

K'

dn0

dk.

Given the full similarity between the dispersion relation~14!and the one found by Landau@29#, the physical origin of thestabilizing effect predicted in the present paper may bescribed by using an analogy with the phenomenon of Landamping. In fact, the above stabilizing effect can be attruted to the ‘‘nonmonochromatic’’ character of the Wignspectrum of the surface gravity waves. Similar to the stdard Landau damping, where the electrons interact indivially with a linear plasma wave and statistically produce atransfer of energy from the wave to the particles, the gravwave train interacts with the perturbation and producetransfer of energy between wave numbers which is msignificant aroundk5V/2mK.

Furthermore, it is worth noting that, on the basis of tabove physical interpretation, the stabilizing effects tcome out also from the more general dispersion relation~9!,may be thought of as an extension, to arbitrary wave nubers, of the Landau-type damping of an ensemble of inherent surface gravity waves. However, this effect is notanalog of the weak Landau damping and cannot be prediwith a Vlasov-like equation. In fact, for arbitraryK, and forthe broad-band spectrum~11!, all the terms of the vonNeumann-Weyl kinetic-like equation~4! contribute to thedispersion relation~13!.

III. NUMERICAL SIMULATIONS AND DISCUSSION

Numerical simulations of Eq.~1! have been computeusing a standard pseudospectral Fourier method. Initial cditions for the free surface elevationz(x,0) have been constructed as the following random process@31#:

z~x,0!5 (n51

N

Cncos~knx2fn!, ~15!

wherefn are uniformly distributed random numbers on tinterval (0,2p), and Ci5A2P(ki)Dki , where P(ki) is thediscretized spectrum given in Eq.~11!. The Hilbert transformis used in order to convert the free surfacez to the complexenvelope variableA of the NLS. The spectrum of the complex envelopeA is nothing other than the unperturbed homgeneous solutionn0(k). The dominant wave number for thnumerical simulation was selected to bek050.1 m21. Thislast choice is not restrictive: the parameters that rule thenamics of the spectrum are its width and the steepnwhich, oncek0 is fixed, are univocally determined byaandg.

In order to investigate the effects of the instability, whave chosen two different initial conditions characterizedvalues ofa andg such that the dispersion relation~13! pre-dicts, respectively, instability and stability. We have consered values in thea-g plane that are far away from thmarginal stability curve~see crosses in Fig. 1!. For the un-

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stable case, we have consideredg53.5 anda50.03 and forthe stable one we have takeng51.5 anda50.005

We now start the discussion of the numerical resultsshowing in Fig. 2 the evolution ofuA(x,t)u in the x-t planefor the unstable case. How is this instability manifesteFrom Fig. 2, we note the presence of a ‘‘coherent structuri.e., a structure~oblique darker zones in thex-t plane! thatpersists in the presence of nonlinear interactions and mtains statistically its shape and velocity during propagat~note that periodic boundary conditions are used!. Every ran-dom realization with the same values ofa and g showssimilar results even though the resulting coherent structumay have different velocity and amplitude. If nonlinearityincreased, more than one coherent structure may appethex-t plane. The nonlinear stage of MI is therefore resposible for the formation of such coherent structures in thex-tplane. Indeed it is possible to show that the NLS equathas periodic solutions such as breathers or unstable m@13,32#. These solutions, which are the result of a linearstability, are nevertheless very robust. Moreover theygrow up to more than three times the initial unperturbsolution and, therefore, have also been addressed as simodels for freak waves@10–13#. In contrast to solitons thahave constant amplitude in time, these unstable modescharacterized by a continuous exchange of energy amongFourier modes. The energy is transferred from one modanother and back again: the process is completely reversand, therefore, coherent structures persist in physical spWe stress that these kinds of solutions appear naturally finitial conditions with random phases. The striking reshere is that even if initial conditions are completely randothe nonlinear interactions generate a strong correlaamong wave numbers resulting in coherent structuresbedded in a random wave field.

We now consider the stable case. We again showspace-time evolution ofuA(x,t)u in Fig. 3. The x-t planeappears as a random field and there is no evidence ofstructure that survives for long periods of time. Numeric

FIG. 2. uA(x,t)u from numerical simulation of the NLS. Theinitial condition is characterized by a Lorenzian spectrum withg53.5 anda50.03. A coherent structure is evident in thex-t plane.Space and time have been scaled, respectively, withk0 and v0

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LANDAU DAMPING AND COHERENT STRUCTURES IN . . . PHYSICAL REVIEW E67, 046305 ~2003!

simulations with initial conditions characterized by the savalues ofa and g but with different random phases areaccordance with the results just shown and are not hereported.

We will now answer to the following two natural quetions: How sharp is the transition region in Fig. 1? Howcoherent structures develop from random phase initial cditions? We have performed a number of numerical simutions with initial conditions characterized by values ofa andg that are close to the marginal stability curve. In particuhere we report numerical simulations that have beentained by settingg53 anda50.005, 0.01, 0.0153, and 0.02Circles in Fig. 1 are located where these last numerical silations have been performed. As is clear from the figupoints are selected in order to cross the margial stabcurve: we move accross the stability curve by changingvalue of a. In Figs. 4~a!–4~d! we show the evolution ofuA(x,t)u in the x-t plane, respectively, forg53 and a50.005, 0.01, 0.0153, and 0.02. Figures 4~a! and 4~b! are theresult of the evolution of initial conditions characterizedvalues ofa and g for which the theory predicts a stabregime, see Fig. 1. While in Fig. 4~a! there is no clear evi-dence of a coherent structure, in Fig. 4~b! a darker coherenregion has already developed. We recall that larger valu

FIG. 3. uA(x,t)u from numerical simulation of the NLS. Theinitial condition is characterized by a Lorenzian spectrum withg51.5 anda50.005. The field in thex-t plane appears to be randowithout any evidence of coherent structure. Space and time hbeen scaled, respectively, withk0 andv05Agk0.

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a implies larger waves~the steepness is increased! and,therefore, an increase in nonlinearity of the initial conditionThe effect of nonlinearity is such that the uncorrelated wanumbers at timet50 sec develop some correlations and aresult coherent structures naturally are formed. As wecreasea, Figs. 4~c! and 4~d!, coherent structures becommore and more well defined.

From the numerical simulations just shown, it seems tcoherent structures can appear from initial conditions chacterized by values ofa and g taken below the marginastability curve. This result is not so surprising: the marginstability curve has been recovered via a linear stability anasis of the kinetic equation which is the result of a statistiapproach to the NLS equation. However, in the natural lotime evolution of a nonlinear wave train, perturbations aregeneral not small. Consequently, perturbations in the simtions should evolve according to the governing equatioOur observation is consistent with some very recent resobtained by Janssen@22#; he was the first one to point outhat the broadening of the spectrum in numerical simulatiof the NLS and Zakharov equation starts for values ofsteepness and spectral width that are lower with respecthe one predicted by the nonhomgeneous theory~see alsoRef. @23#!. In order to explain this result, he has proposedmodification of the Hasselmann-Zakharov kinetic equatby taking into account also nonresonant interactions@22#.

IV. CONCLUSIONS AND REMARKS

In this paper, we have studied the stability of randowave spectra for surface gravity waves in (111) dimension.Theoretical results from a Wigner approach on the Nequation are compared with direct numerical simulationsthe NLS equation. One interesting result of this study cocerns the effect of the instability in physical space: numerisimulations show that, starting with a random wave fiecoherent structures naturally develop as long as the inconditions have sufficient energetics~large a) and are nar-row banded~large g). The theory developed by using thWigner-Moyal transform allows one to isolate the regionthe a-g plane where those structure are more likely to apear. The transition region predicted by the theory doesappear to be sharp. Coherent structures appear for valuea andg lower than predicted by the theory. Our numericsimulations are in one space dimension and moreoverspectrum that we have used in the theory and numersimulations is a narrow-banded approximation of t

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FIG. 4. uA(x,t)u from numerical simulation of the NLS. The initial condition is characterized by a Lorenzian spectrum withg53 anda50.005~a!, 0.01 ~b!, 0.0153~c!, and 0.02~d!. Space and time have been scaled, respectively, withk0 andv05Agk0.

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ONORATO et al. PHYSICAL REVIEW E 67, 046305 ~2003!

JONSWAP spectrum. Needless to say, the theory cannotaken as quantitative. Nevertheless, the marginal stabcurve could give a first qualitative indication of unstabspectra in realistic conditions in infinite water depth~valuesof a and g here considered are typical of ocean wave!.Many physical questions remain open. For example, it wobe interesting to investigate the case of a two-dimensiowave field. It is well known that the NLS in (211) dimen-sion is not integrable and the dynamics of coherent structis still far from being understood. Numerical simulatio

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with the fully nonlinear Euler equations are also under cosideration in order to extend the validity of these results.

ACKNOWLEDGMENTS

We would like to thank P. Janssen, K. Dysthe, andTrulsen for valuable discussions. M.O. was supported bresearch contract from the Universita` di Torino. This workwas financially supported by the Office of Naval Researc

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