Wave Turbulence

0.35 MB·

by  Yeontaek Choi

Wave Turbulence
Yeontaek Choi∗, Yuri V. Lvov†, Sergey Nazarenko∗
5
0 ∗ Mathematics Institute, The University of Warwick,
0
Coventry, CV4-7AL, UK
2
† Department of Mathematical Sciences, Rensselaer Polytechnic Institute,
n
Troy, NY 12180
a
J

5 February 7, 2008
2

2
Abstract
v
5 Inthispaperwereviewrecentdevelopmentsinthestatisticaltheoryofweaklynonlineardispersivewaves,thesubject
4
known as Wave Turbulence (WT). We revise WT theory using a generalisation of the random phase approximation
0
(RPA). This generalisation takes into account that not only the phases but also the amplitudes of the wave Fourier
2
modes are random quantitiesand it is called the“Random Phase and Amplitude”approach. This approach allows to
1
systematicallyderivethekineticequationfortheenergyspectrumfromthethePeierls-Brout-Prigogine(PBP)equation
4
0 for the multi-mode probability density function (PDF). The PBP equation was originally derived for the three-wave
/ systems and in the present paper we derive a similar equation for the four-wave case. Equation for the multi-mode
h
PDF will be used to validate the statistical assumptions about the phase and the amplitude randomness used for
p
WT closures. Further, the multi-mode PDF contains a detailed statistical information, beyond spectra, and it finally
-
h allows to study non-Gaussianity and intermittency in WT, as it will be described in the present paper. In particular,
t we will show that intermittency of stochastic nonlinear waves is related to a flux of probability in the space of wave
a
amplitudes.
m
:
v 1 Introduction
i
X
Imagine surface waves on sea produced by wind of moderate strength, so that the surface is smooth and there is
r
a no whitecaps. Typically, these waves exhibit great deal of randomness and the theory which aims to describe their
statistical properties is called Wave Turbulence (WT). More broadly, WT deals the fields of dispersive waves which
areengagedinstochasticweaklynonlinearinteractionsoverawiderangeofscales invariousphysicalmedia. Plentiful
examplesofWTarefoundinoceans,atmospheres,plasmasandBose-Einsteincondensates[1,2,3,4,5,6,7,10,11,12,
30]. WTtheoryhasalongandexcitinghistorywhichstartedin1929fromthepioneeringpaperofPeierlswhoderived
akineticequationforphononsinsolids[19]. Inthe1960’stheseideashavebeenvigorouslydevelopedinoceanography
[6, 5, 2, 4, 30] and in plasma physics [3, 11, 9]. First of all, both the ocean and the plasmas can support great many
types of dispersive propagating waves, and these waves play key role in turbulent transport phenomena, particularly
thewind-wavefrictioninoceansandtheanomalousdiffusionandthermo-conductivityintokamaks. Thus,WTkinetic
equations where developed and analysed for different types of such waves. A great development in the general WT
theorywasdonebyinthepapersofZakharovandFilonenko[5]. Beforethisworkitwasgenerallyunderstoodthatthe
nonlinear dispersive wavefields are statistical, but it was also thoughtthat such a“gas” of stochastic waves is close to
thermodynamicequilibrium. ZakharovandFilonenko[5]werethefirsttoarguethatthestochasticwavefieldsaremore
likeKolmogorov turbulencewhich isdeterminedbytherateat whichenergy cascades throughscales ratherthanbya
thermodynamic “temperature” describing the energy equipartition in the scale space. This picture was substantiated
by a remarkable discovery of an exact solution to the wave-kinetic equation which describes such Kolmogorov energy
1
cascade. These solutions are now commonly known as Kolmogorov-Zakharov (KZ) spectra and they form thenucleus
of theWT theory.
DiscoveryoftheKZspectrawassopowerfulthatitdominatedtheWTtheoryfordecadesthereafter. Suchspectra
were found for a large variety of physical situations, from quantum to astrophysical applications, and a great effort
wasputintheirnumericalandexperimentalverification. Foralongtime,studiesofspectradominatedWTliterature.
A detailed account of these works was given in [1] which is the only book so far written on this subject. Work on
these lines has continued till now and KZ spectra were found in new applications, particularly in astrophysics [16],
ocean interior [18] and even cosmology [27]. However, the spectra do not tell the whole story about the turbulence
statistics. Inparticular, theydonottell usif thewavefield statistics is Gaussian ornot and,if not,in what way. This
question is of general importance in the field of Turbulence because it is related with the intermittency phenomenon,
- an anomalously high probability of large fluctuations. Such “bursts” of turbulent wavefields were predicted based
on a scaling analysis in [32] and they were linked to formation of coherent structures, such as whitecaps on sea [28]
or collapses in optical turbulence [7]. To study these problems qualitatively, the kinetic equation description is not
sufficient and one has to deal directly with the probability density functions (PDF).
Infact, suchadescriptionintermsofthePDFappearedalready inthethePeierls1929papersimultaneouslywith
the kinetic equation for waves [19]. This result was largely forgotten by the WT community because fine statistical
detailsandintermittencyhadnotinterestedturbulenceresearchersuntilrelativelyrecentlyandalsobecause,perhaps,
this result got in the shade of the KZ spectrum discovery. However, this line of investigation was continued by Brout
andPrigogine[20]whoderivedanevolutionequationforthemulti-modePDFcommonlyknownastheBrout-Prigogine
equation. ThisapproachwasappliedtothestudyofrandomnessunderlyingtheWTclosuresbyZaslavskiandSagdeev
[21]. All of these authors, Peierls, Brout and Prigogine and Zaslavski and Sagdeev restricted their consideration to
the nonlinear interaction arising from the potential energy only (i.e. theinteraction Hamiltonian involves coordinates
but not momenta). This restriction leaves out the capillary water waves, Alfven, internal and Rossby waves, as well
as many other interesting WT systems. Recently,this restriction was removed by considering themost general three-
waveHamiltonian systems[24]. Itwas shownthatthemulti-modePDFstill obeysthePeierls-Brout-Prigogine (PBP)
equationinthisgeneralcase. Thisworkwillbedescribedinthepresentreview. Wewillalsopresent,forthefirsttime,
aderivationoftheevolutionequationforthemulti-modePDFforthegeneralcaseoffourwave-systems. Thisequation
is applicable, for example, to WT of the deep water surface gravity waves and waves in Bose-Einstein condensates or
optical media described by thenonlinear Schroedinger (NLS) equation.
We will also describe the analysis of papers [24] of the randomness assumptions underlying the statistical WT
closures. Previous analyses in this field examined validity of the random phase assumption [20, 21] without devoting
much attention to theamplitude statistics. Such“asymmetry” arised from a common mis-conception that the phases
evolve much faster than amplitudes in the system of nonlinear dispersive waves and, therefore, the averaging may be
madeoverthephasesonly“forgetting” thattheamplitudesarestatistical quantitiestoo(seee.g. [1]). Thisstatement
become less obviousif onetakesintoaccount that weare talkingnot about thelinearphases ωtbutabout thephases
of the Fourier modes in the interaction representation. Thus, it has to be the nonlinear frequency correction that
helpsrandomisingthephases[21]. Ontheotherhand,forthree-wavesystemstheperiodassociated withthenonlinear
frequencycorrectionisofthesameǫ2 orderinsmallnonlinearityǫasthenonlinearevolutiontimeand,therefore,phase
randomisationcannotoccurfasterthatthenonlinearevolutionoftheamplitudes. Onecouldhopethatthesituationis
better for 4-wave systems because the nonlinear frequency correction is still ǫ2 but the nonlinear evolution appears
onlyintheǫ4order. However,inordertomaketheasymptoticanalysisconsist∼ent,suchǫ2correctionhastoberemoved
from the interaction-representation amplitudes and the remaining phase and amplitude evolutions are, again, at the
same time scale (now 1/ǫ4). This picture is confirmed by the numerical simulations of the 4-wave systems [26, 23]
whichindicatethatthenonlinearphaseevolvesatthesametimescaleastheamplitude. Thus,toproceedtheoretically
onehastostartwithphaseswhicharealreadyrandom(oralmostrandom)andhopethatthisrandomnessispreserved
over the nonlinear evolution time. In most of the previous literature prior to [24] such preservation was assumed but
not proven. Below, we will describe the analysis of theextent to which such an assumption is valid made in [24].
We will also describe the results of [22] who derived the time evolution equation for higher-order moments of the
Fourier amplitude, and its application to description of statistical wavefields with long correlations and associated
“noisiness” oftheenergyspectracharacteristictotypicallaboratory andnumericalexperiments. Wewillalsodescribe
the results of [23] about the time evolution of the one-mode PDF and their consequences for the intermittency of
stochastic nonlinear wavefields. In particular, we will discuss the relation between intermittency and the probability
2
fluxesin theamplitude space.
2 Setting the stage I: Dynamical Equations of motion
Wave turbulence formulation deals with a many-wave system with dispersion and weak nonlinearity. For systematic
derivations one needs to start from Hamiltonian equation of motion. Here we consider a system of weakly interacting
waves in a periodic box [1],
∂
ic˙l = ∂Hc¯ , (1)
l
where c is often called thefield variable. It represents theamplitude of theinteracting plane wave. The Hamiltonian
l
is represented as an expansion in powers of small amplitude,
= + + + +..., (2)
2 3 4 5
H H H H H
where H is a term proportional to product of j amplitudes c,
j l
∞
= Tq1q2...qn c¯ c¯ ...c¯ c c ...c +c.c. , n+m=j
Hj p1,p2...pm q1 q2 qm p1 p2 pm
q1,q2,q3,...qnX,p1,p2,...pm=1(cid:0) (cid:1)
where q ,q ,q ,...q and p ,p ,...p are wavevectors on a d-dimensional Fourier space lattice. Such general j-wave
1 2 3 n 1 2 m
Hamiltonian describe thewave-wave interactions where n waves collide to create m waves. Here Tq1q2...qn represents
p1,p2...pm
the amplitude of the n m process. In this paper we are going to consider expansions of Hamiltonians up to forth
→
order in waveamplitude.
Under rather general conditions the quadratic part of a Hamiltonian, which correspond to a linear equation of
motion, can bediagonalised to theform
∞
= ω c 2. (3)
2 n n
H | |
n=1
X
This form of Hamiltonian correspond to noninteracting (linear) waves. First correction to the quadratic Hamiltonian
is a cubic Hamiltonian, which describes the processes of decaying of single wave into two waves or confluence of two
waves into a single one. Such a Hamiltonian has theform
∞
=ǫ Vl c¯c c δl +c.c.,
H3 mn l m n m+n
l,mX,n=1
(4)
whereǫ 1isaformalparametercorrespondingtosmallnonlinearity(ǫisproportionaltothesmallamplitudewhereas
≪
c is normalised so that c 1.) Most general form of three-wave Hamiltonian would also have terms describing the
n n
∼
confluenceof threewaves or spontaneous appearance of three waves out of vacuum. Such a terms would havea form
∞
Ulmncc c δ +c.c.
l m n l+m+n
l,mX,n=1
It can be shown however that for systems that are dominated by three-wave resonances such terms do not contribute
to long term dynamics of systems. Wetherefore choose toomit those terms.
The most general four-wave Hamiltonian will have 1 3, 3 1, 2 2, 4 0 and 0 4 terms. Nevertheless
→ → → → →
1 3,3 1,4 0and0 4termscanbeexcludedfrom Hamiltonian byappropriatecanonicaltransformations, so
→ → → →
that we limit our consideration to only 2 2 terms of , namely
4
→ H
∞
=ǫ2 Wlmc¯c¯ c c .
H4 µν l m µ ν
m,n,µ,ν=1
X
(5)
3
It turns out that generically most of the weakly nonlinear systems can be separated into two major classes: the ones
dominated by three-wave interactions, so that describes all the relevant dynamics and can be neglected, and
3 4
H H
the systems where the three-wave resonance conditions cannot be satisfied, so that the can be eliminated from
3
H
a Hamiltonian by an appropriate near-identical canonical transformation [25]. Consequently, for the purpose of this
paper we are going to neglect either or , and study thecase of resonant three-waveor four-wave interactions.
3 4
H H
Examples of three-wave system include the water surface capillary waves, internal waves in the ocean and Rossby
waves. Themostcommonexamplesofthefour-wavesystemsarethesurfacegravitywavesandwavesintheNLSmodel
of nonlinear optical systems and Bose-Einstein condensates. For reference we will give expressions for the frequencies
and theinteraction coefficients corresponding to these examples.
For thecapillary waves we have[1, 5],
ω =√σk3, (6)
j
and
1 L L L
Vl = (ωω ω )1/2 km,kn kl,−km kl,−kn , (7)
mn 8π√2σ l m n ×(cid:20)(kmkn)1/2kl − (klkm)1/2kn − (klkn)1/2km(cid:21)
where
L =(k k )+k k (8)
km,kn m· n m n
and σ is thesurface tension coefficient.
For theRossby waves [13, 14],
βk
ω = jx , (9)
j 1+ρ2k2
j
and
iβ k k k
Vl = k k k 1/2 ly my ny , (10)
mn −4π| lx mx nx| × 1+ρ2k2 − 1+ρ2k2 − 1+ρ2k2
(cid:18) l m n(cid:19)
where β is the gradient of theCoriolis parameter and ρ is theRossby deformation radius.
The simplest expressions correspond to the NLS waves [15, 7],
ω = k 2, Wlm=1. (11)
j | j| µν
The surface gravity waves are on the other extreme. The frequency is ω = √gk but the matrix element is given by
notoriously long expressions which can be found in [1, 17].
2.1 Three-wave case
When = + we haveHamiltonian in a form
2 3
H H H
∞ ∞
= ω c 2+ǫ Vl c¯c c δl +c.c. .
H n| n| mn l m n m+n
nX=1 l,mX,n=1(cid:16) (cid:17)
Equation of motion ic˙ = ∂H is mostly conveniently represented in theinteraction representation,
l ∂c¯l
∞
ia˙l=ǫ Vmlnamaneiωml ntδml +n+2V¯lmna¯name−iωlmntδlm+n , (12)
mX,n=1(cid:16) (cid:17)
where aj = cjeiωjt is the complex wave amplitude in the interaction representation, l,m,n d are the indices
∈ Z
numbering the wavevectors, e.g. k = 2πm/L, L is the box side length, ωl ω ω ω and ω = ω is
the wave linear dispersion relation.mHere, Vl 1 is an interaction coefficiemntna≡nd ǫklis−intkrmod−ucekdmas a forlmal smklall
mn ∼
nonlinearity parameter.
4
2.2 Four-wave case
Consider a weakly nonlinear wavefield dominated by the 4-wave interactions, e.g. the water-surface gravity waves [1,
6, 29, 28], Langmuir waves in plasmas [1, 3] and the waves described by the nonlinear Schroedinger equation [7]. The
a Hamiltonian is given by(in theappropriately chosen variables) as
∞ ∞
= + = ω c 2+ǫ2 Wlmc¯c¯ c c .
H H2 H4 n| n| µν l m µ ν
n=1 m,n,µ,ν=1
X X
(13)
As in three-wave case the most convenient form of equation of motion is obtained in interaction representation, c =
l
ble−iωlt, so that
ib˙l=ǫ Wµlαν¯bαbµbνeiωµlανtδµlαν (14)
αµν
X
where Wlα 1 is an interaction coefficient, ωlα = ω +ω ω ω . We are going expand in ǫ and consider the
µν ∼ µν l α− µ− ν
long-time behaviour of a wave field, but it will turn out that to do the perturbative expansion in a self-consistent
manner we haveto renormalise thefrequency of (14) as
ia˙l =ǫ Wµlανa¯αaµaνeiω˜µlανtδµlαν −Ωlal, (15)
αµν
X
where al=bleiΩlt ,ω˜µlαν =ωµlαν +Ωl+Ωα−Ωµ−Ων, and
Ω =2ǫ WlµA2 (16)
l lµ µ
µ
X
is the nonlinear frequency shift arising from self-interactions.
3 Setting the stage II: Statistical setup
In this section we are going to introducestatistical objects that shall beused for the description of the wavesystems,
PDF’s and a generating functional.
3.1 Probability Distribution Function.
Let us consider a wavefield a(x,t) in a periodic cube of with side L and let the Fourier transform of this field be
a(t) where index l d marks themode with wavenumber k =2πl/L on the grid in thed-dimensional Fourier space.
l l
∈Z
For simplicity let us assume that there is a maximum wavenumber k (fixed e.g. by dissipation) so that no modes
max
with wavenumbers greater than this maximum value can be excited. In this case, the total number of modes is
N = (k /πL)d. Correspondingly, index l will only take values in a finite box, l d which is centred at 0
max N
andallsidesofwhichareequaltok /πL=N1/3. Toconsiderhomogeneousturbu∈leBnce,⊂thZelargeboxlimitN
max
→∞
will haveto be taken. 1
Let us write the complex a as a = Aψ where A is a real positive amplitude and ψ is a phase factor which
l l l l l l
takes values on 1, a unit circle centred at zero in the complex plane. Let us define the N-mode joint PDF (N)
S P
as the probability for the wave intensities A2 to be in the range (s,s +ds) and for the phase factors ψ to be on
l l l l l
the unit-circle segment between ξ and ξ +dξ for all l . In terms of this PDF, taking the averages will involve
l l l N
∈B
integration overall thereal positive s’s and along all thecomplex unit circles of all ξ’s,
l l
f A2,ψ = ds dξ (N) s,ξ f s,ξ (17)
h { }i  l | l| P { } { }
l∈YBNZR+ IS1
1Itiseasilytoextend the analysisto theinfiniteFourierspace, kmax =∞. Inthis case, the fulljointPDFwouldstillhave tobedefined
as a N →∞ limit of an N-mode PDF, but this limitwould have to be taken in such a way that both kmax and the density of the Fourier
modestendtoinfinitysimultaneously.
5
wherenotationf A2,ψ meansthatf dependsonallA2’sandallψ’sintheset A2,ψ;l (similarly, s,ξ means
s ,ψ;l , {etc). T}he full PDF that contains thelcomplete staltistical infor{maltioln a∈bBoNut}the wavefie{ld a}(x,t) in
l l N
{ ∈ B }
the infinitex-spacecan be understood as a large-box limit
s ,ξ = lim (N) s,ξ ,
k k
P{ } N→∞P { }
i.e. itisafunctionalactingonthecontinuousfunctionsofthewavenumber,s andξ . Inthethelargeboxlimitthere
k k
is a path-integral version of (17),
f A2,ψ = s ξ s,ξ f s,ξ (18)
h { }i D |D |P{ } { }
Z I
The full PDF defined above involves all N modes (for either finite N or in the N limit). By integrating out all
→∞
the arguments except for chosen few, one can have reduced statistical distributions. For example, by integrating over
all theangles and over all but M amplitudes,we havean “M-mode” amplitude PDF,
(M) = ds dξ (N) s,ξ , (19)
Pj1,j2,...,jM  l | m|P { }
l=6 j1,Yj2,...,jMZR+ mY∈BNIS1
 
which dependsonly on theM amplitudes marked by labels j ,j ,...,j .
1 2 M N
∈B
3.2 Definition of an ideal RPA field
Following the approach of [22, 23], we now define a “Random Phase and Amplitude” (RPA) field.2 We say that the
field a is of RPAtypeif it possesses thefollowing statistical properties:
1. All amplitudes A and their phase factors ψ are independent random variables, i.e. their joint PDF is equal to
l l
theproduct of theone-mode PDF’s corresponding to each individual amplitude and phase,
(N) s,ξ = P(a)(s)P(ψ)(ξ)
P { } l l l l
l∈YBN
2. The phase factors ψ are uniformly distributed on theunit circle in the complex plane, i.e. for any mode l
l
P(ψ)(ξ)=1/2π.
l l
Note that RPA does not fix any shape of the amplitude PDF’s and, therefore, can deal with strongly non-Gaussian
wavefields. Suchstudyofnon-GaussianityandintermittencyofWTwaspresentedin[22,23]andwillnotberepeated
here. However, we will studysome new objects describing statistics of the phase.
In[22,23]RPAwasassumedtoholdoverthenonlineartime. In[24]thisassumptionwasexaminedaposteriori,i.e.
based on theevolution equation for themulti-pointPDFobtained with RPAinitial fields. Below we will describethis
work. We will see that RPA fails to hold in its pure form as formulated above but it survives in the leading order so
that theWTclosure builtusing theRPAis valid. Wewill also see thatindependenceofthethephasefactors is quite
straightforward, whereas the amplitude independence is subtle. Namely, M amplitudes are independent only up to a
O(M/N) correction. Based on thisknowledge, andleaving justification for lateron in thispaper,we thusreformulate
RPA in a weaker form which holds over the nonlinear time and which involves M-mode PDF’s with M N rather
≪
than the full N-mode PDF.
2We keep the same acronym as in related “Random Phase Approximation” but now interpret it differently because (i) we emphasise the
amplituderandomnessand(ii)nowRPAisadefinedpropertyof thefieldtobeexaminedandnotanapproximation.
6
3.3 Definition of an essentially RPA field
Wewill say that thefield a is of an “essentially RPA”typeif:
1. The phase factors are statistically independent and uniformly distributed variables up toO(ǫ2) corrections, i.e.
1
(N) s,ξ = (N,a) s [1+O(ǫ2)], (20)
P { } (2π)NP { }
where
(N,a) s = dξ (N) s,ξ , (21)
P { }  | l|P { }
l∈YBNIS1
is theN-mode amplitude PDF.  
2. The amplitude variables are almost independent is a sense that for each M N modes the M-mode amplitude
PDF is equalto the product of theone-mode PDF’s up to O(M/N) and o(ǫ≪2) corrections,
(M) =P(a)P(a)...P(a) [1+O(M/N)+O(ǫ2)]. (22)
Pj1,j2,...,jM j1 j2 jM
3.4 Why ψ’s and not φ’s?
Importantly, RPA formulation involves independent phase factors ψ = eφ and not phases φ themselves. Firstly, the
phases would not beconvenient because themean valueof thephases is evolving with therate equal tothe nonlinear
frequency correction [24]. Thus one could not say that they are “distributed uniformly from π to π”. Moreover
−
the mean fluctuation of the phase distribution is also growing and they quickly spread beyond their initial 2π-wide
interval[24]. Butperhapsevenmoreimportant,φ’sbuildmutualcorrelationsonthenonlineartimewhereasψ’sremain
independent. Letusgiveasimpleexampleillustratinghowthispropertyispossibleduetothefactthatcorrespondence
between φ and ψ is not a bijection. Let N be a random integer and let r and r be two independent (of N and of
1 2
each other) random numberswith uniform distribution between π and π. Let
−
φ =2πN+r .
1,2 1,2
Then
φ =2π N ,
1,2
h i h i
and
φ φ =4π2 N2 .
1 2
h i h i
Thus,
φ φ φ φ =4π2( N2 N 2)>0,
1 2 1 2
h i−h ih i h i−h i
which means that variables φ and φ are correlated. On theother hand,if we introduce
1 2
ψ =eiφ1,2,
1,2
then
ψ =0,
1,2
h i
and
ψ ψ =0.
1 2
h i
ψ ψ ψ ψ =0,
1 2 1 2
h i−h ih i
which means that variables ψ and ψ are statistically independent. In this illustrative example it is clear that the
1 2
difference in statistical properties between φ and ψ arises from the fact that function ψ(φ) does not have inverseand,
consequently,theinformation about N contained in φ is lost in ψ.
Summarising, statistics of the phase factors ψ is simpler and more convenient to use than φ because most of
the statistical objects depend only on ψ. This does not mean, however, that phases φ are not observable and not
interesting. Phases φ can be “tracked” in numerical simulations continuously,i.e. without making jumps to π when
−
the phase value exceeds π. Such continuous in k function φ(k) can achieve a large range of variation in values due to
thedependenceofthenonlinearrotation frequencywithk. Thiskindoffunctionimpliesfastlyfluctuatingψ(k)which
is the mechanism behind de-correlation of thephase factors at different wavenumbers.
7
3.5 Wavefields with long spatial correlations.
Often in WT, studies are restricted to wavefields with fastly decaying spatial correlations [2]. For such fields, the
statistics of the Fourier modes is close to being Gaussian. Indeed, it the correlation length is much smaller than the
size of the box, then this box can be divided into many smaller boxes, each larger than the correlation length. The
Fourier transform over the big box will be equal to the sum of the Fourier transforms over the smaller boxes which
are statistically independent quantities. Therefore, by the Central Limit Theorem, the big-box Fourier transform
has a Gaussian distribution. Corrections to Gaussianity are small as the ratio of the correlation volume to the box
volume. Ontheotherhand,intheRPAdefinedabovetheamplitudePDFisnotspecifiedandcansignificantlydeviate
from the Rayleigh distribution (corresponding to Gaussian wavefields). Such fields correspond to long correlations
of order or greater than the box size. In fact, long correlated fields are quite typical for WT because, due to weak
nonlinearity,wavepacketscanpropagateoverlongdistancepreservingtheiridentity. Moreover,restrictingourselvesto
short-correlated fields would render our study of the PDF evolution meaningless because the later would be fixed at
the Gaussian state. Note that long correlations modify the usual Wick’s rule for the correlator splitting by adding a
singular cumulant, e.g. for theforth-order correlator,
a¯ a¯ a a =A2A2(δjδα+δjδα)+Q δαδαδα,
h j α µ νiψ j α µ ν ν µ α ν µ j
where Q =(A4 2A2). These issues were discussed in detail in [22].
α α− α
3.6 Generating functional.
Introduction of generating functionals simplifies statistical derivations. It can be defined in several different ways to
suit a particular technique. For ourproblem, themost useful form of thegenerating functional is
Z(N) λ,µ = 1 eλlA2lψµl , (23)
{ } (2π)Nh l i
l∈YBN
where λ,µ λ,µ;l is a set of parameters, λ and µ .
l l N l l
{ }≡{ ∈B } ∈R ∈Z
1
(N) s,ξ = δ(s A2)ψµlξ−µl (24)
P { } (2π)N h l− l l l i
X{µ} l∈YBN
where µ µ ;l . This expression can be verified by considering mean of a function f A2,ψ using the
l N
averagi{ng}r≡ule{(17∈) aZnd∈exBpan}dingf in theangular harmonics ψm; m (basis functions on theun{it circ}le),
l ∈Z
f A2,ψ = g m,A ψml, (25)
{ } { } l
{Xm} l∈YBN
where m m ;l are indices enumerating theangular harmonics. Substitutingthis into (17) with PDF
l N
{ }≡{ ∈Z ∈B }
given by (24) and taking into account that any nonzero power of ξ will give zero after the integration over the unit
l
circle,onecanseethatLHS=RHS,i.e. that(24)iscorrect. Nowwecaneasilyrepresent(24)intermsofthegenerating
functional,
(N) s,ξ = ˆ−1 Z(N) λ,µ ξ−µl (26)
P { } Lλ  { } l 
X{µ} l∈YBN
where ˆ−1 standsfor inversetheLaplace transform withrespect to all λ parameters and µ µ ;l are
Lλ l { }≡{ l ∈Z ∈BN}
the angular harmonics indices.
Note that we could have defined Z for all real µ’s in which case obtaining P would involve finding the Mellin
l
transform ofZ with respect toallµ’s. Wewill seebelowhoweverthat,giventherandom-phasedinitialconditions, Z
l
will remain zero for all non-integer µ’s. More generally, themean of any quantitywhich involvesa non-integer power
l
ofaphasefactorwill alsobezero. Expression (26)can beviewedasaresultoftheMellin transform forsuchaspecial
case. It can also beeasily checkedby considering themean of a quantity which involves integer powers of ψ’s.
l
8
By definition,inRPAfieldsallvariables A andψ arestatistically independentandψ’sareuniformly distributed
l l l
on theunit circle. Suchfields imply thefollowing form of the generating functional
Z(N) λ,µ =Z(N,a) λ δ(µ), (27)
l
{ } { }
l∈YBN
where
Z(N,a) λ = eλlA2l =Z(N) λ,µ µ=0 (28)
{ } h i { }|
l∈YBN
isanN-modegeneratingfunctionfortheamplitudestatistics. Here,theKroneckersymbolδ(µ)ensuresindependence
l
of the PDF from the phase factors ψ. As a first step in validating the RPA property we will have to prove that the
l
generating functional remains of form (27) up to 1/N and O(ǫ2) corrections over the nonlinear time provided it has
this form at t=0.
3.7 One-mode statistics
Of particular interest are one-mode densities which can be conveniently obtained using a one-amplitude generating
function
Z(t,λ)= eλ|ak|2 ,
h i
where λ is a real parameter. Then PDF of the wave intensities s = a 2 at each k can be written as a Laplace
k
| |
transform,
1 ∞
P(a)(s,t)= δ(a 2 s) = Z(λ,t)e−sλdλ. (29)
h | k| − i 2π
Z0
For theone-point momentsof theamplitude we have
M(p) a 2p = a2peλ|a|2 =
k ≡h| k| i h| | i|λ=0
∞
Z = spP(a)(s,t)ds, (30)
λ···λλ=0
|
Z0
where p and subscript λ means differentiation with respect to λ p times.
Thefi∈rsNtofthesemoments,n =M(1),isthewaveactionspectrum. HighermomentsM(p) measurefluctuationsof
k k k
the waveaction k-space distributions about their mean values [22]. In particular the r.m.s. value of these fluctuations
is
σ = M(2) n2. (31)
k k − k
q
4 Separation of timescales: general idea
Whenthewaveamplitudesaresmall,thenonlinearityisweakandthewaveperiods,determinedbythelineardynamics,
are much smaller than the characteristic time at which different wave modes exchange energy. In the other words,
weak nonlinearityresultsin atimescale separation andourgoal willbetodescribetheslowly changingwavestatistics
by averaging over the fast linear oscillations. To filter out fast oscillations, we will seek seek for the solution at time
T such that 2π/ω T τ . Here τ is the characteristic time of nonlinear evolution which, as we will see later is
nl nl
≪ ≪
1/ωǫ2 for thethree-wavesystems and 1/ωǫ4 for thefour-wave systems. Solution at t=T can besought at series
∼ ∼
in small small nonlinearity parameter ǫ,
a(T)=a(0)+ǫa(1)+ǫ2a(2). (32)
l l l l
Then we are going to iterate theequation of motion (12) or (15) to obtain a(0), a(1) and a(2) by iterations.
l l l
During this analysis the certain integrals of a type
t
f(t)= g(t)exp(iωt)
Z
0
9
will play a crucial role. Following [2] we introduce
∆l = T eiωml ntdt= (eiωml nT −1),
mn iωl
Z0 mn
∆kl = T eiωmklntdt= (eiωmklnT −1),
mn iωkl
Z0 mn
(33)
and
T
E(x,y)= ∆(x y)eiytdt.
−
Z0
Wewill beinterested in a long time asymptotics of theaboveexpressions, so the following properties will be useful:
1
lim E(0,x)=T(πδ(x)+iP( )),
T→∞ x
and
lim ∆(x)2 =2πTδ(x).
T→∞| |
5 Weak nonlinearity expansion: three-wave case.
Substitutingtheexpansion (32) in (12) we get in thezeroth order
a(0)(T)=a(0),
l l
i.e. the zeroth order term is time independent. This corresponds to the fact that the interaction representation wave
amplitudes are constant in the linear approximation. For simplicity, we will write a(0)(0) = a, understanding that a
l l
quantityis taken at T =0 if its time argument is not mentioned explicitly.
Here we havetaken into account that a(0)(T)=a and a(1)(0)=0.
l l k
The first order is given by
∞
a(1)(T)= i Vl a a ∆l δl
l − mn m n mn m+n
mX,n=1(cid:16)
+2V¯ma a¯ ∆¯mδm , (34)
ln m n ln l+n
Here we have taken into account that a(0)(T)= a and a(1)(0) =0. Perform th(cid:1)e second iteration, and integrate over
l l k
time to obtain To calculate thesecond iterate, write
∞
ia˙(2) = Vl δl eiωml nt a(0)a(1)+a(1)a(0)
l mn m+n m n m n
mX,n=1h (cid:16) (cid:17)
+2V¯mδm e−iωlmnt a(1)a¯(0)+a(0)a¯(1) .
ln l+n m n m n
(cid:16) (cid:17)i (35)
Substitute(34) into (35) and integrate overtime to obtain
∞
a(2)(T) = 2Vl Vma a a E[ωl ,ωl ]δm 2V¯µ a a a¯ E¯[ωlν ,ωl ]δµ δl
l mn − µν n µ ν nµν mn µ+ν − mν n µ ν nµ mn m+ν m+n
m,nX,µ,ν=1h (cid:16) (cid:17)
+2V¯m Vma¯ a a E[ωln, ωm]δm 2V¯µ a¯ a a¯ E[ ωµ , ωm]δµ δm
ln − µν n µ ν µν − ln µ+ν − mν n µ ν − nνl − ln m+ν l+n
(cid:16) (cid:17)
+2V¯m V¯na a¯ a¯ δn E[ ωm , ωm]+2Vµa a¯ a E[ωµl , ωm]δµ δm ,
ln µν m µ ν µ+ν − lνµ − ln nν m µ ν νm − ln n+ν l+n
(cid:16) (cid:17) i (36)
10