Wave turbulence and intermittency

Transcription

1 Physica D 5 53 () 5 55 Wave turbulence and intermittency Alan C. Newell,, Sergey Nazarenko, Laura Biven Department of Mathematics, University of Warwick, Coventry CV4 7L, UK Abstract In the early 96s, it was established that the stochastic initial value problem for weakly coupled wave systems has a natural asymptotic closure induced by the dispersive properties of the waves and the large separation of linear and nonlinear time scales. One is thereby led to kinetic equations for the redistribution of spectral densities via three- and four-wave resonances together with a nonlinear renormalization of the frequency. The kinetic equations have equilibrium solutions which are much richer than the familiar thermodynamic, Fermi Dirac or Bose Einstein spectra and admit in addition finite flux (Kolmogorov Zakharov) solutions which describe the transfer of conserved densities (e.g. energy) between sources and sinks. There is much one can learn from the kinetic equations about the behavior of particular systems of interest including insights in connection with the phenomenon of intermittency. What we would like to convince you is that what we call weak or wave turbulence is every bit as rich as the macho turbulence of 3D hydrodynamics at high Reynolds numbers and, moreover, is analytically more tractable. It is an excellent paradigm for the study of many-body Hamiltonian systems which are driven far from equilibrium by the presence of external forcing and damping. In almost all cases, it contains within its solutions behavior which invalidates the premises on which the theory is based in some spectral range. We give some new results concerning the dynamic breakdown of the weak turbulence description and discuss the fully nonlinear and intermittent behavior which follows. These results may also be important for proving or disproving the global existence of solutions for the underlying partial differential equations. Wave turbulence is a subject to which many have made important contributions. But no contributions have been more fundamental than those of Volodja Zakharov whose 6th birthday we celebrate at this meeting. He was the first to appreciate that the kinetic equations admit a far richer class of solutions than the fluxless thermodynamic solutions of equilibrium systems and to realize the central roles that finite flux solutions play in non-equilibrium systems. It is appropriate, therefore, that we call these Kolmogorov Zakharov (KZ) spectra. Elsevier Science B.V. All rights reserved. Keywords: Wave turbulence; Intermittency; Asymptotic closure; Cumulants. Introduction and motivation Turbulence is about understanding the long-time statistical properties of solutions of nonlinear field equations with sources and sinks, and in particular with calculating transport. Sometimes the transport is in physical space such as the flux of heat carried across a layer of fluid by turbulent convection, or the flux of momentum from a fast moving turbulent stream to a fixed plate or the flux of angular momentum across an annulus from an inner rotating cylinder to slower moving outer one. However, the transport can also be in Fourier space, such as the flux of energy from large energy containing scales to small scales where dissipation occurs, and it is the description of this transport that is the main topic of this lecture. Corresponding author. address: anewell@maths.warwick.ac.uk (A.C. Newell). Also at University of Arizona, Tucson, AZ 857, USA //$ see front matter Elsevier Science B.V. All rights reserved. PII: S67-789()9-

2 A.C. Newell et al. / Physica D 5 53 () The picture we have in mind is this. We have a source of energy at large scales, throughput at middle scales (windows of transparency, inertial ranges) over which the system is essentially conservative and Hamiltonian, and dissipative output at small scales. Among other things, we would like to find solutions of the moment equations for the unforced, undamped system which describe a finite flux of some conserved density, such as energy, across these windows of transparency. This picture derives from the Richardson scenario for the complex vorticity and irregular flow fields encountered in 3D, high Reynolds number hydrodynamics, the granddaddy of turbulent systems. The main role of the energy conserving nonlinear terms (advection and pressure) in the Euler equations is to transfer energy from the large eddies at the integral scale where the system is forced to smaller and smaller eddies and eventually to the viscous sink. If the average kinetic energy E = u is written as E(k)dk, the dissipation rate P = νω as ν k E(k)dk, then the von Karman Howarth equation is E(k, t) t = f(k)+ T(k) νk E(k), (.) where f (k), T (k) and νk E(k) represent forcing, nonlinear transfer and damping, respectively. E(k) is the angle averaged Fourier transform of the two point velocity correlation function. T(k)is the Fourier transform of third-order moments. Since the nonlinear transfer T(k)conserves energy, we may write it as the k derivative of a flux, P / k. Then, in the window of transparency, t b a E dk = P a P b, which can be zero either because P a = P b = or because P a = P b = P, a constant. The first possibility describes an isolated system and leads to an equidistribution of energy in the interval (a, b). The second possibility allows for an energy flux between source and sink, and leads to what is known as the Kolmogorov or finite flux spectrum (see Fig. ). However, there is a fundamental difficulty with the moment (cumulant) hierarchy of which (.) is the first member. The hierarchy is unclosed and infinite, and no cumulant discard approximations work. Three-dimensional, high Reynolds number hydrodynamics affords the theoretician no useful approximations, no separation of scales, and no footholds for analytical traction. It is too difficult. Simple fluids are easier to drink than they are to understand. Nevertheless, some progress in understanding turbulent behavior has come as a result of the formidable insights of Kolmogorov. Arguing that the symmetries of translation and isotropy are restored in the statistical sense and that, in the infinite Reynolds number limit, all small-scale statistical properties depend only on the local scale and the energy dissipation rate P, one can deduce that lim r lim ν lim t (u ( x + r,t) u ( x,t)) n =C n (Pr) n/3, (.) Fig.. Forcing, inertial range and damping regions of k space.

3 5 A.C. Newell et al. / Physica D 5 53 () 5 55 where u = u ˆr and C n are universal constants. The limits correspond to long time and the necessity of staying away from the sink and source scales, respectively. But, for almost all n, (.) is conjecture. From the Navier Stokes equations, it can only be proved rigorously for n = 3 [37]. Moreover, recent experimental evidence seems to indicate that the nth-order structure function, n 4, S n (r) = (u( x + r) u( x)) n has index less than 3 n. If that is the case, the ratio S n/(s ) n/, which in the Kolmogorov theory is a pure constant, diverges as r. The divergence of this ratio indicates that large fluctuations are more probable than Kolmogorov theory would suggest. The present consensus is that fluctuations in the local dissipation rate are responsible for elevating the tails of the probability density function for velocity differences and for what is generally called intermittent behavior. In contrast, the turbulence of a sea of weakly coupled, dispersive wavetrains has a natural asymptotic closure. One can:. find a closed kinetic equation for the spectral energy density;. understand the mechanisms (resonance) by which energy and other conserved densities are redistributed throughout the spectrum; 3. obtain stationary solutions of the kinetic equation analogous to both the thermodynamic and Kolmogorov spectra; 4. test the validity of the weak turbulence approximation; Moreover, the results are not simply a useful paradigm but are of direct interest in their own right in a variety of contexts from optics to oceans, from sound to semiconductor lasers to the solar wind. The natural closure occurs because of two factors, the weak coupling, ε, <ε and the dispersive nature of the waves. The combination means that there is an effective separation of time scales. On the linear time scale t L = ω,ω a typical frequency in the initial spectrum, the higher-order cumulants decay towards a state of joint Gaussianity. On the much longer nonlinear time scale t NL = ε (r 4) ω, nonlinear resonant interactions of order r, r = 3, 4, 5, bring about coherence and a departure from joint Gaussian behavior. The regeneration of the higher-order cumulants by nonlinearity occurs in a special way. For example, in the regeneration of the third-order cumulant (moment), the product of second-order cumulants is more important than the fourth-order cumulant. This pattern continues. The regeneration of cumulants of order N is dominated by products of lower-order ones. This feature leads to the natural closure. The new message of this paper is to suggest that wave turbulence is an even richer paradigm for non-equilibrium systems than previously believed. The reason is simple and dramatic. As the system relaxes to its asymptotic stationary state, the closure equations almost always become non-uniform at some scale we call k NL.Ifk NL lies near the ultraviolet (infrared) end of the spectrum, then for all k, k > k NL (k < k NL ), the dynamics become increasingly dominated by large fluctuation local events which are intermittent and fully nonlinear. Some are shock-like; others are spawned by condensate formation. We give explicit formulae for k NL in terms of properties of the linear dispersion relation, the nonlinear coupling coefficients and the KZ fluxes.. Wave turbulence: asymptotic closure The derivation of the closed equations for the long-time behavior of the statistical moments (cumulants) and a discussion of their properties is carried out in a series of seven steps.. The set up: the equation for the Fourier amplitudes.. Moments, cumulants.

4 A.C. Newell et al. / Physica D 5 53 () The cumulant (BBGKY) hierarchy. 4. The strategy for solution and the dynamics. 5. Resonant manifolds and asymptotic expansions. 6. The closure equations. 7. Properties: conservation laws, reversibility, finite flux solutions and their temporal formation... The set up: equation for Fourier amplitudes Consider a system which in the linear limit admits wavetrain solutions u s = exp(i k x + iω s ( k)t), where ω s ( k) is the dispersion relation and s labels the degree of degeneracy which corresponds to the order of the system or the number of frequencies corresponding to a given wavevector k. Often, s =+, corresponding to second-order systems where waves travel in one of two directions. By an appropriate choice of canonical variables (sometimes suggested by the Hamiltonian structure), we write u s ( x,t) = A s ( k, t) e i k x d k, A s ( k, t) = A s k, As ( k, t) = (π) d u s ( x,t)e i k x d x, (.) and find (for ω s = sω( k), s =+,, ω( k) = ω k ), da s k isω k A s k = ε L ss s A s k A s k δ( k + k k) d k s s +ε L ss s s 3 k 3 A s k A s k A s 3 k3 δ( k + k + k 3 k) d k 3 +. (.) Remark. s s s 3. Since the class of functions we are dealing with are bounded and non-decaying as x,a s k is not an ordinary function but a generalized one. However, as we shall see, for spatially homogeneous systems, ensemble averages of products of the Fourier amplitudes have good properties.. Notation: δ, = δ( k + k k), d k = d k d k, etc. 3. ε, <ε, is a measure of nonlinearity. The system Hamiltonian takes the form H = ω k A s k A s k d k +. (.3) s Examples. Example (a).. Optical waves of diffraction in nonlinear media [3,4,45].. Superfluids [3,53]. These examples are described by the nonlinear Schrödinger equation for a complex field u( x,t), x R d,t R. ( u u a u u ) d x, u t + i u + iau u u = a constant, = i δh t δu, H = u( x,t) = u + = A + k ei k x d k, u ( x,t) = u = A k ei k x d k, A k = A+ k. (.4)

5 54 A.C. Newell et al. / Physica D 5 53 () 5 55 For fluctuations about the zero state, u s =, A s k isk A s k t = ias A s k A s k A s k 3 δ 3, d k 3. Thus, εl ss s =, ε L ss s s 3 k 3 = 3 iasp3 δ s, sδ s,sδ s3,s, ω k = k. In Fourier coordinates (A k = A + k,a k = A k ) H = k A k A k d k a A k A k A k A k3 δ,3 d k 3. For fluctuations about the condensate state u = ρ / e iaρ t, we rewrite (.4) in polar coordinates u = ρ / e iϕ, and obtain ρ ϕ + (ρ ϕ) =, t t + ϕ + ( a)ρ ( ρ) =, (.5) ρ which are the Euler equations for a compressible fluid with velocity field ϕ and pressure p = aρ plus the addition of the extra term (/ p) ( ρ) sometimes called the quantum pressure. Clearly, p/ ρ > only when a<. Take a =. The Hamiltonian is H = (( ρ / ) + ρ( ϕ) + ) ρ d x, and (.5) is ρ/ t = δh/δϕ, δϕ/ t = δh/δρ. Setting ρ = ρ + δρ, ϕ = ρ t + δϕ, one finds to quadratic order that ( ) t δρ + k δϕ = (δρ δϕ), t δϕ + k δρ = k 4 δρ δρ k 4 (δρ) ( (δϕ)), where k = ρ. Let ( ) δρ = ρ k e i k x d k, δϕ = ϕ k e i k x k d k, ρ k = k / A s k ω, k s ( ) / ϕ k = i ω k sa s k, ω k = k k + k 4, k k s and find da s k isω k A s k = ε s s L ss s A s k A s k δ, d k + cubic terms, (.6)

6 where εl ss s = i 4k s k k A.C. Newell et al. / Physica D 5 53 () ( ωω k ω k k ωk k ) / + s k k ( ωω k ω k k ) / ( ) / ( ) ω ω k +ss s k k k k / + k s( k k k ) ωω ω. Note: εl +++ kk,k = i V kk k in Ref. [3]. For k k, L L because leading terms cancel. For k k, L k 3/. In general, for the class of zero mean conservative systems (.) for which A s k and A s k are conjugate variables and A s kt = isδh/δa s k, the following properties hold:. L ss s r kk k r = L ss s r kk k r = L s s s r k k k r.. L ss s r kk k r is symmetric in (,,...,r). 3. L ss s r k k r =, k + + k r = (except for NLS). 4. L s s s s r k k k k r = (s /s)l ss s s r k r when k + + k r = k. Example (b). Water waves [,,4 8,,,9,3,34,36,44,48]. For details, see [] (see Fig. ). Fig.. An ocean of depth h, surface deformation η(x,y,t) and velocity potential ψ(x,y,z,t). ωk η(x,y,t) = η( x,t) = A s k s ν ei k x d k, ψ( x,t) = k s ω k = (gk + σk 3 ) tanh kh, νk = g + σk,g= gravity, σ= S/ρ = Energy = ω k Q s s (k) d k, A s k As k =δ(k + k )Q ss (k ). s iν k s A s cosh k(z + h) k e i k x d k, ωk cosh kh surface tension, density Also εl ss s = i { ( ω ω ν P ν (k ω ν ν s ω + k k ) sω ν ε L ss s s 3 k 3 = i { ( ) ω ω ω 3 ν 3 3 P Ps 3 ω 3 ω ν ν ν 3 6 k 3 + k k 3 [ sω ν 3 k3 + ν k 3ν + 3 P ν ω 3 (k ] k k 3 ) ω ω 3 4 s ω s 3 ω 3 ω ( ) +k 3 h sω 3ν σ ω + ω 3 + s ω s 3 ω 3 k k 3 ν ν 3 s ω s 3 ω 3 ω + ω + s ω s ω k k ν ν s ω s ω ν sω ν ν 3 6ν k + k 3 tanh k ( ( k k )( k k 3 ) 3 k k k 3 ) }, )},

7 56 A.C. Newell et al. / Physica D 5 53 () 5 55 where P 3 is permutation over (, 3), P 3 over (,, 3). Forg σk,l ss s s 3 k 3 is homogeneous of order 3, i.e. L εk = ε 3 L k.forg σk,l ss s is homogeneous of order 4 9. Example (c). Sound waves [3,4,,38]. ( ) ( ρ µ p = p, c = µp, ρ = ρ + ) A s k ρ ρ ei k x d k, s v j = s c k j sω k A s k ei k x d k, ω k = ck, where p, ρ and v j are pressure, density and the jth component of velocity, respectively. ( εl ss s = ic k k + ) k k sω + k k + i 4 s ω s ω s ω s ω 4 (µ )sω, ε L ss s s 3 k 3 = iω k (µ )(µ 3). Each is homogeneous in k of degree. For examples of weak turbulence in semiconductor lasers, magnetohydrodynamics, coupled oscillators, plasmas and atmospheric waves, quantum systems see [3,9,,5 8,,,4,8,3,3,39,4,45,46,49]... Moments, cumulants But Eq. (.) for the Fourier amplitude is only a means to an end. We really want to describe the behavior of moments ss s (N ) MN ( x; r, r r,..., r (N ) ; t) = u s ( x)u s ( x + r) u s(n ) ( x + r (N ) ) (.7) as time t becomes large. At this stage denotes an ensemble average connected with a joint probability density function (jpdf) P(u s,us,...,us(n ) (N ) ). Namely, P(us,...)dus is the probability that the field us at x lies between u s and u s + dus, the field us at x + r lies between u s and us + dus, and so on. However, we now make the assumption of spatial homogeneity. M N only depends on the relative geometry of the configuration and does not depend on the base coordinate x. In this situation, it is convenient to think of as an average over the base coordinate. But one should be cautious with the assumption of spatial homogeneity. It is a property of the equations that if M N is independent of the base point x initially, it will remain so. But suppose M N depends weakly on x initially. Then, the nonlinear dynamics can lead to a steepening of derivatives (cf. Burgers equation) and a possible breakdown of spatial homogeneity into patches separated by shocks. In this essay, we ignore this possibility. Moments are inconvenient for several reasons. In particular M N does not tend to zero as the separations r, r,... tend independently to infinity. This means that it does not possess an ordinary Fourier transform which is the coordinate most relevant when dealing with a field of wavetrains. Therefore, we use cumulants which are related in a : fashion with the moments. M s (t) = us ( x) =R ()s (t) ={u s ( x)}, M ss ( r,t) = us ( x)u s ( x + r) ={u s ( x)u s ( x + r)}+{u s ( x)}{u s ( x + r)}=r ()ss ( r) + R ()s R ()s, M ss s 3 ( r, r,t)= u s ( x)u s ( x + r)u s ( x + r ) =R (3)ss s ( r, r ) + one of all possible partitions, = R (N)ss + one of all possible partitions. M (N)ss N

8 A.C. Newell et al. / Physica D 5 53 () Curly brackets denote cumulants, and angle brackets are moments. We will also study systems where M = u. Then, M ss ( r) = R()ss ( r) = u s ( x)u s ( x + r), M ss s 3 ( r, r ) = R (3)ss s ( r, r ) = u s ( x)u s ( x + r)u s ( x + r ), M ss s s 4 ( r, r, r ) = R (4)ss s s ( r, r, r ) + R ()ss ( r)r ()s s ( r r ) + R ()ss ( r )R ()s s ( r r) +R ()ss ( r )R ()s s ( r r),. R (N)ss s(n ) ( r, r,...,r (N ) ) have the property that R (N) as r, r,...tend to infinity independently and in any directions because it is reasonable to assume, at least at one particular time when the fields are first stirred, that the statistics of widely separated points are uncorrelated. This means that its Fourier transforms Q (N), R (N)ss s(n ) ( r, r,..., r (N ) ) = Q (N)ss s(n ) ( k, k,..., k (N ) ) e i k r+i k r + +i k (N ) r (N ) d k d k d k (N ), Q (N)ss s(n ) ( k, k,..., k (N ) ) = (π) (N )d R (N)ss s(n ) ( r, r,..., r (N ) ) e i k r i k (N ) r (N ) are smooth functions, at least at one time. Remark. d r d r d r (N ) (.8). The dynamics will produce non-smooth components due to nonlinearities. Indeed, understanding these nonsmooth behaviors is the key to understanding the long-time behavior of wave turbulent fields.. The notation of pairing k with r, k with r,...and k (N ) with r (N ) is a matter of future convenience..3. The cumulant hierarchy We now introduce the dynamics by writing down the equations for the Fourier space cumulants Q ()ss (k ), Q (3)ss s (k,k ),...,Q (N)ss s(n ) (k,k,...,k (N ) ). To do this, we first seek a relationship between averages of products of the Fourier amplitudes A s k and the Fourier cumulants Q(N). We know already that A s k = (π) d u s ( x)e i k x d x is not an ordinary function because u s ( x) does not tend to zero as x. It is simply bounded there. If u s ( x) were a finite sum of wavetrains, u s (x) = j As j ei k j x, then A s k = j As j δ( k k j ), namely the sum of complex numbers times Dirac delta functions. Interacting wavetrains transfer energy via three-wave interactions on a time scale of ω t = O(/ε). To see this, write A s k in the above form, solve (.) iteratively and find that the zero denominators first enter when one calculates the first iterate at order ε. If u s ( x) were to belong to a set of smooth fields which decayed sufficiently rapidly as x, then A s ( k, t) would be smooth and the asymptotic expansion A s ( k, t) = a s( k) e isωkt + εa s ( k, t) + would remain uniformly valid in time. This is because the small denominators appear in integrals in the combination iw, (exp(iw,t) ), W, = s ω +s ω sω. Even though W, can be zero, from Section.5 we will see that A s ( k, t) is bounded

9 58 A.C. Newell et al. / Physica D 5 53 () 5 55 in time (we exclude the case of multiple zeros). Thus, if the fields are sufficiently small and decay sufficiently fast as x, for all intents and purposes, (.) behaves as a linear system. The physical reason is that resonant wavetrains are not long enough to have enough time to interact to produce order exchanges of energy. The fields of interest to us here are bounded at infinity and consist of collections of infinitely long wavetrains (or wavepackets). Because they are infinite in extent, they have enough time to exchange energy and produce long-time cumulative effects. However, because they are not collections of discrete wavetrains, there is additional phase mixing and, due to these statistical cancellations, the non-uniformities (coherences) produced by nonlinear interactions take longer to appear and energy is exchanged on the time scale ε ω. To summarize, L functions have wavetrains which do not interact long enough to produce order cumulative changes. Discrete wavetrains interact very strongly and, via r wave resonances, exchange order amounts of energy on the time scale ω t = O(ε r+ ). Continuous spectra of infinite wavetrains, corresponding to bounded, spatially random fields, have additional mixing which slows the rate of order energy exchange via r wave resonances to the time ω t = O(ε (r ) ). Let u s ( x) be a spatially homogeneous random field of zero mean. Then, because A s k is a functional of us ( x), A s k =, A s k As k = (π) d = (π) d u s ( x)u s ( x + r) e i k x i k ( x+ r) d x d( x + r) R ()ss ( r)e i k r d r e i( k+ k ) x d x = δ( k + k )Q ()ss ( k ) = δ( k + k )Q ()ss ( k). Note R ()ss ( r) = R ()s s ( r) because u s ( x)u s ( x + r) = u s ( x r)u s ( x ) =R s s ( r). Likewise A s k As k A s k = δ( k + k + k ) (π) d R (3)ss s ( r, r ) e i k r i k r d r d r = δ( k + k + k )Q (3)ss s ( k, k ), A s k As k A s k A s k =δ( k + k + k + k )Q (4)ss s s ( k, k, k ) +δ( k + k )δ( k + k )Q ()ss ( k )Q ()s s ( k ) +δ( k + k )δ( k + k )Q ()ss ( k )Q ()s s ( k ) +δ( k + k )δ( k + k )Q ()ss ( k )Q ()s s ( k ) For convenience, we often write Q ()ss ( k ) as Q ()ss ( k, k ), remembering k + k = and Q (3)ss s ( k, k ) as Q (3)ss s ( k, k, k ), remembering k + k + k =. This allows one to keep better track of symmetries. Remark. Note A s k is now not a Dirac delta function as it was when us (x) was a sum of discrete wavetrains. Its products involve, after averaging, a Dirac delta function times a smooth (at least at some initial time) cumulant. The fact that the cumulants containing products of δ functions in A s k As k A s k A s k (namely the products of Q () s) have more potency than the fourth-order cumulant makes for a natural asymptotic closure. The regeneration of higher-order cumulants by the dynamics depends only weakly on even higher ones and strongly on products of lower ones.

10 A.C. Newell et al. / Physica D 5 53 () We build the cumulant hierarchy by multiplying (.) by A s k, and (.), with s, k replaced by s,k,bya s k and adding to find, after averaging dq ()ss ( k ) = P ε s s i(sω + s ω )Q ()ss ( k ) L ss s Q (3)s s s ( k, k )δ, d k + P ε +3 P ε Q ()s 3s ( k ) s s s 3 s s s 3 L ss s s 3 k 3 Q (4)s s s 3 s ( k, k, k 3, k )δ,3 d k 3 L ss s s 3 kk k k Q()s s ( k ) d k, k + k =. (.9) In (.9), we have factored out δ(k + k ) and used the symbol P to denote rewriting the following term with s, k replaced by s,k and adding the two contributions together. Multiplying (.) by A s k A s k and applying P,we find to order ε dq (3)ss s ( k, k ) = P ε s s i(sω + s ω + s ω )Q (3)ss s ( k, k ) L ss s Q (4)s ss s s ( k, k, k )δ, d k +ε P L ss3s4 k k k Q ()s 3s ( k )Q ()s 4s ( k ) + o(ε ), (.) s 3 s 4 where k + k + k =. Note: It is convenient to relabel s,s as s 3,s 4 in the second term of the right-hand side of (.). In addition, we have used the properties that L = when k = and is symmetric over the indices,. We continue to build the hierarchy in this way. The reader should write down the equation for Q (4)ss s s ( k, k, k ). Note that the consistency of A s k = requires that Lss s k k =. Proof. d A s k isω k A s k =ε L ss s A s k A s k δ, d k = ε δ(k) s s s s d A s k = when L ss s k k =..4. The strategy for solutions and the dynamics L ss s k k Q ()s s (k ) d k, We solve the cumulant hierarchy of which (.9) and (.) are the first two members as an asymptotic expansion in powers of ε, namely we assume Q (N)ss s(n ) ( k,..., k (N ) (N)ss s(n ) ) = q ( k, k,..., k (N ),t)exp(i(sω + s ω + +s (N ) ω (N ) )t) + εq (N) + ε Q (N) + +ε r Q (N) r + o(ε r ). (.) This means that the difference between Q (N) and any finite sum, say up to ε r, divided by ε r, tends to zero as ε. We will insist that (.) is a valid asymptotic expansion for all time t. To achieve this, we will have to allow q (N) to vary slowly in time.

11 53 A.C. Newell et al. / Physica D 5 53 () 5 55 Remark.. Because of non-smoothness which will appear in Q (3),Q(4) in the long-time limit, it is better to interpret the statement of asymptotic expansions in physical space. Namely, we order Q (N) where it makes sense and, where it does not, we order the corresponding R (N).. We are seeking asymptotic and not convergent series representations of the solutions to (.9) and (.)! We recognize that, because of small denominators which will occur on resonant manifolds M defined by k + k = k, s ω( k ) + s ω( k ) = sω( k), (.) non-uniformities in the asymptotic expansions will arise. For example, we will find lim Q ()ss t ( k ) = t Q ()ss ( k ) + ˆQ ()ss ( k ) ε t fixed will contain terms which grow with t and terms which are bounded. As a result, for times ε t = O(), the asymptotic expansion is not well ordered. We restore the well-ordered asymptotic expansion by recognizing that, because of these secular terms, there will be an order transfer in spectral energy (i.e. in Q ()ss ( k ), s = s) over long times. (N)ss s(n ) This transfer is captured by seeking an asymptotic expansion for the time derivative of q ( k,..., k (N ) ) as (N)ss s(n ) dq ( k,..., k (N ) ) (N)ss s(n ) = εf ( k,..., k (N ) ) +ε (N)ss s(n ) F (k,...,k (N ) ) +, (.3) and choosing F (N),F (N),... so as to remove secular terms in (.) and render it a well-ordered asymptotic expansion uniform in time t. Eq. (.3) is the equation of asymptotic closure. This is closed because N =, s = s, F () =, F () depends only on q ss (k) = n k (t). This is the kinetic equation and we write it as dn k = T [n k ] + T 4 [n k ] +. (.4) N =, s = s and N>, F (N) =, F (N) (N)ss s(n ) = iq ( k,..., k (N ) )(Ωk s + +Ωs(N ) ) which is k solved by frequency renormalization sω k sω k + ε Ωk s +. (.5) We, therefore, achieve a natural asymptotic closure for the cumulant hierarchy. It is natural because it does not assume anything about the initial statistical distributions of the fields. They need not be joint Gaussian. What happens is that the linear behavior of the system brings the system close to a state of joint Gaussianity. Non-joint Gaussianity is regenerated by nonlinear interactions, e.g. dq (3) / depends on Q (4) and products of Q (). However, it is the products of the lower-order cumulants that give a cumulative long-time effect and therefore only they play a role in the long-time dynamics. Hence, one achieves closure. It is asymptotic because the closure equations (.3) (.5) are long-time average equations. They are solved iteratively. To find the behavior for times ω t = o(ε ) one solves the truncated equation (.4) with T 4,T 6 ignored. To find the behavior for ω t = o(ε 4 ) one solves (.4) with T 6,T 8,...ignored, and so on.

12 It is important to make several points. A.C. Newell et al. / Physica D 5 53 () Although T N [n k ],N, is formally of order ε N in (.4) and ε N Ω s Nk,N, is formally of order εn in (.5), the terms in the series are still functions of k. It turns out that the ratios T N+ /T N,ε Ω s N+k /Ωs Nk, formally of order ε, do not in general remain uniformly bounded as k ork when calculated on solutions of the truncated equations. This fact is the origin of the breakdown of wave turbulence at small and large scales. We return to this in Section 4.. The term T is the kinetic equation (.4) contains triad (three-wave) resonances. The term T 4 has been calculated [] and contains four-wave resonances and gradients (with respect to k) of three-wave resonances and principal part integrals. It is not clear how many of these terms can be reabsorbed into T by renormalizing the frequency in the exponent of the Dirac delta function. Often the successive terms in the series are schematically represented in diagram notation. 3. The imaginary part of the renormalized series (.5) for sω k has positive imaginary part at o(ε ) if there are three-wave resonances, at o(ε 4 ) if there are four-wave resonances, and so on. This means that, over long times, and due to resonant interactions, the zeroth-order Fourier space cumulants (which essentially correspond to initial data), N =, s = s, N 3, decay exponentially. Their physical space counterparts decay for two reasons. They decay algebraically in t because of the Riemann Lebesgue lemma and the presence of fast oscillations multiplying the zeroth-order Fourier space cumulants. These remarks must be revisited in those regions of k space where the weak turbulence closure (.4) and (.5) is non-uniform in k = k..5. Resonant manifolds and asymptotic expansions Solve (.9) and (.),...iteratively. Q ()ss ( k,t)= q ()ss ( k,t)e i(sω+s ω )t, k + k =, Q (3)ss s ( k, k,t)= q (3)ss s ( k, k,t)e i(sω+s ω +s ω )t, k + k + k =, ε :. Note: (N)ss s(n ) Q (N)ss s (N ) ( k, k,..., k (N ) ) = q ( k k (N ),t)e i(sω+ +s(n ) ω (N ))t, k + + k (N ) =. For N =, s = s, the oscillatory dependence disappears because ω( k) = ω = ω. q ()s s ( k,t) is proportional to the spectral energy density. In anticipation of non-uniformities in (.), we allow the coefficients q (N) vary slowly in time via (.3). ε : Q ()ss (k,t)= P s s L ss s kk,k q (3)s s s ( k, k ) (W, ) e i(sω+s ω )t δ, d k, where (x) = (e ixt )/ix, W, = s ω + s ω sω. We need to consider long-time behavior of integrals of the form F( k ) eih( k ;k)t d k, ih( k ; k)

13 53 A.C. Newell et al. / Physica D 5 53 () 5 55 where h( k ; k) = s ω( k ) + s ω( k k ) sω( k). The integrals are dominated by values of k near a zero of h( k ; k), the resonant manifold M defined by (.), k + k = k, s ω( k ) + s ω( k ) = sω( k) for some s,s,s. In the neighborhood of M : h( k ; k) =, we coordinatize k (say k x,k y if k is D in a basis locally parallel and perpendicular to M. Let k () M. Then h( k ; k) = h( k () ; k) + ( k k () ) k h( k () ; k) +. In what follows, we will assume that on M, k h( k () ; k) for k () M, and illustrate by example what happens when k h =. Calling the local perpendicular coordinate by x, the above integral can be written as dy Therefore, we study lim t f(y,x) eixt ix f(x) eixt ix dx. dx. In Q (), the function f(x)(proportional to s s q(3)s ( k, k )) is slowly varying in time so the limit is taken in the sense that t,ε r t fixed for some r =,,... The smoothness of q (3)s s s ( k, k ) is also important. At the initial time, we have assumed this. If we are to recalculate beginning at some later time t = O(/ε ),ano(ε) non-smoothness will have developed in the new initial state. We will discuss this and show that for lim t t + it does not contribute whereas for lim t t it does. This is important in resolving an apparent irreversibility paradox. A little calculation (which can be done many ways, e.g. let f(x)= f(x) f() e x + f() e x ) reveals where lim t sgnt = f(x) eixt ix { +, t >,, t <, f(x) dx = π sgnt f() + ip dx, x and P denotes the Cauchy principal value. Schematically ( ) lim (x) = π sgnt δ(x) + ip. t x Returning to Q ()ss ( k,t), we see that it is bounded as t. This is important. A non-zero q (3)s s s ( k, k ) at the initial time (here taken to be t = ) does not affect the uniformity of (.). We can begin with initial conditions far from joint Gaussian. This was first pointed out by Benney and Saffman [4].

14 Next, we calculate Q (3)ss s ( k, k ) = P s s A.C. Newell et al. / Physica D 5 53 () L ss s q (4)s s s s ( k, k, k ) (W, ) e i(sω+s ω +s ω )t δ, d k + s 3 s 4 L ss 3s 4 k k k q ()s 3s ( k )q ()s 4s ( k ) (s 3 ω + s 4 ω sω)e i(sω+s ω +s ω )t + s 3 s 4 L s s 3 s 4 k k k q()s 3s ( k )q ()s 4s ( k) (s 3 ω + s 4 ω s ω ) e i(sω+s ω +s ω )t + s 3 s 4 L s s 3 s 4 k k k q ()s 3s ( k)q ()s 4s ( k ) (s 3 ω + s 4 ω s ω ) e i(sω+s ω +s ω )t. (.6) Note: Because there are no integrals left on some of these terms, and because they contain oscillating factors, their long-time limit cannot be directly taken. We, therefore, return to physical space and examine instead the asymptotic well orderedness of the corresponding physical space cumulant R (3). Schematically, however, we can write lim t (x) = π sgnt δ(x)+ip(/x). The integral in physical space will generally be the product of bounded and oscillatory factors which tend to zero because of the Riemann Lebesgue lemma. However, for s 3 = s,s 4 = s in the first, s 3 = s,s 4 = sin the second, and s 3 = s, s 4 = s in the third, the exponents coalesce ( sω s ω s ω ) e i(sω+s ω +s ω )t = (sω + s ω + s ω ) and these terms survive the t limit. In Section 4, we will calculate these surviving terms for each R (N) and examine their asymptotic expansions for well orderedness. We will find that, consistent with Remark in Section.4, they are not always well ordered. Non-uniformities in the corresponding asymptotic expansions for the structure functions can appear at both small and large separations. This can lead to intermittent behavior dominated by fully nonlinear solutions of the field equations. We will discuss this further in Section 4. As t ±, Q (3)ss s ( k, k ) Q (3)ss s ( k, k ) = ( ( π sgnt δ(sω + s ω + s ω ) + ip sω + s ω + s ω ( P L s s s k k k q () s s ( k )q () s s ( k )). (.7) In our calculation of Q ()ss ( k ), however, we will find that it is Q (3)s s s ( k, k ) e i(sω+s ω )t which appears in the integrand and it is its long-time limit that we will need. Remark. We now, at O(ε), have seen the appearance of irreversibility in a reversible system. The limit process effectively ignores all the fluctuating terms so that information on phase is effectively lost in taking the limit. Solutions of the resulting kinetic equations can be attractors. Before we complete the calculations by identifying the secular terms which appear at O(ε ) and thereby calculate the kinetic equation and renormalization factors (i.e. F (N) for N ), let us return to the study of (.5) and illustrate in Example what happens if h ˆn = (ˆn unit normal to M). Example (ω = k ). Take k = (k, ); k = (k x,k y ), M : h = k + ( k k ) k = ((k x k) + k y 4 k ). The resonant manifold for k is M( k) and it is the set of k for which h =. It is a circle of radius k centered at the middle of the vector k. We can coordinatize M as follows: k x = k cos (θ/), k y = k sin(θ/) cos(θ/), h = (k x k), k y = k(cos θ, sin θ) is never identically zero ( h ˆt =, h ˆn ). ))

15 534 A.C. Newell et al. / Physica D 5 53 () 5 55 Example (Acoustic waves: ω = k ). Here h = s k +s k k s k is zero when k is collinear with k.now h = for all k M. The integral (.5) now depends critically on dimension. In D, f (x)((e iht )/ih) d k f (x)((e ix t )/ix ) dx t /.In3D, f(x,y)(e i(x +y )t /i(x + y )) dx dy ln t (see [3,4,38]). Remark. The web created by the manifolds M( k), M( k ), M( k ),..., where k M( k), k M( k ),...,, has an interesting geometry [4]..6. The closure equations We now come to the final step in the closure calculation, namely the identification of the first terms in (.) which are non-uniform, the choices for F (N),F (N) and the resulting equations for asymptotic closure. The equation for Q ()ss ( k ) is d (Q()ss ( k ) e i(sω+s ω )t ) + F ()ss ( k )= P L ss s (Q (3)s s s ( k, k ) e i(sω+s ω )t )δ, d k. (.8) s s From (.6) (the appearance of the indices, here necessitated the use of the dummy indices 3, 4 in (.6)), Q (3)s s s ( k, k, k ) e i(sω+s ω )t = P L s s 3 s 4 k k 3 k 4 q (4)s s s 3 s 4 ( k, k 3, k 4 ) (W 34, ) e i(s ω +s ω +s ω )t δ,34 d k 34 e i(sω+s ω )t s 3 s 4 + s 3 s 4 L s s 3 s 4 k k k q ()s 3s ( k )q ()s 4s ( k ) (s 3 ω + s 4 ω s ω ) e i(s ω +s ω +s ω )t e i(sω+s ω )t + s 3 s 4 L s s 3 s 4 k k k q ()s 3s ( k )q ()s 4s ( k ) (s 3 ω + s 4 ω s ω ) e i(s ω +s ω +s ω )t e i(sω+s ω )t + s 3 s 4 L s s 3 s 4 k k k q ()s 3s ( k )q ()s 4s ( k ) (s 3 ω + s 4 ω s ω ) e i(s ω +s ω +s ω )t e i(sω+s ω )t. Note: δ( k k 3 k 4 ) combined with δ( k + k + k ) implies δ( k + k + k 3 + k 4 ), the Dirac delta function associated with q (4)s s s 3 s 4 ( k, k, k 3, k 4 ) in the first term of the above equation. For the second term, strong response occurs only when s 3 = s,s 4 = s and s = s. The dominant term is ( ) ( ) δ s sl s s s k k k q () s s ( k )q () s s ( k ) π sgnt δ(s ω + s ω sω) + ip. s ω + s ω sω Recall also k + k =. For the third term, strong response occurs when s 3 = s,s 4 = s (any s, s ); the dominant term is ( ) ( ) L s s s k k k q() s s ( k )q ()ss ( k ) π sgnt δ(s ω + s ω sω) + ip. s ω + s ω 3 sω For the fourth term, strong response occurs when s 3 = s, s 4 = s (any s, s ); the dominant term is ( ) ( ) L s s s k k k q ()ss (k )q () ss (k ) π sgnt δ(s ω + s ω sω) + ip. s ω + s ω sω The only terms in Q () e i(sω+s ω )t which contribute to t growth arise from F ()ss (k ) and the three ( ) terms above. The terms containing q (4)s s s 3 s 4 do not contribute to secular growth. Thus, we do not require any assumption on the initial statistics other than the smoothness of the initial Fourier space cumulants or, equivalently, the decay of the physical space cumulants at large separations. We choose F () to kill the secular growth.

16 For s = s,wefind dq s s ( k) = dq ss ( k) A.C. Newell et al. / Physica D 5 53 () = 4πε sgnt s s { L s s s k k k q () ss ( k) + L s s s k k k q () s s ( k ) + L ss s q () ss ( k)q () s s ( k )q () s s (k ) L s s s k k k q () s s ( k ) } δ(s ω +s ω sω)δ( k + k k) d k d k. (.9) Remark. The principal value terms cancel on the application of P = P. The Dirac delta terms add. Eq. (.9) is the kinetic equation for the redistribution of the spectral density q () ss ( k) via resonant exchange between the three waves k, k, k lying on the manifold M, s ω( k ) + s ω( k ) = sω( k), k + k = k : M (.) for some choices of s,s,s. For s s (thus, s = s since s = s = ), we obtain dq ()ss ( k ) = iε q ()ss ( k )(Ωk s + Ωs k ), (.) where Ωk s = 4 L ss s L s s s k k k q () s s ( k )δ, d k s s ( ) P s ω + s ω sω iπ sgnt δ(s ω + s ω sω). (.) If we had included L ss s s 3 k 3, then would be no change in (.9) but Ω s k = s d k q () s s ( k )G ss s s kk k k Gss s s kk k k = 3iL ss s s kk k k + 4 s One can also show (N)ss s(n ) dq ( k k (N ) ) ( L ss s kk k L s s s k k k P ) iπ sgnt δ(w, ) δ, d k. (.3) W, = iε (N)ss s(n ) q ( k k (N ) )(Ωk s + Ωs k + +Ω s(n ) k (N ) ). (.4) The set of equations (.) and (.4) can be jointly solved for all N by the frequency renormalization sω k sω k + ε Ωk s + ε4 Ω4k s +. (.5) It turns out that Im Ωk s > which means that the zeroth-order Fourier cumulants slowly decay due to resonances. In summary, the asymptotic closure of the equations for wave (weak) turbulence occurs because (a) the linear dynamics causes phase mixing and a relaxation towards the state of joint Gaussianity on the time scale /ε ω t /ω, and (b) the nonlinear regeneration of cumulants of order N, on the time scale /ε ω, which involves cumulants of order higher than N and products of cumulants of order less than or equal to N, is dominated by the latter.

17 536 A.C. Newell et al. / Physica D 5 53 () Properties of the kinetic equation We rewrite the kinetic equation (.9) with q ss ( k) replaced by n s k : = ε sgnt T [n] = 4πε sgnt s s dn s k We list its properties: { L s s s k k k n s k + Ls s s k k k n s k L ss s n s k ns k n s k δ( k + k k)δ(s ω + s ω sω)d k d k + L s s s } k k k n s. (.6) k. The mechanism for energy transfer is one of resonance.. The property L s s s k k k = (s /s)l ss s = (s /s)l s s s k k k shows that n s k = T/ω k, the thermodynamic equilibrium or Rayleigh Jeans spectrum of energy equipartition (recall that s ω kn s k is the spectral energy density), is an exact solution of (.6). This stationary solution has a zero flux of energy. As a result, it is not particularly relevant for non-equilibrium situations. Indeed, since in most applications there is a dissipative sink at ω k =, the effective temperature T of the random wavefield is zero. 3. Conservation of energy E = (/) s ωk n s k d k follows from multiplying (.6) by ω k, summing over s and integrating over k and then interchanging the order of integration. Formally, in the second (third) term in (.6), interchange k ( k ) and k, s (s ) and s, and change the sign of both s and k (s and k ). Using the properties () (4) of Section., we find the integrand of (.6) becomes L ss s L s s s k k k n s k ns k n s k δ, δ(w, ){ω k (s /s)ω (s /s)ω } which vanishes on the resonant manifold. But this result is only formal and the exchange of integration order relies on Fubini s theorem which demands that the double integral (one can integrate out k ) converges before any exchange of order is done. We must check that n s k has the right behavior in k to allow this. It may turn out that convergence fails after a certain time t < because the solution n s k (t) reaches a stationary state in finite time which renders the k integration of the collision integral (the right-hand side of (.6)) divergent. It also turns out that the stationary state is no longer energy conserving. What happens is that energy can be lost to k =. 4. Reversibility [,]. We have shown that if the Fourier space cumulants are initially smooth, say at some time t = t, then n s k evolves according to (.6) with sgnt replaced by sgn(t t ). The graph versus time of n s k is shown in Fig. 3. We note that while the detailed slope of q ss (k, t) will be continuous, the slope of n s k (t) at t = t (found after averaging q ss (k, t) over all oscillations) is discontinuous. There is nothing unusual or surprising in this; the system will flow towards some attracting equilibrium state no matter whether it goes forward or backwards in time. The loss of exact reversibility is the result of a loss of phase information introduced in the mathematical formulation through the limit ω (t t ),ε ω (t t ) finite. However, this result opens the door to an apparent contradiction. Suppose one were to redo the calculation beginning at a later time Fig. 3. The solution η( k, t) is retraceable.

18 A.C. Newell et al. / Physica D 5 53 () t,ω (t t ) = O(ε ). Then (.6) would suggest that the slope of n s k at t would mirror that at t where there is a discontinuity in slope. Namely, in (.6), sgn(t t ) (which is + for t>t ) would be replaced by sgn(t t ). It would be impossible to retrace the solution n s k from t towards t. But this conclusion is false. The reason is that, over long times, long-distance correlations are built up and the Fourier transform of R (N) picks up a non-smooth component at order ε (N ). In particular, Q (3)ss s ( k, k ) picks up an order ε non-smooth behavior given by (.7). Taking account of the fact that at t,q (3) has this order ε behavior leads to additional terms in the kinetic equation when one calculates beginning at t. They are exactly equal to (.6) with the sgn(t t ) (which because t >t is +) replaced by ( sgn(t t )). The effect of the non-smoothness in Q (3) does not change (.6) at all for t>t.fort<t, on the other hand, it changes the evolution along BC (see Fig. 3) (because of sgn(t t )) to one along BA and BD (because sgn(t t ) + sgn(t t ) = = sgn(t t )). The solution n s k (T = ε (t t )) can be retraced on the ε ω time scale. There is no inconsistency or paradox. 5. Finite flux spectrum. The realization that the kinetic equation (.6) has attracting, stationary solutions which describe a constant flux of conserved densities is one of the many important contributions of Zakharov. Before we give his derivation, we shall first deduce the main result from dimensional considerations. We have seen that there is, at least locally in time, energy conservation. We can then write E k = (/) s ω kn s k as the k divergence of an energy flux P k. Assuming isotropy and angle averaging, we find (take t>) t E kk (d ) = ε s ω k k (d ) T [n] = ε P k. (.7) Assuming n k = cp / k x, we have that the k dependence of ω k k (d ) T [n]isk α k (d ) k β c Pk x k α k d (ω k k α,l k β,n c Pk x, δ(ω) k α, d k k d and this should equal Pk. Solving, we find x = β + d. The spectrum n s k = cp/ k (β+d) (.8) has two very interesting properties. First, if we calculate the total energy E = (/) s ωk n k d k, we find this behaves as cp / k (α β) (dk/k). Now imagine that this spectrum is responsible for carrying a finite flux of energy, inserted at a steady rate at some low value of k, say k L. Clearly, the spectrum around k L will not be universal but for k k L, let us imagine it is and approaches (.8). For β > α, the energy integral converges. In this case, the spectrum (.8), if it is indeed the attractor, can only absorb a finite amount of energy from the source. We say it has finite capacity. This has several consequences. If energy is delivered to the system at a finite rate and the universal finite flux (Kolmogorov) spectrum can only absorb a finite amount, then there must be a sink at k = to absorb that energy. Moreover, the front (imagine the initial spectrum of n s k is on finite support, e.g. n s k (t = ) =, k>k ) that sets up the spectrum (.8) must reach k = in a finite time t equal to the time it takes the source energy to fill the finite capacity spectrum (.8). This means that in the absence of a sink at k =, the inviscid system must develop highly irregular behavior at large k. This observation has not yet been exploited in the investigation of possible singularity development in the long-time behavior of nonlinear PDEs of conservative type. For β α, the spectrum (.8) has infinite capacity and can absorb whatever energy is fed into it. In this case, the front that sets up (.8) from spectrum initially on finite support, reaches k = in infinite time. We will examine these fronts in Section 5. There are anomalies and surprises! Second, let us examine the ratio of the linear (t L ) and nonlinear (t NL ) time scales on the spectrum (.8) as a function of k.

19 538 A.C. Newell et al. / Physica D 5 53 () 5 55 Estimating t L t L = n k t NL ω k n k t by ω k and t NL as (/n k) n k / t, we find (we take n + k = n k ), = ε ω k n k k (d ) P k. (.9) The last term behaves as k α P / k (β+d) k ( d) ε Pk = ε P / k (β α).forβ > α(β < α), the ratio of linear to nonlinear time scales, which for the applicability of the weak turbulence approximation, must be small (because of the need to separate scales), is no longer uniformly bounded by ε. It diverges for large (small) k suggesting that when either β α, the small (large) scale behavior of the system becomes more and more fully nonlinear. This finding will be corroborated by a separate calculation on the validity of the weak turbulence approximation carried out in Section 4. There we will find that the ratio of the structure function S N (r) = (v s ( x + r) v( x)) N to (S ) (N/) is a finite series in powers of ε P / r (α β) which is not uniformly bounded at small scales when β>α. Moreover, the coefficients grow with N. Similarly, for large r, the ratios of cumulants R (N) to (R () ) N/ diverge when β<α. Note: When we evaluate the structure functions on the finite energy flux spectrum, we choose v s (x) to be the physical field whose pair correlation has the spectral energy ω k n k as its Fourier transform. The formal Fourier transform of v s ( x) is therefore ω k A s k. For surface tension waves, β = 4 9,α= 3,d=, so that n k = cp / k 7/4. Since α >β>α, we find that the spectrum has finite capacity and also preserves the uniformity of the wave turbulence approximation at high k. We now turn to the Zakharov derivation of (.8). Using property 3 of Section. repeatedly and assuming n s k = n k,s=+,, we rewrite (.6) as dn k where = 4πε L +++ δ(k + k k) d k d k n k n k n k F(ω,ω,ω ), ( F(ω,ω,ω ) = n ) n n ( + n + n n ( δ(ω ω ω ) + ) δ(ω ω ω ), n n + n ) δ(ω ω ω ) where n k = n, n k = n,n k = n. Assuming isotropy, writing n, n,n as functions of ω,ω and ω, respectively, and introducing N ω by N ω dω = n k d k, we find upon averaging over angles in k, k space (denoted by ), dn ω = S[n] = S ωω ω nn n F(ω,ω,ω ) dω dω, (.3) where is the quarter plane ω,ω > and S ωω ω = 4πε L +++ (d ) dk dk dk δ( k + k k) ( ). dω dω dω If k = ω /α and L is homogeneous of degree β, then S is homogeneous of degree τ = ((β + d)/α) 3. (Note: A correction δ must be added to τ when the waves are almost non-dispersive; see [3].) We want to find stationary solutions to (.3) other than the thermodynamic equilibrium ω k n k = T which makes F identically zero. The class of solutions which Zakharov discovered in the late 96s makes use of the scaling symmetries (homogeneity) in both the dispersion relation and the coefficients L +++ and S ωω ω. The idea is to find

20 A.C. Newell et al. / Physica D 5 53 () a transformation (now called the Zakharov transformation) which maps the lines ω + ω = ω, ω = ω + ω and ω = ω + ω on which F is supported into each other. This is achieved by the maps ω ω, ω ωω, (.3) ω ω which sends ω = ω + ω into ω = ω + ω and ω ωω ω, ω ω ω, (.3) which sends ω = ω +ω into ω = ω +ω. Next, let n = cω x for values of x for which the collision integral S[n] exists. This should be checked a posteriori. There is a danger otherwise that divergences in S[n] could be cancelled by application of the Zakharov transformation. S[n = cω x ] = c S ωω ω (ωω ω ) x ˆF(ω,ω,ω ) dω dω, where ˆF = (ω x ω x ωx )δ(ω ω ω ) + (ω x ω x + ωx )δ(ω ω ω ) + (ω x + ω x ωx )δ(ω ω ω ). Apply (.3) to the second term and (.3) to the third and use the facts that S ω(ω /ω )(ωω /ω ) = (ω/ω ) τ S ω ωω = (ω/ω ) τ S ωω ω and the Jacobian of (.3) is (ω/ω ) 3 to rewrite S[n] as ( S[n] = c S ωω ω (ωω ω ) x (ω x ω x ωx ( ) ω ) y ( ω ) y ) δ(ω ω ω ) dω dω, ω ω where y = x τ = x ((β + d)/α) +. Using homogeneity we may also write S[n] as S[n] = c ω (τ x+) I(x,y) = c ω ( y ) I(x,y), (.33) where ω = ξω, ω = ηω and I(x,y) = c S ξη (ξη) x ( ξ x η x )( ξ y η y )δ( ξ η) dξ dη is the line integral along ξ + η = between (ξ =,η = ) and (ξ =,η = ). Observe that S[n] = both for x = Rayleigh Jeans, (.34) y = or x = β + d pure Kolmogorov. (.35) α The constant c may be determined using first part of (.3) and defining P such that ωn ω t and integrating to find = ωs[n] = P ω, (.36) P = c ω( y+) y + I(x,y). Taking the limit of y, we find P = c I y, y=