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Cumulants of order 1 N2 tend to zero on the fast time scale I I is a typical frequency at which the energy is injected. The system approaches a state of exact joint Gaussian statistics. But the systems of interest to us are nonlinear and, therefore, although the cumulants undergo an initial decay, they are regenerated by the nonlinear terms. In particular, the cumulant of the order N is regenerated both by cumulants of higher order and by products of lower order cumulants.

The important terms in the regeneration of the Nth order cumulant are not the terms containing cumulants of order higher then N, but rather those terms which are products of lower order cumulants. Important means that, even though the nonlinear coupling is weak, the effects of these terms persist over long times because of resonant interaction.

Mathematically, these results are obtained by perturbation theory, in which the terms leading to long time cumulative effects can be identified, tabulated, and summed. The method closely parallels that of the Dyson-Wyld diagrammatic approach which will be discussed in Sec. The function h con- 1 tains the fast oscillations of the order of I time t, whereas the other functions in the integrand, here denoted by f kr , only change over much longer times. The exponent of N h is1sr kr where kr is the linear dispersion relation and sr of ten sr1 denotes its multiplicity.

For example, in acoustic waves, a wave vector k has two frequencies corresponding to waves running parallel and antiparallel to k. The maximum contribution to integrals such as Eq. However, the precise form of the asymptotic limit also depends on whether the zeros of h on M are simple or of higher order. The asymptotic equations for the change of the higher order cumulants can be interpreted as a complex frequency modification whose real part describes the expected nonlinear shift in frequency, and whose imaginary part describes a broadening of the resonant manifold along its normal directions.

But acoustic waves are not fully dispersive. The linear dispersion relation 2 2 kckck k, kk ,k 1. As we will see, this changes the asymptotic. Furthermore, three wave resonances occur between wave vectors which are purely collinear. Therefore, since the kinetic equation KE only considers wave interaction on the resonant manifold, there is no way of redistributing energy out of a given direction.

At best, the KE will only describe spectral energy transfer along rays in wave-vector space. Moreover, depending on dimension d, the long time behavior of integrals 1. For a given vector k, the locus of the resonant partners k 1 and kk 1 in a resonant triad is given by the surface in k 1 space defined by hk 1s 1k 1s 2kk 1sk0. For d1 and the appropriate choices of the wave directions s 1 , s 2, and s, this manifold is all k 1.

Therefore the fast oscillations in the integral are of no consequence, and do not cause any decorrelation to occur. All waves moving in the same directions travel with the same speed. Initial correlations are completely preserved. Moreover, we know that for one-dimensional compressible flow, nonlinear terms, no matter how weak initially, eventually lead to finite time multivalued solutions.

Assuming the usual viscous regularization, multivalued solutions are replaced by shocks, namely, almost discontinuous solutions where discontinuities are resolved across very thin viscous layers. Indeed, for d1, while k1 h0 1. In two dimensions, integral 1. In three dimensions, the growth is much weaker. Since that is the case we will look at in detail, we give the exact result. The essential difference from Eqs.

This will not change the kinetic equation for the spectral energy density. If we write the total energy per unit volume E as E2 0c 2 2 ekdk, 1. Inddimensions a little calculation show, that the right-hand side of Eq. Note that for d5, there is no cumulative effect of this resonance. While the extra term in Eq. The calculation of the frequency renormalization is the main result of this paper.

We present two derivations of this result, in the framework of the above analysis and making use of a diagrammatic perturbation approach. Equation 1. In this case three-wave resonant conditions kk 1k 2, kk 1k 2 1. It is unclear a priori that the three-wave kinetic equation can be used in the dispersionless case; is certainly less plausible in the two-dimensional case, where the formal implementation of the kinetic equation leads to stronger divergences.

The derivation presented above is taken from the article of Newell and Aucoin 9, who made the first serious attempt at an analytical description of the dispersionless acoustic turbulence. Newell and Aucoin 9 also argued that a natural asymptotic closure is also obtained in two dimensions because of the relative higher asymptotic growth rates of terms in the kinetic equation involving only the spectral energy. However this is still a point of dispute, it is not yet resolved and will not be addressed further here.

One should also mention that the quantum kinetic equation was applied to a description of a system of weakly interacting dispersionless phonons as long ago as by Landau and Rumer Kadomtsev and Petviashvili 11 criticized this result on the grounds that the kinetic equation in the dispersionless case can hardly be justified because of the special nature of the linear dispersion relation.

They suggested that acoustic turbulence in two and three dimensions was much more likely to have parallels with its analog in one dimension. We have already mentioned in that case that the usual statistical description is inadequate both because there is no decorrelation dynamics and because shocks form no matter how weak the nonlinearity initially is. The equilibrium statistics relevant in that case is much more likely to be a random distribution of discontinuities in the density and velocity fields which lead to an energy distribution of Eq.

Further, Kadomtsev and Petviashvili argued that even in two and three dimensions one would expect the same result, namely, k d1 ekk 2 , 1. However, wave packets traveling in almost parallel directions are not independent. Moreover, the phase mixing, which occurs due to the crossing of acoustic wave beams, occurs on a shorter time scale, a fact that suggests that the resonant exchange of energy is the more important process.

But even then, several very important questions remain. In the latter case, one might argue that shock formation may again become the relevant process, especially if the energy should condense on rays with very different directions. The aim of this paper is to take a very modest first step in the direction of answering these questions. In particular, we present a curious result. At first sight, it would appear that that is indeed the case, that the modified nonlinear dispersion law is kck 1 2 ln 1 2 k , 1.

But a surprising and nontrivial cancellation occurs, which means that the first corrections to the wave speed still keeps the system nondispersive in the propagation direction. While this fact is the principal result of this paper, our approach lays the foundation for a systematic evaluation of the contribution to energy exchange that occurs at higher order. Indeed, we expect that some of the terms found by Benney and Newell 5, involving gradients across resonant manifolds which, in the fully dispersive case, are not relevant because the resonant three-wave interaction gives rise to an isotropic distribution, may be more important in this context.

The paper is written as follows. In Sec. II, we derive the equation of motion for acoustic waves of small but finite amplitude. A second approach discussed in Sec. II B starts from the Hamiltonian formulation of the Euler equations, and again makes use of the small amplitude parameter of the problem to simplify the interaction Hamiltonian. As we will see in Sec. II C, both approaches are equivalent and which approach to use is the question of taste.

Next, in Sec. III we write down the hierarchy of equations for the spectral cumulants and solve them perturbatively. Certain resonances manifest themselves as algebraic and logarithmic time growth in the formal perturbation expansions, and mean that these expansions are not uniformly asymptotic in time. The kinetic equation, describing the long time behavior of the zeroth order spectral energy, and the equations describing the long time behavior of the zeroth order higher cumulants, are simply conditions that effectively sum the effect of the unbounded growth terms.

Under this renormalization, the perturbation series becomes asymptotically uniform. By asymptotically uniform, we mean that the asymptotic expansion for each of the cumulants remains an asymptotic expansion over long times.

All unbounded growths are removed. While this procedure in principle requires one to identify and calculate unbounded terms to all orders, in practice one gains a very good approximation by demanding uniform asymptotic behavior only to that order in the coupling coefficient where the unboundedness first appears. Likewise, it also assumes that there appear no worse secular terms at a higher order, such as, for example, 4 t 3 2.

To achieve uniformity, one requires an intimate knowledge of how unbounded growth appears. This sort of perturbative analysis was first done in the s by Dyson. A technical innovation was to use graph notations, called diagrams, for representing lengthy analytical expressions for high order terms in the perturbation series.

It of ten happens that one can find the principal subsequence of terms just by looking on the topological structure of corresponding diagrams. This method of treating perturbation approaches is called the diagrammatic technique. The first variant of diagrammatic technique for nonequilibrium processes was suggested by Wyld 12 in the context of the Navier-Stokes equation for an incompressible fluid.

This technique was later generalized by Martin, Siggia, and Rose 13, who demonstrated that it may be used to investigate the fluctuation effects in the low-frequency dynamics of any condensed matter system. In fact this technique is also a classical limit of the Keldysh diagrammatic technique 14 which is applicable to any physical system described by interacting Fermi and Bose fields.

IV, we will use this particular method for treating acoustic turbulence. Note that in such a formulation, unbounded growths appear as divergences or almost divergences due to the presence of zero denominators caused by resonances—the very same resonances, in fact, that give rise to unbounded growth in our more straightforward perturbation approach. Moreover, diagrammatic techniques are schematic methods for identifying all problem terms and for adding them up.

If one uses the diagram technique only to the first order at which the first divergences appear, this is called the one-loop approximation, and is equivalent to identifying the first long time nonlinear effects. This is exactly analogous to what we will do in our first approach in this paper, although we will also display the diagram technique. The one loop approximation will give the same long time behavior of the system for times of NL defined earlier.

In Appendix C we analyze twoloop diagrams, and show that some of them gives the same order contribution to k as two-loop diagrams. Nevertheless one may believe that even the one-loop approximation gives a qualitatively correct description of the dynamics of the system. Section V is devoted to some concluding remarks and the identification of the remaining challenges. We now begin with deriving the basic equations of motion for weak acoustic turbulence.

Here v x,t is the Euler fluid velocity, x,t the density, and p r,t the pressure which, in the general case, is a function of fluid density and specific entropy s pp ,s. In ideal fluids where there is no viscosity and heat exchange, the entropy per unit volume is carried by the fluid, i. A fluid in which the specific entropy is constant throughout the volume is called barotropic; the pressure in such a fluid is a single-valued function of the density pp. From Eqs. There are three cases.

Hamiltonian description of acoustic turbulence 1. Equations of motion and canonical variables Consider again the Euler equations for a compressible fluid, Eqs. Asa result of direct differentiation with respect to time, it is readily evident that equations 2. As seen from Eq. Hamiltonian of acoustic turbulence Let us expand the Hamiltonian 2. We also neglected contributions from four-wave and higher terms, because three-wave interaction is the dominant one.

Since we have an almost linear dispersion relation, only almost parallel wave vectors can interact; therefore cos ij with high accuracy can be replaced by 1, and Eq. Canonical equation of motion The Hamiltonian equations of motion 2. After performing a time Fourier transform, one has, instead of 2.

Thus one concludes that the two approaches are equivalent, and that a choice between them is a question of taste. The mean is zero. Indeed, the property of spatial homogeneity affords one a way of defining averages, which does not depend on the presence of a joint distribution. We can define the average x xr as simply an average over the base coordinate, namely, xxr 1 L 2L 3 L xxrdx.

The spectral energy is therefore 2c 2 2 q k. For convenience we denote q k as e k. To leading order in , qss k,k and qsss k,k,k which we may call q ss 0 k,k and sss q0 k,k,k are independent of time. Anticipating, however, that certain parts of the higher order iterates in their asymptotic expansions may become unbounded, we will allow both q ss 0 k,k and sss k,k,k to be slowly varying in time, q 0 dq ss 0 k,k dt 2F ss 2 , dq sss 0 k,k,k dt 2F sss 3 3.

Note that, in Eq. The perturbations method has the advantage that it is relatively simple to execute. However, there is no a priori guarantee that terms appearing later in the formal series cannot have time dependencies, which mean they affect the leading approximations on time scales comparable to or less than 2 e.

The diagram approach, which requires some familiarity to execute, is designed to do this and, both for completeness and the fact that we will have to proceed beyond the one-loop approximation to resolve the questions of the angular redistribution of spectral energy, we include it here. In the denominator we added the term i0 by requirement of causality. This is an infinite series with respect to the bare amplitude V k,q,p , Eq. All of the contributions of the second and fourth orders in V are shown in Fig.

We have not specified the direction of arrows in Fig. For example, diagram a in Fig. The diagram a4 on Fig. Diagrams a from Fig. This is an infinite series with respect to the same objects G k, ,n k, , and V k,q,p. All diagrams of the second and fourth orders are shown in Fig. We also have not specified arrow directions in the diagrams for k, and k,.

In complete analogy with diagrams for G k, , one diagram in Fig. All the rest diagrams for k, reproduce in the same way—one chooses all possible directions of arrows, and discards those which are incompatible with the definition of vertex V see Fig. One-pole approximation 1. Therefore, k, 0k.

First terms in the diagrammatic pertubation expansion for mass operator k,. Double correlation function The same type of approximation may be performed for the correlation function. That is, in the Wyld equation 4. One-loop approximation Let us begin our treatment with the simple one-loop or direct interaction approximation for mass operators and. This approximation corresponds to taking into account just the second order in bare vertex V, Eq. The two-loop approximation will be considered in Appendix C.

We will estimate two-loop diagrams, and will show that some of them give the same order contribution to k as one-loop diagrams. Therefore, the one-loop approximation is an uncontrolled approximation, but we believe that it gives qualitatively correct results. We will see below that this difference is very important in the particular case of acoustic turbulence.

The KE for waves with a linear dispersion law forbids the angular evolution of energy because conservation V. As a result, there exists a cone of allowed angles between k and k 1 in which interactions are allowed. Therefore one has to expect some angle evolution of wave packages within this approximation. Combining Eq. In the resulting expression one can perform the integration over analytically.

The result is Vk 2 ,k,k1 2kk1k 2nk 1nk 2 k 1k 2i 1 2 Vk 0 ,k1 ,k2 2kk1k 2nk 2 k. It follows from Eq. One may evaluate the integral with respect to y as 4. Equation 4. This shows that our above calculations of k is selfconsistent. After substitution of n k, in the one-pole approximation 4.

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