Single-wavenumber Representation of Nonlinear Energy Spectrum in Elastic-Wave Turbulence of {F}öppl-von {K}ármán Equation: Energy Decomposition Analysis and Energy Budget
aa r X i v : . [ n li n . C D ] D ec Single-wavenumber Representation of Nonlinear Energy Spectrumin Elastic-Wave Turbulence of F¨oppl-von K´arm´an Equation:Energy Decomposition Analysis and Energy Budget
Naoto Yokoyama ∗ Department of Aeronautics and Astronautics, Kyoto University, Kyoto 615-8540, Japan
Masanori Takaoka † Department of Mechanical Engineering, Doshisha University, Kyotanabe 610-0394, Japan (Dated: October 2, 2018)A single-wavenumber representation of nonlinear energy spectrum, i.e., stretching energy spec-trum is found in elastic-wave turbulence governed by the F¨oppl-von K´arm´an (FvK) equation. Therepresentation enables energy decomposition analysis in the wavenumber space, and analytical ex-pressions of detailed energy budget in the nonlinear interactions are obtained for the first time inwave turbulence systems. We numerically solved the FvK equation and observed the following facts.Kinetic and bending energies are comparable with each other at large wavenumbers as the weakturbulence theory suggests. On the other hand, the stretching energy is larger than the bendingenergy at small wavenumbers, i.e., the nonlinearity is relatively strong. The strong correlationbetween a mode a k and its companion mode a − k is observed at the small wavenumbers. Energytransfer shows that the energy is input into the wave field through stretching-energy transfer at thesmall wavenumbers, and dissipated through the quartic part of kinetic-energy transfer at the largewavenumbers. A total-energy flux consistent with the energy conservation is calculated directlyby using the analytical expression of the total-energy transfer, and the forward energy cascade isobserved clearly. PACS numbers: 62.30.+d, 05.45.-a, 46.40.-f
I. INTRODUCTION
Energy decomposition analysis helps understanding ofthe mechanism of energy distribution. Exchange betweenkinetic energy and potential energy is observed in os-cillatory or wave motion, while the total energy is con-served. The exchange is seen as elliptic motion, whichcan be distorted by the nonlinearity, in the phase space.In Refs. [1, 2], the energy is decomposed into kinetic,bending and stretching energies to derive the governingequation of the elastic waves.In the relaxation, known as selective decay process,of hydrodynamic turbulent flows, the depression of non-linearity has been often discussed. The strong cor-relations between velocity and vorticity have been re-ported in hydrodynamic turbulence; e.g., parallelizationof velocity and vorticity called Beltramization in three-dimensional flow [3], and negative temperature state suchas the sinh-Poisson state in two-dimensional flow [4].These relaxed states have the correlation among modes.It is in contrast with the weak turbulence, where theindependence among modes are presupposed. In theMajda-McLaughlin-Tabak (MMT) model, which is a one-dimensional mathematical model of wave turbulence,spatially-localized coherent structures are reported [5].Zakharov et al. [6] modified MMT model to fit the weakturbulence theory (WTT) by introducing a nonlinear ∗ [email protected] † [email protected] term that prevents the correlation of modes. We will herereport correlations between each pair of modes at largescales in elastic-wave turbulence, which is consistent withour previous work where the separation wavenumber be-tween the weak and strong turbulence is estimated viathe applicability limit of the random phase approxima-tion (RPA) in WTT [7].Since the coexistence of non-weak and weak turbulencewill be investigated in this paper, we here distinguish wave turbulence and weak turbulence: the former is re-ferred to as a wave turbulent state where the nonlinearinteractions are not necessarily weak, and the latter is awave turbulent state where WTT can be applied. Thus, wave turbulence includes weak turbulence.Fourier spectral representation is widely used in theanalysis of the homogeneous turbulence governed by theNavier-Stokes equation, because one of the most impor-tant study objectives is to clarify energy distributionformed by hierarchical structures over a wide range ofscales. The so-called cascade theory, which was proposedby Kolmogorov [8] as the first statistical theory of turbu-lence, predicts the direction of energy transfer and is welldescribed in the wavenumber space. Also in researchesof the weak turbulence systems, the Fourier spectral rep-resentation is convenient to introduce the complex am-plitudes as elementary waves to apply RPA in WTT.The analysis of the wave turbulence is confronted withthe following difficulties, which stem from the fact thatonly the quadratic quantities of the complex amplitudeshave been considered as energy. More properly, thequadratic energy corresponds to the linear part of thedynamics and the ensemble-averaged quadratic energy isconserved only in the weakly nonlinear limit, even if itsdynamics is governed by a Hamiltonian. Although it isconvenient to use the complex amplitudes in applicationof RPA to derive the kinetic equation, the perturbativeexpansion of the complex amplitudes is inevitable to rep-resent the nonlinearity of the system. The nonlinear en-ergy appears as convolutions of the complex amplitudes,since the complex amplitudes are introduced for the dif-ferent purpose. On the other hand, for example, in theNavier-Stokes turbulence the energy is given by a single-wavenumber representation like | u k | /
2, and this kind ofproblem does not appear, since the energy is simply givenby the quadratic form by its nature.To analyze energy budget, it is indispensable to takeinto account the full Hamiltonian dynamics. A single-wavenumber representation of the higher-order energy isrequired to identify the nonlinear dynamics at each scale.In addition to the nonconservation of the quadratic en-ergy, its transfer in the wavenumber space cannot be ob-tained as a closed expression in the representation of thecomplex amplitudes. If a single-wavenumber representa-tion of the energy can be found, the explicit expressionof the detailed energy budget is obtained, even not in theweakly nonlinear limit.Demanding the constancy of the energy flux and thecomplete self-similarity, the dimensional analysis usinga specific form of the kinetic equation predicts theKolmogorov-Zakharov spectrum as described in Chap. 3of Ref. [9]. While the spectral form can be obtained eas-ily in this approach, the Kolmogorov-Zakharov spectrumcan be obtained also as a stationary solution of the kineticequation with help from the so-called Zakharov trans-formation. The energy flux in the framework of WTTcan be represented by the collision integral. Care shouldbe taken to distinguish WTT and the Hamiltonian dy-namics, since the ensemble-averaged quadratic energy isconserved only under the kinetic equation. Although thequadratic-energy fluxes for a variety of spectral param-eters were numerically obtained in Ref. [10], no total-energy flux has been obtained so far even in the weaklynonlinear limit. We will here report that the flux of the total energy is directly calculated for the first time byusing the analytical expression for the transfer.The energy flux not in the weakly nonlinear limit isdifficult to obtain. The most primitive estimation of theenergy flux P ( k ) through k = | k | is obtained from the cu-mulative energy e E ( k ), the cumulative energy input e F ( k ),and the cumulative energy dissipation e D ( k ) between 0and k by using the scale-by-scale energy budget equa-tion P ( k ) = − ∂ e E ( k ) /∂t + e F ( k ) − e D ( k ) [11]. The energyflux in a statistically-steady state is usually estimatedby measuring the energy injected into the system whenthe dissipation is localized at large wavenumbers [12].The energy flux obtained in Ref. [13], which is definedas e F ( k ) − e D ( k ), is the same as the flux estimated onlyby the energy input for the dissipation localized at thelarge wavenumbers. Their approaches do not contain the expression derived from the nonlinear term of the gov-erning equation. The constancy in the inertial subrangeof the energy flux estimated from e F ( k ) − e D ( k ) is an ob-vious consequence from the localization of the externalforce and dissipation, and the constancy is independentof whether the nonlinear interactions are local or not.The statistical steadiness ∂ e E ( k ) /∂t = 0 should be rigor-ously verified. Furthermore, the energy injected into thesystem is not necessarily in strict accordance with theenergy flux that cascades in the inertial subrange [14].In laboratory experiments of surface waves, the energyflux is estimated indirectly by the energy decay rate af-ter switching off the energy input or by the dissipationspectrum. This estimation requires additional assump-tions, because it is the power spectrum of the displace-ment that can be obtained experimentally [15]. The en-ergy flux may be evaluated by using structure functionsin the real space, though it is a little different from thatdefined in the wavenumber space. Even in direct numer-ical simulations according to dynamical equations, theenergy flux consistent with the energy conservation hasnot been obtained directly [16, 17].The elastic-wave turbulence, which is tractable experi-mentally, numerically and theoretically, exhibits rich phe-nomena: weak turbulence [12, 18], spatio-temporal dy-namics [19], spectral variation [7, 20] and strongly non-linear structures [21]. Among them, the coexistence ofthe weakly nonlinear spectrum and a strongly nonlinearspectrum is one of the most remarkable properties [7, 22].It is an interesting challenge to clarify the energy budgetin the state where the weak turbulence and the strongturbulence coexist. It should be noted here that we use“strong” as short-hand notation to represent the rela-tively strongly nonlinear state whose nonlinearity is not so strong as to break the first-principle dynamical equa-tions, but as strong as to break the weak nonlinearityassumption in WTT.In this paper, we analyze the wave turbulence in a thinelastic plate by numerical simulations according to theF¨oppl-von K´arm´an equation. The single-wavenumberrepresentation of the nonlinear energy spectrum opensa way for the above difficulties. It enables the energydecomposition analysis and the investigations of the en-ergy budget due to the nonlinear interactions. The nextsection is devoted to the formulation of the problem withfocusing on the Fourier representation of the system. InSec. III, two kinds of numerical results are shown. One isthe energy decomposition analysis, and the other is theenergy budget. The last section is devoted to concludingremark. II. FORMULATIONA. Governing equation and numerical scheme
The dynamics of elastic waves propagating in a thinplate is described by the F¨oppl-von K´arm´an (FvK) equa-tion for the displacement ζ and the momentum p via theAiry stress potential χ [1, 2]. Under the periodic bound-ary condition, the FvK equation is written as dζ k dt = p k ρ , dp k dt = − ρω k ζ k + X k + k = k | k × k | ζ k χ k , (1a) χ k = − Y k X k + k = k | k × k | ζ k ζ k , (1b)where ζ k , p k and χ k are the Fourier coefficients of thedisplacement, of the momentum, and of the Airy stresspotential, respectively. The Young’s modulus Y and thedensity ρ are the material quantities of an elastic plate.The frequency ω k is given by the linear dispersion rela-tion: ω k = s Y h − σ ) ρ k , (2)where σ and h are respectively the Poisson ratio and thethickness of the elastic plate.The complex amplitude is defined as a k = ρω k ζ k + ip k √ ρω k . (3)The complex amplitude is used as the elementary waveof the wavenumber k in WTT. Then, the variables inEq. (1) are given as ζ k = 1 √ ρω k ( a k + a ∗− k ) , (4a) p k = − i r ρω k a k − a ∗− k ) , (4b) χ k = − Y ρk X k + k = k | k × k | √ ω k ω k ( a k + a ∗− k )( a k + a ∗− k ) , (4c)where a ∗ represents the complex conjugate of a . Equa-tion (1) is reduced to a single equation for a k as da k dt = − iω k a k − iY ρ X k + k + k = k | k × k | | k × k | | k + k | × ( a k + a ∗− k )( a k + a ∗− k )( a k + a ∗− k ) √ ω k ω k ω k ω k . (5)The first term in the right-hand side corresponds to thelinear harmonic oscillation, and the second one to thenonlinear interactions.Direct numerical simulations (DNS) according toEq. (5) are performed with the parameter values as ρ = 7 . × kg/m , Y = 2 . × Pa, σ = 0 .
30, and h = 5 . × − m. The plate is supposed to have the pe-riodic boundary of 1m × k is discretized as k ∈ (2 π Z ) . Thepseudo-spectral method is employed and the number ofthe aliasing-free modes is 512 × × F k and the dissipation D k are addedto the right-hand side of Eq. (5) to make statistically-steady non-equilibrium states. The external force F k areadded so that a k ’s at the small wavenumbers | k | ≤ π have a magnitude constant in time, while the phases of a k ’s are determined by Eq. (5). The dissipation is addedas D k = − ν | k | a k , where ν = 1 . × − . As we canrecognize from Figs. 1 and 3, which appear below, the dis-sipation is effective in the wavenumber range | k | ' π .The exponential decay of the energy spectra shown inFig. 1 at the large wavenumbers gives the assurance ofour DNS with this mode number. Details of the numeri-cal scheme are explained in Ref. [22].It is preferable for the external force and the dissipa-tion to be localized in scales to achieve a large inertialsubrange of turbulence spectra. Although it is reportedthat broadly-affecting Lorentzian dissipation successfullyreproduces the experimentally-observed spectrum [21],we are interested in the properties in the inertial sub-range in the FvK turbulence. According to the deriva-tion of the equation, it might be realized and examinedin laboratory, if one could perform the experiment in thevacuum environment to reduce drags acting on the thinplate, e.g., induced mass, by using much less dissipativeplates to reduce internal friction. B. Hamiltonian and energy decomposition
The FvK equation (1) can be written as a canonicalequation: dζ k dt = δ H δp ∗ k , dp k dt = − δ H δζ ∗ k , when we introduce the Hamiltonian H as H = X k (cid:18) ρ | p k | + ρω k | ζ k | (cid:19) + Y X k + k − k − k = | k × k | | k × k | | k + k | ζ ∗ k ζ ∗ k ζ k ζ k , (6)where δ/δζ ∗ k and δ/δp ∗ k express the functional derivativeswith respect to ζ ∗ k and p ∗ k , respectively. Use has beenmade of ζ k = ζ ∗− k to rewrite the second term in right-hand side into the symmetric form. Note that ζ k ( p k )and ζ ∗ k ( p ∗ k ) are not independent of each other. The rela-tion to the conventional representation with the complexamplitudes in WTT is given in Appendix A.The Hamiltonian consists of three kinds of energies,i.e., the kinetic energy, the bending energy, and thestretching energy [2]. The bending energy derives fromthe out-of-plane displacement, while the stretching en-ergy comes from the in-plane strain.The total energy of each mode E k is the sum of thekinetic energy K k and the potential energy V k , i.e., E k = K k + V k . The potential energy of each mode is the sumof the bending energy V b k and the stretching energy V s k ,i.e., V k = V b k + V s k . Here, K k = 12 ρ | p k | = ω k (cid:0) | a k | + | a − k | − a k a − k ) (cid:1) , (7a) V b k = ρω k | ζ k | = ω k (cid:0) | a k | + | a − k | + 2Re( a k a − k ) (cid:1) , (7b) V s k = k Y | χ k | = Y ρ k X k + k = kk + k = k | k × k | | k × k | √ ω k ω k ω k ω k × ( a ∗ k + a − k )( a ∗ k + a − k )( a k + a ∗− k )( a k + a ∗− k ) . (7c)The quadratic energy of each mode is given as the sumof the kinetic energy and the bending energy, i.e., E (2) k = K k + V b k , because both energies are O ( | a | ). On the otherhand, the quartic energy E (4) k is the stretching energy V s k , which is O ( | a | ). The Hamiltonian (6) can also bewritten in terms of these energies as H = X k E k = X k ( E (2) k + E (4) k ) = X k ( K k + V b k + V s k ) . (8)It should be emphasized that usage of the Fourier co-efficient of the Airy stress potential χ k , given as Eq. (1b)enables the representation of the nonlinear energy for asingle-wavenumber mode as Eq. (7c) in this system. Thecomplex amplitude a k is introduced as the elementarywave in WTT. When the system’s Hamiltonian is ex-panded in terms of a k , it leads to the nonlinear energy inthe form of a convolution consisting of the four wavenum-bers as shown in Eq. (7c). We here consider ζ k , p k and χ k as elementary waves in the representation of the en-ergies, K k , V b k and V s k .In the framework of WTT, the energy of k is de-fined as the quadratic energy: E WTT k = ω k h| a k | i , where h·i denotes the ensemble averaging. The quadratic en-ergy in our notation and the energy in WTT are relatedas h E (2) k i = h K k + V b k i = E WTT k + E WTT − k . The en-ergy in WTT P k E WTT k is not conserved under the FvKequation regardless of the ensemble-averaging, since itlacks the stretching energy V s k in the Hamiltonian (8),i.e., P k E WTT k = P k h E (2) k i = hH i 6 = H , where H represents the quadratic part of the Hamiltonian. Itshould be noted that E WTT − k is independent of E WTT k ,but E (2) k = E (2) − k as well as E k = E − k , because E (2) k and E k are given by the Fourier coefficients of the real-valuedfunctions. PSfrag replacements k (m − ) E n e r g y Sp e c t r a ( J m − ) E (2) K V b V s E V
FIG. 1. (Color online) Energy spectra of the total energy E ,the quadratic energy E (2) , the kinetic energy K , the potentialenergy V , the bending energy V b , and the stretching energy V s . III. RESULTS
We will show the numerical results for the moderateenergy level, which corresponds to EL3 in Ref. [7]. Thisenergy level is chosen so as to realize the coexistence ofthe weak and strong energy spectra. The number of themodes are twice those in Ref. [7] in each direction to ob-tain larger inertial subrange. The FvK equation is appli-cable for this energy level, because the root mean squareof the gradient of the displacement h|∇ ζ | i / ≈ .
15 [23].Furthermore, this energy level looks intermediate be-tween the two fields reported in Fig. 2 of Ref. [21], i.e.,much smaller than the energy level at which the dynamiccrumpling appears.
A. Decomposed energy spectra and correlationbetween companion modes
The azimuthally-integrated energy spectra of the to-tal energy E ( k ), the quadratic energy E (2) ( k ), the ki-netic energy K ( k ), the potential energy V ( k ), the bend-ing energy V b ( k ), and the stretching energy V s ( k ) areshown in Fig. 1. The azimuthally-integrated spec-trum of the total energy, for example, is defined as E ( k ) = (∆ k ) − P k − ∆ k/ ≤| k ′ | 0) and k x ∈ [10 π, π ]) nary parts of the companion modes are defined as C R k = h Re( a k )Re( a − k ) i p h (Re( a k )) ih (Re( a − k )) i , (10a) C I k = h Im( a k )Im( a − k ) i p h (Im( a k )) ih (Im( a − k )) i . (10b)In Fig. 2, the correlations between companion modesat k = ( k x , 0) and − k = ( − k x , C C( k x , , C R( k x , ,and C I( k x , are drawn in the range k x ∈ [10 π, π ] toavoid the influence from the artificially-added externalforce and dissipation.At the large wavenumbers, where the nonlinearity isweak, the correlations, C C k , C R k and C I k , are almostzero. It is consistent with RPA. At the small wavenum-bers, where the nonlinearity is relatively strong, C C k ≈− C R k ≈ − C I k ≈ 1. It indicates a k ≈ − a ∗− k ,which is confirmed by the time series of a k and a − k ,though the graphs are omitted here. This fact is con-sistent with the results in Ref. [7], where it is shownthat the separation wavenumber which forms the divi-sion between the weakly and strongly nonlinear spectraagrees with the critical wavenumber at which the non-linear frequency shift is comparable with the linear fre-quency. Namely, it means that RPA, which is the basisof WTT, becomes inapplicable below the vicinity of theseparation wavenumber.In all the wavenumbers, Re( C C k ) ≈ C R k ≈ − C I k . Thecurve for Re( C C k ) is smoother than C R k and C I k , sincethe former consists of the latter two elements, i.e., thetwice ensemble number. If we decrease the amplitude ofthe external force, the range of the wavenumbers whereWTT holds becomes larger. It is consistent with the re-sults in Ref. [19]. The weak nonlinearity which results in h a k a k ′ i = 0 at the large wavenumbers and the stronglynonlinear correlation a k ≈ − a ∗− k at the small wavenum-bers make Im( C C k ) ≈ a k ≈ − a ∗− k at the smallwavenumbers appears as K ( k ) ≫ V b ( k ) in Fig. 1, which is –6–4–202 10 ( × ) PSfrag replacements k (m − ) T r a n s f e r s ( J m − s − ) T T T (4) K T V s T (2) K T V b –4–20245 × × ( × ) PSfrag replacements k (m − )Transfers (J m − s − ) TTT (4) K T V s T (2) K T V b FIG. 3. (Color online) Energy transfers of the total energy,of the quadratic and quartic parts of the kinetic energy, andof the bending and stretching energies. The abscissa is log-arithmically scaled. The inset shows the enlargement at thelarge wavenumbers. consistent with Eqs. (7a) and (7b). Because of Eq. (4a),this correlation makes ζ k small. It leads to depressionof the summand in the nonlinear term (see Eq. (5)),which reminds us of the depression in the relaxation pro-cesses [3, 4] as written in the introduction. It seems thatthis kind of the correlated states will survive in contrastwith the fast cascade of the uncorrelated modes.One might think that this correlation, a k ≈ − a ∗− k , con-tradicts to the strong nonlinearity at the small wavenum-bers, since it appears to suppress the nonlinear term, thesecond term in the right-hand side of Eq. (5). The non-linearity can be large at the small wavenumbers owing tothe convolution, which is the summation of the productsof ( a k + a ∗− k ), ( a k + a ∗− k ), and ( a k + a ∗− k ) at allwavenumbers, because ( a k i + a ∗− k i ) for k i ( i = 1 , , 3) atthe large wavenumbers are not small. Namely, the non-linearity at a wavenumber is not determined only by theelementary wave at the wavenumber. This fact is alsoconfirmed in Fig. 1. While the amplitudes of the linearenergies, E (2) , K and V b , decay at the small wavenum-bers, those including the nonlinear energy, E , V and V s ,do not and are almost constant k ≤ π . B. Energy budget To investigate the energy budget in detail, our anal-ysis here starts with energy transfer. We define the en-ergy transfer of k as T k = ˆ dE k / ˆ dt , where the operatorˆ d/ ˆ dt expresses the time derivative neglecting the exter-nal force and the dissipation. According to the energydecomposition in Sec. II, the total-energy transfer is also decomposed as T k = ˆ dK k ˆ dt + ˆ dV b k ˆ dt + ˆ dV s k ˆ dt = T K k + T V b k + T V s k . (11)Corresponding to the linear and nonlinear terms in dp k /dt , the transfer of the kinetic energy T K k consistsof the quadratic and quartic parts, T (2) K k and T (4) K k , i.e., T K k = ˆ dK k / ˆ dt = T (2) K k + T (4) K k . From Eqs. (1) and (7), T (2) K k = − ω k p ∗ k ζ k + c . c ., (12a) T (4) K k = p ∗ k ρ X k + k = k | k × k | ζ k χ k + c . c ., (12b) T V b k = ˆ dV b k ˆ dt = ω k p ∗ k ζ k + c . c ., (12c) T V s k = ˆ dV s k ˆ dt = − χ ∗ k ρ X k + k = k | k × k | p k ζ k + c . c . (12d)Although the kinetic energy is represented as aquadratic function of the complex amplitude, its trans-fer has both quadratic part T (2) K k and quartic part T (4) K k .While the transfer of the bending energy T V b k is aquadratic function of the complex amplitude, that of thestretching energy T V s k is a quartic function.Apparently, T (2) K k and T V b k cancel each other, repre-senting the harmonic exchange between the kinetic andbending energies for a wavenumber. Thus, the quadraticparts of the transfer do not contribute to the cascade be-tween different scales. In this sense, to be exact, T (2) K k and T V b k are not transfers but transmutations from oneform of the energy to the other. Nonetheless, we naivelyuse the word “transfers” both for transfers and for trans-mutations. The quartic-energy transfers, T (4) K k and T V s k ,are the energy transfers due to the nonlinear interactionsamong modes in the wavenumber space as known fromEqs. (12b) and (12d). They are of the same quartic orderof the complex amplitude. However, only T (4) K k has beentaken into account for the energy transfer in WTT as T WTT k = ˆ dE WTT k / ˆ dt , because it comes from the quadraticenergy. Namely, T WTT k + T WTT − k = h T K k + T V b k i = h T (4) K k i .It should be emphasized that the energy conservationholds only for the total energy, which is the sum of thekinetic, bending and stretching energies, but each de-composed energy is not conservative separately. Namely, P k T k = 0, but P k T (2) K k , P k T (4) K k , P k T V b k , P k T V s k =0. Moreover, P k T WTT k = 0.We here further decompose the quartic-energy trans-fers. Let us introduce the triad interaction functions cor-responding to Eqs. (12b) and (12d) as T (4) K kk k = | k × k | ρ p k ( ζ k χ k + χ k ζ k ) δ k + k + k , + c . c ., (13a) T V s kk k = − | k × k | ρ χ k ( p k ζ k + ζ k p k ) δ k + k + k , + c . c ., (13b)which represent the transfer of each energy to k due toa triad with one leg k and the other k . To symmetrizethe triad interaction functions and to make a triad in theform k + k + k = , we use ζ k = ζ ∗− k , p k = p ∗− k and χ k = χ ∗− k . Then, the quartic-energy transfers can berepresented as the sum of these terms: T (4) K k = X k , k T (4) K kk k , T V s k = X k , k T V s kk k . (14)The triad interaction function of the total energy is de-fined as T kk k = T (4) K kk k + T V s kk k . The triad interac-tion function T kk k is interpreted as the temporal rate ofthe energy increment at k due to the interaction amongthe three wavenumbers k + k + k = . The triad inter-action function of the total energy satisfies the detailedenergy balance: T kk k + T k k k + T k kk = 0 . (15)Namely, the triad interaction function shows the inter-changes of the energy among wavenumbers keeping thesum of the energies of the three wavenumbers.The triad interaction functions have high symmetry.If we define the triad interaction functions in a piecewiseway as e T (4) K kk k = | k × k | ρ p k ζ k χ k δ k + k + k , , (16a) e T V s kk k = − | k × k | ρ χ k p k ζ k δ k + k + k , , (16b)then another detailed energy balance holds: e T (4) K kk k + e T V s k kk = 0 . (17)This represents the gain of the kinetic energy at k andthat of stretching energy at k have the same absolutevalue with the opposite signs through the triad interac-tion atomized as Eqs. (16). It indicates the exchangebetween the kinetic energy and the stretching energythrough the triad interaction. The atomized triad in-teraction function of the total energy is then defined as e T kk k = e T (4) K kk k + e T V s kk k , (18)and the detailed energy balance that is the same asEq. (15) holds also for e T kk k . The detailed energy balances hold via the triad interac-tion functions among the Fourier coefficients of the phys-ical variables, ζ k , p k and χ k . It suggests that the presentrepresentation by using these Fourier coefficients is suit-able for the analysis of energy budget. Since χ k is givenby the convolution as defined in Eq. (1b), it is consistentwith the fact that the nonlinear interactions occur amongfour waves when the complex amplitudes are used for thegoverning equation (5).The azimuthally-integrated energy transfers, whichare defined in the similar way to the energy spectra,are drawn in Fig. 3. The azimuthally-integrated en-ergy transfer T ( k ), for example, is defined as T ( k ) =(∆ k ) − P k − ∆ k/ ≤| k ′ | In this paper, the energy is decomposed into the ki-netic, bending and stretching energies in the elastic-waveturbulence governed by the F¨oppl-von K´arm´an (FvK)equation. The Fourier coefficient of the Airy stress po-tential appropriately gives the nonlinear energy, i.e., thestretching energy, for a single wavenumber in the elasticwaves. The complex amplitude a k has been introduced asan elementary wave to apply the random phase approxi-mation in researches of weak turbulence. In fact, a k hasclear physical meaning in analogy with the wave action,and gives the sophisticated formalism in the weak tur-bulence theory (WTT). However, the use of the Fouriercoefficients of physical variables, ζ k , p k and χ k , is nat-ural for evaluation of energy, since the nonlinear energyexpressed by the complex amplitude a k is given by theconvolution.By the energy decomposition analysis, it was foundthat the kinetic energy and the stretching energy aremuch larger than the bending energy in the (relatively)strongly nonlinear regime, while the bending energy com-parable with the kinetic energy is much larger than thestretching energy in the weakly nonlinear regime. Theimbalance between the kinetic and bending energies re-sults from the strong correlation between a k and a − k . Infact, a k ≈ − a ∗− k in the strongly nonlinear regime. Al-though one may expect a distinctive structure in the realspace due to this correlation, it is not so easy to identifyit because of the cumulative effect of all active modes.Namely, the summation of the all active modes includingphase correlation makes the real-space structure. It isour future work to clarify such properties.The S-theory is developed to explain the strong pair-ing between a k and a − k in the spin waves under strongparametric excitation [25]. In this case, the interactionsamong pairs are more essential than those among ele-mentary waves. The external force in the present studyis not parametric, though the pairing plays an importantrole in the strongly nonlinear regime. Independently ofthe S-theory, the pairing itself might be essential for theenergy budget, because not a k but ζ k , p k and χ k arethe basic elements, and the nonlinear terms appear as( a k + a ∗− k ) in the governing equation.As a result of the single-wavenumber representationof the nonlinear energy, the analytical expression of theenergy budget was obtained for the first time in wave tur-bulence systems. The quadratic-energy transfers, whichare the quadratic part of the kinetic-energy transfer andthe bending-energy transfer, transmute the energies fora wavenumber. Since the quartic part of the kinetic-energy transfer and the stretching-energy transfer are thesame quartic order of the complex amplitude, both en-ergy transfers should not be discriminated even in theweakly nonlinear limit. The analytical expression ofthe energy budget shows that the total-energy transfer,which is sum of the quartic-energy transfers, satisfies thedetailed energy balance. These facts indicate that thestretching energy has equal essentiality to the kinetic en-ergy in considering the energy budget, though the or-der of the stretching energy ( O ( | a | )) is higher than thatof the kinetic energy ( O ( | a | )) in the complex-amplituderepresentation. It was numerically found in the presentsystem that the energy is input into the system throughthe stretching-energy transfer at small wavenumbers, anddissipated through the quartic part of the kinetic-energytransfer at large wavenumbers. The energy transfer is defined as the rate of changeof the energy, and it holds independently from the total-energy conservation. On the other hand, the energy fluxis defined based on the continuity equation of energy.Therefore, while the decomposed-energy transfer can re-flect the energy budget, the decomposed-energy flux can-not. It follows that only the total-energy flux is the actualflux. It is indispensable to include the nonlinear energyproperly to satisfy the energy conservation and to obtainthe total-energy flux. In order to compare with previ-ous researches, we introduced and examined the pseudo-fluxes as well, though they are not actual but spuriousfluxes, since the conservation of energy which the fluxesrely on does not hold for each decomposed energy.We have succeeded for the first time to evaluate thewell-defined total-energy flux directly by using the ana-lytical expression of the total-energy transfer due to thenonlinear interactions. The total-energy flux evaluatedby the nonlinear terms is positively constant in the in-ertial subrange, and it indicates the forward energy cas-cade. The fluxes of the quadratic energies reported invarious wave turbulent systems [16, 17] have physicalmeaning only in the weakly nonlinear limit. Because theexternal force used in Ref. [13] directly excites only thelinear energy, which is the kinetic energy, the expressionof the cumulative energy input e F ( k ) is indistinguishablefrom the one where the nonlinear energy is not consid-ered. This approach conceals the energy budget in theinertial subrange, and loses the distinction between thequadratic and quartic energies. For a general externalforce that may excite the nonlinear energy directly, thestretching-energy transfer should be taken into accountas pointed out above in the present paper. Note that e F ( k ) − e D ( k ), which is used as a total-energy flux in thesame reference [13], is always constant in the inertial sub-range when both the external force and the dissipationare localized in the wavenumber space, and hence theenergy cascade cannot be examined by such flux. Theanalytical expression of the energy flux obtained fromthe nonlinear terms in the governing equation is neces-sary to investigate the wave turbulence statistics in theinertial subrange.Although one may expect to evaluate the energy fluxby using the expression based on the two-points structurefunctions in the real space as usually done in analyses ofhydrodynamic turbulence, it may be difficult to evalu-ate those for the nonlinear energy in wave systems. It isbecause the nonlinearity in such systems appears as thehigher-order expansion of the complex amplitudes in con-trast with the success of the K´arm´an-Howarth relationin the Navier-Stokes turbulence where the total energyis represented in the quadratic form. One might be ableto find alternative ways to go beyond in this direction byintroducing adequate modes of physical quantities.It is of interest that the total-energy fluxes are nearlyequal in both weak and strong turbulence regimes whilethe two regimes coexist in the inertial subrange. It mayshow another mechanism than those considered in the0critical balance, e.g., turning of the energy transfer inquasi-geostrophic turbulence, since the present system isstatistically isotropic in contrast with those where thecritical balance is predicted [26]. Appendix A: Hamiltonian structure expressed interms of complex amplitude The complex amplitude a k introduced in Eq. (3) playsa role as a canonical variable: i da k dt = δ H δa ∗ k , because the Hamiltonian can be rewritten in terms of thecomplex amplitude as H = X k ω k | a k | + X k + k − k − k = W kk k k a k a k a ∗ k a ∗ k + X k − k − k − k = (cid:0) G kk k k a k a ∗ k a ∗ k a ∗ k + c . c . (cid:1) + X k + k + k + k = ( R kk k k a k a k a k a k + c . c . ) . (A1) The second, third, and fourth terms respectively show the2 ↔ 2, 1 ↔ ↔ W kk k k (= W k k ∗ kk ), G kk k k and R kk k k are the matrix elements of the interactions. Note thatthe interactions include both resonant and non-resonantinteractions. Only under the kinetic equation of WTT,where only the resonant terms are retained, the quadraticenergy is conserved.The third and fourth terms of the Hamiltonian (A1)are rarely taken into account in the literature [9], becausethese terms can often be reduced by a canonical trans-formation in the weak turbulence regime of most waveturbulence systems [27]. In the elastic-wave turbulence,the fourth term can be reduced, but the third term can-not be as known from the linear dispersion relation (2),which allows the 1 ↔ ↔ ↔ ↔ ACKNOWLEDGMENTS Numerical computation in this work was carried out atthe Yukawa Institute Computer Facility. This work waspartially supported by KAKENHI Grant No. 25400412. [1] L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Butterworth–Heinemann, Oxford, 1986).[2] B. Audoly and Y. Pomeau, Elasticity and geometry (Ox-ford University Press, Oxford, 2010).[3] R. H. Kraichnan and R. Panda, Phys. Fluids , 2395(1988); R. B. Pelz, V. Yakhot, S. A. Orszag, L. Shtilman,and E. Levich, Phys. Rev. Lett. , 2505 (1985); W. T.Ashurst, A. R. Kerstein, R. M. Kerr, and C. H. Gibson,Phys. Fluids , 2343 (1987).[4] W. H. Matthaeus, W. T. Stribling, D. Martinez,S. Oughton, and D. Montgomery, Phys. Rev. Lett. ,2731 (1991); S. Riyopoulos, A. Bondeson, and D. Mont-gomery, Phys. Fluids , 107 (1982); D. Montgomery,W. H. Matthaeus, W. T. Stribling, D. Martinez, andS. Oughton, ibid . , 3 (1992).[5] D. Cai, A. J. Majda, D. W. McLaughlin, and E. G.Tabak, Physica D , 551 (2001).[6] V. E. Zakharov, O. A. Vasilyev, and A. I. Dyachenko,JETP Lett. , 63 (2001).[7] N. Yokoyama and M. Takaoka, Phys. Rev. E , 012909(2014).[8] A. N. Kolmogorov, Dokl. Akad. Nauk. SSSR , 301(1941).[9] V. E. Zakharov, V. S. L’vov, and G. Falkovich, Kolmogorov Spectra of Turbulence I: Wave Turbulence (Springer-Verlag, Berlin, 1992).[10] D. Resio and W. Perrier, J. Fluid Mech. , 603 (1991).[11] U. Frisch, Turbulence: the legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, 1995).[12] A. Boudaoud, O. Cadot, B. Odille, and C. Touz´e, Phys.Rev. Lett. , 234504 (2008); N. Mordant, ibid . ,234505 (2008).[13] B. Miquel, A. Alexakis, and N. Mordant, Phys. Rev.E , 062925 (2014); One of the referees informed usthis reference, which was published during the reviewingprocess.[14] T. Humbert, O. Cadot, G. D¨uring, C. Josserand, S. Rica,and C. Touz´e, Europhys. Lett. , 30002 (2013).[15] P. Denissenko, S. Lukaschuk, and S. Nazarenko, Phys.Rev. Lett. , 014501 (2007); L. Deike, M. Berhanu, andE. Falcon, Phys. Rev. E , 023003 (2014).[16] B. Rumpf and L. Biven, Physica D , 188 (2005).[17] A. N. Pushkarev and V. E. Zakharov, Physica D , 98(2000); A. I. Dyachenko, A. O. Korotkevich, and V. E.Zakharov, Phys. Rev. Lett. , 134501 (2004); S. Y.Annenkov and V. I. Shrira, ibid . , 204501 (2006).[18] G. D¨uring, C. Josserand, and S. Rica, Phys. Rev. Lett. , 025503 (2006).[19] P. Cobelli, P. Petitjeans, A. Maurel, V. Pagneux, andN. Mordant, Phys. Rev. Lett. , 204301 (2009);N. Mordant, Eur. Phys. J. B , 537 (2010).[20] B. Miquel and N. Mordant, Phys. Rev. Lett. , 034501(2011).[21] B. Miquel, A. Alexakis, C. Josserand, and N. Mordant,Phys. Rev. Lett. , 054302 (2013).[22] N. Yokoyama and M. Takaoka, Phys. Rev. Lett. , × × points:1024 independent realizations, 4 different times at an in-terval sufficiently longer than the longest linear period,and 512 grid points.[24] The spectra are obtained by averaging over 4096 =1024 × , 656 (1971).[26] S. Nazarenko, Wave Turbulence (Springer, Heidelberg,2011).[27] V. P. Krasitskii, J. Fluid Mech.272