Kolmogorov spectra of weak turbulence in media with two types of interacting waves

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1 3 Decemer 2001 Physics Letters A 291 (2001) Kolmogorov spectra of wea turulence in meia with two types of interacting waves F. Dias a P. Guyenne V.E.Zaharov c a Centre e Mathématiques et e Leurs Applications Ecole Normale Supérieure e Cachan 61 avenue u Présient Wilson Cachan ceex France Department of Mathematics Statistics McMaster University 1280 Main Street West L8S 4K1 Hamilton ON Canaa c Lau Institute for Theoretical Physics 2 Kosygin str Moscow Russia Receive 2 July 2001; accepte 22 Octoer 2001 Communicate y A.P. Fory Astract A system of one-imensional equations escriing meia with two types of interacting waves is consiere. This system can e viewe as an alternative to the moel introuce y Maja McLaughlin Taa in 1997 for assessing the valiity of wea turulence theory. The preicte Kolmogorov solutions are the same in oth moels. The main ifference etween oth moels is that coherent structures such as wave collapses quasisolitons cannot evelop in the present moel. As shown recently these coherents structures can influence the wealy turulent regime. It is shown here that in the asence of coherent structures wea turulence spectra can e clearly oserve numerically Elsevier Science B.V. All rights reserve. PACS: D; m; g Keywors: Kolmogorov spectra; Dispersive waves; Wea turulence 1. Introuction Wea turulence theory is an efficient tool for escriing turulence in systems ominate y resonant interactions etween small-amplitue waves. One of the ey ingreients to the theory of wea wave turulence is the so-calle Kolmogorov spectrum [1]. Kolmogorov wea-turulence spectra have een oserve in several physical systems (for example in a sea of win-riven wealy couple ispersive water waves). We elieve that Kolmogorov wea-turulence * Corresponing author. aress: ias@cmla.ens-cachan.fr (F. Dias). spectra are a useful theoretical tool for explaining various complex wave phenomena oserve in nature. Only a few attempts have een mae to compare preictions of wea turulence theory with numerical results. One can mention the results of Pusharev Zaharov [2] who numerically solve the threeimensional equations for capillary water waves oserve a power-law spectrum close to that erive y Zaharov Filoneno [3]. Maja McLaughlin Taa [4] introuce a moel the so-calle MMT moel for assessing numerically the valiity of wea turulence theory. Since their results inicate a failure of the preictions of wea turulence theory more computations have een carrie out recently to get a etter unersting of wave turulence in the MMT moel [5 7]. The present unersting is that co /01/$ see front matter 2001 Elsevier Science B.V. All rights reserve. PII: S (01)

2 140 F. Dias et al. / Physics Letters A 291 (2001) herent structures can strongly affect wea turulence. These coherent structures essentially are wave collapses quasisolitons. Wave collapses in the form of sporaic localize events represent a strongly nonlinear mechanism of energy transfer towars small scales. Quasisolitons or envelope solitons enote approximate solutions of the MMT moel which ten to classical solitons in the limit of a narrowe spectrum. The presence of such quasisolitons may explain the eviation from wea turulence leaing to the appearance of a steeper spectrum (the so-calle MMT spectrum) in some cases [67]. Recently Biven et al. [8] aresse the prolem of reaown of wave turulence y intermittent events associate with nonlinear coherent structures. In the present Letter we consier a moel which is quite similar to the MMT moel. However coherent structures cannot evelop. Our moel taes the form of a system of equations escriing the interactions of two types of waves. This is a fairly wiesprea case which inclues for example the interaction of electrons with photons or the interaction of electromagnetic waves with Langmuir waves [19]. The main conclusion of this Letter is that numerical results ase on the present moel are in agreement with the preictions of wea turulence theory. Agreement etween numerical simulations wea turulence theory was also recently otaine y Zaharov et al. [10] who examine a moifie version of the MMT moel that allows for one to three wave interactions. Of course it will e necessary in the future to perform numerical computations on the full equations escriing the physical phenomena of interest. However we elieve that a lot of information can e otaine from the solution of simplifie moels. Since the theory of wea turulence is quite general its main statements can e teste with simple moels for which numerical simulations can e performe more easily. 2. Moel equations Kolmogorov spectra We consier the system of equations propose in [7] i a = ω a T 123 a δ( ) i = sω T a 2 a 3 δ( ) (1) where a enote the Fourier components of two types of interacting wave fiels asteris sts for complex conjugation. Lie the MMT moel this moel is etermine y the linear ispersion relation ω = α the interaction coefficient T 123 = β/4. Thus ω sω T 123 are homogeneous functions of their arguments. The three parameters s α β are real with the restriction sα > 0. If we set α = 2β = 0 Eqs. (1) correspon to couple nonlinear Schröinger equations. The system possesses two important conserve quantities the positive efinite Hamiltonian H which we split into its linear part H L its nonlinear part H NL H = H L H NL ( = ω a 2 s 2) T 123 a a δ( ) the total wave action (or numer of particles) N = ( a 2 2). Note that oth iniviual wave actions a 2 2 are conserve in the system. Eqs. (1) escrie four-wave resonant interactions satisfying 1 2 = 3 ω 1 sω 2 = sω 3 ω. (2) It is well nown that when s = 1 conitions (2) have nontrivial solutions only if α<1. The case s = 1 α = 1/2 which mimics gravity waves in eep water was treate in some recent stuies [4 7]. In particular Zaharov et al. [7] showe that the MMT moel with α<1exhiits coherent structures which strongly affect the wealy turulent regime. Here accounting for s 1 allows the resonance conitions (2) to e satisfie for any α. Ifα = 2 we can solve explicitly Eqs. (2) to otain 3 = 1 2( 1 s 2 ) 1 s

3 F. Dias et al. / Physics Letters A 291 (2001) = 2 2( 1 s 2 ). (3) 1 s It is clear that Eqs. (3) with s = 1 give the trivial solution 3 = 2 = 1. As a general rule for a given α nontrivial families of resonant quartets oeying Eqs. (2) can e foun for all values of s 1. In the framewor of wea turulence theory we are intereste in the evolution of the two-point correlation functions a a = n a δ( ) = n δ( ) where represents ensemle averaging. Uner the assumptions of rom phases quasi-gaussianity it is then possile to write a system of inetic equations for n a n as n a n with = 2π T U a 123 δ(ω 1 sω 2 sω 3 ω ) δ( ) (4) = 2π T U123 a δ(sω 1 ω 2 ω 3 sω ) δ( ) (5) U123 a = na 1 n 2 n 3 na 1 n 2 na na 1 n 3 na n 2 n 3 na. The stationary power-law solutions of Eqs. (4) (5) can e foun explicitly. To o so let us examine Eq. (4) only since the prolem is similar for Eq. (5) y permuting n a n as well as ω sω. Looing for solutions of the form n a ω γ n (sω ) γ applying Zaharov s conformal transformations the inetic equation (4) ecomes N a ω ω y 1 I a sαβγ where Nω a = na /ω Isαβγ a = [ S (sξ3 ) γ (sξ 2 ) γ ξ γ ] 1 (6) δ(1 sξ 3 sξ 2 ξ 1 ) δ ( 1 ξ 1/α 3 ξ 1/α 2 ξ 1/α ) 1 [ 1 (sξ 3 ) y (sξ 2 ) y ξ y ] 1 ξ1 ξ 2 ξ 3 (7) with ={0 <ξ 1 < 1 0 <sξ 2 < 1 ξ 1 sξ 2 > 1} S 123 = 2π α 4 s 2γ (ξ 1ξ 2 ξ 3 ) (β/21)/α 1 γ y = 3γ 1 2β 3. α The nonimensionalize integral Isαβγ a in Eq. (6) results from the change of variales ω j ω ξ j (j = 1 2 3). Thermoynamic equilirium solutions (γ = 0 1) given y n a = const n a ω 1 (8) are ovious. In aition there exist Kolmogorov-type solutions (y = 0 1) n a n a ω ( 2β/3 1α/3)/α ω (2β/31)/α (9) which correspon to a finite flux of wave action Q energy P respectively. We point out that Eqs. (8) (9) are also steay solutions of Eq. (5) they are ientical to those erive from the MMT moel [4]. The fact that the inetic equation epens on the parameter s implies that the fluxes the Kolmogorov constants also epen on s (see elow). However there is no s-epenence on the Kolmogorov exponents ecause of the property of scale invariance. As foun in [7] the criterion for appearance of the Kolmogorov spectra (9) is β< 3 or β>2α 3 (10) 2 2. This means physically that a flux of wave action towars large scales (inverse cascae with Q<0) a flux of energy towars small scales (irect cascae with P>0) shoul occur in the system. The full expressions of Eq. (9) can e otaine from imensional analysis yieling n a = ca 1 Q1/3 a ω ( 2β/3 1α/3)/α n = c 1 Q1/3 (sω ) ( 2β/3 1α/3)/α (11) n a = ca 2 P a 1/3 ω (2β/31)/α n = c 2 P 1/3 (sω ) (2β/31)/α (12)

4 142 F. Dias et al. / Physics Letters A 291 (2001) where c a 1 = ( Ia ) sαβγ 1/3 y y=0 ( I a ) c a sαβγ 1/3 2 = y y=1 (13) enote the imensionless Kolmogorov constants. These can e compute irectly y using integral (7) its analogue for N ω /. In the numerical computations we will fix α = 3/2 > 1 in orer to prevent the emergence of coherent structures such as wave collapses quasisolitons reveale in [7]. Our goal is to chec the valiity of the Kolmogorov spectra which are relevant in several real wave meia as alreay sai in the introuction [1]. We will restrict our stuy to solutions (12) associate with the irect cascae. 3. Numerical results Numerical experiments were carrie out to integrate Eqs. (1) y use of a pseuospectral coe with perioic ounary conitions. The metho inclues a fourthorer Runge Kutta scheme in comination with an integrating factor technique which permits efficient computations over long times [47]. Resolution with up to 2048 e-aliase moes in a omain of length 2π is achieve here ( max = 1024). To generate wealy turulent regimes source terms of the form ( f a i ) e iθ i f ( a ) [ (ν a ν ) ( ) 2 ( ν a ν ) ] ( ) 2 (14) were ae to oth right-h sies of Eqs. (1). The first term in Eq. (14) enotes a white-noise forcing where 0 θ < 2π is an uniformly istriute rom numer varying in time. The term in square racets consists of a wave action sin at large scales an energy sin at small scales. The rom feature of the forcing maes it uncorrelate in time with the wave fiel. Consequently it is easier to control the input energy with a rom forcing than with a eterministic forcing. For the results presente elow the forcing region is locate at small wave numers i.e. { f a 6 3 if8 12 = 0 otherwise. Parameters of the sins are { ν a = if ( = 5) 0 otherwise { ν a = if ( = 550) 0 otherwise. Using this in of selective issipation ensures large enough inertial ranges at intermeiate scales where solutions can evelop uner the negligile influence of amping. Accoring to criterion (10) we focuse on β = 2s = 1/10 as a typical case for testing wea turulence preictions. Simulations are run from lowlevel initial ata until a quasisteay state is reache then averaging is performe over a sufficiently long time to compute the spectra. The time step set equal to t = has to resolve accurately the fastest harmonics τ 1/ω max of the meium or at least those from the inertial range. Time integration with such a small time step leas to a computationally time-consuming proceure espite the oneimensionality of the prolem. This explains why we chose α = 3/2 rather than a greater integer value (e.g. α = 2) as well as s = 1/10 rather than a value s>1. Otherwise the constraint on t woul e more severe. There is apriorino special requirement in the choice of the value of s except s 1. Figs. 1 2 show the temporal evolution of the wave action N the quaratic energy H L over the winow 80 t 100. At this stage the stationary regime is clearly estalishe since the wave action the quaratic energy fluctuate aroun some mean values N 0.5H L 5.3. Typically the time interval for oth the whole computation the time averaging must excee significantly the longest linear perio. In orer to monitor the level of turulence we efine the average nonlinearity ɛ as the ratio of the nonlinear part to the linear part of the Hamiltonian i.e. ɛ = H NL. H L As in [27] this quantity provies a relatively goo estimate of the level of nonlinearity once the system reaches the steay state. We can see in Fig. 3 that

5 F. Dias et al. / Physics Letters A 291 (2001) Fig. 1. s = 1/10 α = 3/2 β = 2. Evolution of wave action N vs. time in the stationary state. Fig. 2. s = 1/10 α = 3/2 β = 2. Evolution of quaratic energy H L vs. time in the stationary state. the average nonlinearity fluctuates aroun some mean value ɛ 0.14 which inicates that the conition of wea nonlinearity hols in our experiments. However it shoul e emphasize that ɛ coul not e impose too small (y ecreasing the forcing) otherwise the ifferent moes woul not e excite enough to generate an effective flux of energy. This prolem is particularly important in numerics ue to the iscretization which restricts the possiilities for four-wave resonances. Since the effects of nonlinearity are assume to e small in wea turulence it is sufficient to consier only the quantity H L which contains the main part of the energy. We euce the conservation of the total Hamiltonian from the conservation of H L ɛ as illustrate in Figs. 2 3 ecause H = H L (1 ɛ). Fig. 4 isplays the stationary isotropic spectra n a realize in the present situation. By comparison we also plotte the preicte Kolmogorov solutions given y Eq. (12). For α = 3/2β = 2 they rea n a = ca 2 P a 1/3 ω 14/9 = c2 a P a 1/3 7/3 n = c 2 P 1/3 (sω ) 14/9 = c 2 P 1/3 s 14/9 7/3 s = 1/10 (15) (16)

6 144 F. Dias et al. / Physics Letters A 291 (2001) Fig. 3. s = 1/10 α = 3/2 β = 2. Evolution of average nonlinearity ɛ vs. time in the stationary state. Fig. 4. s = 1/10 α = 3/2 β = 2. Compute spectra (n a for the lower one n for the upper one) preicte Kolmogorov spectra C a 7/3 with C a = c2 ap a 1/3 C = c2 P 1/3 s 14/9 (ashe lines). where c2 a = c 2 = are numerically calculate from Eq. (13). The mean fluxes of energy P a in Eqs. (15) (16) can e expresse as P a = 2 ν( a ) 2ω n a > P = 2 ν( ) 2sω n > with the cutoff of ultraviolet issipation [17]. Then it is straightforwar to get their values P a = 0.86 P = 0.56 from simulations. We can oserve in Fig. 4 that for oth wave fiels the spectra are well approximate y the Kolmogorov power-laws over a wie range of scales (say 20 <<300). Here the agreement etween theory numerics is foun with respect to oth the slope the level of the spectra. 4. Conclusion We have stuie a simplifie one-imensional moel escriing meia with two types of interacting waves. The regime of parameters has een chosen such

7 F. Dias et al. / Physics Letters A 291 (2001) that wea turulence theory can e applie correctly coherent structures cannot evelop. This way we avoi any interference etween wea wave turulence coherent structures. Our numerical results show the appearance of a pure wea turulence state with the formation of a complete Kolmogorov spectrum. This suggests the general relevance of wea turulence even in one-imensional systems. In the future it will e of interest to exten the present wor to higher imensions y still consiering simplifie moels such as the present one to perform computations on the full equations escriing the physical phenomena of interest. Recall that Pusharev Zaharov [2] successfully oserve wea turulence for capillary water waves in three imensions. However their numerical simulations ase on the truncate asic equations were very time-consuming the Kolmogorov spectrum that they measure extene only over a small range of scales. Acnowlegements This wor was performe in the framewor of the NATO Linage Grant OUTR.LG V.E.Z. also acnowleges support of the US Army uner the grant DACA C-0044 of ONR uner the grant N References [1] V.E. Zaharov V.S. Lvov G. Falovich Kolmogorov Spectra of Turulence Springer [2] A.N. Pusharev V.E. Zaharov Phys. Rev. Lett. 76 (1996) [3] V.E. Zaharov N.N. Filoneno J. Appl. Mech. Tech. Phys. 4 (1967) 506. [4] A.J. Maja D.W. McLaughlin E.G. Taa J. Nonlinear Sci. 6 (1997) 9. [5] D. Cai A.J. Maja D.W. McLaughlin E.G. Taa Proc. Natl. Aca. Sci. 96 (1999) [6] P. Guyenne V.E. Zaharov A.N. Pusharev F. Dias C. R. Aca. Sci. Paris II 328 (2000) 757. [7] V.E. Zaharov P. Guyenne A.N. Pusharev F. Dias Physica D (2001) 573. [8] L. Biven S.V. Nazareno A.C. Newell Phys. Lett. A 280 (2001) 28. [9] M. Lyutiov Phys. Lett. A 265 (2000) 83. [10] V.E. Zaharov O.A. Vasilyev A.I. Dyacheno JETP Lett. 73 (2001) 63.