Weak turbulence of Kelvin waves in superfluid He

Physics of small-scale quantum turbulence in superfluids is essentially based on the knowledge of the energy spectrum of Kelvin waves, Ek₋. In our paper, we derive a new type of kinetic equation for Kelvin waves on quantized vortex filaments with random large-scale curvature which describes a step-b...

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Дата:	2010
Автори:	L’vov, V. S., Nazarenko, S.
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Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2010
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Теми:	К 80-летию со дня рождения В.Г. Барьяхтара
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Цитувати:	Weak turbulence of Kelvin waves in superfluid He / V. S. L’vov, S. Nazarenko // Физика низких температур. — 2010. — Т. 36, № 8-9. — С. 986–993. — Бібліогр.: 23 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1174452025-02-09T23:20:08Z Weak turbulence of Kelvin waves in superfluid He L’vov, V. S. Nazarenko, S. К 80-летию со дня рождения В.Г. Барьяхтара Physics of small-scale quantum turbulence in superfluids is essentially based on the knowledge of the energy spectrum of Kelvin waves, Ek₋. In our paper, we derive a new type of kinetic equation for Kelvin waves on quantized vortex filaments with random large-scale curvature which describes a step-by-step energy cascade over scales caused by five-wave interactions. This approach replaces the previously used six-wave theory, which was recently shown to be inconsistent due to nonlocality. Solving the four-wave kinetic equation, we found a new local spectrum with a universal (curvature-independent) exponent, Ek∝k⁻⁵/³, which must replace the nonlocal spectrum of the six-wave theory, Ek∝k⁻⁷/⁵ in future theory, e.g., in finding the quantum turbulence decay rate, found by Kosik and Svistunov under wrong assumption of the locality of energy transfer in the sixwave interactions. We thank J. Laurie and O. Rudenko for help in the evaluation of the effective interaction coefficient. We acknowledge support of the US-Israel Binational Scientific Foundation administrated by the Israeli Academy of Science, and of the EC — Research Infrastructures under the FP7 Capacities Specific Programme, MICROKELVIN project number 228464. 2010 Article Weak turbulence of Kelvin waves in superfluid He / V. S. L’vov, S. Nazarenko // Физика низких температур. — 2010. — Т. 36, № 8-9. — С. 986–993. — Бібліогр.: 23 назв. — англ. 0132-6414 PACS: 67.25.dk, 47.37.+q, 45.10.Hj, 47.10.Df https://nasplib.isofts.kiev.ua/handle/123456789/117445 en Физика низких температур application/pdf Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	К 80-летию со дня рождения В.Г. Барьяхтара К 80-летию со дня рождения В.Г. Барьяхтара
spellingShingle	К 80-летию со дня рождения В.Г. Барьяхтара К 80-летию со дня рождения В.Г. Барьяхтара L’vov, V. S. Nazarenko, S. Weak turbulence of Kelvin waves in superfluid He Физика низких температур
description	Physics of small-scale quantum turbulence in superfluids is essentially based on the knowledge of the energy spectrum of Kelvin waves, Ek₋. In our paper, we derive a new type of kinetic equation for Kelvin waves on quantized vortex filaments with random large-scale curvature which describes a step-by-step energy cascade over scales caused by five-wave interactions. This approach replaces the previously used six-wave theory, which was recently shown to be inconsistent due to nonlocality. Solving the four-wave kinetic equation, we found a new local spectrum with a universal (curvature-independent) exponent, Ek∝k⁻⁵/³, which must replace the nonlocal spectrum of the six-wave theory, Ek∝k⁻⁷/⁵ in future theory, e.g., in finding the quantum turbulence decay rate, found by Kosik and Svistunov under wrong assumption of the locality of energy transfer in the sixwave interactions.
format	Article
author	L’vov, V. S. Nazarenko, S.
author_facet	L’vov, V. S. Nazarenko, S.
author_sort	L’vov, V. S.
title	Weak turbulence of Kelvin waves in superfluid He
title_short	Weak turbulence of Kelvin waves in superfluid He
title_full	Weak turbulence of Kelvin waves in superfluid He
title_fullStr	Weak turbulence of Kelvin waves in superfluid He
title_full_unstemmed	Weak turbulence of Kelvin waves in superfluid He
title_sort	weak turbulence of kelvin waves in superfluid he
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2010
topic_facet	К 80-летию со дня рождения В.Г. Барьяхтара
url	https://nasplib.isofts.kiev.ua/handle/123456789/117445
citation_txt	Weak turbulence of Kelvin waves in superfluid He / V. S. L’vov, S. Nazarenko // Физика низких температур. — 2010. — Т. 36, № 8-9. — С. 986–993. — Бібліогр.: 23 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT lvovvs weakturbulenceofkelvinwavesinsuperfluidhe AT nazarenkos weakturbulenceofkelvinwavesinsuperfluidhe
first_indexed	2025-12-01T16:37:05Z
last_indexed	2025-12-01T16:37:05Z
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fulltext	© Victor S. L’vov and Sergey Nazarenko, 2010 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9, p. 986–993 Weak turbulence of Kelvin waves in superfluid He Victor S. L’vov1,3 and Sergey Nazarenko2 1Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel 2Mathematics Institute, Warwick University, Coventry, CV4 7AL, UK 3Theoretical Department, Institute for Magnetism, National Academy of Sciences of Ukraine, Kiev, Ukraine E-mail: victor.lvov@weizmann.ac.il Received January 21, 2010 Physics of small-scale quantum turbulence in superfluids is essentially based on the knowledge of the energy spectrum of Kelvin waves, .kE In our paper, we derive a new type of kinetic equation for Kelvin waves on quantized vortex filaments with random large-scale curvature which describes a step-by-step energy cascade over scales caused by five-wave interactions. This approach replaces the previously used six-wave theory, which was recently shown to be inconsistent due to nonlocality. Solving the four-wave kinetic equation, we found a new local spectrum with a universal (curvature-independent) exponent, 5/3 kE k−∝ , which must replace the nonlocal spectrum of the six-wave theory, 7/5 kE k−∝ in future theory, e.g., in finding the quantum turbulence decay rate, found by Kosik and Svistunov under wrong assumption of the locality of energy transfer in the six- wave interactions. PACS: 67.25.dk Vortices and turbulence; 47.37.+q Hydrodynamic aspects of superfluidity; quantum fluids; 45.10.Hj Perturbation and fractional calculus methods; 47.10.Df Hamiltonian formulations. Keywords: turbulence, superfluid, Kelvin wave. 1. Physical background Turbulence in superfluids [1,2] is one of most fascina- ting natural phenomena where transition from the laws of classical physics to the quantum laws occurs gradually as energy passes from large to small scales along the turbu- lent cascade. Such a coexistence of the classical and quan- tum physics in the same system and their interplay is a fundamental consequence of absence of viscosity, the role of with in classical turbulence is to quench the energy cas- cades at scales which are still large enough to be classical. In superfluids, on the other hand, when temperature is close to the absolute zero, such quenching mechanism is absent, and the energy flux unavoidably reaches the scales where the quantization of the vortex circulation (disco- vered by Feynman [3]) is essential. Recently, there have been significant advances in experimental techniques al- lowing studies of turbulence in various systems such as 3He [4,5], 4He [6,7] and Bose–Einstein condensates of supercold atoms [8,9]. Often, experimental devices are not small enough to probe the transitional and quantum scales directly. For this reason, an impressive progress in numeri- cal simulations [10,11] is very important because they give access to characteristics of turbulence yet unavailable ex- perimentally. In zero-temperature limit, one of the most interesting questions is the nature of the energy dissipation, namely the mechanisms of transfer the energy down to the tiny (almost atomic) scales where vortices can radiate their energy away by emitting phonons. A commonly accepted model of superfluid turbulence comprises a randomly moving tangle of quantized vortex lines which can be characterized by the mean intervortex distance and the vortex core radius .a The vortex core radius has an atomic size and the conventional de- scription used for fluid media fails within such a core. There are two approaches to deal with the vortex core. First one is a «microscopic» model in which the core is resolved: it is based on the Gross–Pitaevski equation, 2 2\| \| = 0, t ∂Ψ +∇ Ψ −Ψ Ψ ∂ (1) where Ψ is so-called condensate wave function. This model is systematically derived for the Bose–Einstein con- densates in super-cold atoms, but not for the liquid helium. Nevertheless, it is frequently used for describing superfluid flows in helium because it contains several essential fea- tures of such superfluids, i.e., vortex quantization, acoustic Weak turbulence of Kelvin waves in superfluid He Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9 987 waves (phonons) in presence of a condensate, and it de- scribes a gradual (nonsingular) reconnection of vortex lines. However, the Gross–Pitaevski equation can be costly to study, and one often resorts to using so-called Biot–Savart formulation in the Euler equations for the ideal classical fluids, exploiting the fact that far away from the vortex cores the Gross–Pitaevski dynamics is isomorphic to the ideal classical flow via the Madelung transformation. In the Biot–Savart model, the vortices are postulated via a cutoff in the equations for the vortex line elements. Name- ly, the equations used are 3 ( )= , 4 \| \| dκ × − π −∫ s r sr r s (2) with a cutoff at the core radius a , i.e., integrating over the range \| \|> .a−r s Here κ is the circulation quantum. In what follows, we will base on the Biot–Savart model. Naturally, at scales L the discreteness is unimpor- tant and they can be described classically with the ener- gy flux toward smaller scales by the celebrated Richard- son–Kolmogorov cascade. Then the energy is transferred through the crossover scale by some complicated me- chanisms [17–19], thereby exciting smaller scales < < aλ which propagate along the individual vortex filaments as waves. These were predicted by Lord Kelvin more than one century ago [12] and experimentally observed in su- perfluid 4He about 50 years ago. It is believed that Kelvin waves (KWs) play a crucial role in superfluid dynamics, transferring energy from to a much smaller scale, where it can dissipate via emission of bulk phonons. In a wide range of scales KWs are weakly nonlinear and can be treated within the theory of weak-wave turbulence [16]. Such an approach for KWs was initiated in [13] where a six-wave kinetic equation (KE) was presented, and a KW spectrum was obtained from this equation based on a di- mensional analysis, 7/5( )KSE k k−∝ . Dimensional analysis of the KE is based on the assumption that all integrals in the collision term are converges. Physically it means that the energy transfer over scales can be considered as step- by-step cascade, in which energy to a given range of wave- vectors k comes from the smaller k ′ of the same order of magnitude and is transferred toward larger ,k ′′ again of the order of .k This assumption, firstly suggested in 1941 by Kolmogorov for hydrodynamical turbulence is often called «locality of the energy transfer». Spectrum 7/5( )KSE k k−∝ was subsequently used in theoretical con- structions in superfluid turbulence, e.g., to describe the classical-quantum crossover range of scales and to explain the dissipation rate in the superfluid turbulence [17–20]. However, it was recently shown in [14] that this spectrum is nonlocal and, therefore, nonrealizable. This crucial lo- cality check was only possible after a highly nontrivial calculation of the six-wave interaction coefficient done in Ref. 14 which took into account previously omitted impor- tant contributions and which yielded explicit relations for this coefficient in relevant asymptotical limits. In this paper, we exploit the consequences of the nonlo- cality of the 6-wave theory, and replace the latter with a new local 5-wave theory of KW turbulence. Our 5-wave theory arises from the 6-wave theory (completed in [14]) in the strongly nonlocal case, when one of the waves in the sextet is much longer than the other five and corres- ponds to the outer scale — infra-red (IR) cutoff. We derive a new spectrum of the KW turbulence which is local, and which must be used in future for revising the parts of the superfluid turbulence where the nonlocal spectrum of the 6-wave theory has previously been used. 2. On statistical description of weak-wave turbulence Weak-wave turbulence refers to a class of strongly non- equilibrium statistical systems consisting of a large number of excited weakly nonlinear waves in nondissipative (Ha- miltonian) dispersive media [16]. Such systems comprise a unique example where strongly nonequilibrium statistics can be addressed systematically, and states analogous to Kolmogorov–Richardson cascades of classical turbulence can be obtained analytically. Let us briefly overview the theory of weak-wave turbulence with application to the five- and six-wave systems (three- and four-wave processes are absent for KWs) starting from a classical Hamiltonian equation for the complex canonical amplitude of waves ( , )a a t≡k k and a∗k (classical analogues of the Bose-operators of particle creations and annihilation) with a wavevector :k = . a i t a∗ ∂ δ ∂ δ k k H (3) Here H is a Hamiltonian which for the wave systems is free int free= ; = ,k a a d∗+ ω∫ k k kH H H H (4) where kω is the wave frequency. For KWs 2= / 4k kω Λκ π where κ is the circulation quantum. intH is an effective interaction Hamiltonian for KWs, propagating along straight vortex line, that is equal to 2,3,4 2,3,4 1 3 1 4 1 2 3 41 1 1= [ c.c.] 6 d d V a a a a∗ ∗ ∗ ↔ δ +∫ k k…H (5) for four-wave systems or 4,5,6 4,5,6 3 3 1 6 1 2 3 4 5 61,2,3 1,2,3 1= 36 d d W a a a a a a∗ ∗ ∗ ↔ δ∫ k k…H (6) for six-wave systems. Here we use shorthand notations: ,j j a a≡ k = \| \|,j jk k 3,4,5 1 2 3 4 51,2 ( )δ ≡ δ + − − −k k k k k and 4,5,6 1 2 3 4 5 61,2,3 ( ).δ ≡ δ + + − − −k k k k k k These equations effectively describe weakly nonlinear waves of any nature [16], using only relevant dynamical Victor S. L’vov and Sergey Nazarenko 988 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9 information, that presents in the system Hamiltonian .H The main technical problem is to find H for a particular complicated physical system. Fortunately, in case of KWs this cumbersome job was done for us and for general read- er in Ref. 14. Statistical description of weakly interacting waves can be reached [16] in terms of the KE ( , ) = St( , ),n t t t ∂ ∂ k k (7) for the waveaction spectrum ( , )n tk , defined by ( , ) ( , ) = ( , ) ( ),a t a t n t∗ ′ ′δ −k k k k k where … stands for the ensemble averaging. The colli- sion integral St ( , )tk can be found in various ways [16], including the Golden Rule widely used in quantum me- chanics. For the five- and six-wave processes we have, respectively, { 1,2,3 1,2,3 1,2,32 1 3 1 3St = \| \| 12 d d V↔ π δ ×∫ k k kk k… N 1 2 3( )×δ ω −ω −ω −ω +k },2,3 ,2,3 ,2,32 1 2 31 1 13\| \| ( ) ,V+ δ δ ω −ω −ω −ωk k k kN (8) 2,3,4 1 1 1 1 1 2 3 4 1 2 3 41 ( );n n n n n n n n− − − −≡ − − −N 4,5,6 4,5,62 3 3 1 5 ,1,3 ,1,3St \| \| 12 d d W↔ π = δ ×∫ k kk k… 1 2 3 4 5 1 2 3 4 5( )n n n n n n×δ ω +ω +ω −ω −ω −ω ×k k 1 1 1 1 1 1 1 2 3 4 5 6( ) .n n n n n n− − − − − −× + + − − − (9) Scaling solutions of these KE's can be found under two conditions satisfied for various wave systems, e.g., gravity and capillary waves on the fluid surface, Langmuir and ion-sound waves in plasma, etc. [16]. Scale-invariance of the wave system, when the fre- quency of waves and the interaction coefficients are ho- mogeneous functions of wave vectors: 2 1 2 3 4 5( ) = ( ), ( , ; , , )k k Vαω λ λ ω λ λ λ λ λ ≡k k k k k 5 1 2 3 4 5( , ; , , ),Vα≡ λ k k k k k and a similar relationship for 4,5,6 1,2,3W with an index 6α . Interaction locality, in a sense that the main contribu- tion to the energy balance of a given k-wave (with wave- vector )k originates from its interaction with k′-waves with .k k′∼ Mathematically it means that all integrals over 1,k 2 ,k etc. in the KE’s (7)–(9) converge, and there- fore in the scale-invariant case the leading contribution to the collision integral indeed originates from the regions 2 ,k k∼ 3 ,k k∼ etc. Note that nonlocal spectra are not solutions of the KE’s (7)–(9) and, therefore, physically irrelevant. In the scale-invariant wave systems one seeks for the scale-invariant solutions of the KE’s: ( ) = ,xn k Ak− (10) where A is a dimensional number. To find the scaling index x for turbulent spectra with a constant energy flux over scales, we note that all KE’s (7)–(9) conserve the total energy of the wave system, = 0, , .dE E E d E n dt ≡ ≡ ω∫ k k k kk Therefore the k-space energy density, ,Ek satisfies a con- tinuity equation: = 0 .k kE t k ∂ ∂ε + ∂ ∂ Here kε is the energy flux over scales, expressed via an integral over sphere of radius :k < = St( , ) .k k k k d k t′ ′ ′ ′ε ω∫ k Under the assumption of the interaction locality, one es- timates the d-dimensional integral d∫ k as dk , the interac- tion coefficients 2,3,4 , , 4 1 ,k k k kV V Vkα∼ ∼ 4,5,6 , , 6 1,2,3 , , ,k k k k k kW W Wkα∼ ∼ (11) and = xp k pn A k − (for the p-wave interactions). There- fore: 4 2 45 55( ) ( ) , 2 3 scattering; xd k k V k A kα −ε ⇔∼ 5 2 56 66( ) ( ) , 3 3 scattering .xd k k W k A kα −ε ⇔∼ (12) For the spectra of turbulence with a constant energy flux = =kε ε const, i.e., 0.k kε ∝ For the p-wave process this gives the scaling exponent of ( ),n k ,px and the energy scaling exponent ,py ( ) y pE k k − ∝ : 2 2 = , = . 1 p p p px d y x p α + −α − (13) In fact, these expressions are valid for any > 2p . For the three- and the four-wave processes (with p = 3 and p = 4) this gives the well-known results, see, e.g., Ref. 16. Note however, that the 4-wave 1↔3 is considered here for the first time, and it is different from the previously considered standard 2↔2 processes. Weak turbulence of Kelvin waves in superfluid He Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9 989 3. Kelvin-wave turbulence with six-wave interaction To consider the KW system, one has to start with the Biot–Savart equations (2), consider an equilibrium state corresponding to an infinitely long straight vortex line and perturb it with small angle disturbances. This will corres- pond to a setup of weakly nonlinear KWs which are dis- persive, and for description of which the weak-wave turbu- lence theory can be used. For this, one has to parametrize the transverse displacement vector of the perturbed like by the distance along the unperturbed line, pass to Fourier space and expand in small perturbation angles and small parameter 1/ Λ , where = ln( / ) 1aΛ . Such expansion in two small parameters is not easy. This is because in the leading order in 1/Λ the model is integrable, i.e., noncas- cading, and to describe the leading order of the energy transfers one has to go to next order in 1/ Λ . Second diffi- culty is that the lowest order process, the four-wave reson- ances, are absent for such one dimensional systems with concave up dispersion relation. Thus one has to go to the next order in the small nonlinearity too. Combination of these two facts makes finding of the effective interaction Hamiltonian intH for KWs a hard task. For the six-wave process, which assumes that the underlying vortex is per- fectly straight, this task was accomplished only recently [14]. Effective 3↔3-interaction coefficient W was shown to have a form 4,5,6 4,5,6 1 2 3 4 5 61,2,3 1,2,3 3= , 4 W k k k k k k F− πκ (14) where F is a nonsingular dimensionless function of 1 6, ,k k… close to unity in the relevant region of its argu- ments. In particular, 1F → when one or several k’s are much less than the maximum wavenumber in the sextet. Equations (3), (6) and (14) provide general reader with all necessary information about KWs required for further developments in this paper. Those interested in further details about the derivations of these equations can find them in [14]. Notice that the form of Eq. (14) could be expected be- cause it demonstrates a very simple physical fact: long KWs (with small k’s) can contribute to the energy of a vor- tex line only when they produce curvature. The curvature, in turn, is proportional to wave amplitude ka and, at fixed amplitude, is inversely proportional to their wave-length, i.e., .k∝ Therefore in the effective motion equation each ja has to be accompanied by ,jk if .jk k Exactly this statement is reflected by Eq. (14). One can say, that cum- bersome calculations [14] support these reasoning, and additionally provide with explicit numerical factor 3 / 4− π and give an explicit expression for F which can be impor- tant in further research, required for careful comparison with future experiments or numerics. Equation (14) estimates 4,5,6 1,2,3W as 6Wk . Thus, Eq. (12) reproduces the Kozik–Svistunov (KS) scaling for the 3↔3 processes, which for further discussion is writhen with a dimensionless constant KSC : 2/5 1/5 7/5 1/5 17/5 7/5= , = .KS KS KS KS C C n E k k κ ε κ ε Λ (15) We repeat that KS spectrum (15) would be valid only if it was local, i.e., if all integrals (9) converged and thus one could estimate 3 3St ( )k↔ as in Eq. (12). However, detailed analysis (given in Ref. 14 and shortly reproduced below) shows the KS spectrum is nonlocal and therefore physical- ly nonrealizable. In order to find the valid spectrum of tur- bulent KW we will briefly reproduce this analysis, using the Eq. (14). 4. Nonlocality of the energy transfer with the six-wave interactions Let us now check if the KS spectrum (15) is local or not. For this, we consider the 3 3↔ collision term (9) for KW with the interaction amplitude 4,5,6 1,2,3W as in (14) and ( )n k as in Eq. (10). In this case jd∫ k are one- dimensional integrals jdk ∞ −∞∫ . In the IR region 1 , jk k k , =j 2, 3, 4, 5, we have 1F ≈ and the integral over 1k scales as 2 2 1 1 1 1 1 1/ 1/ 2 2( ) = .xAk n k dk k dk−Ψ ≡ κ κ∫ ∫ (16) Lower limit 0 in Eq. (16) is replaced by 1/ , where is the mean inter-vortex separation , at which approxima- tion of noninteracting vortex lines fails and one expects a cutoff of the power like behavior (10). Prefactor 2 in Eq. (16) reflects the fact that the ranges of positive and nega- tive 1k give equal contributions, and factor 1/ κ is intro- duced to make parameter Ψ dimensionless. Ψ has a meaning of the mean-square angle of the deviation of the vortex lines from straight. Therefore 1;Ψ for highly polarized vortex lines 1.Ψ Clearly, integral (16) IR-diverges if > 3x , which is the case for the KS spectrum (15) with 6 = 17 / 5.x Note that all the similar integrals over 2 ,k 3,k 4 ,k and 5k in Eq. (9) also diverge exactly in the same manner as integral (16). Moreover, when two of the wavenumbers belonging to the same side in the sextet tend to zero simultaneously then each of such wavenumbers will yield an integral as in (16), and the net result will be the product of these inte- grals, i.e., a stronger singularity than in the case of just one small wavenumber. On the other hand, small wavenumbers which are on the opposite sides of the resonant sextet do not lead to a stronger divergence because of an extra smallness arising in this case in Eq. (9) from 1 1 1 1 1 1 1 2 3 4 5 6 .n n n n n n− − − − − −+ + − − − Victor S. L’vov and Sergey Nazarenko 990 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9 Divergence of the integrals in Eq. (9) means that KS spectrum (15) is not a solution of the KE (9) and thus non- realizable. One should find another, self-consistent solu- tion of this KE. 5. Effective four-wave theory of KW turbulence Nonlocality of the six-wave theory is a serious problem. It indicates that dominant sextets contributing to the 3↔3- scattering are those for which two of the wavenumbers from the same side of the six-wave resonance conditions 1 2 3 4 5= ,kω +ω +ω ω +ω +ω 1 2 3 4 5= ,+ + + +k k k k k k (17) are very small, 1 /jk . Thus these equations effectively become 1 2 3 1 2 3= , = ,+ + + +k k k k k k k k 2 1 3 3 2 1= , or = ,+ + + +k k k k k k k k (18) and respective conditions for the frequencies, which im- plies a 4-wave process of the 1↔3 type. In the other words, one can interpret such nonlocal sextets on straight vortex lines as quartets on curved vortices, with the slow- est modes in the sextet responsible for the large-scale cur- vature R of the underlying vortex line in the 4-wave ap- proach. To derive an effective 4-wave KE, let us start with the 6-wave collision integral (9) and find the leading contribu- tions to it when the spectrum kn is steeper than 3k− in the IR region. There are four of them. The first one originates from the region where 1k and 2k are much smaller than the rest of 'jk s. The three other contributions originate from the other side of the sextet: regions where either 3k and 4,k or 3k and 5k , or 4k and 5k are small. These contributions are equal and we may find only one of them and multiply the result by three. Notably, the sum of the four contributions can be written exactly in the form of the 1↔3-collision term (8) with the effective 1↔3-interaction amplitude 2,3,4 1 2 3 41 = 3 / (4 2),V − Ψ πk k k k (19) because, as shown in Ref. 14, 1 2 3 4 5 6 01 lim ( , , \| , ) = 1. k F → k k k k k k Deriving Eq. (8) with 2,3,4 1 ,V Eq. (19), we took only lead- ing contributions in the respective IR regions, factorized the integrals over these wavevectors like in Eq. (16) and took only the zeroth order terms with respect to the small wavevectors (by putting these wavenumbers to zero) in the rest of the expression (9). Equation (8) with 2,3,4 1V as in Eq. (19) is an effective 4-wave KE, which we were aiming to obtain. This KE cor- responds to interacting quartets of KWs propagating along a vortex line having a random large-scale curvature R . Equation (19) estimates 2,3,4 1V as 4kV with ΨV ∼ . Us- ing this scaling in Eq. (12) , we arrive at a spectrum for the 1↔3 processes with scaling exponents 4 = 11/ 3x and 5 = 5 / 3y , 1/3 1/3 2/3 11/3 2/3 5/3= = ,LN LN LN LN C C n E k k ε Λκε ⇒ Ψ Ψ (20) Local (1↔3) L’vov–Nazarenko (LN) spectrum. 6. Local step-by-step energy transfer with four-wave 1↔3 interactions Mathematically, locality of the energy transfer in the 1↔3-wave processes means convergence of the multi- dimensional integral in the corresponding collision term (8). Here we will show that proof of convergence in Eq. (8) is a delicate issue and cannot be done only on the basis of power counting because the latter would give a divergent answer. 6.1. Proof of the infrared convergence Let us show that in the IR region, when at least one of the wave vectors, say 2k , is much smaller then ,k only a quadrupole cancelation of the largest, next to the largest and the two further sub-leading contributions appear to result in the final, convergent result for the collision term (8). Three integrations in Eq. (8) are restricted by two con- servation laws, namely by 2 2 2 2 1 2 3 1 2 31 3 : = , =k k k k→ + + + +k k k k (21) in the first term, and by 2 2 2 2 2 3 1 2 3 13 1: = , =k k k k→ + + + +k k k k (22) in the second term. Therefore, only one integration, say with respect to 2 ,k remains in each term. In the IR region 2 1k k k , we find from Eqs. (21), (22) for the 1 3→ and the 3 1→ terms: 2 2 2 2 1 1 2 1 3 : = , k k → − ≈ − + k k k k k k 1 2 3 2 1 2 1 3 : = ,→ − ≈ − + k k k k k k (23) 2 2 2 2 1 2 3 1: = , k k → + ≈ + + k k k k k k 2 3 2 2 3 1: = .→ − ≈ − + kk k k k k (24) Weak turbulence of Kelvin waves in superfluid He Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9 991 These equations demonstrate three important facts: 1) in both cases in the leading order 3 2 ,k k i.e., when 2k k then 3k is small as well; 2) the difference between 1k and k is of the second order in small 2k : 2 1 2\| \| / ;k k−k k 3) these leading contributions to 1 −k k have the same modulus and different sign in the 1↔3 term and in the 3↔1 term. Therefore in the leading order the expressions for N in Eq. (8) can be written as 3 2(1 )1,2,3 2 2 2 3 2( 2)( / ) ,x kk x x Ax k k n n n k k − + − −N (25) 3 2(1 ),2,3 2 2 2 31 2( 2)( / ) ,xk k x x Ax k k n n n k k − + + +N (26) where we substituted jn from Eq. (10). Importantly, these estimates (in the leading order) have the same magnitude and different signs. Next step is to compute integrals 1 3 1 3 1 2 3( )I d d→ ≡ δ − − − ×∫ k k k k k k 2 2 2 2 2 1 2 3 2 2 2 2 \| \| 1( ) = ; 22 \| 2 \| k k k k kk k + ×δ − − − → + − k k kk (27) 3 1 1 3 2 3 1( )I d d→ ≡ δ + + − ×∫ k k k k k k 2 2 2 2 2 3 1 1 1 1( ) = , 2 \| \| 2 k k k k k ×δ + + − → +k k (28) i.e., in the leading order these results coincide and do not contain the smallness. Now we can find the contributions to St1 3↔ , given by Eq. (8), from the region 2 .k k According to Eq. (19) we can write 1,2,3 ,2,3 1 2 31= = .k kV V Vkk k k Using our esti- mates (25), (26) for N and Eqs. (27), (28) we have 2 3 2(3 )2 21 3 211 3 : St ; 24 k k x x x V A k d k − → − π → ≈ − ∫ k (29) 2 3 2(3 )2 23 1 21 33 1: St . 24 k k x x x V A k d k − → − π → ≈ + ∫ k (30) One can see that, in spite of the deep cancelations in the estimates for ,N the integrals (29), (30) diverge if 3.5x ≥ , which is satisfied for LN-scaling exponent = 11/ 3.x Nevertheless on has to take into account the following: the 1↔3 contribution to the collision integral has three identical divergent regions: 2 3 1k k k k≈∼ , 1 3 2k k k k≈∼ and 2 1 3k k k k≈∼ , and Eq. (29) estimates only the first one. Therefore the total contribu- tion is 2 3 2(3 )2 1 3 21 3 21 3St = 3St , 24 IR k k x x x V A k d k − → → − π ≈ − ∫ k (31) while the 3↔1 contribution has only one divergent region 1 .≈k k Therefore, 2 3 2(3 )2 1 3 21 3 21 3St = St , 24 IR k k x x x V A k d k − → → − π ≈ + ∫ k (32) i.e., exactly the same result as in Eq. (29), but with the dif- ferent sign. Therefore the divergent contributions (29), (30) cancel each other and one has to take into account the next order. Notice that next order terms in the expansion over 2k k results in the already convergent integral (6 2 ) 1 3 2 22 0 St , IRk k IR xk d− ↔ ∝ ∫ k k (33) with the LN exponent = 11/ 3x . Moreover, typically exci- tation of KWs is symmetrical in ↔−k k . In this case, this integral has an odd integrand and, therefore, it is equal to zero. Then the leading contribution to the 1↔3-collision term in the IR region can be summarized as follows: 2 3 9 22(4 ) 1 3 221 0 St . IRk k IR xx IRx V A k d k k −− ↔ + ∝∫ k∼ (34) The IR convergence require: < 9 / 2 .x With LN exponent = 9 / 3x this gives 1 3 5 / 3St .IR IR IR IRk k→ δ∝ ≡ Here we introduce an «IR convergence reserve»: = 5 / 3.IRδ 6.2. Proof of the ultraviolet convergence Convergence of the integral (8) in the UV region, when one of the wavevectors, say 2k k , can be established in a similar manner. Notice first of all that in the 1 3→ term in Eq. (8), there is no UV region, because by the 2nd of Eq. (21) we have jk k≤ . In the 3 1→ term to satisfy Eq. (22) in the leading order we can take 2 1 2; UVk k k≥k k (case 3 1 3; UVk k≥k k gives an identical result). Using para- metrization 2 1 2 2 3 2 2= / ( ), = / ( )k+ + − +k k k k k kk k k (cf. (26)) we get some cancelations in ,2,3 1 kN and the leading order result is 2 2 ,2,3 2 2 1 ( 1) . x x k k k x x k k − − − ⎛ ⎞ ⎛ ⎞∝ + −⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ N (35) Victor S. L’vov and Sergey Nazarenko 992 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9 Further, similarly to Eqs. (27), (28), one gets 3 1 21/ .I k→ As before, the interaction coefficient 2 2V k∝ or 2 4 2V k∝ . Counting the powers of 2k one gets: 1 3St , = max( 2 4, 2) . UV UV yk y x x↔ ∝ − + − + (36) The UV convergence require: < 0 > 2 .y x⇒ One concludes that in the case = 11/ 3,x 1 3 5 / 3St ,UV UV UV UVk k→ −δ−∝ ≡ where we introduce an «UV convergence reserve» = 5 / 3 .UVδ 6.3. Counterbalanced interaction locality Notably, = .IR UVδ δ This equality is not occasional. Observed «counterbalanced» IR-UV locality is a conse- quence of the scale-invariance of the problem. Indeed, for a given values of IR UVk k k the IR-energy flux IRk k⇒ (from the IR region IRk k≤ toward the region k∼ ) should scale with ( / )IRk k exactly in the same manner as the UV-energy flux UVk k⇒ (from the -regionk toward the UV-region UVk k≤ ) scales with / UVk k . This is because the UV-flux UVk k⇒ from -regionk can be considered as the IR flux toward -region.UVk Remembering that the IR-energy flux IRk k⇒ scales like ( / ) ,IR IRk k δ while the UV-flux UVk k⇒ is proportional to ( / ) ,UVUVk k δ one immediate- ly concludes that IRδ should be equal to UVδ . The overall conclusion is that the collision term St1 3↔ is convergent in both the IR and the UV regions for = 11/ 3x and the energy transfer in the 1↔3 kinetic equa- tion is local. 7. Discussion — In this paper we have revised the theory of superflu- id turbulence in the quantum range of scales where the turbulent cascade is due to nonlinear interaction of weak Kelvin waves on quantized vortex lines. In particular, we have addressed the problem that the previously used KS spectrum is nonlocal, i.e., an invalid mathematically and irrelevant physically solution. — We have presented a new effective theory of Kelvin wave turbulence consisting of wave quintets interacting on vortex lines with random large-scale curvature. This four- wave theory replaces the nonlocal six-wave theory. We derived an effective four-wave kinetic Eqs. (8), (19), and solved it to obtain a new wave spectrum (20). We proved that this new spectrum is local, and therefore it is a valid solution of the kinetic equation, which should replace the nonlocal (and therefore invalid) Kosik–Svistunov spectrum (15) in the theory of quantum turbulence. In particular, it is now necessary to revise the theory of the classical- quantum crossover scales and its predictions for the turbu- lence dissipation rate Refs. 17–20. Further, a similar revi- sion is needed for the analysis of laboratory experiments and numerical simulations of superfluid turbulence, which have been done over the last five years with reliance on the un-physical KS spectrum (15). — The difference between the LN-exponent –5/3 (see (20)) from the KS-exponent –7/5 (see (15)) is 4/15 which is rather small. This may explain why the previous numeri- cal experiments seem to agree with the KS spectrum, ob- tained numerically in Ref. 15. However, by inspection one can also see that these results also agree with the LN slope. The different physics results in different expressions for the dimensional pre-factors in the KS and LN spectra, in particular the different dependence on the energy flux ,ε as well as an extra dependence on the large-scale behavior (through Ψ ) in (20). Careful examination of such pre- factors is necessary in future numerical simulations in or- der to test the predicted dependencies. Such numerical si- mulations can be done efficiently with the local nonlinear equation (LNE) suggested in Ref. 14 based on the detailed analysis of the nonlinear KW interactions: 4 1 = 0. 4 4 w w wi t z z z ⎧ ⎫⎡ ⎤∂ κ ∂ ∂ ∂⎪ ⎪⎢ ⎥+ Λ −⎨ ⎬⎢ ⎥∂ π ∂ ∂ ∂⎪ ⎪⎣ ⎦⎩ ⎭ (37) The LNE model is similar but not identical to the Trun- cated LIA model of [15] (these models become asymptoti- cally identical for weak KWs). — Both pre-factors in the KS spectrum (15) and in the LN spectrum (20) contain very different numerical con- stants C: an order one constant in LN ( LN 1C ∼ , yet to be found) and a zero constant in KS ( KS 0C ≡ as a formal consequence of its nonlocality). Also we should note a mysterious very small numerical factor 510− in formula (16) for the energy flux in Ref. 13, that has no physical justification. Actually, nonlocality of the energy transfer over scales means that this number should be very large, rather than very small. This emphasizes the confusion, and highlights the need for numerical re-evaluation of the spec- trum’s prefactor. To conclude comparison between KS and LN approach notice that the drastic difference in the numerical pre- factors constitutes an important difference between the KS and the LN spectra for a practical analysis of experimental date, while the difference between the underlying physics of the local and nonlocal energy cascades, that results in the difference between spectral indices, is important from fundamental, theoretical viewpoint. — In this work, the effective local five-wave kinetic equation was derived from the six-wave kinetic equation by exploiting nonlocality of the latter. Strictly speaking, this derivation is valid only when the six-wave kinetic equ- ation is valid, i.e., when all the scales are weakly nonlinear, including the ones at the infra-red cutoff. However, the resulting five-wave kinetic equation is likely to be applica- ble more widely, when only the small scales, and not the large scales, are weak. A similar picture was previously Weak turbulence of Kelvin waves in superfluid He Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 8/9 993 observed for the nonlocal turbulence of Rossby/drift waves in Ref. 22 and for nonlocal MHD turbulence in Ref. 23. In future we plan to attempt derivation of the five-wave kinet- ic equation directly from the dynamical equations for the Kelvin waves, which would allow us to extend its applica- bility to the case with strong large scales. — Finally we note that the suggested here theory can potentially be useful for other one-dimensional physical systems, including optical fibers, where nonlinear interac- tions of one-dimensional wave packages becomes impor- tant with increase in network capacity. Acknowledgments We thank J. Laurie and O. Rudenko for help in the evaluation of the effective interaction coefficient. We ac- knowledge support of the US-Israel Binational Scientific Foundation administrated by the Israeli Academy of Science, and of the EC — Research Infrastructures under the FP7 Capacities Specific Programme, MICROKELVIN project number 228464. 1. R.J. Donnelly, Quantized Vortices in He II, Cambridge University Press, Cambridge (1991). 2. W.F. Vinen and J.J. Niemela, J. Low Temp. 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Weak turbulence of Kelvin waves in superfluid He

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