Comment on "Amplitude of waves in the Kelvin-wave cascade"
aa r X i v : . [ c ond - m a t . o t h e r] J u l Comment on “Amplitude of waves in the Kelvin-wave cascade”
E. B. Sonin
Racah Institute of Physics, Hebrew University of Jerusalem, Givat Ram, Jerusalem 9190401, Israel (Dated: July 6, 2020)
Eltsov and L’vov [1] derived the relation between theamplitude of Kelvin waves and the energy flux in theKelvin-wave cascade. This returns us to the rather old,but still unresolved dispute on the role of the tilt sym-metry and the locality in the Kelvin-wave cascade (seeSec. 14.6 of the book [2] for references).Kozik and Svistunov [3] investigated the Kelvin wavecascade using the Boltzmann equation for the Kelvinmodes. They took into account the weak 6-waves in-teraction and used the locality condition similar to thatin the classical Kolmogorov cascade: the energy flux inthe space of wave numbers k depends only on the energydensity at k of the same order of magnitude. L’vov andNazarenko [4] challenged their analysis arguing that thecascade is connected to the 4-wave interaction despitethe latter breaks the rotational invariance and does de-pend on the tilt of the vortex line with respect to somedirection. In the general case of the n -wave interactionthe expression connecting the energy flux ǫ in the k spaceand the the energy density E k is [2, 5] E k ∼ κ Λ (cid:16) ǫκ (cid:17) n − k − n +1 n − . (1)Here κ is the circulation quantum and Λ = ln ℓa is thelarge logarithm, which depends on the ratio of the inter-vortex distance or the vortex line curvature radius ℓ andthe vortex core radius a . We use notations of Eltsov andL’vov [1] and their energy normalization. Here and fur-ther on we ignore all numerical factors in our expressionsas not important for our qualitative analysis.At n = 6 Eq. (1) gives the spectrum E k ∝ k − / ofKozik and Svistunov [3], while at n = 4 one obtains E k ∼ κ Λ (cid:16) ǫκ (cid:17) k − / . (2)This agrees with the spectrum E k ∝ k − / of L’vov andNazarenko [4].However, L’vov and Nazarenko denied not only sym-metry arguments, but also the assumption of locality.Since they believed that the Kelvin mode-mode inter-action must depend on the tilt of the vortex line, theyconcluded that the interaction vertices in the Boltzmannequation are determined by divergent integrals and thelocality assumption is invalid. Meanwhile, Eq. (1), aswell as its particular case Eq. (2), was derived assuminglocality. Instead of Eq. (2), the nonlocal scenario of L’vovand Nazarenko yields [6] E k ∼ κ ΛΨ / (cid:16) ǫκ (cid:17) k − / . (3) Here the dimensionless parameterΨ ∼ κ ∞ Z k min E k dk (4)takes into account the effect of nonlocality since it is anintegral over the whole Kelvin-wave cascade interval inthe k space. The lower border of this interval is k min .From Eqs. (3) and (4) one obtains thatΨ ∼ (cid:18) ǫκ k min (cid:19) . (5)So the nonlocality does not affect the dependence on k but does change the dependence on the energy flux ǫ .The outcome of the nonlocal scenario is not clear with-out an evaluation of the minimal wave number k min .In the theory of quantum turbulence k min is the wavenumber √L , at which the crossover from the classicalKolmogorov cascade to the Kelvin-wave cascade occurs.Here L is the vortex line length per unit volume in Vi-nen’s theory of the 3D vortex tangle. On the other hand,in agreement with Eltsov and L’vov [1], the parameterΨ determines also the ratio of the vortex line length in-creased by the Kelvin waves participating in the cascadeto the length of the straight vortex in the ground state.The crossover is determined by the condition that thisratio is on the order of unity [2]. If Ψ ∼ k ).In contrast to the case of the 3D vortex tangle, innumerical simulations of the Kelvin-wave cascade in astraight vortex, the condition Ψ ∼ k min and the energy flux ǫ can be chosen inde-pendently. All scenarios of the Kelvin-wave cascade dis-cussed above used the theory of weak turbulence validstrictly speaking only if Ψ ≪
1. If Ψ ≫ n → ∞ .This is the spectrum E k ∼ /k predicted by Vinen et al. [9]. However, the condition Ψ ∼ [1] V. B. Eltsov and V. S. L’vov, JETP Letters (2020).[2] E. B. Sonin, Dynamics of quantised vortices in superfluids (Cambridge University Press, 2016).[3] E. Kozik and B. Svistunov, Phys. Rev. Lett. , 035301(2004).[4] V. S. L’vov and S. Nazarenko, Pis’ma Zh. Eksp. Teor. Fiz. , 464 (2010), [ JETP Lett. , , 428–434 (2010)].[5] E. B. Sonin, Phys. Rev. B , 104516 (2012).[6] L. Bou´e, R. Dasgupta, J. Laurie, V. L’vov, S. Nazarenko,and I. Procaccia, Phys. Rev. B , 064516 (2011).[7] G. Krstulovic, Phys. Rev. E , 055301 (2012).[8] A. W. Baggaley and J. Laurie, Phys. Rev. B , 014504(2014).[9] W. F. Vinen, M. Tsubota, and A. Mitani,Phys. Rev. Lett.91