Pis’ma v ZhETF, vol. 111, iss. 10, pp. 705 - 706

May 25

Comment on “Amplitude of waves in the Kelvin-wave cascade”

(Pis’ma v ZhETF 111, 462 (2020))

E. B. Sonin¹⁾

Racah Institute of Physics, Hebrew University of Jerusalem, Givat Ram, 9190401 Jerusalem, Israel

Submitted 9 April 2020

Resubmitted 9 April 2020

Accepted 16

April 2020

DOI: 10.31857/S1234567820100109

Eltsov and L’vov [1] derived the relation between

However, L’vov and Nazarenko denied not only sym-

the amplitude of Kelvin waves and the energy flux in

metry arguments, but also the assumption of locality.

the Kelvin-wave cascade. This returns us to the rather

Since they believed that the Kelvin mode-mode inter-

old, but still unresolved dispute on the role of the tilt

action must depend on the tilt of the vortex line, they

symmetry and the locality in the Kelvin-wave cascade

concluded that the interaction vertices in the Boltzmann

(see Sec. 14.6 of the book [2] for references).

equation are determined by divergent integrals and the

Kozik and Svistunov [3] investigated the Kelvin wave

locality assumption is invalid. Meanwhile, Eq. (1), as

cascade using the Boltzmann equation for the Kelvin

well as its particular case Eq. (2), was derived assum-

modes. They took into account the weak 6-waves inter-

ing locality. Instead of Eq. (2), the nonlocal scenario of

action and used the locality condition similar to that

L’vov and Nazarenko yields [6]

in the classical Kolmogorov cascade: the energy flux in

κ²Λ

( ǫ

the space of wave numbers k depends only on the en-

E_k ∼

³ k^-5/3.

(3)

Ψ^2/3

κ³

ergy density at k of the same order of magnitude. L’vov

and Nazarenko [4] challenged their analysis arguing that

Here the dimensionless parameter

the cascade is connected to the 4-wave interaction de-

∫

∞

spite the latter breaks the rotational invariance and does

Ψ∼

E_k dk

(4)

depend on the tilt of the vortex line with respect to

Λκ²

kmin

some direction. In the general case of the n-wave inter-

action the expression connecting the energy flux ǫ in the

takes into account the effect of nonlocality since it is an

k space and the the energy density E_k is [2, 5]

integral over the whole Kelvin-wave cascade interval in

)

the k space. The lower border of this interval is k_min.

( ǫ

E_k ∼ κ²Λ

^n-1 k-n−1 .

(1)

From Eqs. (3) and (4) one obtains that

κ³

(

)

Here κ is the circulation quantum and Λ = ln^ℓ is the_a

Ψ∼

⁵ .

(5)

κ³k²

large logarithm, which depends on the ratio of the in-

min

tervortex distance or the vortex line curvature radius ℓ

So the nonlocality does not affect the dependence on k

and the vortex core radius a₀. We use notations of Eltsov

but does change the dependence on the energy flux ǫ.

and L’vov [1] and their energy normalization. Here and

The outcome of the nonlocal scenario is not clear

further on we ignore all numerical factors in our expres-

without an evaluation of the minimal wave number k_min.

sions as not important for our qualitative analysis.

In the theory of quantum turbulence k_min is the wave

√

At n = 6 Eq. (1) gives the spectrum E_k ∝ k^-7/5 of

number

L, at which the crossover from the classical

Kozik and Svistunov [3], while at n = 4 one obtains

Kolmogorov cascade to the Kelvin-wave cascade occurs.

)

Here L is the vortex line length per unit volume in Vi-

( ǫ

E_k ∼ κ²Λ

³ k^-5/3.

(2)

nen’s theory of the 3D vortex tangle. On the other hand,

κ³

in agreement with Eltsov and L’vov [1], the parameter

Ψ determines also the ratio of the vortex line length in-

This agrees with the spectrum E_k ∝ k^-5/3 of L’vov and

creased by the Kelvin waves participating in the cascade

Nazarenko [4].

to the length of the straight vortex in the ground state.

¹⁾e-mail: sonin@mail.huji.ac.il

The crossover is determined by the condition that this

Письма в ЖЭТФ том 111 вып. 9 - 10

2020

705

706

E. B. Sonin

ratio is on the order of unity [2]. If Ψ ∼ 1 the energy

pendently. All scenarios of the Kelvin-wave cascade dis-

density Eq. (3) obtained from the nonlocal scenario does

cussed above used the theory of weak turbulence valid

not differ from the energy density Eq. (2) derived under

strictly speaking only if Ψ ≪ 1. If Ψ ≫ 1 the turbu-

the assumption of locality. Maybe the reason for the in-

lence is strong and the spectrum is given by Eq. (1) at

sensitivity of the Kelvin-wave cascade to nonlocal effects

n → ∞. This is the spectrum E_k ∼ 1/k predicted by

deserves a further investigation, but at least it is pre-

Vinen et al. [9]. However, the condition Ψ ∼ 1 should

mature to discard the locality assumption as physically

be imposed on the simulation parameters if one wants

irrelevant.

to reach better imitation of processes in the 3D vortex

Also I would like to comment the statement of Eltsov

tangle.

and L’vov [1] that “finally the L’vov-Nazarenko model

In summary: (i) The analysis of Eltsov and L’vov [1]

got supported by numerical simulations [7, 8]”. This can

demonstrates that the possible nonlocality of the energy

be interpreted as persisting on the previous claims of the

flux in the Kelvin-wave cascade has no essential effect

proponents of the L’vov-Nazarenko scenario that the

on the Kelvin-wave cascade in the 3D vortex tangle ex-

scenario is universal despite it breaks the tilt symmetry.

pected by L’vov and Nazarenko. (ii) There is no conflict

The author of the present Comment thinks that it is a

between the Kozik-Svistunov and the L’vov-Nazarenko

bad idea to check the laws of symmetry experimentally

scenarios. They are valid for different external condi-

or by numerical simulations. If an experiment or a nu-

tions.

merical simulation is in conflict with the symmetry law

I thank Vladimir Eltsov and Victor L’vov for useful

(e.g., the energy conservation law based also on symme-

remarks.

try) the experiment or the simulation must be reconsid-

ered, but not the other way around. Suppose that there

1. V. B. Eltsov and V. S. L’vov, Pis’ma v ZhETF 111, 462

is a theory based on the crystal cubic symmetry, but

(2020) [JETP Lett. 111 (2020), to be published].

they do experiments with parallelepiped samples. Dis-

2. E. B. Sonin, Dynamics of quantised vortices in superflu-

agreement with the original theory does not mean that

ids, Cambridge University Press, Cambridge (2016).

the theory must be discarded. This does mean that one

3. E. Kozik and B. Svistunov, Phys. Rev. Lett. 92, 035301

should do experiments at the conditions when the sam-

(2004).

ple shape is not important, e.g., at spatial scales much

4. V. S. L’vov and S. Nazarenko, Pis’ma v ZhETF 91, 464

less than the sample size. In numerical simulations [7, 8]

(2010) [JETP Lett. 91, 428 (2010)].

the tilt symmetry was broken since the simulations dealt

5. E. B. Sonin, Phys. Rev. B 85, 104516 (2012).

with the vortex stretched between two parallel surfaces.

6. L. Boué, R. Dasgupta, J. Laurie, V. L’vov, S. Nazarenko,

Probably the spectrum compatible with the tilt invari-

and I. Procaccia, Phys. Rev. B 84, 064516 (2011).

ance could be observed at shorter scales (larger wave

7. G. Krstulovic, Phys. Rev. E 86, 055301 (2012).

numbers k).

8. A. W. Baggaley and J. Laurie, Phys. Rev. B 89, 014504

In contrast to the case of the 3D vortex tangle, in

(2014).

numerical simulations of the Kelvin-wave cascade in a

9. W. F. Vinen, M. Tsubota, and A. Mitani, Phys. Rev.

straight vortex, the condition Ψ ∼ 1 is not obligatory,

Lett. 91, 135301 (2003).

since k_min and the energy flux ǫ can be chosen inde-

Письма в ЖЭТФ том 111 вып. 9 - 10

2020