Pis’ma v ZhETF, vol. 111, iss. 7, pp. 462 - 463

April 10

Amplitude of waves in the Kelvin-wave cascade

V. B. Eltsov⁺¹⁾, V. S. L’vov^∗

⁺Department of Applied Physics, Aalto University, POB 15100, FI-00076 AALTO, Finland

^∗Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel

Submitted 5 March 2020

Resubmitted 5 March 2020

Accepted 5 March 2020

DOI: 10.31857/S0370274X20070061

In quantum turbulence, velocity fluctuations and

∑

E_kw =

ν_sL_k = L

ν_sA^2kk² =

vortex reconnections drive oscillating motion of quan-

k=±kmin

k=kmin

tized vortices - Kelvin waves [1]. Kelvin waves inter-

∫ ∞

ν_s

act non-linearly and support a cascade of energy to-

A^2kk² dk.

(3)

wards smaller length scales and larger wave numbers [2].

min kmin

The theory of the Kelvin-wave cascade was the subject

Comparing this result to the expression of the energy

of controversy, until finally the L’vov-Nazarenko model

via the Kelvin-wave frequency ω_k and the combined oc-

[3, 4] got supported by numerical simulations [5, 6]. Re-

cupation number N_k for modes with ±k [4]

cently, progress in experimental techniques [7-9] enables

∫ ∞

controllable excitation of waves on nearly straight vor-

κΛ

E_kw = ρ_sL

E_k dk, E_k = ω_kN_k, ω_k =

k²,

tices and potential observation of the Kelvin-wave cas-

4π

kmin

cade. In this work we provide relation of the energy flux

(4)

carried by the cascade to the amplitude of the excited

we find

k_min

Kelvin waves, which is important for analysis of such

A^2k =

N_k.

(5)

experiments.

The L’vov-Nazarenko spectrum is [4]

We assume that the Kelvin-wave cascade on a vortex

of length L (cm) carries the energy flux ǫ (erg/s) and

κΛǫ^1/3

starts from the wave number k_min (cm^-1). Our goal is

E_k = C_LN

C_LN ≈ 0.304,

(6a)

Ψ^2/3k^5/3

to find the amplitude A_k (cm) of the Kelvin wave with

∫ ∞

8π

the wave number k (cm^-1). We start by noting that in

Ψ=

E_kdk.

(6b)

Λκ²

kmin

the local induction approximation the energy of a vor-

tex line E_v is given by the product of its length L and

Here ǫ is the energy flux per unit length and per unit

the vortex tension ν_s

mass. It is related to the flux ǫ as

)

cm⁴

κ²Λ

( ℓ

ǫ=

[ǫ] =

(7)

E_v = ν_sL, ν_s = ρ_s

Λ = ln

(1)

Lρ_s

s³

4π

a₀

Solving Eq. (6) for Ψ we get

Here ρ_s is the superfluid density, κ is the circulation

quantum, a₀ is the vortex core radius and ℓ is the mean

(12πC_LN)^3/5ǫ^1/5

intervortex spacing or the size of the enclosing volume,

Ψ=

(8)

κ^3/5k^2/5

min

in the case of a single vortex. For a spiral Kelvin wave of

the radius A_k and wavelength λ_k = 2π/k, the increase

and from Eq. (5) finally

of the length compared to that of the straight vortex is

(√

)

(2π³C^3LN )^1/5 k^19/15minǫ^1/5

A^2k = 2

≈

2π²A^2k

κ^3/5k^11/3

L_k = λ^2k + (2πA_k)² - λ_k

≈L

(2)

λ_k

λ²

(

)_1/5

19/15

min

≈ 1.4

(9)

where we assumed that A_k ≪ λ_k. Thus the total energy

κ^3/5k^11/3 Lρ_s

due to Kelvin waves is

Note that A_k ∝ ǫ^1/10. Thus determination of the ampli-

¹⁾e-mail: vladimir.eltsov@aalto.fi

tude from the energy flux should be relatively reliable,

462

Письма в ЖЭТФ том 111 вып. 7 - 8

2020

Amplitude of waves in the Kelvin-wave cascade

463

while the reverse procedure is bound to be very uncer-

Together with Eq. (11) this results in

tain.

L_kw

The total increase of the vortex line length due to

〈tan² θ(z)〉 ≃ 2

= Ψ,

(15)

Kelvin waves can be found from the energy as L_kw =

= E_kw/ν_s, where E_kw is given by Eqs. (4), (6a) and

where Ψ is given by Eq. (8).

(8):

To conclude, we have found the dependence of the

1/5

E_kw

2^1/5(3πC_LN)^3/5ǫ

amplitude of the Kelvin waves, of the length increase of

L_kw =

(10)

ν_s

the vortex, and of the average vortex tilt on the energy

κ^3/5k^2/5

min

flux carried by the Kelvin-wave cascade. The results are

Thus for the relative increase we get a simple formula

applicable in the regime of weak turbulence of Kelvin

L_kw

E_kw

(11)

waves, which is uniform along the vortex. We stress that

E_v

the amplitude of the Kelvin waves, generated when a

In cases, where instead of a single vortex, one con-

vortex is mechanically agitated, does not necessary co-

siders a vortex array with the total length L occupy-

incide with the amplitude of the motion of the agitator.

ing volume V with the density L = L/V

= ℓ^-2, it

Solving the problem of excitation of Kelvin waves in a

might be more convenient to operate with the standard

realistic experimental geometry remains a task for fu-

3-dimensional energy flux ε = ǫL per unit mass and unit

ture research.

volume, [ε] =cm² s^-3. Then for the increase L_kw of the

The work has been supported by the European

vortex-line density due to Kelvin waves, we find using

Research Council (ERC) under the European Union’s

Eqs. (8) and (11)

Horizon

2020

research and innovation programme

(Grant Agreement # 694248).

)_1/5

L_kw

[2(3πC_LN)³ε]1/5

( ε

≈ 2.2

Full text of the paper is published in JETP Letters

b²L²κ³

(12)

journal. DOI: 10.1134/S0021364020070012

where we introduced

b = k_minℓ ∼ 1.

(13)

1. W. F. Vinen, Philos. Trans. R. Soc. A 366, 2925 (2008).

We note that the numerical value of the prefactor in

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

Eqs. (9) and (12) should be taken with caution. In the

(2004).

calculations we assume that the total energy of Kelvin

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

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

waves can be found by the integral (4) limited from be-

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

low by k_min with the scale-invariant spectrum (6). In

S. V. Nazarenko, and I. Procaccia, Phys. Rev. B 84,

reality this spectrum was derived for k ≫ k_min while

064516 (2011).

the main contribution to E_kw is coming from the re-

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

gion k ≃ k_min. Behavior of the Kelvin-wave spectrum

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

in this long-wavelengths region may be different and, in

(2014).

general, is not universal.

7. A. M. Guénault, A. Guthrie, R. P. Haley, S. Kafanov,

In some applications, the tilt θ of a vortex carry-

Yu. A. Pashkin, G. R. Pickett, M. Poole, R. Schanen,

ing Kelvin waves with respect to the direction of the

V. Tsepelin, D.E. Zmeev, E. Collin, O. Maillet, and

straight vortex is of interest. The averaged tilt angle

R. Gazizulin, Phys. Rev. B 100, 020506(R) (2019).

can be connected to the length increase

8. C. S. Barquist, W. G. Jiang, P. Zheng, Y. Lee, and

∫L

√

H. B. Chan, J. Low Temp. Phys. 196, 177 (2019).

L_kw =

1 + tan2 θ(z)dz-L ≃

〈tan² θ(z)〉L. (14)

9. T. Kamppinen and V. B. Eltsov, J. Low Temp. Phys.

196, 283 (2019).

Письма в ЖЭТФ том 111 вып. 7 - 8

2020