shoji fujiyama and makoto tsubota- vortex line density fluctuations of quantum turbulence

8/3/2019 Shoji Fujiyama and Makoto Tsubota- Vortex line density fluctuations of quantum turbulence

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Vortex line density uctuations of quantum turbulence

Shoji Fujiyama · Makoto Tsubota

Received: date / Accepted: date

Abstract We investigate the vortex line density uctuations of quantum turbulence. The

scenario of quantum turbulence experimentally suggested by the Lancaster group is con- rmed in the numerical simulation. The spectrum of the vortex line density uctuations withrespect to frequency obeyed a -5/3 power law, which is consistent with the experiment of the Lancaster group. Based on the argument of time scales experienced by vortex rings withdifferent sizes and on the power spectrum, the connection between self-similar structure of the vortex tangle and the power spectrum is discussed.

Keywords numerical simulation · vortex line density · uctuations · power spectrum

PACS 67.30.he · 67.30.hb · 47.32.C- · 47.37.+q

1 Introduction

The universality of quantum turbulence has attracted considerable interests in recent years.Super uid, in which quantum turbulence exists, has special hydrodynamic features suchas lack of viscosity and quantization of vorticity. The former one provides the differentmechanism for the transition from laminar to turbulent state. The latter one distinguishesquantum turbulence from classical turbulence in the way that the vorticity can be de nedwithout ambiguity, and simpli es the mathematical descriptions of turbulence properties.Beyond these hydrodynamic differences, however, quantum and classical turbulence areconnected with universal laws. For example, Kolmogorov law, a statistical law found inclassical turbulence, has been con rmed in quantum turbulence both experimentally andnumerically [1–5]. Furthermore, it is con rmed that a drag force on an oscillating objectthat creates quantum turbulence exhibits the similarity with the classical one [6,7].

Recently, the experiments on quantum turbulence, which suggests another possibility of the universality, were conducted. The Lancaster group [8] measured the vortex line density uctuations of quantum turbulence in super uid 3He at ultimately low temperatures where

Shoji Fujiyama · Makoto TsubotaDepartment of Physics, Osaka City University, Sumiyoshi-Ku, Osaka 558-8585, JapanE-mail: [email protected]



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mutual friction can be neglected. They showed that the spectrum of the line density uc-tuations | L( f )|2 with respect to frequency obeys a power law with exponent -5/3. On theother hand, Roche et al. [9] performed the second sound measurement of the vortex linedensity uctuations in super uid 4He at relatively high temperatures where the normal uid

component signi

cantly remains. They also con

rmed the spectrum of the line density

uc-tuations obeys the same power law. At such high temperatures, the turbulences in super uidand the normal uid components are tightly locked by mutual friction in length scale largerthan the mean vortex line spacing of vortex tangle. It is remarkable that the same powerspectrum was obtained in different temperature regimes, and these experiments suggest thata universal feature is underlying over these turbulences.

Taking exponent -5/3 as a key feature for the universality, in this paper, we report thenumerical simulation supporting the experiments of the Lancaster group [8,10,11], and drawsome conclusions from the power law. Section 2 consists of three parts: the model describingthe experiments of the Lancaster group, the formulation for calculating the vortex dynamics,and the obtained results. In Sect. 3, the power law is linked with a self-similar structure of the vortex tangle, based on the vortex ring emission from the vortex tangle.

2 Numerical Simulation

Model: We start from modelling the experiments of the Lancaster group [8,10,11]. Theycreated quantum turbulence by transversely oscillating a grid at T

∼0.2 Tc where a normal

uid component is negligible. According to [12], remnant vortices are pinned on the grid. Anoscillatory ow around the grid induces Kelvin waves on the remnant vortices when the fre-quency matches the resonance of the vortices, resulting in the ampli cation of Kelvin waves.The elongated vortices nally reconnect to themselves and produce vortex rings, leaving theoriginal vortices on the grid. The production of vortex rings continues as long as the gridkeeps oscillating. It is expected that the production rate of the vortex rings is proportional tothe grid velocity. At low grid velocity, the vortex rings propagate independently and causeno turbulence. At high grid velocity, in contrast, the vortex rings are no longer independent,and collide with and reconnect to each other. The reconnection of two rings forms a larger

ring with low propagation velocity. The larger vortex ring intercepts subsequent rings, and nally forms a vortex tangle.

First we con rm this scenario in numerical simulation and establish the equilibriumstate of the vortex tangle. The scheme of the simulation is as follows. We set a numericalcell with periodic boundaries in transverse directions and with open ends in longitudinaldirection. The cross section of the cell is 200 µ m × 200 µ m and the length is 600 µ m. Tomimic the production of vortex ring, we inject vortex ring from the left side of the cell (seeFig. 1). The size of the injected rings is 20 µ m in diameter, which is estimated from thedispersion relation of Kelvin wave using the grid oscillation frequency

∼1300 Hz. They are

injected at random position and at random angle within a∼

20◦ cone around the forwarddirection. The time interval ! of the injection is 2 ms.

Formulation: Vorticity in B phase of super uid 3He is quantized with the circulation quan-

tum " = h/ 2m3 , where h is Planck’s constant and m3 is mass of 3He atom. Core size a0 of a quantized vortex is

∼10 nm, which is far smaller than any length scale appearing in this

paper, thus we accept the vortex lament model. Since the experiments were performed atlow temperatures enough to neglect normal component, we omit the contribution of mutualfriction. In the absence of mutual friction, a vortex moves with super uid, so the vortex



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Fig. 1 Snapshot of the simulation. t = 20 mson the top, t = 250 ms on the middle, andt = 500 ms on the bottom

108

109

0 1 2

v o r t e x

l i n e

l e n g

t h d e n s i t y

( m - 2 )

Time from switching on grid (s)

Fig. 2 Vortex line length density

dynamics is equivalent to calculating the super uid velocity at every point on the vortex. Inthe present case, super uid velocity consists purely of the velocity produced by the vorticesthemselves. The velocity on a position rrr by the vortex laments is described in the form of the Biot-Savart integration

vvv# (rrr) ="

4$ L

(sss111 − rrr ) ×dsss111

|sss111 −rrr |3, (1)

where sss111 indicates a position vector on a vortex, and the integral is performed all over thevortices L . The velocity eld in (1) diverges when sss111 →rrr since we adopt the vortex lamentmodel. To evade this dif culty, we take cut-off parameter a0 . Calculating the velocity of eachpoint on the vortices leads us to follow the trajectory of the vortices.

It is known that two vortices reconnect when they become close and cross each other[13]. In the vortex lament approximation, however, the structure of the vortex core is ne-glected, so such a process needs to be handled manually. The reconnection process is in-cluded in our numerical code in the way that two vortices reconnects when they becomeclose within the numerical resolution [14]. Because of the numerical resolution, Kelvinwaves of wavelength smaller than the resolution is neglected, and vortex rings smaller thanthe resolution are removed from the dynamics.

Result: Figure 1 is the time evolution of the vortex dynamics. As soon as several vortexrings are injected, they reconnect to themselves, forming larger rings (top of Fig. 1). Largervortex rings with the low propagation velocity trap subsequent vortex rings (middle of Fig.1), ending up with vortex tangle (bottom of Fig. 1). The vortex line density L(t ) is plottedas a function of time in Fig. 2. After 0.5 s, the vortex line density reaches a statistically

equilibrium state at 6 ×108

m−2. Looking at the line density of the experiment 1 ×10

8m−

2

[11], we can conclude the vortex tangle in numerical simulation is established as dense asthat of the experiments.

Since we obtain an equilibrium state from 0.5 s to 1.5 s, we conduct Fourier transformof the vortex line density with respect to frequency (Fig. 3). The spectrum is de ned as



4

108

109

1010

1011

1012

1013

1014

1015

1 10 100 1000

L i n e D e n s i t y

P o w e r

S p e c t r u m

( m - 4

H z - 1

)

Frequency (Hz)

Fig. 3 Spectrum | L( f )|2 of the vortex line density uc-tuations with respect to fre-quency. The spectrum isplotted on a log-log scale.Dashed line indicates %

f −5/ 3

L( f ) = T int

0 L(t )

e−i2$ f t

√T intdt , (2)

where T int is a time period in which the line density uctuations are measured. The powerspectrum agrees well with a f −5/ 3 pro le, consistent with the experiment of the Lancastergroup.

3 Discussion and Conclusion

The origin of the line length density uctuations can be clasi ed as follows:

1. length change by the reconnection (inducing Kelvin waves)2. injection of vortex rings3. escape of vortex rings from the computational cell4. elimination of vortex rings smaller than the computational resolution5. expansion and contraction of vortices due to the velocity eld created by vortices.

The vortex rings are injected in the time interval 2 ms which corresponds to 500 Hz in fre-quency. Thus the injection of vortex rings contributes only on the peak at 500 Hz, not on theentire power spectrum. In the simulation, vortex rings with various sizes—from the compu-tational resolution to the size of the cell—escape from the cell, so the escape of vortex rings

is considered to affect the change of the line length most. In this paper, we pick up the con-tribution of the vortex rings that escape from the computational cell from the contributionslisted above.

The emergence of the power spectrum implies the existence of a self-similar structurewith respect to time. Focusing on the vortex rings, we draw the conclusion that the power



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spectrum is linked with the self-similar structure of the vortex tangle as follows. Each am-plitude of the spectrum means how many vortex rings are emitted from the cell with thecorresponding frequency. Since the propagation velocity of the ring with the radius R iswritten as vring ∼

" / R and the turnover time [15] of the vortex ring is R/ vring ∼ R2/ " , the

smaller vortex ring evolves in smaller time scale. Thus large vortex rings contribute to theamplitude of low frequency in the spectrum, and vice versa. Here, let us consider the numberdensity of vortex rings n( R) with the radius R

∼ R+ dR . Because the number of the emit-

ted vortex rings is considered to be proportional to n( R), the decrease of the vortex lengthdensity is described as

|cem n( R)dR · 2$ R| T int L( f )d f , (3)

where cem is a dimensionless constant and T int is the time period in which the line density uctuations are measured. 1 Since | L( f )|2 % f −5/ 3 , (3) is rewritten as

n( R) =d f dR

·A f −5/ 6

cem2$ R, (4)

where A is a normalisation constant with dimension [ L−2T 2/ 3 ]. As the frequency is givenby f ∼

" / R2 [16], then Eq. (4) becomes

n( R)" 1/ 6 A

2$ cem √T int R−7/ 3 . (5)

The time period T int should not be involved in the nature of the structure, thus A should bewritten as B√T int to cancel out, where B has the dimension [ L−2 T 1/ 6 ]. The number densityn( R) should have something to do with the time interval of vortex injection ! , which shouldbe involved in B. Since the nature of vortex tangle is determined by " , B can be constructedwith ! and " , then one obtains B = C " −1 ! −5/ 6 from dimensional analysis, where C is adimensionless constant. Finally Eq. (5) becomes

n( R) =C

2$ cem (" ! )5/ 6 R−7/ 3 . (6)

Equation (6) represents the self-similar structure in terms of the number density with respectto vortex ring radius, having exponent -7/3. Although we only consider the contribution of the vortex ring emission and the argument is based on the time scale in which the vortexrings experience, it is worth remarking the connection between the uctuation spectrum andthe self-similar structure of quantum turbulence.

In conclusion, the grid experiments conducted by the Lancaster group are modelled andthe numerical simulation is performed. As a result, the scenario proposed by them is con- rmed, and the vortex line length density of the numerical simulation qualitatively agreeswith that of the experiments. Furthermore the vortex line density uctuations in equilibriumquantum turbulence are calculated and found to obey the power spectrum | L( f )|2 % f −5/ 3 ,consistent with the experiment. Finally, considering the contribution of the vortex ring emis-

sion to the power spectrum and assuming time scales experienced by the vortex rings, weformulate the number density n( R), which unveils the connection between the spectrum andthe self-similar structure of quantum turbulence.

1 The time period T int appears in the formulation because the de nition (2) of the spectrum includes T int .



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Acknowledgements S. F. acknowledges the support of JSPS Research Fellowships for Young Scientists(Grant No. 217762). M. T. acknowledges the supports of Grant-in-Aid for Scienti c Research from JSPS(Grant No. 21340104) and Grant-in-Aid for Scienti c Research on Priority Areas (Grant No. 17071008)from MEXT.

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shoji fujiyama and makoto tsubota- vortex line density fluctuations of quantum turbulence

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