Vortex emission from quantum turbulence in

PHYSICAL REVIEW B 87, 024511 (2013)

Vortex emission from quantum turbulence in superfluid 4 He Y. Nago, A. Nishijima, H. Kubo, T. Ogawa, K. Obara, H. Yano,* O. Ishikawa, and T. Hata Graduate School of Science, Osaka City University, Osaka 558-8585, Japan (Received 29 November 2012; revised manuscript received 4 January 2013; published 22 January 2013) An oscillating object can stretch quantized vortices attached to it in superfluids due to the relative superflow, steadily generating quantum turbulence, even in the zero-temperature limit. We report the emission and propagation of quantized vortices from quantum turbulence generated in superfluid 4 He at low temperatures. A vortex-free vibrating wire enables us to detect the first collision of vortex rings and therefore to measure the time-of-flight of a vortex emitted from a generator to a detector. The detection times from the start of turbulence generation exhibit an exponential distribution, suggesting that the detection is a Poisson process. Vortices are emitted continuously, but each vortex has a random flight velocity and direction. We estimated the nondetection time and mean detection period from the distribution for two flight distances. By estimating the flight velocity, we find that only vortices with velocities lower than the detector velocity can be detected, even if the sizes of the emitted vortices are smaller than the wire thickness or the vibration amplitude. The ratio of the detection rate as a function of vortex velocity suggests anisotropic emission of vortices from the quantum turbulence. DOI: 10.1103/PhysRevB.87.024511

PACS number(s): 67.25.dk, 47.32.cf, 47.27.Cn

I. INTRODUCTION

Quantum turbulence in superfluids has attracted considerable research interest due to the fact that it has a simpler form than classical turbulence in viscous fluids.1 Turbulence consists of a disordered tangle of quantized vortices that all have the same quantized circulation κ with a thin nonsuperfluid core. Since a superfluid is inviscid, its flow causes no mutual friction against quantized vortices. This simplicity enables the study of energy transfer in turbulence with respect to the scale of eddy motion. At large scales, flow energy in superfluid turbulence transfers from large scales to smaller scales by reconnection of vortices.2 At scales smaller than the inter-vortex-line distance, the energy goes into generating Kelvin waves on vortex lines,3,4 cascading down to shorter wavelengths, eventually dissipating as phonons or other thermal excitations.2 The diffusion process of vortex loops from a turbulent region may also cause the decay of turbulence.5 These cascades have been experimentally observed in the decay of the vortex line density in superfluid helium,6,7 even at very low temperatures where the normal-fluid component is almost absent. The energy is released from a turbulent region by excited quasiparticles for superfluid 3 He-B.8 The energy transfer outside a turbulent region is still unsolved for superfluid 4 He however. To experimentally generate quantum turbulence in superfluids, it is necessary to drive superflows and remanent vortices.9 Superflows are easily driven by thermal counterflows,10,11 by superfluid steady flows using grids12 and propellers,13 or by oscillating objects such as spheres,14 grids,15,16 wires,17–19 and forks.20,21 In superfluid 4 He, vortex seeds nucleate during the superfluid transition, and remain attached to superfluid boundaries.9 Superflows along the boundaries may stretch the remanent vortices at a sufficiently high velocity, resulting in the generation of turbulence.9,15,18 Using an oscillating object, quantum turbulence can be generated continuously: stretched vortices are entangled in the oscillation path, colliding with the object repeatedly. In a previous study,17 we found that an oscillating object can generate quantum turbulence with a steady density 1098-0121/2013/87(2)/024511(9)

of vortex lines in the oscillation path. During turbulence generation, the creation of vortex lines appears to be balanced by dissipation or evaporation in the oscillation path. The emission of vortices from an oscillating object has indeed been observed in superfluids.19,22 Numerical simulations of turbulence generation by an oscillating sphere22,23 suggest that stretched vortex lines form tangles in the path of the oscillation and vortex rings are emitted in the ambient superfluid. These results indicate that the generation power of turbulence is transferred from a turbulent region to the surroundings by the creation of vortex rings. This work is concerned with vortex emission from quantum turbulence in superfluid 4 He. In this system, a vortex ring travels in a superfluid sea due to self-induced Magnus forces.24 The motion of vortex rings in superfluid 4 He has been investigated by a visualization technique using tracer particles.25 However, this technique has been limited to studies at high temperatures, and the tracer particles may affect vortex motion. To study vortex motion in superfluid 4 He at low temperatures, we used a vortex-free vibrating wire developed in a previous study.9 Such a wire cannot generate turbulence even at high vibration velocities. However, the vibrating wire starts to generate turbulence when a vortex ring collides with it. Using this technique, we can detect a vortex ring emitted from the quantum turbulence with no tracer particles. This technique is an efficient way to explore vortex motion in superfluid 4 He at very low temperatures. We used a vortex-attached vibrating wire and a vortex-free vibrating wire as a turbulence generator and a vortex detector, respectively, to study the flight of vortex rings between the wires. In a previous paper, we reported the experimental setup and preliminary results.26 In this former setup, we measured only the time for detection to occur after turbulence generation. To obtain the flight time of emitted vortex rings in the present study, we set up another wire27 and measured the detection times for two flight distances simultaneously. These measurements enable us to study vortex emission from steady quantum turbulence. In the present paper, we report time-of-flight results for emitted vortex rings in Sec. III and

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discuss the characteristics of vortex detection in Sec. IV. In Sec. V, we report vortex emission from steady quantum turbulence.

well as for various injection powers, in order to investigate the efficiency of the detector and the isotropy of vortex emission.

II. EXPERIMENTAL

III. TIME-OF-FLIGHT MEASUREMENTS

A vortex-free vibrating wire with few remanent vortices is useful as a detector of vortex rings.22 In order to reduce the number of remanent vortices, a thin wire with smooth surfaces is a suitable material.28,29 In the present study, we prepared a NbTi superconducting wire with a diameter of about 2.4 μm stretched from a commercial multifilament wire with a die. This technique produces wires with relatively smooth surfaces. The experimental setup, which is identical to that used in a previous work,27 is shown in Fig. 1. Three vibrating wires A, B, and C are placed in parallel to each other with a spacing of 1.13 mm between them. The vibrating wires are covered with a copper box with a helium-inlet pin hole, which is also effective in reducing the number of remanent vortices.9 Helium liquid was cooled down to 50 mK using a 3 He–4 He dilution cryostat. The vibrating wires were driven by ac electric currents I rms in a magnetic field of B = 25.0 mT, obtaining a resonance frequency of 3 kHz in vacuum. We measured the Faraday voltage V rms induced by wire oscillation in the field B simultaneously using a phase locked loop technique as used in a previous work.17 The√peak velocity of the apex of each wire is calculated by v = 2V rms /cBd, where d is the distance between the legs of the wire loop and c is a geometrical constant. We assume the wire loop is semicircular, which gives c = π/4. The injection power in a turbulent state for a current I rms and a Faraday voltage V rms is calculated by V rms × (I rms − ILrms ), where ILrms indicates the current in a laminar state for the Faraday voltage V rms . We slowly filled 4 He liquid in the cell through a filling line.9,22 In the filled cell, it was found that wires A and B could generate turbulence at several hundred mm/s while wire C showed no turbulence generation up to 1 m/s. This may be due to the flow of liquid 4 He during filling, though the precise origin is not yet clear. Vibrating wires A and B were used as generators of turbulence and “vortex-free” wire C as a detector. We measured the time-of-flight of vortex rings repeatedly for a flight distance of 1.13 mm using wires B and C and for a flight distance of 2.26 mm using wires A and C. We also measured the time-of-flight for different velocities of the detector as

A. Detection of vortex emission

velocity (mm/s) velocity (mm/s)

(a)

The time-of-flight measurements were performed using a similar method as in previous studies,26,30 but the analysis method was improved. In previous studies, we defined the start of vortex emission as the onset of the velocity drop due to the turbulent transition (see Fig. 3 in Ref. 30). Just before the transition, we occasionally observed a metastable laminar state in which the detector velocity exceeds the critical velocity vc of the turbulence state. The metastable state is unstable for the generator to produce vortices stochastically.14,17 In fact, the onset velocity of dissipation due to turbulence generation is not constant. In the metastable state, the generator may produce vortex rings, because it enters the turbulent state after a short period. This leads to an ambiguity in estimating the onset of vortex emission. To avoid this ambiguity, we chose only data indicating no metastable laminar state. Figure 2 shows a typical case of the time difference between turbulence generation and vortex detection measured using generator B and detector C. The driving force of the generator was increased, so that its velocity increased through the critical velocity vc up to 100 mm/s in a turbulent state. The data shown in Fig. 2(a) do not show the metastable laminar state, in contrast to those observed in Ref. 30. We define the start of vortex emission (t = 0 s) at the critical velocity vc as shown in Fig. 2(a). After a while, the velocity of the detector suddenly drops to about 100 mm/s, as shown in Fig. 2(b). This result indicates that vortices are emitted from the generator and propagate to the detector in the surrounding superfluid, colliding with the detector and causing a dissipation of the detector vibration.

(b )

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FIG. 1. (Color online) Schematic drawings of the experimental setup: (a) vibrating wires mounted in a copper chamber and (b) top view of the wires. The distance between the legs of each wire is 0.7 mm. The wires are placed in parallel with a spacing of 1.13 mm.

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FIG. 2. (Color online) Time series of velocities of (a) generator wire B and (b) detector wire C (flight distance 1.13 mm). The origin of the time axis is determined by the critical velocity vc of the turbulent transition for the generator. The horizontal line in (b) shows the average of the detector velocities in the laminar state. Dissipation occurs in the detector vibration with a delay time t after turbulence generation by the generator at vc (see text).

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FIG. 3. (Color online) Histogram of detection time t, obtained by repeated measurements, as shown in Fig. 2.

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FIG. 5. (Color online) Nondetection probability 1 − P for a flight distance of 2.26 mm. The solid line indicates an exponential distribution expressed by Eq. (1).

B. Distribution of detection times

The detection times t from the beginning of turbulence generation are distributed as shown in Fig. 3, measured with wires B and C for a flight distance of 1.13 mm, an injection power of 25 pW, and a detector velocity of 200 mm/s. These results are similar to those observed in a preliminary study.26 It is considered that vortex rings are emitted continuously in any direction from the generator,22 and the detector can respond only to reachable rings. To study the distribution, we plot in Fig. 4 the nondetection probability 1 − P for a time t estimated from Fig. 3. Here, P is the detection probability within a period of time t. The data are well fitted to an exponential distribution t − t0 1 − P = exp − , (1) t1 indicating a Poisson process; detection of vortex rings occurs independently and continuously at irregular intervals with a mean detection period t1 , estimated to be t1 = 6.1 ms for the case of Fig. 4. The Poisson process is associated with emission from steady turbulence. We discuss the details in Sec. V. The nondetection time t0 , which is equivalent to the detection time including the generation and flight times of a detectable vortex tmin

t0

1−P

1

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ring, is estimated to be 13.1 ms. Note that a vortex ring is detected even at t < t0 ; the data deviate from the fitting line expressed by Eq. (1) as shown in the regime of tmin < t < t0 in Fig. 4. This result indicates that vortex rings are also emitted from nonfully developed turbulence that appears just before the steady turbulence. It is possible that there are fewer vortex rings emitted during this developing process, resulting in a low rate of vortex detection at tmin < t < t0 . The nondetection time t0 includes the time of turbulence generation and the time-of-flight of an emitted vortex ring. To estimate the time-of-flight, we measured the distribution of the delay times for a different flight distance using the A and C wires, estimating 1 − P as shown in Fig. 5. In this measurement, vibrating wire B between the generator and the detector might disturb the flights of vortex rings: vortex rings colliding with wire B may split into several segments and reconnect to form a vortex tangle. A vortex tangle around wire B is likely to screen the vortex rings traveling towards the detector. Figure 5 shows that the data are well fitted to an exponential distribution expressed by Eq. (1), similar to the 1.13-mm case, except for the short time regime corresponding to a nonfully developed turbulence. In practice, it appears that wire B does not act as a screen. Apparently, the three wires are not aligned straight and even if vortex rings collide with wire B, they will not get entangled but will slide on the wire surface and detach from the wire. This process does not explicitly affect the detection rate for the detector. The nondetection time t0 and the mean detection period t1 are estimated from Fig. 5 to be 19.2 and 23.2 ms, respectively, which are larger than those estimated for the 1.13-mm case. This may be because the flight times of vortex rings increase and the densities of the vortex rings decrease as the flight distance increases. C. Time-of-flight of a vortex ring

0

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FIG. 4. (Color online) Nondetection probability 1 − P estimated from the detection time distribution shown in Fig. 3. Here, P is the detection probability in a period of time t. The data are well fitted to an exponential function as indicated by the solid line. The arrows show the observed minimum detection time tmin and the nondetection time t0 (see text).

Since a vortex ring travels straight with a constant translational velocity vring in a superfluid at 50 mK, vortex rings emitted from steady turbulence toward the detector will always collide with the detector. The high-amplitude oscillation of the detector then stretches the first incoming ring to form a vortex tangle, resulting in a turbulent flow around the detector. Vortices are considered to expand within the detector oscillation period of ∼0.3 ms, much lower than the observed

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time-of-flight of vortex rings. We can therefore assume that the nondetection time t0 corresponds to the sum of the generation period tg of fully-developed steady turbulence and the minimum time-of-flight tf of detectable vortex rings. Thus the nondetection time t0 (l) for a flight distance l is given by

t0 (ms)

(2)

We estimate the flight velocity vring of the detected vortex rings from t0 obtained in Figs. 4 and 5 as follows. Since the timeof-flight was measured repeatedly under the same generator conditions, the generation period tg of steady turbulence is expected to be constant. Therefore the difference between the flight times t0 measured for a flight distances of l and l + l is equal to the time-of-flight of a vortex ring traveling a distance l, i.e., t0 ≡ t0 (l + l) − t0 (l) = tf (l) estimated from Eq. (2). Consequently, the velocity vring of a ring traveling a distance l is estimated by vring = l/t0 .

2.26mm 1.13mm

where κ and a0 are the circulation quantum and the radius of the vortex core, respectively. Assuming the detected vortex ring has a circular shape, the velocity vring = 159 mm/s corresponds to a vortex ring with a radius of 0.49 μm estimated from Eq. (4). This size is smaller than the wire diameter, the wire vibration amplitude, and the Kelvin wavelength,24 which are 2.4 μm, 8.7 μm, and 13.3 μm for 3.15 kHz, respectively. At a low injection power, as in the present case, the turbulent region is considered to be restricted in the swept path of an oscillating object.17,22 An emitted vortex ring should be smaller than the turbulent region, consistent with the present result. An emitted vortex ring may have a somewhat distorted form. A wavy vortex ring can be created by an external perturbation, e.g., Kelvin waves are excited thermally31 or due to vortex reconnection.32 The results of a previous timeof-flight study at a high temperature30 imply the possibility of generation of wavy vortex rings. It is plausible that wavy vortex rings are also generated due to vortex reconnection at low temperatures. Since a wavy vortex ring travels more slowly than a circular one,33,34 the effective radius of the detected vortex rings is expected to be smaller than the estimated value of 0.49 μm for a circular ring. In either case, it is an open question as to what determines the minimum size of detectable vortex rings. A vortex ring colliding with the detector can cause turbulent flow if it is stretched sufficiently without detaching during a few periods of the detector oscillation. Therefore the detector oscillation conditions could be principally associated with the minimum size of detectable vortex rings. The results of time-of-flight measurements for various detector velocities will be discussed in the next section.

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FIG. 6. (Color online) Nondetection time t0 as a function of the peak velocity of the detector for each flight distance and an injection power of 12 pW. The blue circles are for a flight distance of 1.13 mm and the red squares for 2.26 mm.

(3)

In the present study, the difference between the flight distances is l = 1.13 mm and the difference between the nondetection times is t0 = 7.1 ms. Thus the highest velocity of the detected vortex rings is estimated to be vring = 159 mm/s. The velocity of a vortex ring is associated with the quantized circulation around a vortex core. The velocity of a circular vortex ring with a radius R is expressed by24 8R κ 1 c ln , (4) vring = − 4π R a0 2

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IV. CHARACTERISTICS OF DETECTION

It is possible that the minimum size of vortex rings to which the detector can respond depends on the oscillation velocity of the detector, because vortices colliding with the detector grow to form a tangle in the relative superflow with a sufficient velocity due to the detector oscillation. The time-of-flight of vortex rings was measured for various detector velocities and the nondetection probability 1 − P was estimated for each. We find that the detection times have a similar distribution to those shown in Fig. 3 but tend to decrease with increasing detector velocity. The distributions are exponential [see Eq. (1)] for each detection velocity. Nondetection times t0 are obtained by fitting the data for an injection power of 12 pW to Eq. (1), shown in Fig. 6. The nondetection time t0 decreases with increasing detector velocity for both flight distances. The highest velocity of detected vortex rings is estimated as discussed in the previous section, shown in Fig. 7. We find that the highest velocity of detected vortex rings increases with increasing detector velocity, indicating that a detector with a higher-velocity oscillation can respond to a faster vortex ring. Assuming the vortex rings to have a circular shape, the minimum size of the vortex rings can be estimated from 1000 800 vring (mm/s)

t0 (l) = tg + tf (l) = tg + l/vring .

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v det (mm/s) FIG. 7. Flight velocity vring of the minimum vortex ring estimated from Fig. 6 using Eq. (3) as a function of the peak velocity vdet of the detector. Here, the data for vdet = 1000 mm/s is omitted because the induced velocity cannot be estimated√ accurately due to a large error. The solid line represents vring = vdet / 2.

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FIG. 8. Radius R of detected minimum vortex rings estimated from Fig. 7 using Eq. (4), as a function of the peak velocity vdet of the detector.

Eq. (4), as shown in Fig. 8. This result indicates that the detector can detect submicron vortex rings, depending on its oscillation velocity. Since the injection power has a constant value of 12 pW, the size distribution of the emitted vortex rings should be the same for each measurement. It is therefore clear that vortex rings with different submicron sizes are emitted randomly from the turbulent region, which is consistent with the results of previous numerical simulations,22 and the size of a detectable ring is largely determined by the detector velocity. The detector can respond only to those incoming vortex rings which the detector oscillation can cause to undergo unstable expansion to produce turbulence. If a vortex ring with a radius R reaches the detector, it attaches forming a loop with its edges separated by no more than 2R. In a superfluid boundary flow, attached vortices with this curvature are considered to undergo unstable expansion due to Glaberson-Donnelly (GD) instability29 at a critical flow c velocity equal to the flight velocity vring of a vortex ring with a curvature radius R given by Eq. (4). When a vortex ring with a velocity lower than the oscillation velocity of the detector collides with the detector and becomes attached, the vortex is extended unstably by oscillatory flows due to GD instability, forming a tangle. Consequently, we can observe the detection of a vortex ring as a turbulent transition. A vortex ring with a velocity higher than the oscillation velocity, however, will not undergo a turbulent transition, even if it collides and attaches to the detector. The vortex will quickly detach or move on the detector surface, deforming its shape to balance the Magnus force due to oscillatory flow and eddy flow.22 We find that vring √ is nearly equal to the rms value of the detector velocity vdet / 2, as shown by the solid line in Fig. 7. Here, vdet is the peak velocity of the apex of the detector wire. √ This result suggests that only vortex rings for vring vdet / 2 cause GD instability due to detector oscillation. Note that the above discussion is based on the assumption that a vortex ring is perfectly circular. Nevertheless, the present results are well described by such a simple model. Kelvin waves seem to be excited on a vortex ring mainly by the reconnection of rings.22,32 The flight velocity of a vortex ring perturbed by a Kelvin wave is expressed by the following form modified from Eq. (4):35 w vring

=

c vring

A2 3 1 − K2 NK2 − , R 4

(5)

where AK and NK are the amplitude and number of the Kelvin waves, respectively. The condition for stability of a wavy vortex ring should satisfy A2K /R 2 1.34 Since the minimum size of the detected vortex rings, of the order of 0.1 μm as seen in Fig. 8, is much lower than the wire thickness (∼2 μm) and the Kelvin wavelength, which the wire vibration may generate (13.3 μm), it is not likely that a large number of waves will be excited (i.e., the wavelength should be less than ∼0.1 μm) on such a small ring. Therefore the number of waves NK is sufficiently low and, as a result, the extra term due to the contribution of excited waves in Eq. (5) is negligible for the minimum (fastest) vortex ring. This may be why a circular-vortex-ring model describes the present results well, as seen in Fig. 7. From Eq. (2), the generation period tg of fully developed turbulence can be estimated by t0 (l = 1.13 mm) − tf (l = 1.13 mm) for each velocity, and is found to be 6 ms and independent of the detector velocity. This result suggests that vortex rings of any size observed here may be emitted from the start of turbulence generation. Consequently, we find that the oscillation velocity of the detector strongly determines the detectable minimum vortex ring size. Measurements with the same detector velocity are necessary to investigate the injection-power dependence of vortex-ring generation. We will discuss vortex emissions for different injection powers for a detector velocity of 200 mm/s in the next section. V. VORTEX EMISSION A. Detection distribution vs injection power

In a steady turbulent state, the injection power due to boundary oscillation seems to be balanced by the dissipation power due to vortex emission17 and phonons created in energy cascades.2 Therefore a larger injection power can yield a higher vortex line density, creating more vortex rings. In Fig. 9, we plot the nondetection probability 1 − P for different injection powers for a flight distance of 1.13 mm and a detector velocity of 200 mm/s. We can see that the detection period behaves similarly for each injection power on the whole, but tends to decrease with increasing power, indicating that the number of emitted vortex rings increases and therefore the detection rate increases. 1 9 pW 25 pW 50 pW

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FIG. 9. (Color online) Nondetection probability 1 − P with different injection powers for a flight distance of 1.13 mm. Each solid line indicates an exponential function expressed by Eq. (1).

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FIG. 10. (Color online) Nondetection time t0 for flight distances of 1.13 mm (blue circles) and 2.26 mm (red squares) as a function of injection power.

FIG. 12. Generation period tg of steady turbulence as a function of injection power. Each plot has the same error range but error bars are shown only for the 50-pW plot for clarity.

The results for a flight distance of 2.26 mm are similar to those shown in Fig. 9. The data for both flight distances can be well fitted to Eq. (1) in the same way as shown in Fig. 4. However, the data for higher injection power tend to show somewhat curved characteristics, being upwardly convex. For a high-power oscillation, many vortex rings are emitted continuously, reconnecting and creating large vortex rings or possibly a vortex tangle36 near the generator. A high density of large vortex loops may affect the detection rate of vortex rings, i.e., the upwardly convex curve displayed in Fig. 9. Such a deviation from the exponential function is qualitatively consistent with previous numerical simulation results for a large rate of vortex injections,26 though the mechanism is not yet clear. For both flight distances, the nondetection time t0 and the mean detection period t1 are estimated from a fit by Eq. (1). Figure 10 shows the nondetection time t0 for both flight distances as a function of the power injected by the generator. We can see that t0 decreases with increasing injection power for both cases; however, the difference of t0 , which corresponds to the time-of-flight for the flight distance l = 1.13 mm, is almost constant for each power. Consequently, the traveling velocity of the detected minimum ring vring is found to be almost independent of injection power, as shown in Fig. 11. The radius of the vortex ring is estimated from Eq. (4) to be R = 0.45 μm. This is consistent with the conclusion discussed in the previous section (see Sec. IV) that the

minimum size of detectable vortex rings is dominated by the velocity of the detector oscillation. These results indicate that submicron vortex rings are emitted from turbulence regardless of the injection power. The generation periods tg of steady turbulence are plotted in Fig. 12. The period tg decreases with increasing injection power, indicating that a large injection power generates steady turbulence more quickly. Even for the shortest period tg ≈ 4 ms at 50 pW, the generator vibrates ten times before entering the steady turbulent state.

vring (mm/s)

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FIG. 11. Flight velocity vring of the detected vortex ring estimated from Fig. 10 using Eq. (3), as a function of injection power.

B. Detection rate of vortex emission

The exponential distributions shown in Figs. 4, 5, and 9 indicate a Poisson process where detection events occur randomly but continuously at irregular intervals with a mean period of t1 , as shown in Eq. (1). The turbulent region produced by the generator is much smaller than the distance between the generator and the detector. The detector can detect vortex rings with a velocity lower than its own velocity, as discussed in Sec. IV. Hence, the random detections suggest that vortex rings are emitted from a turbulent region with a random flight velocity and direction. In the steady state, vortex emissions are expected to be continuous. Nevertheless the mean detection rate t1−1 is associated with the emission rate from the turbulent region in the detector direction, which is parallel to the oscillation direction. Figure 13 shows the mean detection rate t1−1 as a function of injection power. At injection powers lower than 20 pW, the rate t1−1 seems to be proportional to the injection power for each flight distance, shown by the solid lines in Fig. 13. All the vortex emission events occur independently because the nondetection probability is described by a Poisson process. Therefore the detection rate of vortex rings is proportional to the emission rate for the generator in the case of low injection powers. The proportionality relation observed here suggests that the number of vortex rings increases proportionally with increasing power. The detection rate t1−1 , however, deviates downward from the proportional line in Fig. 13 at injection powers above 20 pW. As mentioned in the previous subsection, a high injection power generates many vortex rings, creating a large vortex ring, or a vortex tangle by reconnection.36 In this case, Kelvin waves may be excited on a large vortex loop: collision

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FIG. 13. (Color online) Mean detection rate for flight distances of 1.13 mm (blue circles) and 2.26 mm (red squares) as a function of injection power. The plots for injection powers lower than 20 pW are well fitted to a proportional function shown by the solid lines.

among small rings leads to perturbation on the subsequently created large loop and this local distorted part of the loop gives rise to a self-induced Magnus force on the other parts, causing helical waves.32 If a vortex ring of the same size as the wire diameter or larger is created by collision of several small rings, it is possible that this large ring has Kelvin waves with several numbers of waves. According to Eq. (5), a vortex ring with Kelvin waves travels slowly due to contributions from the second term. Therefore a large ring emitted from the generator screens small faster rings flying from behind, resulting in a decrease of the rate that vortex rings can reach the detector. These processes may be why the mean detection rate t1−1 at high injection powers is lower than the value expressed by the proportional line in Fig. 13. New qualitative and quantitative approaches to account for these processes are eagerly anticipated in future studies. C. Anisotropy of vortex emission

The mean detection period t1 increases with increasing flight distance, as shown in Fig. 13. For the present range of injection power, the ratio t1 (l = 2.26 mm)/t1 (l = 1.13 mm) is found to be nearly constant at 3.7 ± 0.3. Assuming that vortex rings spread isotropically from the source, the density of vortex rings at a distance r from the source is proportional to r −1 in the case of a line source, while proportional to r −2 in a case of a point source. The present result reveals that the ratio of the mean detection periods t1 (l = 2.26 mm)/t1 (l = 1.13 mm) = 3.7 can be approximated as (2.26/1.13)−2 = 4 for a flight distance increased by a factor of two. Therefore we can model vortex rings as being emitted radially from the top of the wire as a point source, though the wire is nearly line shaped. This is because a vortex tangle is created at the apex part of the wire, which oscillates at the highest velocity. We also measured the detection rate for a finite injection power as a function of detector velocity, because the detector velocity affects the detection ability. As discussed in Sec. IV, the detection range of vortex rings can be extended by increasing the detector velocity. The detection area of the detector wire is also extended at higher oscillations. Figure 14(a) shows the mean detection period t1 for an injection power

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FIG. 14. (Color online) (a) Mean detection period t1 measured at an injection power of 12 pW for flight distances of 1.13 mm (blue circles) and 2.26 mm (red squares) as a function of detector velocity. (b) Ratio of mean detection periods t1(2.26 mm) /t1(1.13 mm) .

of 12 pW. It is seen to decrease with increasing detector velocity for both flight distances, consistent with expectations. The ratio t1 (l = 2.26 mm)/t1 (l = 1.13 mm), however, is no longer constant; it decreases with increasing detector velocity, as shown in Fig. 14(b). The ratio changes from four to two as the detector velocity increases, implying that the turbulent region extends to a line source. The injection power is, however, fixed at a small value through these measurements, so that the turbulence is restricted to a small region. Therefore the turbulent region is considered to be a point source under these conditions. Nevertheless, the ratio change is associated with vortex emission. Since the turbulence is driven by the oscillating object, vortex rings are considered to be emitted anisotropically with respect to the direction parallel or perpendicular to the oscillation. In fact, numerical simulations37 predict that the emission rate of vortex rings varies with size and emission direction: the rate perpendicular to the oscillation is lower than that parallel to the oscillation for large vortex rings. In the present experiments, the detection area is widened by increasing the detector velocity and therefore the detector becomes more sensitive to the direction distribution of the vortex emissions. Furthermore, the detector can catch faster vortex rings at high vibration velocities. Hence, the results shown in Fig. 14(b) suggest that the vortex emissions are anisotropic. In the experimental setup, however, the detector is mounted only in the direction of the generator oscillation and therefore we cannot currently study the direction distribution of the vortex emissions more precisely. Further experiments with alternative arrangements of vibrating wires are needed to clarify the anisotropy of vortex emissions. VI. CONCLUSIONS

We performed time-of-flight measurements on vortex rings emitted from steady turbulence created by a vibrating wire in superfluid 4 He, made possible by recent developments in vortex detection methods. The detection periods of emitted

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vortex rings were measured using thin vibrating wires as generators and a detector of vortex rings, and the periods were found to have an exponential distribution. This indicates that vortex rings with different sizes are emitted continuously and independently from the steady turbulence, and stochastically collide with the detector. The flight times of vortex rings measured for different flight distances revealed that rings smaller than the wire thickness collide with the detector and stretch to form a vortex tangle due to the detector oscillation. The threshold size for detectable rings is strongly influenced by the detector velocity. We found that the number of emitted vortex rings increases proportionally with the power injected into the turbulence. However, a higher injection power creates steady turbulence more quickly and emits more vortex rings, which collide with each other so that large tangled loops are created. The mean detection period increases by about four times by increasing the flight distance by a factor of two, suggesting that vortex rings are emitted radially from the top of the wire as a point source. However, the ratio of the detection periods decreases with increasing detector velocity,

suggesting that the vortex emissions are anisotropic. Thus the vortex detection using a vortex-free vibrating wire is an efficient way to observe the flight of a quantized vortex ring, immune from tracer particles. By applying this method, one can investigate the energy transfer from quantum turbulence via vortex emission. The present experimental results allowed a preliminary view of vortex emission from steady turbulence, but an outstanding issue remains in clarifying the anisotropic emission of vortex rings. Further experiments using thin vibrating wires are required to more fully understand vortex emission from turbulence in superfluid 4 He.

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ACKNOWLEDGMENTS

We would like to thank M. Tsubota, S. Yamamoto, and A. Nakatsuji for many stimulating discussions, and K. J. Thompson for his input on the analysis. This research is supported by a Grant-in-Aid for Scientific Research (B) (Grant No. 23340108) from the Japan Society for the Promotion of Science.

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