Superdiffusion of quantized vortices uncovering scaling behavior of quantum turbulence
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n Superdiffusion of quantized vortices uncoveringscaling behavior of quantum turbulence
Yuan Tang a,b , Shiran Bao a,b , and Wei Guo a,b,* a National High Magnetic Field Laboratory, 1800 East Paul Dirac Drive, Tallahassee, Florida 32310, USA; b Mechanical Engineering Department, Florida State University,Tallahassee, Florida 32310, USA
Generic scaling laws, such as the Kolmogorov’s 5/3-law, are mile-stone achievements of turbulence research in classical fluids. Forquantum fluids such as atomic Bose-Einstein condensates, super-fluid helium, and superfluid neutron stars, turbulence can also existin the presence of a chaotic tangle of evolving quantized vortex lines.However, due to the lack of suitable experimental tools to directlyprobe the vortex-tangle motion, so far little is known about possiblescaling laws that characterize the velocity correlations and trajectorystatistics of the vortices in quantum-fluid turbulence (QT). Acquiringsuch knowledge could greatly benefit the development of advancedstatistical models of QT. Here we report an experiment where a tangleof vortices in superfluid He are decorated with solidified deuteriumtracer particles. Under experimental conditions where these tracersfollow the motion of the vortices, we observed an apparent superdif-fusion of the vortices. Our analysis shows that this superdiffusionis not due to Lévy flights, i.e., long-distance hops that are knownto be responsible for superdiffusion of random walkers. Instead, apreviously unknown power-law scaling of the vortex-velocity tempo-ral correlation is uncovered as the cause. This finding may motivatefuture research on hidden scaling laws in QT. quantum turbulence | superfluid | superdiffusion | particle tracking ve-locimetry | scaling laws Q uantum fluids, such as superfluids, superconductors, andBose-Einstein condensates (BECs), exhibit macroscopicquantum coherence that is responsible for their dissipationlessmotion (1). In these quantum fluids, all rotational motion issustained by quantized vortex lines, i.e., line-shaped topologi-cal defects characterized by a circulating flow of particles witha discrete circulation κ = h/m , where h is Planck’s constantand m is the mass of the particle.Turbulence in quantum fluids, i.e., quantum turbulence(QT), can be induced by a tangle of interacting vortex lines(2). These vortex lines evolve chaotically under their self-and mutually induced velocities and can reconnect when theymove across each other (3). The underlying science of QT isbroadly applicable to a variety of coherent physical systems,such as coherent condensed matter systems (e.g., superfluid He and He (4), atomic and polariton BECs (5), and type-IIsuperconductors (6)), cosmic systems (e.g., neutron-pair su-perfluid in neutron stars (7, 8), cosmic strings in the Abelian-Higgs model (9), and possible axion dark matter BECs ingalactic halos (10)), and even complex light field (11). Insightinto the generic scaling laws that characterize the evolutionof quantized vortex tangles can inform statistical models ofQT, which could have a broad significance spanning multiplebranches of physics.QT research has been conducted mostly in superfluid Heand He due to the material’s accessibility and the widerange of length scales involved in their turbulence behaviors(12). Nevertheless, despite extensive theoretical and numer- ical studies of the vortex-line dynamics in superfluid helium(13–15), past experimental research has largely been limited tothe measurements of spatially-averaged quantities such as thevortex-line density L (i.e., length of vortices per unit volume)(16–18) or local pressure and temperature variations (19, 20).Important statistical properties of a fully-developed vortextangle, such as the vortex-velocity correlations and their tra-jectory statistics, remain largely unexplored due to the lackof experimental tools for probing the vortex-line motion.A breakthrough has been made in recent years with the de-velopment of quantitative flow visualization techniques (21).In particular, by decorating the vortices in superfluid He (HeII) with solidified hydrogen particles, Bewley et al. demon-strated direct vortex-line visualization (22). Since then,vortex-line reconnections and Kelvin-wave excitations on in-dividual vortices have been filmed (23–25). Nevertheless, vi-sualization data showing the real-time evolution of a complexvortex tangle are still lacking, which impedes the developmentof reliable statistical models for describing QT (26).In our recent experiment on He II QT driven by an appliedheat current, we seeded the fluid with solidified deuterium(D ) tracer particles and observed that a group of particlescould remain trapped on the tangled vortices (27–30). Byapplying a separation scheme in data analysis (28), we wereable to track solely these trapped particles and therefore coulddirectly probe the vortex-tangle dynamics. In this paper, wediscuss our study of the apparent diffusion of these trappedparticles under experimental conditions where they faithfullyfollow the motion of the evolving vortices.We report the first observation of a superdiffusion of thevortices in the tangle when their root mean squared dis-placement (RMSD) is less than the mean inter-vortex dis-tance ℓ = L − / . Surprisingly, our analysis shows that thissuperdiffusion is not due to Lévy flights (i.e., randomized,long-distance hops) that are known to be responsible for su-perdiffusion in various physical and non-physical systems (31).Instead, we reveal that a previously unknown power-law scal-ing of the vortex-velocity temporal correlation is the cause.The derived power-law exponent appears to be temperatureand vortex-line density independent, suggesting that the ob-served scaling behaviors may be generic properties of a fully-developed random vortex tangle. These findings may excitefuture research on hidden scaling laws in QT. W.G. designed the research; Y.T. performed the research; Y.T., S.R.B., and W.G. analyzed the dataand wrote the paper.The authors declare no competing interest. * To whom correspondence should be addressed. E-mail: [email protected] (a) A schematic diagram of the experimental setup. (b) Representative trajectories obtained at T = q =
38 mW/cm in He II for G1 particles trapped onvortices (blue) and G2 particles entrained by the normal fluid (red). (c) The corresponding streamwise particle-velocity distribution, where the solid and the dashed curvesrepresent Gaussian fits to the data. Results
Experimental setup and procedures.
Our experimental setupis shown schematically in Fig. 1 (a). A 400-Ω planar resistiveheater is installed at the bottom of a vertical flow channel(1.6 × ×
33 cm ) inside a He II bath. The temperature T of the He II in the bath can be controlled by regulating thevapor pressure. When a DC voltage is applied to the heater,a counterflow of the two interpenetrating fluid componentsof He II establishes in the flow channel (32): the viscousnormal-fluid component that consists of thermal quasiparti-cles in He II (i.e., phonons and rotons) flows away from theheater at a mean velocity given by v n = q/ρsT , where q is theheat flux, and ρ and s are the He II density and specific en-tropy, respectively; while the inviscid superfluid component(i.e., the condensate) moves in the opposite direction at a ve-locity v s = − v n ρ n /ρ s , where ρ n /ρ s is the density ratio of thetwo fluids.It has been known that above a small threshold heat flux ofthe order 10 mW/cm (33), turbulence appears spontaneouslyin the superfluid as a random tangle of quantized vortex lines,each carrying a quantized circulation κ ≃ − cm /s aroundits angstrom-sized core (3). A mutual friction between thetwo fluids arises due to scattering of the thermal quasiparti-cles off the vortices (34). Above a heat flux of the order 10 mW/cm , the normal fluid can also become turbulent (35–37),rendering a complex doubly turbulent system (38–41). Ourcurrent research focuses on the low heat flux regime whereonly the superfluid is turbulent.To probe the flow, we adopt a particle tracking velocimetry(PTV) technique using solidified D particles as tracers. Dueto their small sizes (i.e., about 4 µ m in diameter (42)), theseparticles have a small Stokes number in the normal fluid andhence are entrained by the viscous normal-fluid flow (43). Butwhen they are close to the vortex cores, a Bernoulli pressuredue to the superfluid flow induced by the vortex cores canpush the particles toward the cores (3), resulting in the trap-ping of the particles on the quantized vortex lines. Thesetracer particles are illuminated by a thin continuous-wavelaser sheet and their positions are recorded by a video camera at 90 Hz. We have also installed a pair of second-sound trans-ducers for measuring the vortex-line density using a standardsecond-sound attenuation method (44). More details can befound in the Method section.As we reported in Ref. (28–30), two distinct groups ofparticles can be observed at q below about 10 W/cm (seeFig. 1 (b)). The G1 group includes particles entrapped onvortices, resulting in irregular trajectories. The G2 groupincludes untrapped particles entrained by the up-moving lam-inar normal fluid, resulting in relatively straight trajectories.The streamwise particle velocity distribution based on theanalysis of all trajectories exhibits two nearly separated peaks(see Fig. 1 (c)), which allows us to distinguish these two groupsof particles for separately analyzing their motion (28). Themean velocity of the G2 particles equals the expected normal-fluid velocity. The G1 particles are carried by the vortex tan-gle which drifts at v s towards the heater at small q (45), butin general the G1 particles may also slide along the vorticesdue to the viscous drag from the normal fluid.Nevertheless, at q less than a few tens of mW/cm , we findthat the mean velocity of the G1 particles is about v s (28),in agreement with the observations of Paoletti et al. (46, 47).This observation suggests that the viscous drag effect on thetrapped G1 particles should be mild in the low heat-fluxregime. Furthermore, a recent theoretical work suggests thatmicron-sized tracers trapped on quantized vortex lines in HeII are indeed immobilized along the lines due to an effectivefriction originated from the breakdown of the vortex coher-ence (48). Therefore, in the low heat flux regime, it is feasibleto explore the genuine vortex-tangle dynamics by trackingthe motion of the trapped G1 particles. Specifically, we focuson studying the apparent diffusion of the G1 particles in thehorizontal direction so as to keep the viscous drag influenceminimal. Vortex diffusion statistics.
For the data obtained at each tem-perature and heat flux, we first calculate the horizontal meansquared displacement of the G1 particles h ∆ x ( t ) i = h [ x ( t ) − x (0)] i , where the diffusion time t starts from the moment Representative data showing the horizontal mean squared displacement h ∆ x ( t ) i of the G1 particles as a function of the diffusion time t . The solid and thedashed lines are power-law fits to the data. when a particle is first observed along its trajectory, and theangle brackets denote an ensemble average over at least 10 trajectories. In general, a power-law scaling h ∆ x ( t ) i ∝ t γ is expected, where the exponent γ is often used to identifydifferent types of diffusions, i.e., normal diffusion ( γ =1), su-perdiffusion ( γ> γ<
1) (49). Fig. 2 showsa representative result obtained at T =1.7 K and q =38 W/cm .The data exhibit two power-law scaling regimes: a superdiffu-sion regime with γ ≃ p h ∆ x ) anda nearly normal diffusion regime with γ ≃ p h ∆ x i c ≃ . µ m. Due totheir irregular trajectories, the G1 particles seldom stay inthe thin laser plane for long time. Therefore, we have rela-tively few long trajectories to study the G1 particle diffusionat large t , which limits the range of the observed γ -scalingregime. We would also like to comment that at sufficientlysmall diffusion times, the vortex segments are expected tomove ballistically at the local superfluid velocity (3), whichshould lead to a t scaling of h ∆ x i . However, this distinctregime likely would occur only below a few milliseconds forthe vortex-line density examined in our experiments, which isbeyond the resolution of typical He II PTV measurements.Our analysis of the data sets obtained at other heat fluxesand temperatures also show similar two power-law scalingregimes. The derived γ , γ and p h ∆ x i c are collected inFig. 3. Surprisingly, in the explored temperature range of 1.7K to 2.0 K where ρ n /ρ s varies from 0.3 to 1.24 (50), the γ value is always around 1 . − . γ is close to unity,regardless of the applied heat fluxes. This suggests that theobserved diffusion scalings could be generic properties of anevolving random vortex tangle. We have also examined thetransition RMSD that separates the two diffusion regimes andfind that p h ∆ x i c increases with decreasing the vortex-linedensity L . Considering the fact that the diffusion occurs inthree-dimensional (3D) space, we compare p h ∆ x i c withthe measured inter-vortex distance ℓ in Fig. 3 (c). These twoquantities agree reasonably, suggesting that the transition oc-curs when the RMSD of the vortices is greater than ℓ .It is worthwhile noting that the spatial spreading of a de-caying isolated vortex tangle at T =0 K either near a solid sur-face (51) or in bulk He II (52) has been simulated. The growthof the tangle diameter d as reported in Ref. (52) exhibits a Fig. 3. (a) and (b) show, respectively, the obtained scaling exponents γ and γ for the two diffusion regimes of the G1 particles. (c) A comparison of the transitionRMSD with the mean inter-vortex distance ℓ . The vertical error bars represent theuncertainties in the power-law fits as shown in Fig. 2, and the horizontal error barsdenote the standard deviation of the measured ℓ . At small vortex-line density L (i.e.,large ℓ ), the increased measurement uncertainty leads to large horizontal error bars. normal diffusion regime at large t and a clear superdiffusionregime at small t with a fitted scaling of about d ∝ t . (notethat the authors interpreted this latter regime as the ballisticregime). This similarity is encouraging, although our work fo-cuses on the trajectory statistics of vortices in a steady tangleat finite temperatures. Furthermore, following Ref. (52), wecan use our data in the normal diffusion regime to evaluatethe effective diffusion coefficient ν ′ /κ =3 h ∆ x ( t ) i / t . For thedata shown in our Fig. 2, we get ν ′ /κ ≃ .
3, which is close tothe simulated values (51, 52).
Vortex-displacement distribution.
Naturally, one would won-der about the cause of the observed vortex-line superdiffu-sion and why there is a transition to normal diffusion at p h ∆ x i c ∼ ℓ . Indeed, superdiffusion has been observedin a wide range of systems, such as the motion of cold atomsin an optical lattice (53), the chaotic drifting of tracers inrotating flows (54), the cellular transport in biological sys-tems (55), and even the search patterns of human hunter-gatherers (56). A useful function for characterizing superdif-fusion is the distribution function P (∆ x, t ) of the particledisplacement ∆ x at time t , whose time evolution is oftendescribed by a fractional diffusion equation (57). A generalproperty of P (∆ x, t ) is the existence of a self-similar scaling P (∆ x, t )=( t/t ′ ) − γ · P (∆ x · ( t/t ′ ) − γ , t ′ ), where the scaling ex-ponent γ should be identical to the diffusion exponent of themean squared displacement (31). To test whether this prop- (a) Representative particle-displacement distribution function P (∆ x, t ) forthe data obtained at 1.7 K and 29 mW/cm . (b) The measure of the self-similarity m versus the scaling exponent γ . (c) The rescaled profiles of P (∆ x, t ) . erty holds for the apparent diffusion of the G1 particles, weexamine the P (∆ x, t ) profiles at different t for the data takenat 1.7 K and 29 mW/cm (see example profiles in Fig. 4 (a)).We use the profile at t = τ as the reference, where τ ≃
11 ms isthe time step set by the camera frame rate. To determine theoptimal exponent γ opt that gives the best match among the P (∆ x, t ) profiles after the rescaling, we minimize the profiledifference by calculating the standard L m ( γ ) = Nτ P t = τ R | ( t/τ ) γ · P (∆ x · ( t/τ ) γ ,t ) − P (∆ x,τ ) | dx R P (∆ x,τ ) dx , [1]where the summation goes over all the P (∆ x, t ) profiles ob-tained at t ∈ [ τ, Nτ ], where Nτ is the maximum diffusion timein the γ -scaling regime. Fig. 4 (b) shows the calculated vari-ance m as a function of γ . The minimum m is achieved at γ opt =1.65, which is indeed close to the diffusion scaling expo-nent γ =1.57 for the chosen data set. As shown in Fig. 4 (c),the rescaled P (∆ x, t ) profiles overlap very well except perhapsin the tail region.Indeed, for superdifussion systems involving random walk-ers, another important property of P (∆ x, t ) is its non-Gaussian tails. It has been identified that superdiffusion inthose systems is caused by long-distance hops of the walkers(53–57), i.e., the so-called Lévy flights (31). These flights leadto asymptotic power-law tails of the step-displacement distri-bution P (∆ x, τ ) ∝ | ∆ x | − α with α < P (∆ x, t ) converges to a Lévy distribution withsimilar power-law tails. The variance h ∆ x i for such a heavy-tailed distribution diverges, but a pseudo-variance behavior h ∆ x ( t ) i∝ t γ with γ = α − can be derived through a scalingargument (31, 57, 58), resulting in an apparent superdiffusion(i.e., γ > α < α ≥ h ∆ x i would converge, which then leads to a Gaussian distri-bution of P (∆ x, t ) and hence a normal diffusion of the walkers Fig. 5. (a) An example G1 particle trajectory that exhibits one large step displace-ment. (b) The tail profiles of P (∆ x, t ) , normalized by the standard deviation σ of ∆ x , at different t . The data were taken at 1.7 K and 29 mW/cm . according to the central limit theorem (58).Interestingly, the trapped G1 particles do exhibit occa-sional long-distance hops over the time step τ . An example G1trajectory that contains an exceptionally large step displace-ment is shown in Fig. 5 (a). The origin of these long-distancehops has been understood as due to the particles carried byvortex segments that are close to locations of vortex reconnec-tions (29). As revealed by Paoletti et al. (46), vortex reconnec-tions result in local high vortex-velocity occurrences, whichlead to non-Gaussian | v | − tails of the vortex-line velocity dis-tribution. Therefore, when τ is small, the step-displacementdistribution of the vortex lines should acquire similar power-law tails P (∆ x, τ ) ∝ | ∆ x | − . However, since the velocities ofthe reconnecting vortex segments become high only withina short time window centered at the moment of reconnec-tions, over longer time t the total displacement of a vortexsegment ∆ x = R t v x ( t ′ ) dt ′ would rarely exhibit exceptionallylarge values. Therefore, the tails of the resulted P (∆ x, t ) aresuppressed. To see this effect, we show the tails of P (∆ x, t )at different t in Fig. 5 (b) for the data taken at 1.7 K and29 mW/cm . Obviously, as t increases from τ to 20 τ , thetail changes from close to | ∆ x | − to nearly a Gaussian form.This observation is similar in nature to what was reportedin Ref. (59). Therefore, despite the existence of some long-distance hops of the G1 particles at small time steps, theirstatistical weight is not sufficient to render the observed su-perdiffusion. Vortex-velocity correlation.
Without invoking Lévy flights, su-perdiffusion may still emerge if the motion of the particles isnot completely random but instead exhibits extended tempo-ral correlations (58, 60). For quantized vortices in a vortextangle, the chaotic motion of the vortex segments is drivenby their self- and mutually induced velocities (3). There isno existing knowledge on whether this motion is completelyrandom or indeed has a certain temporal correlation.Mathematically, the mean squared displacement h ∆ x ( t ) i of a vortex-line segment can be evaluated based on its velocity v x ( t ) as (61): h ∆ x ( t ) i = 2 Z t dt Z t − t dt ′ h v x ( t ) v x ( t + t ′ ) i , [2]where the horizontal-velocity temporal correlation function The calculated horizontal-velocity temporal correlation function R x ( t ′ ) forthe data taken at 1.7 K and 38 mW/cm . The solid line represents a power-law fit. R x ( t ′ , t )= h v x ( t ) v x ( t + t ′ ) i for statistically steady and ho-mogeneous systems would only depend on the lapse time t ′ ,i.e., R x ( t ′ )= h v x (0) v x ( t ′ ) i . In this situation, if a power-lawscaling R x ( t ′ ) ∝ ( t ′ ) − β exists, one can easily derive from equa-tion (2) that the mean squared displacement will scale as h ∆ x ( t ) i∝ t − β . On the other hand, if R x ( t ′ ) drops rapidlywith t ′ , a normal diffusion can be obtained. In Fig. 6, weshow the calculated R x ( t ′ ) for the representative data set in-cluded in Fig. 2. At small lapse time t ′ , the data do exhibit apower-law scaling with β of about 0.4. This scaling exponentleads to h ∆ x ( t ) i∝ t . , which agrees nicely with the observedsuperdifussion. Furthermore, R x ( t ′ ) drops sharply beyond atransition time that coincides with the transition to the nor-mal diffusion as seen in Fig. 2, which naturally explains thistransition. Similar R x ( t ′ ) scaling behaviors are also observedfor other data sets. These observations provide a direct evi-dence showing the existence of a possible generic power-lawscaling of the vortex-velocity temporal correlation at scalesless than ℓ for an random vortex tangle. Discussion
The analyses that we have presented support the followingsimply physical picture. The trapped G1 particles move withthe quantized vortices in the tangle whose velocities exhibita power-law temporal correlation. This correlation leads toan apparent superdiffusion of the vortex lines in 3D space.But when their RMSD becomes greater than the mean inter-vortex distance ℓ , the vortices are expected to move acrosseach other and hence would undergo reconnections. Follow-ing the reconnections, the resulted vortex lines move aparttowards directions that are distinct from their original direc-tions (23, 25), a process that effectively randomizes the mo-tion of the vortices. This randomization then leads to a sharpdrop of the vortex-velocity temporal correlation and hence re-sults in the normal diffusion of the vortices at large lengthand time scales.Note that the spatial velocity correlation functions of vor-tices in atomic condensates have been simulated (62, 63),where the authors reported a rapid decay of the correlationover a length scale comparable to ℓ . But to test the physi-cal picture we have outlined, numerical simulations similar to Ref. (64) need to be conducted so that the temporal correla-tion of the vortex velocity in steady counterflow turbulencecan be examined. Indeed, our communication with the au-thors of Ref. (64) has returned encouraging news that theirrecent simulation does reproduce the power-law scaling of thevortex-velocity correlation as depicted in Fig. 6. These au-thors also notice that the derived diffusion exponent γ isnearly temperature independent, thereby supporting our ob-servation about the generic nature of this diffusion scaling.In summary, our work demonstrates that examining thevelocity correlations and trajectory statistics of individual vor-tices in a vortex tangle could uncover hidden scaling proper-ties of QT. Along the lines, many intriguing questions maybe raised. For instance, what is the mechanism underlyingthe observed power-law scaling of the vortex-velocity tempo-ral correlation for a random tangle? Does this scaling alsohold for a vortex tangle with large-scale polarizations? Howdoes superfluid parcels undergo apparent diffusion and disper-sion in QT? We hope that these questions will stimulate morefuture researches on vortex and superfluid dynamics. Materials and Methods
Particle tracking velocimetry.
We use solidified deuterium (D ) par-ticles as tracers in He II. These tracer particles are produced byslowly injecting a mixture of 5% D gas and 95% He gas directlyinto the He II bath via a gas injection system similar to what Fonda et al. reported (65). Upon the injection, the D gas forms small iceparticles with a mean diameter of about 4 µ m, as determined fromtheir settling velocity in quiescent He II (42). Following the particleinjection, we then turn on the heater and wait for 10 to 20 s for asteady counterflow to establish in the flow channel. A continuous-wave laser sheet (thickness: 200 µ m, height: 9 mm) passes throughthe geometric center of the channel to illuminate the particles. Thepositions of the particles in the illuminated plane are captured by avideo camera at 90 frames per second. At a give temperature andheat flux, we took a sequence of 720 images and would typicallyrepeat this data acquisition three times to obtain enough particletrajectories for statistical analyses. A modified feature-point track-ing routine (66) is adopted to extract the trajectories of the tracerparticles from the sequence of images. The velocity of a particlecan be determined by dividing its displacement from one frame tothe next by the frame separation time. Separation data analysis scheme.
To determine whether a tracerparticle belongs to the G1 group (i.e., particles that are trappedon vortices) or the G2 group (i.e., untrapped particles that are en-trained by the normal fluid), a separation data analysis scheme isadopted (28). As shown in Fig. 1 (c), the vertical-velocity distribu-tion based on the analysis of all particle trajectories exhibits twonearly separated peaks at low heat fluxes. Through Gaussian fits tothese two peaks, we can determine their respective mean velocities(i.e., ¯ v and ¯ v ) and the corresponding standard deviations (i.e., σ and σ ). Then, for a particle with a vertical velocity v y < ¯ v − a σ ,it is categorized as a G1 particle. Otherwise, if v y > ¯ v + a σ , theparticle is treated as a G2 particle. Depending on how far the G1and the G2 peaks are separated, the coefficients a and a are ad-justed in the range of 2 to 6 to better distinguish the two groups.Most of the particle trajectories can be identified as either the G1type or the G2 type. At relatively large heat fluxes, some trajecto-ries may appear to be partly the G1 type and partly the G2 type.This is due to the particles originally moving with the normal fluidlater getting trapped by vortex lines (i.e., G2 to G1) or the trappedparticles getting released during vortex reconnections (i.e., G1 toG2). In the current work, we focus on analyzing the whole andpartial trajectories that are identified as the G1 type. Second-sound attention.
We measure the volume-averaged vortex-line density L in the flow channel using the standard second-sound able 1. Measured vortex-line density L T(K) q (mW / cm ) L (cm − )1.70 74 (22 . ± . × (9 . ± . × (5 . ± . × (1 . ± . × (28 . ± . × (2 . ± . × (34 . ± . × attenuation method (44). Due to its two-fluid nature, He II sup-ports two distinct sound modes: an ordinary pressure-density wave(i.e., the first sound) where both fluids move in phase, and atemperature-entropy wave (i.e., the second sound) where the twofluids move out of phase. The second-sound waves can be gener-ated and picked up by oscillating superleak transducers (44). Thesetransducers are essentially parallel plate capacitors with one fixedplate and one flexible plate made of a thin porous membrane coatedwith an evaporated gold layer. By applying an alternating currentto one transducer as shown in Fig. 1 (a), a standing second-soundwave across the channel can be established, whose amplitude can bemeasured by the other transducer installed on the opposite channelwall. In the presence of quantized vortices, the amplitude of thesecond-sound wave is attenuated, and the degree of this attenua-tion can be used to calculate the vortex-line density L (42). Table 1lists our measurement results under various temperatures and heatfluxes. Data Availability.
The analysis results together with the flow visual-ization data and the second-sound data can be obtained from thecorresponding author upon request.
ACKNOWLEDGMENTS.
The authors would like to acknowledgethe valuable discussions with W. F. Vinen and D. Kivotides. Theauthors also thank S. Yui, H. Kobayashi, and M. Tsubota for com-municating their recent simulation results. This work is supportedby the National Science Foundation (NSF) under Grant No. DMR-1807291 and the U.S. Department of Energy under Grant No. DE-SC0020113. The experiment was conducted at the National HighMagnetic Field Laboratory at Florida State University, which issupported through the NSF Cooperative Agreement No. DMR-1644779 and the state of Florida.
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