Statistical laws and self-similarity of vortex rings emitted from a localized vortex tangle in superfluid 4 He
SStatistical laws and self-similarity of vortex rings emitted from a localized vortextangle in superfluid He Tomo Nakagawa, Sosuke Inui, Makoto Tsubota,
1, 2, 3 and Hideo Yano
1, 2 Department of Physics, Osaka City University, 3-3-138 Sugimoto, 558-8585 Osaka, Japan Nambu Yoichiro Institute of Theoretical and Experimental Physics(NITEP),Osaka City University, 3-3-138 Sugimoto, 558-8585 Osaka, Japan The Advanced Research Institute for Natural Science and Technology(OCARINA),Osaka City University, 3-3-138 Sugimoto, 558-8585 Osaka, Japan (Dated: May 1, 2020)We numerically simulated quantum turbulence in superfluid He to investigate the emission ofvortex rings from a localized vortex tangle. Turbulence is characterized by some universal statisticallaws. Although there are a lot of studies on statistical laws in bulk quantum turbulence, studies ininhomogeneous or localized turbulence is scarce. We first investigate the statistical laws of localizedquantum turbulence, referring to two statistical laws deduced from the vibrating wire experimentsin [Yano et al. , J. Low Temp. Phys. ,
184 (2019)]. The first law is the Poisson process for thedetection of vortex rings; the vortex tangle emits vortex rings with frequencies depending on theirsizes. The second law is the power law between the frequency and the size of the emitted vortexrings, showing the self-similarity of the tangle. To study these statistical laws numerically, wedeveloped a system similar to experiments. First, we generate a localized statistically steady vortextangle by injecting vortex rings from two opposite sides and causing collisions. We investigated theconditions that aid the formation of the tangles and the anisotropy of the emission of vortex ringsfrom the tangle. Second, from the data on emitted rings, we reconstruct the two statistical laws.Results from our numerical investigations are consistent with the known self-similarity of emittedvortex rings and localized tangles.
I. INTRODUCTION
Quantum turbulence refers to turbulent states in quan-tum condensed fluids. It is an important phenomenonin low temperature physics and fields such as fluid me-chanics, and non-equilibrium physics. Superfluid He isa typical system wherein quantum turbulence is studied.A lot of researchers have investigated superfluid He forover half a century [1–4]. Some statistical laws are of-ten investigated to determine the universal properties ofturbulence. In classical turbulence, an important sta-tistical law is the Kolmogorov’s law that indicates thatthe energy spectrum follows the − / He, vortices are stable topologicaldefects and their circulation is conserved by quantiza-tion. The quantum circulation is given by κ = h/m ,where h and m are Planck’s constant and the mass of a He atom [7, 8]. Because quantum turbulence consistsof well-defined elements, studies can provide a shortcutto investigate turbulence. The self-similarity of quan-tum turbulence in a wave number space such as Kol-mogorov’s law was studied [9–17]. However, studies on self-similarity in a real space are scarce; an example ofsuch a study is [13, 18, 19]. We focus on the statisticallaws in a real space assuming that quantum turbulencehas some self-similarity.Liquid He changes to superfluid phase at tempera-tures below T λ = 2 .
17 K, and its hydrodynamics can bedescribed by the two-fluid model. This implies the super-fluid He is a mixture of a viscous normal fluid compo-nent and an inviscid superfluid component [20, 21]. Thedensity and velocity of the superfluid component are ρ s ,and v s , respectively and those of the normal fluid com-ponent are ρ n , and v n , respectively. The total densityis ρ = ρ s + ρ n . The ratio ρ s /ρ increases with decreas-ing temperature. Particularly, below approximately 1 K,the ratio is ρ s /ρ (cid:39)
1. At finite temperatures, mutualfriction acts between the two components through quan-tized vortices. Mutual friction can significantly shrink avortex ring that moves in the fluid.There are several methods to generate quantum tur-bulence in a superfluid He [1–3], and experiments usingoscillating objects have been recently conducted [22–33].A vibrating wire is a typical oscillating object. Thin wiresare vibrated by a Lorentz force under a static magneticfield, which generates turbulence around the wire. Yano et al. conducted a series of experiments using vibratingwires [24, 29, 33, 34]. From the Yano group two kindsof vibrating wires, namely, a generator of turbulence anddetector of vortices were discovered. A generator wire hasremnant vortices, whereas a detector wire has no remnantvortices. Although the wire velocity increases with thedriving force, the two kinds of wires have different behav-iors. If the driving force exceeds some critical value, the a r X i v : . [ c ond - m a t . o t h e r] A p r velocity of the generator wire decreases immediately anda vortex tangle is generated around it. However, a detec-tor wire does not generate turbulence by itself becauseof the success of removing remnant vortices around it. Ifa vortex ring approaches a detector wire, it generates avortex tangle around it using the ring as a trigger there-after decreasing the wire velocity. Accordingly, Yano etal. performed experiments using a detector wire to detectthe vortex rings emitted from a vortex tangle made by agenerator wire.An important feature of the experiments is that it ispossible to manage the minimum size of detectable vortexrings by altering the temperature. At 0 K, a vortex ringmoves with its self-induced velocity without shrinking.At finite temperatures, a vortex ring shrinks in its flightand can disappear by mutual friction. The flight distance l for a vortex ring with an initial radius R disappearsis given by l = R /α , where α is the mutual frictioncoefficient described later. Therefore, the diameter 2 R of a detectable vortex ring satisfies 2 R > αD , where D is the distance between the detector and the generatorwires.Using the setup, Yano et al. recently observed someself-similarity of vortices emitted from a vortex tangle.This experiment discovered two important laws. First,the time of flight of vortex rings from the vortex tan-gle to the detector wire follows exponential distributionsfor any detectable minimum size. Particularly, the de-tection of vortex rings follows a Poisson process. Thismeans that vortex rings are detected with frequenciesdepending on their sizes; hence, a vortex tangle is in astatistically steady state. Second, the vortex tangle hasself-similarity. From the experiment, the relationship be-tween the detection frequency and the minimum size ofthe detectable vortex rings that satisfies the power lawwas determined. The vortex ring size should reflect thevortex line spacings in the tangle. Therefore, the vortextangle can have a self-similar structure in a real space.These results show the statistical laws of a localizedvortex tangle. Although there are a lot of studies onstatistical laws in bulk quantum turbulence, studies ininhomogeneous or localized turbulence is scarce.For experiments on quantum turbulence generated byoscillating objects, several numerical simulations havebeen conducted. The purpose of previous simulationswas to investigate the processes of growth and decay of alocalized vortex tangle or the anisotropy of the emissionof vortex rings from a tangle [34–37]. This study focuseson the statistical laws and the self-similarity of vortexrings emitted from a localized tangle, which differs fromthe previous works.Using the vortex filament model, we numerically ex-amine the dynamics and statistics of vortices emittedfrom a localized vortex tangle. Our goal is to examinethe statistical properties of this system and to comparewith the experimental results. First, we obtain a local-ized statistically steady vortex tangle as the source ofemitted vortex rings. Second, we study the statistics of the vortex rings emitted from the tangle. In Section II,we introduce the vortex filament model and the systemtreated in this study. Thereafter, we describe the forma-tion of vortex tangles in Section III. In Section IV, wediscuss statistically steady vortex tangles and introducesome theoretical concepts. Furthermore, we present thestatistical laws and compare the exponents of the powerlaws with those of the experimental results in Section V.Finally, Section VI presents the conclusions. II. THE MODEL AND SYSTEMA. Vortex filament model
Quantized vortices in superfluid He are stable topo-logical defects with quantized circulation and thin coresof order 1 ˚A. Therefore, we can use the vortex filamentmodel wherein vortices are treated as filaments. The su-perfluid velocity field obtained owing to quantized vor-tices is given by the Biot-Savart law [38] v s , BS ( r , t ) = κ π (cid:90) L s (cid:48) ( ξ, t ) × ( r − s ( ξ, t )) | r − s ( ξ, t ) | dξ , (1)where s ( ξ, t ) denotes the position of the vortex filamentsrepresented by the parameter ξ , and s (cid:48) = ∂ s ∂ξ . The inte-gration is performed over the whole vortex filaments L .At finite temperatures, mutual friction affects the motionof vortices. If there are neither boundaries nor appliedsuperfluid flow, the equation of motion becomes d s dt = v s,BS + α s × ( v n − v s,BS ) − α (cid:48) s (cid:48) × [ s (cid:48) × ( v n − v s,BS )] , (2)where α and α (cid:48) are the coefficients of friction dependingon the temperature. In particular, α, α (cid:48) = 0 at T = 0 K.The vortex lines are discretized into a number of pointsheld at a minimum space resolution ∆ ξ = 0 . µ m.The integration in time is performed using the fourth-order Runge-Kutta scheme, wherein the time resolutionis ∆ t = 10 µ s. We use the traditional method to arti-ficially reconnect two vortices that approach each otherwithin ∆ ξ [39]. We delete the vortex rings whose lengthsare shorter than 6∆ ξ .Such reconnection of quantized vortices can be relatedto the dissipation mechanism of quantum turbulence atvery low temperatures with negligible mutual friction.The numerical simulation of the Gross-Pitaevskii modelshows that reconnections emit phonons of short wave-lengths comparable to the coherence lengths and causesthe dissipation [40]. However, the vortex filament modelcannot describe the phonon emission. The change ofvortex length at each artificial reconnection is negligi-ble compared to the vortex dynamics in the large scales.This study focuses on the statistical laws at large scaleswherein details of each reconnection is not relevant. B. The system
The motivation of this study is to reproduce the sta-tistical laws observed by the experiment and reveal theself-similarity of the system. To achieve this, we firstobtain a localized stationary vortex tangle as the sourceand thereafter observe the emission of vortex rings fromthe tangle.The method of generating a localized vortex tangle isa key problem in our simulation. We are predominantlyinterested in the emission of vortex rings from a local-ized vortex tangle. We use a novel method that differsfrom those in previous simulations [34–37] to generate adense localized vortex tangle that can emit many vor-tex rings. As shown in Fig. 1, we prepare two parallel100 µ m × µ m square vortex sources that inject vor-tex rings of some size at a fixed frequency from randompositions in each square. The distance between the twoparallel sources is 240 µ m. The parameters of this sim-ulation are the injection frequency f and the diameter2 R of the injected vortex rings. Now, let f be of order1 kHz corresponding to the frequency of the vibratingwire and 2 R be of order 10 µ m corresponding the am-plitude of the vibration [33]. To be later described in de-tail, we maintain injecting vortices from the two sourcesand generating a localized vortex tangle. These tanglesexpand orthogonally to the direction of the injection, asshown in Fig. 2. These tangles are regarded as sourcevortex tangles formed from a vibrating object.Furthermore, we simulate the detection of vortices bythe detector. One detector was used in the experiment.The experiment was repeated severally to use the obser-vations to obtain the statistical law [33]. However, oursimulation allocates many detectors around the tangle.The vortex tangle and the emitted vortex rings shouldbe symmetric about the azimuthal angle φ . The vortexrings are emitted orthogonally to the direction of injec-tion, as described in the next section. We collect thedata on the vortex rings emitted from a vortex tangle ata fixed distance of 400 µ m from the origin and eliminatefrom the vortices we follow. III. PROPERTIES OF THE LOCALIZEDVORTEX TANGLE
Comparing to the experiments [33], the success of oursimulation depends on obtaining statistically steady lo-calized vortex tangles by the method described in the lastsection. In this section, we describe the development ofthe vortex tangle and show that a statistically steady vor-tex tangle can be generated. Thereafter, we describe thestatistics of the observations of the vortex rings emittedfrom the vortex tangle.
FIG. 1. The coordinate system of this system. We set the x -axis as the injection direction of vortex rings. Vortex ringsare injected from two parallel 100 µ m × µ m square vortexsources at a fixed frequency.FIG. 2. Development of vortex tangle in f = 1000 Hz and2 R = 30 µ m at time (a) t = 0 .
02 s (b) t = 0 .
06 s (c) t = 0 .
16 s(d) t = 0 .
40 s, respectively. The black rectangular box refersto the box in Fig. 1. Although the vortex tangles of (a) and(b) grow, those of (c) and (d) are saturated inside the box.
A. Development of vortex tangle
Figure 2 shows a typical development of a vortex tan-gle. If vortex sources begin to inject vortex rings, theyform a small nucleus of a vortex tangle around the originif they frequently collide (See Supplemental Material).Thereafter, the nucleus absorbs subsequent vortex ringsand develops a vortex tangle. Although this explanationis satisfactory, the statistical steadiness or unsteadinessof the resulting localized vortex tangle is nontrivial.Figure 3 shows the vortex line density distribution af-ter the tangle develops significantly and is statisticallysteady. The density decreases with increasing distancefrom the origin. The density is concentrated around θ = π because of the symmetry of the system.Thereafter, we directly investigate the properties of thetangle. The vortex distribution in Fig. 3 includes emit- FIG. 3. The time averaged( t = 0 . . − ) of the vortex tangle in a r ( µ m) − θ (rad) plane in the log scale. The condition is f =1000 Hz and 2 R = 30 µ m. Because the vortex tangle issymmetric around the azimuthal angle φ , the distribution isobtained by integrating over φ .FIG. 4. The vortex line length in the cylindrical volume in2 R = 30 µ m and f = 1000 Hz. The cylindrical volume hasits height 160 µ m and its radius 250 µ m. This cylinder coversthe vortex tangle. The black box in the left figure is as same asthat in Fig. 2. The figure on the right shows development ofthe vortex line length. The vortex tangle becomes statisticallysteady after approximately t = 0 . ted vortex rings and a localized tangle. From Fig. 3, weknow that the tangle expands orthogonally to the x –axisand can estimate the approximate size of the tangle. Weassume a cylindrical volume with height 160 µ m and ra-dius 250 µ m that covers the vortex tangle and reflectsits symmetry. The centroid of the volume is placed atthe origin, and the bottom is orthogonal to the x –axis.Figure 4 shows the development of the total vortex linelength in the cylindrical volume. The vortex line lengthincreases with time and is statistically steady after ap-proximately t = 0 . L ( s ) of the vortex line density in the hollowcylindrical volumes with height, inner radius, and outerradius as 160 µ m, s − ds , and s , respectively. The centroidof the volume is placed at the origin, and the bottomis orthogonal to the x –axis. Fig. 5(a) shows the time-averaged distribution L ( s ). Because the distribution doesnot change significantly after approximately t = 0 . FIG. 5. (a) The time averaged vortex line density distributionin f = 1000 Hz and 2 R = 30 µ m. The colors represent thedistribution averaged over the different time intervals. (b)The time development of s c in f = 1000 Hz. The colorsrepresent the different injected vortex ring sizes 2 R . Thedensity is averaged for the time interval between t − . t + 0 . From the distribution of the vortices, we estimate thesize of the tangle, although there is some arbitrarinessfor defining the size of the tangle. We define the tanglesize s c such that the density L ( s ) decreases to 10 cm − in the volume.Figure 5(b) shows the time development of s c for2 R = 30 , , µ m and f = 1000 Hz. The tanglesize converge to a finite value in each condition. The sizeof tangle in the statistically steady state increases with2 R .The subsequent steadiness or unsteadiness of a devel-oped vortex tangle is nontrivial. Injected vortex ringsare shuffled to form a localized vortex tangle. The vortextangle emits vortex rings that operate as the dissipationfor the tangle. The statistically steady state is sustainedby the equilibrium of the vortex ring injection, the dele-tion of small rings, and the vortex ring emission from thetangle. We do not know if such statistical steady statesare consistently obtained for arbitrary values of 2 R and f . This can be investigated in future studies. B. The vortex line length of vortex tangle
If the injection frequency f and the size 2 R of theinjected vortex rings are reduced, a statistically steadyvortex tangle may not be generated because vortex ringsdo not frequently collide and generate no nucleus of atangle.We calculate the dynamics with varying f and 2 R tostudy the conditions for the generation of a statisticallysteady vortex tangle. Figure 6 shows that the statisticallysteady vortex line density in the cylinder increases with f and 2 R .The appearance (or no appearance) of a nucleus ofa vortex tangle determines its growth. If no nucleus isformed, no vortex tangle occurs. Investigating the condi-tions for the formation of a nucleus aids the determina-tion of the characteristic vortex lengths. If injected vor-tex rings collide and interact, the vortex length increases.We consider the vortex length when counter-propagating FIG. 6. The mean vortex line density in the cylindricalvolume. The horizontal axis is the frequency f .FIG. 7. The vortex line length normalized by L n of Eq. (3).Here 2 R and l are the size of injected vortex rings and theheight of the cylindrical volume, respectively. rings pass through and no nucleus is formed. This con-sideration yields the characteristic vortex length used tonormalize the vortex line length in the cylinder. Thecharacteristic length can be determined from the geom-etry of Fig. 1. If counter-propagating vortex rings nevercollide, we can obtain the time 4 πlR / ( κ log( R /r c )) thattaken for an injected ring to pass through the cylindricalvolume because of the self-induced velocity of the vor-tex ring v ∼ κ/ πR × log( R /r c ). Here, l is the heightof the cylindrical volume. Because the length of an in-jected vortex ring is 2 πR , the total vortex line length L n required is L n = 2 × f × πlR κ log( R /r c ) × πR = 16 π lf R κ log( R /r c ) . (3)The vortex line length normalized by L n is shown in Fig.7. This quantity is the amplification factor of the vortexline length. If the normalized length exceeds unity, the mere group of ballistic vortex rings develops to a vortextangle. Increasing the injected vortex ring size and thefrequency of the injection increases the length that con-verges to approximately 3 . R . Thismeans that the vortex line length is proportional to f R . C. Emission of vortex rings from a vortex tangle
FIG. 8. The probability density function, PDF ( θ ), of thedirection of the emitted vortex rings about θ . The PDF isobtained from the number of vortex rings received by thedetectors within 2 π sin θdθ . Although the distribution of the emitted vortex ringsis isotropic about φ , that in the direction θ is anisotropic.The probability density function (PDF) in the case of thedirection θ of the vortex rings emitted from the vortextangle is shown in Fig. 8. The data of the vortex ringsare collected by the detectors placed at 400 µ m from theorigin. The emission of the vortex rings is concentratedaround θ = π . IV. STATISTICALLY STEADY TANGLE
A statistically steady vortex tangle should emit vortexrings of each size with the corresponding frequency. Par-ticularly, the emission frequency of some size is governedby a function f ( R ) that depends only on the vortex ringsize R . This statistically steady concept is essential inthis study. From our simulation, f ( R ) is a power of R ,that is, the emission of the vortex rings from a tanglehas some self-similarity. We introduce some theoreticalconcepts of this self-similarity in this section. A. Poisson process
To investigate the statistics of emitted vortex rings,the experiment is conducted assuming a Poisson process[33]. The Poisson process is a stochastic process basedon an exponential distribution. First, we describe thederivation of the process considering the conditions ofthe experiment. We make the following assumptions. Adetector catches n vortex rings in the time interval [0 , T ]that is partitioned into smaller intervals ∆ t . Thus, theprobability that a vortex ring is detected in ∆ t is ∆ t nT .This is based on the assumption that ∆ t is significantlysmall such that the number of vortex rings received in∆ t is 0 or 1. The probability that a vortex ring is notdetected in [0 , t ] and detected in [ t , t + ∆ t ] is (cid:16) − ∆ t nT (cid:17) t ∆ t × ∆ t nT . (4)If P ( t ) is the probability that a vortex ring is detected in[0 , t ], we have P ( t + ∆ t ) − P ( t ) = (cid:16) − ∆ t nT (cid:17) t ∆ t × ∆ t nT . (5)The PDF F ( t ) is defined by the probability F ( t ) dt that avortex ring is detected in [ t , t + ∆ t ]. Hence, F ( t ) is givenby F ( t ) = lim ∆ t → P ( t + ∆ t ) − P ( t )∆ t = 1 t exp (cid:18) − tt (cid:19) (6)where t ≡ Tn is the mean detection interval. Finally,integrating F ( t ) yields P ( t ) = (cid:90) t F ( t (cid:48) ) dt (cid:48) = − exp (cid:18) − tt (cid:19) + 1 (7)and 1 − P ( t ) = exp (cid:18) − tt (cid:19) . (8)This indicates a Poisson process. If this relation is con-firmed, the interval t will be constant indicating thatthe vortex tangle is statistically steady and emits vortexrings continuously and steadily. Yano et al. observed thisrelation that indicates that the generator wire generatesa statistically steady vortex tangle [33]. This is a motiva-tion for our investigation to obtain a statistically steadyvortex tangle in the present simulation. B. The self-similarity
Suppose that the vortex tangle has self-similarity inreal space. We find the power law between the emis-sion frequency and vortex ring size. The power law wasdeduced from experiments in [33]. This self-similarityis understood from the following discussions. We definethe number n ( x, t ) dx of vortex rings with diameter in[ x, x + dx ] emitted from a tangle in [0 , t ]. The number N R> R C ( t ) of vortex rings with diameters larger than2 R C emitted in [0 , t ] is N R> R C ( t ) = (cid:82) ∞ R C n ( x, t ) dx . Be-cause the vibrating wire experiments observe only vortex rings larger than some minimum size, we also consider thenumber of vortex rings larger than some minimum size.When the vortex tangle is statistically steady, the num-ber of emitted rings n ( x, t ) dx is shown by g ( x ) tdx andthe vortex tangle emits vortex rings with various sizes.The g ( x ) dx is the emission frequency of vortex rings withsizes in [ x, x + dx ]. Therefore, the frequency f R> R C ofthe emission of rings larger than 2 R C is given by f R> R C = (cid:90) ∞ R C g ( x ) dx. (9)The distribution of vortices in the tangle should be de-termined by that of the emitted vortex rings. Several lit-erature have reported that the size distribution of vortexrings in a tangle, that shows that the number of vortexrings decreases with ring sizes by some power law [13, 18].Assuming the emission of the vortex rings is self-similar,the frequency g ( x ) can be written as g ( x ) = x − α suchthat f R> R C = (cid:90) ∞ R C x − α dx ∝ (2 R C ) − α +1 . (10)Thus, the power law comes from the self-similarity of thesize distribution of the vortex rings in the tangle. V. STATISTICAL LAWS
We describe the statistical laws of the vortex ringsemitted from a vortex tangle. The first law is the Pois-son process of the detection of the vortex rings emittedfrom a tangle. The second is the power law between thefrequency of the emission and the vortex ring size.The statistically steady vortex tangle emits vortexrings, as mentioned in Section IV. In this section, wenumerically investigate the probability that detectors re-ceive vortex rings with diameters larger than some mini-mum diameter 2 R . Experiments are performed using onedetector. The experiment is repeated to determine thestatistics. However, our simulation involved one simula-tion with 2000 detectors. A. Poisson process
We position detectors at a fixed distance of 400 µ mfrom the origin and orthogonal to the x -axis as shownin Fig. 9. In this study, the number of detectors N det is 2000. The simulation indicates the number N ( t ) ofdetectors that receive at least one vortex ring in [0 , t ][41]. The probability P ( t ) is given by P ( t ) = N ( t ) /N det . (11)This relation is a key idea that relates our simulation tothe experiment. FIG. 9. The schematic figure of the arrangement of the de-tectors around the vortex tangle. The detectors are arrangedsymmetrically with the width 270 µ mFIG. 10. The time development of 1 − P ( t ) in f = 1000 Hzand 2 R = 30 µ m. Here, 2 R refers to the minimum size ofdetectable vortex rings. Figure 10 shows the results of the simulation with f = 1000 Hz and 2 R = 30 µ m. They satisfy Eq. (8);hence, our simulation reproduces the Poisson process ob-served experimentally. These slopes indicate the detec-tion frequencies t − .To directly investigate the property of a vortex tan-gle, we examine the emission frequency. Figure 11 showsthe number N emi ( t ) of vortex rings that have diameterlarger than 2 R and emitted in [0 , t ]. We can confirmthe linear relationship N emi ( t ) = t/t with the emissionfrequency t − . This figure shows that the frequency be-comes constant indicating that the vortex tangle becomesstatistically steady. B. The power law
The power law between t − and 2 R indicates the self-similarity of the vortex rings emitted from the localizedvortex tangle. From Fig. 12, the emission frequency t − satisfies the power law of 2 R for three different values of2 R . For f = 1000 Hz and 2 R = 30 µ m, the powerlaw t − = (2 R ) − . ± . is obtained by the least squaresmethod. Therefore, we can obtain results similar to the FIG. 11. The number N ( t ) of the vortex rings emitted in[0,t]. The frequency of emission converges at finite number ineach minimum size. Here, 2 R refers to the minimum size ofdetectable vortex rings.FIG. 12. The relationship between t − and 2 R in the log-logscale. experiments. Here, we determine the slope in the rangeup to 60 µ m. They show power laws, however, they de-viate for 2 R > µ m. We propose two reasons for thedeviation. The first may come from the rare events catch-ing such large vortices. Second, there may be a differencebetween the emission mechanism of vortex rings smallerand larger than the size of the tangle. The large rings canbe emitted only from the surface of the tangle, whereasthe small ones can be emitted from the surface or insidethe tangle. This result shows that the distribution ofvortex rings emitted from the localized vortex tangle hasself-similarity, that may reflect the self-similarity of thevortex size distribution of a vortex tangle.Finally, we compare the power exponent obtained fromthis study with that obtained from the experiments. Inthe experiments, the exponent depends on the turbu-lence generation power. Yano et al. observed that theexponents increased to − . − .
6, and − . ,
150 pW and 1000 pW, re-spectively. Whereas our simulation shows power expo-nents − . ∼ − .
03, that differ from the experimentalresults because the vortices become significantly dense tobe calculated numerically.The difference in the exponents between the simula-tion and the experiments may connected with the emis-sion power. The energy (cid:15) of the vortex filaments perunit length is (cid:15) = ρ s κ π ln (cid:16) R cv r c (cid:17) ∼ . × − J / m.Thus, the energy injected per unit time is 2 πR × f × (cid:15) ∼ − pW. The order of injected energy in this study issignificantly lower than that in the experiments. How-ever, it is numerically difficult to increase the power tomake it comparable with the experiments. VI. CONCLUSION
We numerically investigate the emission of vortex ringsfrom a statistically steady localized vortex tangle thatwas formed by colliding vortex rings. Our study be-gins the study of statistical laws of localized quantumturbulence. We developed a system similar to the ex-periments and investigated the two statistical laws. Wesucceeded in obtaining the laws, although the exponentsin the power laws were different from the experimentalresults.In this study, although we performed simulations at T = 0 K, the experiments were performed at finite tem-peratures. An advantage of performing the simulation at 0 K is that, the vortex rings do not shrink spontaneously;hence, the sizes of the vortex rings emitted from the tan-gles can be easily fixed. However, at finite temperatures,the mutual friction shrinks the vortex rings whose sizescannot easily determined.Therefore, it is ideal to perform the simulation at finitetemperatures. There are predominantly two methods toachieve this. The first method is traditional, namely, fol-lowing the vortex dynamics under the prescribed normalfluid velocity [38]. This can be easily calculated, andwe will consider in subsequent research. The second isto consider the fully coupling dynamics between a nor-mal fluid component and a superfluid component [42–45].This is better than the first method. However, it is dif-ficult to calculate the fully coupled dynamics if used forthe present problem.Our subsequent research will investigate the self-similar structure of vortex tangles, such as a fractal di-mension [46], and associate with the statistical laws ofthe emitted vortex rings addressed in this paper. Wewill adjust the method used to inject vortex rings andinvestigate its effects on the statistical laws. For exam-ple, we can inject trains of vortex rings with expectedturbulence generated by moving ions [47, 48]. ACKNOWLEDGMENTS
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