Mutual friction in superfluid 3 He-B in the low-temperature regime
aa r X i v : . [ c ond - m a t . o t h e r] F e b Mutual friction in superfluid He-B in the low-temperature regime
J.T. M¨akinen and V.B. Eltsov Low Temperature Laboratory, Department of Applied Physics, Aalto University, FI-00076 AALTO, Finland (Dated: September 29, 2018)We measure the response of a rotating sample of superfluid He-B to spin-down to rest in the zero-temperature limit. Deviations from perfect cylindrical symmetry in the flow environment cause theinitial response to become turbulent. The remaining high polarization of vortices along the rotationaxis suppresses the turbulent behavior and leads to laminar late-time response. We determine thedissipation during laminar decay at (0 . − . T c from the precession frequency of the remnantvortex cluster. We extract the mutual friction parameter α and confirm that its dependence ontemperature and pressure agrees with theoretical predictions. We find that the zero-temperatureextrapolation of α has pressure-independent value α ( T = 0) ∼ · − , which we attribute toa process where Kelvin waves, excited at surfaces of the container, propagate into the bulk andenhance energy dissipation via overheating vortex core-bound fermions. I. INTRODUCTION
The superfluid phases of He were the first experimen-tally accessible macroscopic quantum systems where themulti-component order parameter supports a variety ofquantized vortices with non-singular cores.
Such vor-tices can possess hard cores with radius of the order of thecoherence length, filled with a superfluid phase differentof that in the bulk. The vortex-core-bound fermions playan important role in the dynamics of the vortices. Theyinteract with the bulk thermal excitations, leading to aforce, called mutual friction, acting on a vortex in an ex-ternal flow field. The mutual friction is well understoodat higher temperatures ( T & . T c ) both theoretically and experimentally, but the quantitative experimen-tal confirmation in the zero temperature limit has beenabsent. Furthermore, non-zero extrapolation of dissipa-tion to T = 0 has been observed both in He and in He. In He experiments the remnant dissipation canbe attributed to He impurities in the sample. On theother hand, superfluid He is isotopically pure, but finitezero-temperature extrapolation is observed nonetheless.Spin-down measurements, where a steadily rotatingcontainer is abruptly brought to rest, provide well con-trolled access to superfluid vortex dynamics. Duringthe steady rotation the quantized vortices form a well-defined lattice in which the vortex density is controlledby the angular velocity. When the container is broughtto rest, the normal component imposes a force on vor-tices. In He the post-spin-down dynamics are alwaysturbulent, while in He in a cylindrical container theresponse is found to be laminar at least down to 0 . T c . Deviations from perfect cylindrical symmetry or intro-duction of dissipative AB phase boundary lead to (atleast partially) turbulent response to spin-down also in He-B.In turbulent spin-down the dissipation is greatly en-hanced by vortex reconnections, in particular. Theresulting time scales are generally much faster than forlaminar vortex motion, where the scale is determined bymutual friction. In this work we apply nuclear magnetic resonance (NMR) techniques and Andreev scattering ofquasiparticles to probe the vortex dynamics afterspin-down to rest in He-B in the T → II. MUTUAL FRICTION
Vortex motion r = r + ( v L − v n ) t with respect tothe normal fluid motion v n leads to pumping of core-bound fermions along the zero-crossing branch in the en-ergy spectrum. This phenomenon is known as spectralflow. Here v L is the velocity of the vortex, r = r ( t = 0)its initial position, and t is time. The energy levels of thecore-bound fermions are separated approximately by en-ergy ~ ω ∼ ∆ /E F ≪ ∆, called the minigap. Here E F isthe Fermi-energy and ∆ is the superfluid gap. Relaxationof the core-bound fermions towards the thermal equilib-rium with time constant τ leads to a net force acting ona vortex F N = D ( v n − v L ) ⊥ + D ′ ˆz × ( v n − v L ) . (1)Parameters D and D ′ are given by D = ρκ ω τ ω τ tanh ∆( T )2 k B T (2)and D ′ = ρκ (cid:20) − ω τ ω τ tanh ∆( T )2 k B T (cid:21) − ρ s κ, (3)where τ is the average lifetime of Bogoliubov quasi-particles at the Fermi surface, − ρ s κ is the Iordanskiiforce, ρ = ρ s + ρ n is the total fluid density where ρ s and ρ n are the densities of the superfluid and the normalcomponents, respectively, and T is temperature.If the vortex mass is neglected, the total force actingon a vortex, which includes mutual friction and Magnusforces, should be zero. The force balance can be writtenas ( v n − v L ) × ˆ z + α ( v n − v s ) ⊥ + α ′ ˆz × ( v n − v s ) = 0 , (4)where the first term is the Magnus force. The mutualfriction parameters α and α ′ are defined as α = D/κρ s ( D/κρ s ) + (1 − D ′ /κρ s ) (5)and α ′ = 1 − − D ′ /κρ s ( D/κρ s ) + (1 − D ′ /κρ s ) . (6)In the T → α ′ ∼ α cansafely be neglected. In the same limit the dissipativeterm becomes α ∼ ω τ (7)and the temperature dependence is dominated by thequasiparticle lifetime τ ∝ exp (cid:18) ∆ k B T (cid:19) . (8) III. LAMINAR SUPERFLOW AND RESPONSETO SPIN-DOWN
The coarse-grained hydrodynamic Hall-Vinen-Bekarevich-Khalatnikov equation for the superfluidvelocity v s is ∂ v s ∂t + ∇ µ − ( v s ·∇ ) v s = − α ˆ ω × [( v s − v n ) × ( ∇× v s )] , (9)where µ is the chemical potential and ˆ ω is a unit vectoralong the vorticity. Assuming that vortices remain highlypolarized along the rotational axis as generally is the casefor rotating superflow, vortex reconnections can beignored. The superfluid mimics laminar solid-body-likemotion and quantized vortices create combined super-fluid velocity field v s = Ω s ˆz × r , where Ω s is the angularvelocity of rotation around axis ˆz . With this form of v s Eq. (9) transforms after taking curl of both sides todΩ s ( t )d t = 2 α Ω s ( t )[Ω − Ω s ( t )] , (10)where Ω is the angular velocity of the normal component,assumed to be equal to the drive. If a step-like changefrom Ω = Ω to 0 is performed at t = 0, the response at t > s ( t ) = Ω t/τ , (11)where τ = (2 α Ω ) − . In reality, the step is performedat finite rate − ˙Ω. During the deceleration, i.e., for − Ω / ˙Ω < t <
0, Eq. (10) has solutionΩ s ( t ) = p ˙Ω exp[ α ( t + Ω / ˙Ω)(Ω − ˙Ω t )] τ p ˙Ω + √ πα exp( α Ω / ˙Ω)erf (cid:16)p α ˙Ω t (cid:17) , (12) Magnon BEC5.85 mm mm Quartztuning-forkHeat exchanger NMRpick-up coils
FIG. 1: Experimental setup used in the measurements. Thecontainer (not to scale) is a quartz-glass cylinder with smoothwalls to avoid vortex pinning. It is rotated about its verticalaxis. The NMR pick-up coils, located close to the top ofthe container, are used to probe the vortex dynamics alongwith two quartz tuning forks located at the bottom close tothe heat exchanger. The quartz tuning forks are additionallyused for thermometry since they are sensitive probes for localquasiparticle density. The bottom of the container is opento a heat exchanger volume with rough surfaces covered withsintered silver. where τ = Ω − + q α/ ˙Ω exp( α Ω / ˙Ω)erf (cid:18)q α/ ˙ΩΩ (cid:19) .At low temperatures the typical time scales of the vor-tex dynamics are much longer than those of the deceler-ation in our experiments, i.e. α ≪ ˙Ω / Ω , and at the endof the deceleration Ω s ( t = 0) ∼ = Ω . It is thus justified touse Eq. (11) at all times t > IV. EXPERIMENTAL SETUP
The He-B sample is contained in a 150 mm longsmooth-walled cylindrical quartz-glass container with5 .
85 mm inner diameter, illustrated in Fig. 1. The bot-tom of the container is open to silver-sintered surface act-ing as a heat exchanger. The pressure in the sample isvaried between 0 and 29 bar and the sample can be cooleddown to 0 . T c . The sample is rotated with angular ve-locities up to 2 rad/s. The maximum rate of decelerationis − ˙Ω = − .
03 rad/s . The axial symmetry is brokenby two quartz tuning forks, used as thermometers, and by vortex pinning to the sintered surface at the bot-tom. The inner surfaces of the quartz glass cylinder weretreated with hydrofluoric acid to avoid vortex pinningelsewhere.In the ballistic regime the forks’ resonance width is pro-portional to the Boltzmann factor exp( − ∆ /k B T ). Inthe presence of a superfluid flow field, created for exampleby a nearby vortex bundle, the forks’ resonance width be-comes a function of the surrounding vortex structure.
Owing to Andreev reflection the vortices shadow part of
Time, s0 500 1000 1500 2000 2500 F o r k w i d t h , H z Time, s400 450 500 550 ∆ f , H z ∆ f , H z Ω , r a d / s FIG. 2: Response of the resonance width of the thermometerfork recorded during spin-down from Ω = 1 .
02 rad/s to rest.The initial overshoot is caused by heat produced by turbulentdissipation of the vortex cluster after the spin-down. Time t =0 corresponds to the moment when the drive Ω reaches zero.The insets show zoomed view of the late-time response. Theperiodic oscillations originate from precession of a remnantvortex cluster. The increase of the oscillation period withtime is used to extract the dissipation. the heat flow emanating from the walls of the container. After a spin-down we see oscillations with increasing pe-riod in the resonance width of the fork, see Fig. 2. Weinterpret these oscillations as caused by a precessing vor-tex cluster which develops some rotational asymmetry asa result of the spin-down.Our setup also includes a set of NMR pick-up coils,used to probe the spatial distribution of the order-parameter, called texture. We apply rf pulses thatexcite transverse spin waves, or magnon quasiparticles.Pumped magnons quickly form Bose-Einstein condensate(BEC) in the magneto-textural trap close to the axis ofthe sample.
The competing effect of the axial mag-netic field and of the boundary conditions for the orderparameter imposes smooth variation to the order param-eter in the radial direction, forming an effective poten-tial well for magnons. In the axial direction magnonsare trapped by a shallow minimum in the magnetic field,created by an external solenoid. The textural part ofthe trap is modified in the presence of vortices due tocontributions from their cores and the associated super-fluid velocity field. Magnetization of the magnon BECcoherently precesses with a frequency which depends onthe trapping potential. Therefore, the NMR measure-ments allow us to probe the evolution of vortex distribu-tion within the trap by periodic application of excitationpulses. After a spin-down to rest the measured frequencyof coherent precession oscillates with increasing period,see Fig. 3. This observation further supports the inter-pretation about a precessing nonuniform vortex clusterafter the spin-down. In the measurements the period p ( t ), extracted fromtemporal separation of the local maxima in the NMR orfork response, is converted into the angular velocity ofthe precessing vortex cluster usingΩ s ( t ) = 2 πp ( t ) . (13)Here we assume that the local maximum of either type ofsignal is related to position of some identifiable featurein the precessing vortex cluster and thus the temporalseparation of two subsequent maxima corresponds to asingle round of vortex precession in the container. V. MEASUREMENTS ON LAMINAR DECAYOF PRECESSING VORTEX CLUSTER
The initial vortex density is controlled by the angularvelocity Ω , so that the aerial density of vortices is equalto solid-body-rotation value n v = 2Ω /κ . To preparethe initial state the sample is rotated at velocity Ω > Ω before returning to Ω to ensure enough vortices arecreated. In some measurements this step is done at about0 . T c , where vortex dynamics is fast. Afterwards thesample is cooled down to the desired temperature overa time period of the order of an hour. Alternatively,similar procedure is done at lower temperatures. In thiscase steady rotation at Ω is maintained for a few hoursbefore the spin-down. We ensure that the dissipationof the magnon BEC, proportional to vortex density, hasreached constant value before the spin-down. After thespin-down the response Ω s ( t ) is monitored for as long asthe oscillations are seen, typically for a few hours.A short turbulent burst, seen as an initial overshootof the fork width in Fig. 2, is observed as soon as thedeceleration starts. The first oscillations are typicallyseen right after the turbulent t − / decay of vortex linedensity, some ∼
100 s after the container is at rest. Weuse the initial angular velocity Ω s ( t = 0) ≡ Ω i and thetime constant τ as fitting parameters in Eq. (11). We findthat Ω i ∼ (0 . − . in all our measurements. Thus,we estimate that 20 −
40% of vortices are lost during theinitial turbulent burst. The mutual friction parameter α is extracted from α = 12Ω i τ (14)as a function of temperature at three different pressures,see Fig. 4. We find that α has, within the accuracy of ourmeasurements, a linear dependence on the width of thequartz tuning fork as expected in the T → α ≡ α ( T → α can be written as a function of the forkresonance width as α = α + B ∆ f = α + BC exp( − ∆ /k B T ) , (15) Time, s0 1000 2000 3000 4000 5000 f − f L , H z Time, s1000 1050 1100 1150 1200 f − f L , H z f − f L , H z Ω , r a d / s FIG. 3: Temporal evolution of the ground state of magnonBEC after a spin-down from Ω = 1 .
02 rad/s to rest. The ini-tial increase in the frequency is caused by the drop in the vor-tex density and decreased polarization during the turbulentburst. The insets show zoomed view of late-time behavior,where oscillations in the ground state are caused by periodicmodulation of spatial distribution of the order parameter bya precessing vortex cluster. where the coefficient B is the slope in Fig. 4. Parameter C is a geometrical factor specific to the type of the res-onator, which in our case has been determined to havethe value C = 10 . ± . . T c in He-B. The geometrical factor scales as C ∝ p F as afunction of pressure. We compare the measured value B ∆ f with the expected behavior ∼ ( ω τ ) − as a func-tion of pressure. The results are shown in Fig. 5. We uselow temperature minigap values from Ref. 32, interpo-lated in ∆ p − using quadratic fit. The results show theexpected pressure dependence. While the absolute valueagrees with earlier measurements at 29 bar pressure, its magnitude is a factor of 6 smaller than the value of( ω τ ) − .We use the weak-coupling-plus bulk gap with strongcoupling correction. We have found no need for the gaprenormalization, contrary to the fits at T > . T c pre-sented in Ref. 4. Thus, we believe that the measuredvalues of α can be directly compared with the theory. VI. POSSIBLE SOURCES OF FINITEFRICTION AT T → Finite dissipation in quantum turbulence in the zerotemperature limit has previously been observed in su-perfluid He, and in He-B.
The microscopicsources of dissipation, as well as the role of the normalcomponent, are quite different for the two superfluids.In superfluid He the normal component has indepen-dent dynamics, which couples to the dynamics of the su- ∆ f , mHz468101214 α · T /T c (29 bar) FIG. 4: Dissipative mutual friction parameter α as a functionof the quartz tuning fork resonance width ∆ f . The dashedline follows Eq. (15) at 0.5 bar pressure, assuming α = 5 · − and B ∆ f = (6 ω τ ) − , and the solid lines are fits to the sameequation at different pressures. The fork width is convertedto T /T c scale at 29 bar at the top axis. perfluid component via mutual friction. At large drivesthe dynamics of normal component in He may be tur-bulent. Non-zero density of the normal component andthus friction may exist even in the T → Heimpurities are present. Otherwise, the zero-temperaturedissipation is believed to originate from acoustic emis-sion by rapidly oscillating vortices, which terminatesthe Kelvin-wave cascade. So far the experimental veri-fication of this scenario is absent. In He-B the normalcomponent is practically always laminar and its densityvanishes exponentially towards lower temperatures. Herewe consider a few possible dissipation mechanisms as can-didates for the observed zero-temperature dissipation inlaminar motion in He-B.One possibility is surface friction in the presence ofrough surfaces like the silver-sintered ones in the heatexchanger. The authors in Ref. 11 studied the responseof He-B to spin-down by measuring the magnitude ofcounterflow in a cylinder with smooth walls and possibil-ity to introduce a slab with high dissipation. The regionwith high dissipation could be created in the middle ofthe sample by using magnetic field to stabilize a layer ofsuperfluid He-A. At low temperatures the mutual fric-tion coefficients in the A-phase are orders of magnitudelarger than those in the B-phase. In the presence of theA-B phase boundary the response in the B-phase was al-ways turbulent. Additionally, the flow profile during thedecay was clearly different from solid-body like.In the absence of the phase boundary laminar behaviorwas observed down to the lowest measured temperature0 . T c , with Ω i τ = 740. According to Eq. (14) thiscorresponds to α ≃ · − , which is in good agreementwith our current work. This observation suggests thatthe surface friction can not be accounted for by simple Pressure, bar0 5 10 15 20 25 30 ( α − α ) e x p ( ∆ / k B T ) FIG. 5: Coefficient ( α − α ) exp(∆ /k B T ) as a function ofpressure. Blue squares correspond to measurements in thiswork and the red circle is extracted from data in Ref. 11,measured at 0 . T c , assuming the same geometrical factor C and α = 5 · − . Error bars correspond to the inaccuracyof the determination of the geometrical factor C in Eq. (15).The lines follow (6 ω τ ) − and correspond to (0 . , . , and0 . T c from top to bottom, respectively. increase of the mutual friction coefficient α but it leadsto qualitatively different behavior.Consider a vortex moving along a dissipative surface.The energy dissipation from the motion is limited by thevortex tension T v = κ ρ s (4 π ) − ln ba , where b is the in-tervortex separation and a is the vortex core size. Theexistence of this limit has been previously observed pre-viously in spin-up measurements on He. Assuming maximum pulling force, the surface dissipa-tion power can be calculated as W = Z R T v n v | v ex ( r ) | πr d r = 13 κρ s Ω R ln ba . (16)Here n v = 2Ω s /κ is the vortex density and v ex = Ω s r isvelocity of the vortex ends relative to the surface.Using Eqs. (10) and (16) we can write an equation fordecay of total kinetic energy E after a step-like spin-downd E d t = − β κ ln ba πRH E − α sfm √ M R E / . (17)Here E = M ( R Ω s ) , M = πR Hρ s is the mass of the su-perfluid in the container, H is the height of the container,and 0 ≤ β ≤ E ( t ) fromthe experiments using Ω s ( t ) from Eq. (13) and use theinitial condition E ( t = 0) = M ( R Ω sfm ) . Parameters α sfm , β , and Ω sfm are used as fitting parameters. Thesubscript sfm refers to surface friction model , i.e. to thefitted parameter value in this model. We find β ≪ α sfm and Ω sfm agree with the previously fitted α Time, s Ω S , r a d / s ∆ f = 0.08 Hz, Ω =1.02 rad/sFit to Eq. (9)Fit to Eq. (16) with fi xed α sfm FIG. 6: The dots show an example decay at 9.5 bar pressureusing data from Fig. 3. The solid red line is a fit to laminardecay model, Eq. (10), which only includes mutual friction.The dashed black line shows the behavior of the extendedmodel, Eq. (17), with fixed α sfm and fitted β , see details inthe text. and Ω i . Thus, the model including the surface friction,Eq. (16), when applied to our data effectively reduces tothe one without, Eq. (11). Alternatively, we tried fixing α sfm to α (∆ f ) − α , where α (∆ f ) is taken from Eq. (10).We find that we can not reproduce the observed decay ofΩ s this way, see Fig. 6. The finite zero-temperature dis-sipation α can not thus be replaced by surface friction.Another possible source for finite zero-temperature dis-sipation is proposed by Silaev in Ref. 38. In this model,the vortex-core-bound fermions interact with the flow,obtain additional energy, and escape the core if the vor-tex is in accelerating motion. The process is effectivedown to the absolute zero temperature. There are atleast two sources of accelerating vortex motion in ourmeasurements, whose effects we will now consider.The first source for acceleration is the centrifugal mo-tion of the remnant vortex cluster around the rotationaxis after the spin-down. Consider a cluster of vorticesmoving with angular velocity Ω s with respect to the con-tainer. The vortex-core-bound fermions approximatelyfollow heat balance equation A ξ/v = ∆ k B T loc exp (cid:18) − ∆ k B T loc (cid:19) , (18)where A = h u x ˙ u y − u y ˙ u x i t / ξ is the coherence length,∆ is the superfluid energy gap at zero temperature, v c = ∆ p − is the superfluid critical velocity, and T loc is the temperature inside the vortex core. The brack-ets denote time average and u x and u y are the x and y components of the vortex velocity u ( t ), respectively. Forcircular periodic motion at distance R from the axis, weget A ∼ R Ω . (19)We estimate the dissipation by substituting 29 bar pa-rameter values at T = 0 and using typical experimen-tal values Ω s = 1 rad/s and R = 3 mm. We get k B T loc / ∆ ≈ .
02, which is equal to T loc ∼ . T c . Thedissipation caused by the centrifugal motion thus onlydominates the vortex dynamics below 0 . T c , which ismuch lower than the lowest temperatures in our mea-surements. It seems unlikely that the centrifugal motionis responsible for the extra dissipation.The second source for acceleration is the presence ofvortex waves, such as Kelvin waves. Estimation for A fortypical Kelvin waves was done by Silaev in Ref. 38. Ac-cording to this estimate the presence Kelvin waves over-heats the cores to temperature T loc ∼ . T c . We notethat α ≈ α ( T = 0 . T c ) − α ( T = 0). In other words, theunaccounted dissipation at zero temperature correspondsto temperature increase of roughly 0 . T c , in agreementwith Silaev’s estimate for T loc . The Kelvin waves are nat-urally expected to be created immediately after the spin-down during the initial turbulent burst. However, thedissipation related to Kelvin waves should decrease andeventually cease as the initial small scale vortex wavesdecay. Laminar vortex motion is not a likely candidatefor providing energy input to small scales since there areno vortex reconnections.There is, however, at least one persistent source ofKelvin waves. We suggest that vortex motion along arough surface, such as that of the heat exchanger, gener-ates Kelvin waves that then propagate along the vortices.In principle, one expects that generation of Kelvin wavesis more prominent in areas where vortex motion with re-spect to the surface is faster, i.e. at larger distance fromthe container axis. We note, however, that vortex waveshave been seen to transfer to nearby vortices. Eventu-ally this process can bring the whole volume to a quasi-uniform state with all vortices having similar Kelvin-wavespectrum. Generation of Kelvin waves from vortex mo-tion along a dissipative surface could thus effectively leadto enhanced dissipation in the whole volume via Silaev’smechanism.
VII. CONCLUSIONS
We have studied the spin-down response of superfluid He-B in a cylindrical container at T = (0 . − . T c at 0 . , . , and 29 bar pressures. Deviations from cylin- drical symmetry in our setup lead to an initial turbulentburst, followed by hours-long laminar decay. We extractmutual friction parameter α from the evolution of an-gular velocity of the remnant vortex cluster. We findthat α depends exponentially on the temperature as the-oretically expected. The observed pressure dependenceis explained by the behavior of the minigap separatingthe energy levels of the vortex-core-bound fermions. Theabsolute values of the friction coefficient are a factor of 6lower than the theoretical estimation.The zero-temperature extrapolation of α reveals apressure-independent finite value α ≈ · − . Weconsider surface friction and a mechanism proposed bySilaev in Ref. 38 as possible sources for zero-temperaturedissipation. The latter requires that vortices are in accel-erating motion. We rule out surface friction and Silaevfriction from precessing motion of vortices as possiblesources of the observed extra dissipation, while oscillatingvortex motion from sufficiently developed Kelvin wavescould provide enough dissipation. The Kelvin waves pro-duced in an initial turbulent burst after the spin-downdecay during laminar motion at later times. Thus theycould not support time-independent α seen in our obser-vations. We propose that vortex motion along the roughsurface of the heat exchanger at the bottom of the exper-imental container generates Kelvin waves, which propa-gate into the bulk along the vortices effectively enhancingdissipation in the whole volume via mechanism proposedby Silaev.In future it would be interesting to measure mu-tual friction in a system where vortex interactions withboundaries can be neglected. One possibility may, inprinciple, be provided by freely propagating vortex rings,which can be created in the experiments i.e. by a movinggrid. One should be careful, though, that even in suchsystem Kelvin waves, excited on the rings at the momentof formation, can significantly affect further dynamics. Acknowledgments
We thank M. A. Silaev for useful discussions. Thiswork was supported by the Academy of Finland (grantno. 284594, LTQ CoE). Our research made use of theOtaNano Low Temperature Laboratory infrastructureof Aalto University, that is part of the European Mi-crokelvin Platform. P. J. Hakonen, M. Krusius, M. M. Salomaa, J. T. Simola,Y. M. Bunkov, V. P. Mineev, and G. E. Volovik, Phys.Rev. Lett. , 1362 (1983) R. Blaauwgeers, V. B. Eltsov, M. Krusius, J. J. Ruohio,R. Shanen, and G. E. Volovik, Nature , 471 (2000) N. B. Kopnin and M. M. Salomaa, Phys. Rev. B , 9667(1991) T. D. C. Bevan, A. J. Manninen, J. B. Cook, H. Alles, J. R. Hook, and H. E. Hall, Journal of Low TemperaturePhysics , 423 (1997) V. B. Eltsov, R. de Graaf, P. J. Heikkinen, J. J. Hosio,R. H¨anninen, M. Krusius, and V. S. L’vov, Phys. Rev.Lett. , 125301 (2010) G. W. Rayfield and F. Reif, Phys. Rev. , A1194 (1964) V. B. Eltsov, J. J. Hosio, M. Krusius, and J. T. M¨akinen,Journal of Experimental and Theoretical Physics , P. M. Walmsley, A. I. Golov, H. E. Hall, W. F. Vinen,and A. A. Levchenko, Journal of Low Temperature Physics , 127 (2008) P. M. Walmsley, A. I. Golov, H. E. Hall, A. A. Levchenko,and W. F. Vinen, Phys. Rev. Lett. , 265302 (2007) J. J. Hosio, V. B. Eltsov, M. Krusius, and J. T. M¨akinen,Phys. Rev. B , 224526 (2012) P. M. Walmsley, V. B. Eltsov, P. J. Heikkinen, J. J. Hosio,R. H¨anninen, and M. Krusius, Phys. Rev. B , 184532(2011) V. B. Eltsov, R. de Graaf, M. Krusius, and D. E. Zmeev,Journal of Low Temperature Physics , 212 (2011) M. Blaˇzkov´a, M. ˇCloveˇcko, V. B. Eltsov, E. Gaˇzo,R. Graaf, J. J. Hosio, M. Krusius, D. Schmoranzer,W. Schoepe, L. Skrbek, et al., Journal of Low Temper-ature Physics , 525 (2007) D. I. Bradley, P. Crookston, S. N. Fisher, A. Ganshin,A. M. Gu´enault, R. P. Haley, M. J. Jackson, G. R. Pickett,R. Schanen, and V. Tsepelin, Journal of Low TemperaturePhysics , 476 (2009) R. Blaauwgeers, M. Blazkova, M. ˇCloveˇcko, V. B.Eltsov, R. Graaf, J. Hosio, M. Krusius, D. Schmoranzer,W. Schoepe, L. Skrbek, et al., Journal of Low TemperaturePhysics , 537 (2007) N. B. Kopnin, G. E. Volovik, and ¨U. Parts, EPL (Euro-physics Letters) , 651 (1995) R. J. Donnelly,
Quantized Vortices in Helium II (1991). E. B. Sonin, Rev. Mod. Phys. , 87 (1987) H. Hall, Advances in Physics , 89 (1960) D. Einzel and P. W¨olfle, Journal of Low TemperaturePhysics , 19 (1978) S. Iordansky, Annals of Physics , 335 (1964) C. Wexler, Phys. Rev. Lett. , 1321 (1997) D. E. Zmeev, P. M. Walmsley, A. I. Golov, P. V. E. Mc-Clintock, S. N. Fisher, and W. F. Vinen, Phys. Rev. Lett. , 155303 (2015) J. J. Hosio, V. B. Eltsov, P. J. Heikkinen, M. Krusius, andV. S. L’vov, Nature Communications (2013) G. A. C. M. Spierings, J. Mater. Sci. (1993) D. I. Bradley, S. N. Fisher, A. M. Gu´enault, M. R. Lowe,G. R. Pickett, A. Rahm, and R. C. V. Whitehead, Phys.Rev. Lett. , 235302 (2004) V. Tsepelin, A. W. Baggaley, Y. A. Sergeev, C. F.Barenghi, S. N. Fisher, G. R. Pickett, M. J. Jackson, andN. Suramlishvili, Phys. Rev. B , 054510 (2017) P. J. Heikkinen, S. Autti, V. B. Eltsov, J. J. Hosio, M. Kru-sius, and V. V. Zavjalov, Journal of Low TemperaturePhysics , 3 (2013) S. Autti, Y. M. Bunkov, V. B. Eltsov, P. J. Heikkinen,J. J. Hosio, P. Hunger, M. Krusius, and G. E. Volovik,Phys. Rev. Lett. , 145303 (2012) Y. M. Bunkov, S. N. Fisher, A. M. Gu´enault, and G. R.Pickett, Phys. Rev. Lett. , 3092 (1992) E. V. Thuneberg, Journal of Low Temperature Physics , 657 (2001) M. A. Silaev, E. V. Thuneberg, and M. Fogelstr¨om, Phys.Rev. Lett. , 235301 (2015) S. Davis, P. Hendry, and P. McClintock, Physica B: Con-densed Matter , 43 (2000) D. I. Bradley, D. O. Clubb, S. N. Fisher, A. M. Gu´enault,R. P. Haley, C. J. Matthews, G. R. Pickett, V. Tsepelin,and K. Zaki, Phys. Rev. Lett. , 035301 (2006) V. B. Eltsov, A. I. Golov, R. de Graaf, R. H¨anninen,M. Krusius, V. S. L’vov, and R. E. Solntsev, Phys. Rev.Lett. , 265301 (2007) W. F. Vinen, Proc. R. Soc. Lond. A , 114 (1957). P. W. Adams, M. Cieplak, and W. I. Glaberson, Phys.Rev. B , 171 (1985) M. A. Silaev, Phys. Rev. Lett. , 045303 (2012) R. H¨anninen, Phys. Rev. B , 184508 (2015) N. Hietala, R. H¨anninen, H. Salman, and C. F. Barenghi,Phys. Rev. Fluids , 084501 (2016) D. I. Bradley, D. O. Clubb, S. N. Fisher, A. M. Gu´enault,R. P. Haley, C. J. Matthews, G. R. Pickett, V. Tsepelin,and K. Zaki, Phys. Rev. Lett. , 035302 (2005) J. L. Helm, C. F. Barenghi, and A. J. Youd, Phys. Rev. A83