Dual cascade and dissipation mechanisms in helical quantum turbulence
DDual cascade and dissipation mechanisms in helical quantum turbulence
Patricio Clark di Leoni , Pablo D. Mininni , & Marc E. Brachet Departamento de F´ısica, Facultad de Ciencias Exactas y Naturales,Universidad de Buenos Aires and IFIBA, CONICET,Ciudad Universitaria, 1428 Buenos Aires, Argentina. Laboratoire de Physique Statistique, ´Ecole Normale Sup´erieure,PSL Research University; UPMC Univ Paris 06, Sorbonne Universit´es; Universit´e Paris Diderot,Sorbonne Paris-Cit´e; CNRS; 24 Rue Lhomond, 75005 Paris, France. (Dated: October 21, 2018)While in classical turbulence helicity depletes nonlinearity and can alter the evolution of turbulentflows, in quantum turbulence its role is not fully understood. We present numerical simulations ofthe free decay of a helical quantum turbulent flow using the Gross-Pitaevskii equation at highspatial resolution. The evolution has remarkable similarities with classical flows, which go as faras displaying a dual transfer of incompressible kinetic energy and helicity to small scales. Spatio-temporal analysis indicates that both quantities are dissipated at small scales through non-linearexcitation of Kelvin waves and the subsequent emission of phonons. At the onset of the decay,the resulting turbulent flow displays polarized large scale structures and unpolarized patches ofquiescense reminiscent of those observed in simulations of classical turbulence at very large Reynoldsnumbers.
I. INTRODUCTION
From the oceans to the solar wind, turbulence is widelyfound in nature. It is also observed in quantum flu-ids such as superfluids and Bose-Einstein condensates(BECs) [1]. Unlike classical flows, quantum flows haveno viscosity and vorticity is concentrated along topolog-ical line defects with quantized circulation [2, 3]. Whilesimilarities between these two types of turbulence exist(e.g., both display Kolmogorov spectrum [4, 5]), thereare also differences [6, 7].In classical turbulence helicity is an ideal invariantwhich measures how tangled vorticity field lines are [8].It is known to deplete nonlinearities and energy transfer[9], slow down the onset of dissipation in decaying tur-bulence and affect its dissipation scale [10], play a role inconvective storms [11], and display a dual direct cascadewith the energy [12, 13]. Its role in quantum turbulence isless clear. Efforts focus on determining if it is conservedby studying simple configurations of reconnecting vor-tex knots [14–18]. Numerical evidence indicates that inthis case helicity is transferred from large to small scales[14, 16], and that reconnection or the transfer of helicitycan excite non-linear interacting Kelvin waves [17, 19],which eventually may lead to a loss of helicity by soundemission. Research into the role of helicity in more com-plex quantum flows has been lacking, partly due to thedifficulties of quantifying helicity in fully developed tur-bulent flows. However, new developments in 3D vortextracking in Helium experiments [20] and in knot genera-tion in BECs [21] provide hopeful opportunities to tacklethis problem.We present massive numerical simulations of helicalquantum turbulence using the Gross-Pitaevskii equation(GPE) as a model. A quantum version of the classi-cal Arnold-Beltrami-Childress (ABC) flow is introducedand used as initial condition to create a helical flow. We use different methods to quantify helicity, including theregularized helicity [17] which was shown to give resultsequivalent to the centerline helicity for simple knots, andto the classical helicity for flows with scale separation.We show that helicity is depleted as the incompressiblekinetic energy. As in the classical case [12], both the in-compressible energy and the helicity follow a Kolmogorovspectrum down to the intervortex distance. At smallerscales a bottleneck in the energy spectrum is followedby another Kolmogorov spectrum associated to a Kelvinwaves cascade. Energy and helicity dissipation at coher-ent length scales is related to Kelvin waves damping byphonon emision. In physical space, the flow displays po-larized large scale structures formed by a myriad of smallscale knots, and unpolarized quiescent patches mimick-ing what is observed in isotropic and homogeneous clas-sical flows at large Reynolds. These results open up newquestions about helicity in quantum flows. In particular,while successful theories for the energy spectrum exist[22], this is not yet the case for the helicity spectrum.
II. THE GROSS-PITAEVSKII EQUATION
The GPE describes the evolution of a zero-temperaturecondensate of weakly interacting bosons of mass m , i (cid:126) ∂ t Ψ = − (cid:126) (2 m ) − ∇ Ψ + g | Ψ | Ψ , (1)where g is related to the scattering length. Madelungtransformation Ψ = (cid:112) ρ/m exp ( imφ/ (cid:126) ) relates the wave-function Ψ to a condensate of density ρ and veloc-ity v = ∇ φ . Linearizing Eq. (1) around a con-stant Ψ = ˆΨ yields the Bogoliubov dispersion rela-tion ω B ( k ) = ck (1 + ξ k / / for sound waves (orphonons) of speed c = ( g | ˆΨ | /m ) / , with dispersiontaking place at lengths smaller than the coherence length a r X i v : . [ phy s i c s . f l u - dyn ] M a y ξ = [ g (cid:126) | ˆΨ | / (2 m )] / . The Onsager-Feynman quan-tum of velocity circulation around the Ψ = 0 topologicaldefect lines is h/m , and the vortex core size is of order ξ [23].The GPE conserves the total energy E , which can bedecomposed as [24, 25]: E = E kin + E int + E q , (2)with the kinetic energy E kin = (cid:104)|√ ρ v | / (cid:105) , the inter-nal energy E int = (cid:104) c ( ρ − / (cid:105) and the so-called quan-tum energy E q = (cid:104) c ξ |∇√ ρ | (cid:105) . The kinetic energy E kin can be also decomposed into compressible E ckin and in-compressible E ikin components, using ( √ ρ v ) = ( √ ρ v ) c +( √ ρ v ) i with ∇ · ( √ ρ v ) i = 0. A. Helicity in quantum flows
The definition of helicity in a classical flow is H = (cid:90) v · ω dV, (3)where ω = ∇ × v is the vorticity. It follows fromMadelung transformation that ω ( r ) = hm (cid:90) ds d r ds δ ( r − r ( s )) , (4)where r ( s ) denotes the position of the vortex lines and s the arclength. Thus vorticity in a quantum flow is adistribution concentrated along Ψ = 0 topological linedefects where v is ill-behaved. In spite of this, some au-thors [15] compute H by filtering fields to the largestscales or relying on the regularization introduced by thenumerics. Other authors compute the “centerline helic-ity” by calculating the writhe and link, two topologicalquantities which quantify how knotted vortex lines are,but which require detailed extraction of all centerlines ofthe quantized vortices in the flow [14, 16, 26, 27]. Someauthors suggest to also add the twist of equal-phase sur-faces (or else just the torsion) to this definition, but thenthe total helicity vanishes identically (or else smoothlyformed inflection points change the helicity discontinu-ously) [18]. A new method which yields the same resultsas the “centerline helicity” was introduced in [17] by us-ing the fact that the velocity parallel to the quantizedvortex has only an apparent singularity. The regularsmooth velocity oriented along the vortex line is definedas v reg = v (cid:107) w / √ w j w j , where v (cid:107) = (cid:126) w j (cid:2) ( ∂ j ∂ l Ψ) ∂ l Ψ − ( ∂ j ∂ l Ψ) ∂ l Ψ (cid:3) im √ w l w l ( ∂ m Ψ)( ∂ m Ψ) , (5)and w = (cid:126) im ∇ Ψ × ∇ Ψ (6) t H r , E i k H r E ik GPE, N = 2048GPE, N = 1024Navier-Stokes ‘ v / L t . . . . H, H r FIG. 1. (Color online)
Evolution of the incompressible energy E ik and of the regularized helicity H r in the 1024 and 2048 GPE runs, and in Navier-Stokes. Note the early “inviscid”phase in which quantities are approximately constant. Thesolid black line shows the total vortex length in the 2048 GPErun.
Inset: H r ( t ) (dashed blue line) and the non regularizedhelicity H ( t ) (solid red line) in the 2048 GPE run. (see Appendix A for a detailed derivation). The regular-ized helicity thus reads H r = (cid:90) v reg · ω dV, (7)and is well defined in the sense of distributions [28], asthe test function v reg is smooth. This expression wasproven useful even in flows with hundreds of thousandsof knots. B. Numerical simulations
The GPE is solved in its dimensionless form and allquantities presented here are dimnesionless (see [25, 29]for more details). All numerical simulations in this paperhave mean density ρ = 1. Physical constants in Eq. (1)are determined by ξ and c = 2, and the quantum of circu-lation h/m = cξ/ √
2. Simulations were performed using512 , 1024 , and 2048 grid points, in a domain of lin-ear size L = 2 π . The largest 2048 GPE simulation has ahealing length ξ ≈ . × − and an intervortex distance (cid:96) ≈ × − (computed as in [24, 25]). As a compar-ison, in He experiments the size of the vortex core is ≈ − m, the intervortex distance is ≈ − m, and thesystem size is of order 10 − m [1]. Scale separation issmaller for BECs, which are also better modeled by theGPE. Although proper scale separation in a simulation iscurrently out of reach, the 2048 run is a significant im-provement over most simulations of quantum turbulencewhich have one order or magnitude difference between L and ξ .To compare the GPE simulations with classical ABCflows we also simulated the incompressible Navier-Stokes(NS) equation ∂ t u + ( u · ∇ ) u = −∇ p + ν ∇ u , (8) k − − − − − H ( k ) η (cid:15) − / k − / (b) t = 0.3t = 2t = 5t = 8t = 10 − − − − − − E i k ( k ) (cid:15) / k − / ∼ k − / (a) FIG. 2. (Color online)
Spectrum of the (a) incompressiblekinetic energy, and (b) helicity in the 2048 GPE run. Atlarge scales both follow a scaling compatible with a classicaldual cascade (thick dashed lines). At scales smaller than theintervortex scale ( k (cid:96) ≈
80) a second range compatible witha Kelvin wave cascade is observed in E ik (thick dash-dottedline). The helicity spectrum broadens in time indicating adirect transfer. k − H ( k ) η − (cid:15) / k / t = 6t = 7t = 8t = 9Average FIG. 3. (Color online)
Compensated helicity spectra inthe 2048 GPE run. The spectrum is compatible with η(cid:15) − / k − / scaling. The time-averaged spectrum is alsoshown. with ∇ · u = , using 512 points and viscosity ν = 6 . × − . All equations were integrated using GHOST [30], apseudospectral code with periodic boundary conditions,fourth order Runge-Kutta to compute time derivatives,and the 2 / k = 1and k = 2 basic ABC flows: v ABC = v (1)ABC + v (2)ABC , with v ( k )ABC = [ B cos( ky ) + C sin( kz )] ˆ x + [ C cos( kz )+ A sin( kx )] ˆ y + [ A cos( kx ) + B sin( ky )] ˆ z, (9)and ( A, B, C ) = (0 . , , . / √
3. The basic ABC flowis a 2 π -periodic stationary solution of the Euler equa-tion with maximal helicity. To build its quantum ver-sion we first take the flow with B = C = 0 anduse Madelung transformation to obtain a wavefunctionΨ x,y,zA,k = exp { i [ A sin( kx ) m/ (cid:126) ] y + i [ A cos( kx ) m/ (cid:126) ] z } ,where [ a ] stands for the nearest integer to a to enforce pe-riodicity. The wavefunction of the quantum ABC flow is k ω (a) | ρ | k (d) | ρ | − − − − ω (b) E ik k ω (c) E ck (e) E ik k (f) E ck − − − − FIG. 4. (Color online)
Spatio-temporal power spectra for the512 GPE run between t = 0 and 2 for (a) the mass density ρ ,and zooms between k = 0 and 100 for (b) the incompressibleand (c) compressible velocity. Same for late times ( t ∈ [8 , then obtained as Ψ ( k )ABC = Ψ x,y,zA,k × Ψ y,z,xB,k × Ψ z,x,yC,k . Finally,Ψ ABC = Ψ (1)ABC × Ψ (2)ABC corresponds to the initial flow v ABC . In practice, to correctly set the initial density withdefects along the vortex lines and to correct frustrationerrors arising from periodicity, following [24, 25] we firstevolve Ψ
ABC using the advected real Guinzburg-Landauequation [31], whose stationary solutions are solutions ofthe GPE with minimal emission of acoustic energy.
III. RESULTS
In Fig. 1 we show the evolution of the incompressiblekinetic energy E ik and of the regularized helicity H r forthe 1024 and 2048 GPE runs, and for the NS equa-tion (with H r ≡ H ). In all cases, E ik and H r remainapproximately constant for the first turnover times whileturbulence develops (the so-called “inviscid” phase in thedecay of classical flows). The total vortex length (cid:96) v peaksat the end of this phase, which ends slightly earlier for H r than for E ik . Afterwards, H r and E ik decrease in whatseems a self-similar decay, with different rates in each sys-tem. As total energy is conserved in GPE, the decay of E ik is accompanied by a growth of the other componentsof the energy, in particular of E ck . Indeed, in quantumturbulence the decay of E ik is expected to result from theemission of phonons [32], and thus from classical results FIG. 5. (Color online)
Three-dimensional rendering of vortexlines at the onset of the decay in the 2048 GPE run of (a) aslice of the full box, and succesive zooms in (b) and (c) intothe regions indicated by the (red) rectangles. (d) Sketch ofthe transfer of helicity from writhe to twist in a bundle ofvortices, and for an individual vortex. [33] the decay in E ik should also produce a decay in H r .The inset in Fig. 1 compares H (non regularized) and H r in the 2048 GPE run. Both are in good agreement,but H r is smoother and less noisy, making it a betterfit to study the global evolution of helicity in quantumturbulence. However, the agreement between H r and H allows us to use H to compute spectra.Figure 2 shows spectra of E ik and H at different timesin the 2048 GPE run. The spectra build up rapidly fromthe initial conditions, and the energy and helicity excitelarger wavenumbers as time increases. At t = 5 bothalready display inertial ranges. At large scales they followa power law compatible with a classical dual energy andhelicity cascade [12], with Kolmogorov constant C K ≈ E ( k ) ≈ (cid:15) / k − / , H ( k ) ≈ η(cid:15) − / k − / , (10)and with (cid:15) and η calculated directly using (cid:15) = − dE ik /dt, η = − dH/dt, (11)from the data in Fig. 1 after the onset of decay. Aroundthe mean intervortex scale ( k (cid:96) ≈ E ik diplays a bot-tleneck compatible with predictions in [22]. This bottle-neck is followed by an inertial range ∼ k − / predictedfor a Kelvin wave cascade [22, 34] and which below isconfirmed for the lower resolution run. Figure 3 showscompensated helicity spectra, which corroborates the be-havior observed in Fig. 2. d/ξ − − C ( d / ξ ) t = 0 t = 2 t = 5 t = 8 t = 10 0 20 40 601234 λ max /λ min π/d FIG. 6. (Color online)
Correlation function of ρ in the 2048 GPE run. At t = 0 it decays rapidly in units of the healinglength ξ , but then quickly develops long-range correlations.Inset: Ratio of eigenvalues λ max /λ min as a function of 2 π/d ,with d the size of the box used for the average (blue triangles:regions with structures, red triangles: regions of quiescence). Of particular interest is the evolution of H ( k ). Forearly times H is concentrated at low wavenumbers, as ex-pected for the initial condition. But later it is transferredto larger wavenumbers as the cascade-like spectrum de-velops. While there is no rigourous proof of conserva-tion of helicity in quantum flows, note that using theHasimoto transformation [35] the evolution of a vortexline can be mapped into a nonlinear Schr¨odinger equa-tion which conserves momentum. But momentum of avortex line (e.g., the translation of the centerline in thedirection of vorticity) can result in net helicity. Thus,vortex lines evolution could indeed conserve helicity (ex-cept for depletion by emission of phonons). At smallscales H ( k ) displays wild fluctuations (in amplitude andsign), which is to be expected as the non-reguralized he-licity is ill-behaved at those scales. The fact that thehelicity dynamics, at least at large scales, of a quantumflow mimics those of a classical one is remarkable. But italso begs the question of what happens to the helicity atscales smaller than the intervortex distance. Indicationsexist that Kelvin waves carry helicity [14, 17], but sucha possibility requires confirmation of their presence.To verify this, as well as phonons being the dissipa-tion mechanism for E ik and H , we must detect Kelvinand sound waves. To do this we use the spatio-temporalspectrum [29], i.e., the four-dimensional power spectrumof a field as a function of wave vector and frequency. Thespectrum allows quantification of how much power is ineach mode ( k , ω ), and waves can be separated from therest as they satisfy a known dispersion relation ω ( k ). Asits computation requires huge amounts of data (i.e., stor-age of fields resolved in space and time), we compute itfor the 512 GPE run. Figure 4 shows this spectrum (af-ter integration in k using isotropy to obtain dependencyon k and ω ) for ρ , and zooms for small k for E ik and E ck , for early and late times (respectively, t ∈ [0 ,
2] and t ∈ [8 , ω K ( k ) [36]and sound waves ω B ( k ) are shown. Note that, comparedwith the 2048 run, ξ in this run is 4 times larger, andvalues of k are 4 times smaller.At early times, power in the spatio-temporal spectrumof ρ is broadly spread over modes that do not correspondto waves. E ik shows some excitations compatible withKelvin waves, and E ck has little energy with no phononexcitation. At late times power in fluctuations of ρ movestowards the Kelvin wave dispersion relation up to k ≈ E ik and E ck confirm this picture,with power concentrating in E ck in the vicinity of thesound dispersion relation. Exploration of these spectrafor different time ranges shows that as time evolves moreenergy goes from Kelvin wave modes to phonons. Whilethis analysis is performed at lower resolution and thuswavenumbers for the transition are smaller than in the2048 run, the spectra confirm the dynamics in Figs. 1and 4: with time, energy and helicity go from large tosmaller scales in which Kelvin waves are excited, and theyare finally dissipated into phonons.This can be further confirmed by visualizing vortices inreal space at the onset of decay. Figure 5 shows a three-dimensional rendering of quantum vortices at t ≈ . GPE run. Large-scale eddies, formed upby a myriad of small-scale and knotted vortices, emerge.Similar behavior has been observed at finite temperaturequantum turbulence simulations, where the bundle wascorrelated with high vorticity in the normal fluid com-ponent [37, 38]. At zero temperature two results [39–41]also hinted at this behavior, but in none the fine structureof the vortex bundle was resolved. More importantly, thelarge scale flow shows inhomogeneous regions with highdensity of vortices and quiet regions with low density.These large-scale patches were not present in the initialconditions (which have homogeneously distributed vor-tices) and are created by the evolution as shown below.The patches are reminiscent of those observed in isotropicand homogeneous turbulence at high resolution in non-helical [42] and helical [43] flows, further confirming thesimilarity between quantum and classical turbulence atscales larger than the intervortex separation.The spontaneous emergence of large-scale correlationsin the system can be confirmed by the spatial correlationfunction C ( d ) = (cid:104) ( ρ ( x + d ˆ x ) − ρ )( ρ ( x ) − ρ ) (cid:105) , (12)shown in Fig. 6. This correlation function is related to theinternal energy spectrum by the Wiener-Khinchin the-orem. At t = 0, C ( d ) decays rapidly in units of thehealing length ξ , and it is dominated by the vortex coresize. But the system rapidly develops long-range correla-tions (up to ≈ ξ ) and later C decays in a self-similarway. Furthermore, computing the ratio of eigenvalues τ = λ max /λ min for the tensor (cid:104) ∂ i ρ∂ j ρ (cid:105) averaged in boxesof size 1 /
10 of the linear domain size, typically yields τ ≈ τ ≈ IV. CONCLUSIONS
The results indicate that helicity can be conserved inquantum turbulence at large-scales and as it is trans-ferred towards smaller scales (see Fig. 2), but eventuallyit decays through the emission of phonons produced by aKelvin wave cascade (Fig. 4). We can draw a comparisonwith the classical case, where now a bundle of quantumvortices (as seen in Fig. 5) would play the role of classicalvortex tubes. Tubes, in contrast to lines, add an extradegree of freedom to the helicity: their twist. Thus inthe classical and quantum cases, large scale helicity canbe transformed from writhe to twist for a bundle of vor-tices (see Fig. 5.d). But for individual quantum vortices,the transfer (e.g., through reconnection) would result inthe excitation of a Kelvin wave which can eventually bedamped. This indicates that individual quantum vortexlines behave like classical vortex tubes with a mechanismto relax the twist, and as such, the correct analogy be-tween classical and quantum flows only holds for scaleslarger than (cid:96) for which bundles of quantum vortices be-have as classical vortex tubes.
Appendix A: Derivation of the regularized velocity
To calculate the helicity in a quantum flow we needinformation of both the velocity and the vorticity alongthe vortex lines. This is problematic as both quantitieshave singularities along those lines. Therefore, we needto regularize one of them in order to have a well-behavedintegral for the helicity (in the sense of distributions [28]).Although in principle it may seem possible to regularizeany of the two fields, the choice of regularizing the veloc-ity and not the vorticity is not arbitrary. In the Gross-Pitaevskii equation, the vorticity is correctly describedby a distribution. Instead, the only component of the ve-locity that is not well behaved is the one perpendicular tothe vortex line. But for the calculation of the helicity weneed the parallel component, whose problem is to havea 0 / v = (cid:126) im ¯Ψ ∇ Ψ − Ψ ∇ ¯ΨΨ ¯Ψ . (A1)Without loss of generality we can suppose that there isa vortex line going through r = 0 (the radial cylindricalvector) in the direction of the z -axis. Let us define theunit vector basis ( ˆe x , ˆe y , ˆe z ). The existence of a vortexline passing through r = and pointing in the z -directionimplies that Ψ(0) = 0, ¯Ψ(0) = 0, ˆe z · ∇ Ψ(0) = 0, and ˆe z · ∇ ¯Ψ(0) = 0. Thus ∇ Ψ(0) and ∇ ¯Ψ(0) are linear com-binations of ˆe x and ˆe y . Taylor-expanding to first orderthe numerator and denominator of the above expressionfor v ( r ) around r = 0 one findsΨ( x, y, z ) = x∂ x Ψ(0) + y∂ y Ψ(0) + O ( r ) , (A2)¯Ψ( x, y, z ) = x∂ x ¯Ψ(0) + y∂ y ¯Ψ(0) + O ( r ) , (A3) ∇ Ψ( x, y, z ) = ∇ Ψ(0) + r · ∇ ( ∇ Ψ)(0) + O ( r ) , (A4) ∇ ¯Ψ( x, y, z ) = ∇ ¯Ψ(0) + r · ∇ ( ∇ ¯Ψ)(0) + O ( r ) . (A5)After replacing the above expressions in Eq. (A1) anddropping quadratic terms, the perpendicular ( x and y )components of the velocity diverge in the limit r →
0, as v ⊥ reads v ⊥ ( r ) = (cid:126) im (cid:18) ∇ Ψ(0) x∂ x Ψ(0) + y∂ y Ψ(0) − c.c (cid:19) . (A6)On the other hand, the velocity component parallel tothe centerline vorticity v (cid:107) ( r ) = v ( r ) · ˆe z reads v (cid:107) ( r ) = (cid:126) im (cid:18) x ( ∂ xz Ψ)(0) + y ( ∂ yz Ψ)(0) + z ( ∂ zz Ψ)(0) x∂ x Ψ(0) + y∂ y Ψ(0) − c.c. (cid:19) , (A7)which is finite in the limit r →
0. This last expressionfor v (cid:107) ( r ) can be seen as resulting from l’Hˆopital’s ruleapplied to the limit of v (cid:107) ( r ) when r → x, y, z ). The limit obviously depends on the directionas, in deriving the above formulae, the only hypotheseswe have made are that Ψ is sufficiently differentiable andhas a zero-line directed toward z .In order to turn the above expression into a workableansatz for v (cid:107) (0), we need to pick a reasonable direction along which Ψ will have a significant variation. The sim-plest vectors we have at point r = 0, perpendicular tothe vortex line and satisfying the condition, are ∇ Ψ(0)and ∇ ¯Ψ(0). Thus we can multiply the first term in ther.h.s. of Eq. (A7) by ∇ ¯Ψ, and its complex conjugate by ∇ Ψ in order to maintain the reality of the velocity field.In this way we arrive to the following expression v (cid:107) (0) = (cid:126) m i ( ∂ x ¯Ψ ∂ xz Ψ + ∂ y ¯Ψ ∂ yz Ψ + ∂ z ¯Ψ ∂ zz Ψ ∂ x ¯Ψ ∂ x Ψ + ∂ y ¯Ψ ∂ y Ψ + ∂ z ¯Ψ ∂ z Ψ − c.c. ) . (A8)A first check that this ansatz is reasonable is to plugin Ψ ∼ ( x + iy ) e izU z m/ (cid:126) and explicitly verify that thisgives v (cid:107) (0) = U z . Further validations were performedin [17], where it was shown that the helicity computedwith the regularized velocity agrees with the topologicaldefinitions of writhe, link, and twist. Also, in [17] itwas shown that this expression gives the correct valueof helicity for different knots, and that in quantum flowswith helicity it gives a value that matches the helicity inthe equivalent classical large-scale helical flow.As a final remark, it is important to note that for arbi-trarily aligned vortex lines, the direction parallel to thevortex line (ˆ z in the particular case considered above) canbe easily obtained by doing the vector product between ∇ Ψ and ∇ ¯Ψ. ACKNOWLEDGMENTS
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