Helicity in Superfluids: existence and the classical limit
Hridesh Kedia, Dustin Kleckner, Martin W. Scheeler, William T. M. Irvine
HHelicity in Superfluids: existence and the classical limit
Hridesh Kedia, Dustin Kleckner, ∗ Martin W. Scheeler, and William T. M. Irvine † James Franck Institute and Department of Physics,The University of Chicago, 929 E 57th St, Chicago, IL 60637, USA ‡ James Franck Institute, Enrico Fermi Institute and Department of Physics,The University of Chicago, 929 E 57th St, Chicago, IL 60637, USA
In addition to mass, energy, and momentum, classical dissipationless flows conserve helicity, ameasure of the topology of the flow. Helicity has far-reaching consequences for classical flows fromNewtonian fluids to plasmas. Since superfluids flow without dissipation, a fundamental question iswhether such a conserved quantity exists for superfluid flows. We address the existence of a “su-perfluid helicity” using an analytical approach based on the symmetry underlying classical helicityconservation: the particle relabeling symmetry. Furthermore, we use numerical simulations to studywhether bundles of superfluid vortices which approximate the structure of a classical vortex, recoverthe conservation of classical helicity and find dynamics consistent with classical vortices in a viscousfluid.
I. INTRODUCTION
Our understanding of fluid flow is built on fundamen-tal conservation laws such as the conservation of mass,energy, and momentum [1]. In particular, these give riseto the Euler equations of dissipationless fluid mechanicswhich capture many fluid phenomena including vortexdynamics [2], instabilities [3] and play a key role in thestudy of turbulence [4, 5].Hidden within the Euler equations for isentropic flows,is a less familiar conservation law [6–8]: conservation ofhelicity H Euler = (cid:82) d x u · ωωω , ωωω = ∇ × u . As a measureof the average linking of vortex lines [7, 8], helicity con-servation places a topological constraint on the dynamicsof classical inviscid isentropic flows . Helicity has furtheryielded new insights into viscous flows, from vortex recon-nection events [9, 10], to the study of coherent dynamicalstructures generated by turbulent flow [11–13].Superfluids display striking similarities with classicalfluids in their vortex dynamics [14, 15] and turbulencestatistics [16–18]. Since superfluids flow without dissipa-tion, it is natural to ask whether a conserved quantityanalogous to helicity also exists in superfluid flows. Nat-ural candidates for a “superfluid helicity” are: (i) theexpression for the classical helicity H Euler which is notconserved in superfluid flows [9, 19], and (ii) a Seifert-framing based helicity which vanishes identically [9, 20–22]. However, it has been challenging to establish theirconnection to the fundamental notion of conservation. Ithas thus remained unclear whether additional conserved ∗ Current address: University of California, Merced, 5200 N. LakeRoad Merced, CA 95343, USA † [email protected] ‡ Current address: Physics of Living Systems Group, Mas-sachusetts Institute of Technology, Cambridge, MA 02139, USA From here on, we refer to classical inviscid isentropic flows asEuler flows We shall only consider superfluids with a complex scalar orderparameter as in He and atomic Bose-Einstein condensates. quantities akin to helicity and circulation exist in super-fluids, and how a “classical limit” of superfluid helicitymight behave.In this letter, we use an analytical approach based onthe particle relabeling symmetry, which underlies helicityconservation and Kelvin’s circulation theorem in classi-cal inviscid fluids, to address the question of a “superfluidhelicity”. We find that the conserved quantities associ-ated with the particle relabeling symmetry in superfluidsvanish identically, yielding only trivial conservation lawsinstead of the conservation of helicity and circulation.This raises the question of a “classical limit” in whicha relevant notion of helicity is recovered which has dy-namics akin to helicity in classical flows. To answer thisquestion, we study bundles of superfluid vortices thatmimic the structure of classical vortices and are robustlong-lived structures [23, 24]. Our numerical simulationsshow that the centerline helicity [9] of superfluid vortexbundles behaves akin to helicity in classical viscous flows.
II. SUPERFLUID VORTEX DYNAMICS ANDCONSEQUENCES FOR HELICITY
FIG. 1. A three-fold helical superfluid vortex and a section ofits phase isosurface clipped at a fixed distance from the vortex.The volume occupied by the superfluid naturally separatesinto such surfaces of constant phase.
To simplify our discussion, we consider superfluids atzero temperature, i.e. weakly interacting Bose conden-sates described by a complex order parameter ψ (“wavefunction of the condensate” [25]) obeying the Gross- a r X i v : . [ c ond - m a t . qu a n t - g a s ] O c t Pitaevskii equation [26, 27]: i (cid:126) ∂ t ψ = − (cid:126) m ∇ ψ + g | ψ | ψ (1)where the constant g captures the inter-atomic interac-tion strength [28]. The Gross-Pitaevskii equation (GPE)captures qualitatively important features of superfluidbehavior at low temperatures [14, 29], including the dy-namics of vortices—lines where the complex order pa-rameter ψ vanishes, and around which its phase windsaround by an integer multiple of 2 π (see Fig. 1).Interestingly, the Gross-Pitaevskii equation can bemapped to an Euler flow in the region excluding vor-tices via the Madelung transformation [30, 31]: ψ = (cid:112) ρ/m exp( iφ/ (cid:126) ), by rewriting Eq. (1) in terms of thefluid density ρ = m | ψ | , and velocity u = ∇ φ/m . Themapping between superfluid flow and Euler flow makes ittempting to conclude that classical helicity is conservedin superfluids just as in Euler flows. However, numericalsimulations show that the expression for helicity in Eulerflows: H Euler = (cid:82) d x u · ωωω , ωωω = ∇ × u is not conservedin superfluid flows [9, 19, 21]. H Euler evaluated for sin-gular vortex lines has two contributions: (a) the Gausslinking integral for pairs of vortex lines, giving the linkingbetween them, and (b) the Gauss linking integral evalu-ated for each vortex line and itself giving its writhe [32].Since the writhe of a vortex line is not conserved [9] evenin the absence of reconnections, H Euler is not conservedfor superfluid flows.This disparity between Euler flows and superfluid flowsstems from two key differences: (i) Superfluids have sin-gular vorticity distributions, concentrated on lines ofsingular phase (see Fig. 1), and quantized circulationΓ = (cid:72) u · d l = n h/m , unlike classical vortices whichhave smooth vorticity distributions. (ii) Vortex lines ina superfluid can reconnect [33–35], in contrast to vortexlines in Euler flows which can never cross.The singular nature of superfluid vortices and the pres-ence of vortex reconnections make it challenging to carryover the derivation of helicity conservation [8] in Eulerflows, and suggest that a fundamentally different ap-proach is required to address the question of a “super-fluid helicity”. Previous approaches [21, 36, 37] to seekinga conserved quantity analogous to helicity in superfluidflows have focused on adapting the expression for classi-cal helicity H Euler to superfluids, as opposed to startingfrom a symmetry and seeking conservation laws.We now begin with the fundamental symmetry thatgives rise to helicity conservation in Euler flows viaNoether’s theorem, and carry this over to superfluids.
III. HELICITY AS A NOETHER CHARGE FOREULER FLUIDS AND SUPERFLUIDS
The conservation of helicity in Euler flows [38–47] is aspecial conservation law, arising from the particle rela- beling symmetry via Noether’s second theorem [42, 50].The particle relabeling symmetry arises from an equiv-alence between the Lagrangian description of a flow interms of the positions x ( a , τ ) and velocities ∂ τ x ( a , τ ) offluid particles labeled by a at time τ , and the Euleriandescription of a flow in terms of the velocity u ( x , t ) anddensity ρ ( x , t ) at each point in space. The action forEuler flow is [40, 43, 45]: S Euler = (cid:90) dτ d a (cid:20)
12 ( ∂ τ x ( a , τ )) − E ( ρ ) (cid:21) (2)where τ is time, d a = ρ d x is the mass of a fluid ele-ment, ∂ τ x ( a , τ ) is the velocity, E ( ρ ( a )) is the internal en-ergy density, and the co-ordinate frames ( a , τ ) and ( x , t )are related as follows: ∂ τ = ∂ t + u · ∇ . Note that theEuler flow action in Eq. (2) depends only on the flowvelocity u = ∂ τ x ( a , τ ), and the density ρ : ρ − ( a ) =det (cid:0) ∂x i ( a ) /∂ a j (cid:1) .Particle labels can be interpreted as the initial co-ordinates of the fluid particles, and the relabeling trans-formation as a smooth reshuffling (diffeomorphism) of theparticle labels, akin to a passive co-ordinate transforma-tion, which leaves the fluid velocity and density unaf-fected and hence leaves the action invariant.Relabeling transformations are changes of the parti-cle labels: a i → ˜ a i = a i + (cid:15) η i , where η i satisfies: (i) ∂η i /∂τ = 0 which ensures that the velocity is unchanged,and (ii) ∂η i /∂a i = 0 which ensures that the density ρ = det ( ∂ x /∂ a ) − is invariant. The positions of the fluidparticles remain unchanged under such a transformation,i.e. ˜ x (˜ a , τ ) = x ( a , τ ). The conserved charge associatedwith relabeling transformations [40–44] is: Q Euler = (cid:90) d a u i ∂x i ∂a j η j (3)where u i = ∂x i /∂τ .The conservation of Q Euler gives both Kelvin’s cir-culation theorem, and helicity conservation for differ-ent choices of ηηη . Evaluating Q Euler for the relabelingtransformation η j = (cid:72) C : a ( s ) ds δ (3) ( a − a ( s )) ∂a j ( s ) /∂s which infinitesimally translates particle labels along aloop C [42, 43, 51] gives the circulation along the loop C : Γ C = (cid:72) C u · d x ( s ). Evaluating Q Euler for the relabel-ing transformation η j = (cid:15) jkl ( ∂u p /∂a k )( ∂x p /∂a l ) whichinfinitesimally translates the particle labels a along vor-tex lines, gives the helicity H Euler = (cid:82) u · ω d x [40–44]. Conservation of helicity follows as a special caseof Kelvin’s circulation theorem: from the conservation ofthe sum of circulations along all the vortex lines in thefluid, weighted by the flux of each vortex line.We seek conserved quantities analogous to helicity andcirculation in superfluids, by seeking analogs of the re-labeling symmetry transformations. The action for the For more details on Noether’s second theorem, see [48, 49].
FIG. 2. Vortex lines C , and closed curves C (cid:48) constructedby offsetting vortex lines along a phase isosurface for: (a)a writhing (coiling) vortex line C , (b) a pair of linked rings C , C . Notice that the presence of either writhe or linkingin vortex lines leads to the twisting of the phase isosurfacearound the vortex lines. The circulation around a closed loop γ encircling a vortex line is equal to the change in phase φ asthe loop is traversed, giving a multiple of 2 π . Gross-Pitaevskii superfluid in terms of the hydrodynamicvariables ρ = m | ψ | , and φ = (cid:126) arg ψ is: S gpe = − (cid:90) dt ρ d x (cid:16) ∂ t φm + ( ∇ φ ) m + g m ρ + (cid:18) (cid:126) ∇√ ρm √ ρ (cid:19) (cid:17) where the last term: ( ∇√ ρ/ √ ρ ) is known as the “quan-tum pressure” term, and has no classical analogue. Itsprimary effect is to regularize the size of the vortexcore [52–54] and enable vortex reconnections [28], andis negligible when the typical length scale of densityvariations is much larger [28] than the “healing length” ξ = (cid:112) (cid:126) / (2 m g ρ max ). We make the Thomas-Fermi ap-proximation [25, 28, 55] which neglects the “quantumpressure” term and captures well, the dynamics of super-fluid vortices [28, 55–57]. Within this approximation, weseek to express the action for the Gross-Pitaevskii super-fluid in terms of Lagrangian co-ordinates ( a , τ ), where a is the particle label, and τ is time. To this end, we rewrite ∇ φ as the fluid velocity ∇ φ/m = u = ∂ x ( a , τ ) /∂τ ,and use the relation ∂ τ = ∂ t + u · ∇ to rewrite ∂ t φ as ∂ τ φ − u · ∇ φ . The superfluid action then becomes: S gpe = (cid:90) dτ d a (cid:20)
12 ( ∂ τ x ( a , τ )) − E ( ρ ) − m ∂ τ φ ( a , τ ) (cid:21) where E ( ρ ) = g ρ/ (2 m ), ρ d x = d a as for Eu-ler flow. Note that the action S gpe differs fromthe Euler flow action in Eq. (2) by an extra term: (cid:82) dτ d a ( − ∂ τ φ ( a , τ ) /m ). This extra term is needed toensure Galilean invariance of the action S gpe , and haskey consequences for the conservation of helicity. as described in [58, 59], under a Galilean transformation: { x → x (cid:48) = x − v t, t → t (cid:48) = t } , the phase transforms as: φ ( x , t ) → φ ( x (cid:48) , t ) = φ ( x , t ) − ( v · x − ( v · v ) t/ m = (cid:126) = 1. Particle relabeling transformations of the form a i → ˜ a i = a i + (cid:15) η i , ˜ x (˜ a , τ ) = x ( a , τ ) , ˜ φ (˜ a , τ ) = φ ( a , τ ), where ∂η i /∂τ = 0 , ∂η i /∂a i = 0, leave the velocity, the phase,and the density unchanged, and hence are symmetries ofthe action. Using Noether’s theorem, the correspondingconserved charge is: Q gpe = Q Euler + Q phase = (cid:90) d a u i ∂x i ∂a j η j + (cid:90) d a (cid:18) − m ∂φ∂a j (cid:19) η j = 0(4)where Q Euler is the contribution from the Euler flow partof the action S Euler , and Q phase = (cid:82) d a (cid:0) − ∂φ/∂a j (cid:1) η j is the contribution from S phase . The classical conservedcharge Q Euler is simply the superfluid conserved charge Q gpe in the absence of Q phase since the phase of thecomplex order parameter φ ( a , τ ) is absent from the de-scription of classical flow. Since the superfluid velocity is u = ∇ φ/m , Q Euler and Q phase cancel each other exactly.Hence, the conserved charge Q gpe vanishes identically forall relabeling transformations, instead of giving conser-vation of helicity and circulation.Our calculation shows that even in the absence ofa “quantum pressure” term, the relabeling symmetryyields a vanishing conserved quantity, instead of conser-vation of circulation and helicity. This vanishing of “su-perfluid helicity” is consistent with an alternative calcu-lation based on helicity as a Casimir invariant [40, 43](see SI for details). IV. SUPERFLUID HELICITY—A GEOMETRICINTERPRETATION
The vanishing of superfluid helicity and circulation Q gpe , is a consequence of a relation between the geom-etry of superfluid vortex lines and phase isosurfaces, aswe now illustrate.For a relabeling transformation along a closed loop γ encircling a vortex line as shown in Fig. 2, the van-ishing of the conserved charge comes from a cancellationbetween the circulation (cid:72) γ u · d l and the change in phase (cid:72) γ ( −∇ φ ) · d l . We note, however, that by judiciouslychoosing the shape of the loop, so that it lies entirelyon a phase isosurface as depicted in Fig. 2, it is possi-ble to make the contribution Q phase vanish identically.The vanishing of Q gpe then acquires a simple geometricinterpretation, which we elucidate below.A curve along which Q phase vanishes identically is con-structed by offseting the vortex line C i along a phase iso-surface by a distance ∆ (see Fig. 2) to give a new closedcurve C (cid:48) i (∆) : a (cid:48) ( s ) = a ( s )+∆ ˆ n ( s ), where a ( s ) ∈ C i , andˆ n ( s ) is perpendicular to the vortex line and tangent to the ηηη γ = (cid:72) γ ds δ (3) ( a − a ( s )) d a ( s ) / d s , where a ( s ) ∈ γ FIG. 3. A three-fold helical superfluid vortex bundle (shownin (a)) evolving as a coherent structure, rotating as it travelsforward, akin to a single three-fold helical vortex (shown in(b)). A cross-section of the three-fold helical superfluid vortexbundle, reveals a central vortex and 5 equally spaced vorticesarranged around the central vortex at distance 6 ξ (where ξ is the healing length). After a long time, the helical vortexbundle disintegrates (symbolized by the grey dots) and losesits bundle-like structure. phase isosurface. The quantum pressure term is negligi-ble on the new closed curve C (cid:48) i (∆) as long as the distance∆ is large compared to the healing length ξ . The con-served charge Q gpe evaluated for a relabeling transforma-tion ηηη (∆) which translates particle labels along C (cid:48) i (∆)has no contribution from Q phase , and becomes the circu-lation along the curve C (cid:48) i (∆): Q gpe = (cid:72) C (cid:48) i (∆) u · d l . Thiscirculation can be evaluated by substituting the Biot-Savart flow field for u , since the compressible part of u does not contribute. Q gpe then becomes the linking of the loop C (cid:48) i withall the vortex lines in the superfluid, i.e. Q gpe = (cid:80) j (cid:54) = i Γ j L i (cid:48) j + Γ i L i (cid:48) i = 0 where L i (cid:48) j denotes the link-ing between the vortex line C j , and we have used theGauss linking integral [60]. The vanishing of the con-served charge Q gpe follows as result of the linking L i (cid:48) i between the offset line C (cid:48) i and the vortex line C i can-celing the linking L i (cid:48) j between the offset line C (cid:48) i and allthe other vortex lines C j , j (cid:54) = i . Furthermore, assumingthat the section of the phase isosurface bounded by thetwo loops C (cid:48) i , C i can be considered as a smooth ribbon,we can use the Cˇalugˇareanu-White-Fuller theorem [61–64] to express L i (cid:48) i as the sum of the writhe (Wr i ) andthe twist (Tw ∗ i ) of the ribbon (see Fig. 2), giving: Q gpe = (cid:88) j (cid:54) = i Γ j L ij + Γ i Wr i + Γ i Tw ∗ i = 0 (5)The vanishing of the conserved charge Q gpe is thus re-lated to the vanishing of the sum of: the linking of avortex line C i with all other vortex lines (cid:80) j (cid:54) = i L ij , its ηηη (∆) = (cid:72) C (cid:48) i (∆) ds δ (3) ( a − a (cid:48) ( s )) d a (cid:48) ( s ) / d s writhe Wr i , and the twist Tw ∗ i of a ribbon formed by aphase isosurface ending on it.The vanishing of these geometric quantities was firststudied in the context of helicity of framings of magneticflux tubes [20], and is a consequence of the fact that aphase isosurface is an orientable surface which has as itsboundary, all the vortex lines in the superfluid, i.e. itis a Seifert surface [20, 65–67] for the vortex lines in thesuperfluid. This relation between linking and writhing ofvortex lines and the twisting of phase isosurfaces has beenused in superfluid simulations [9, 68] to calculate the cen-terline helicity (linking and writhing of vortex lines), andwas elaborated on in recent efforts to define a superfluidhelicity [21, 22]. FIG. 4. A superfluid vortex bundle in the shape of a trefoilknot evolving as a coherent structure, akin to a single trefoilknot vortex. (a) A trefoil knotted vortex bundle reconnects toform a smaller three-fold distorted ring bundle, and a largerthree-fold distorted ring bundle, which lose their bundle-likestructure over time. A cross-section of the initial trefoil knot-ted vortex bundle, shows 3 equally spaced vortices arrangedon the circumference of a disk of radius 5 ξ . (b) A single tre-foil knotted vortex reconnects to form a smaller three-folddistorted ring, and a larger three-fold distorted ring, whichundergoes further reconnections to give a large distorted ringat long times. V. CLASSICAL HELICITY—THE SINGULARLIMIT AND DISSIPATION
We now address the question of whether a classicalnotion of helicity can be recovered in superfluids and ifits dynamics are akin to that in Euler flows or viscousflows.While vorticity in superfluids is necessarily concen-trated on lines of singular phase, vorticity in classicalfluids can be continuously distributed and indeed mustbe to avoid a physical singularity in the flow. Follow-ing [8, 69, 70], a natural way of recovering a “classical”notion of helicity is to consider a continuous vorticitydistribution as made up of an infinite collection of vortexlines. The centerline helicity H c of a collection of singular FIG. 5. Helical vortex bundles (N=6) at different stages of evolution (top row), with the corresponding points in the graphsindicated by colored circles (bundle-like structure preserved), and grey circles (bundles disintegrate). (a) 2-fold helical vortexbundles with aspect ratio 0.35, (b) 3-fold helical vortex bundles with aspect ratio 0.25, and (c) 4-fold helical vortex bundleswith aspect ratio 0.2. The rescaled helicity h (middle row) for superfluid vortex bundles having the same overall shape (writhe)but different amounts of twist, trends towards their initial average writhe (horizontal grey band), before eventually decayingtowards zero (grey dotted lines). After a vortex bundle disintegrates at time T (=min t (cid:48) : N ( t (cid:48) ) /N (0) > . t (cid:48) to the initial numberof vortex filaments: N ( t (cid:48) ) /N (0). For each helical vortex bundle configuration, multiple ( >
10) simulations are performed withrandom Gaussian noise (r.m.s is 2% of the r.m.s. radius) added to the initial bundle. The mean rescaled helicity is indicatedby the solid lines, and the width of the shaded band around the solid line indicates the standard deviation (2 σ ). vortex lines is: H c = (cid:88) i (cid:88) j Γ i Γ j L ij = (cid:88) i (cid:88) j (cid:54) = i Γ i Γ j L ij + (cid:88) i Γ i L ii = (cid:88) i (cid:88) j (cid:54) = i Γ i Γ j L ij + (cid:88) i Γ i Wr i (6)where Γ i is the circulation around the i th vortex line,Wr i is the writhe of the i th vortex line, and L ij is thelinking between the i th and j th vortex lines. Since theabove expression includes the writhe of vortex lines whichis not a topological invariant, the centerline helicity of acollection of singular vortex lines is not conserved [9].Assuming that the circulation of each vortex line is Γ,the centerline helicity rescaled by the square of the totalcirculation ( N Γ) becomes: H c ( N Γ) = 1 N (cid:88) i (cid:88) j (cid:54) = i L ij + 1 N (cid:88) i Wr i (7)In the limit N → ∞ , the contribution from the writheterm in Eq. (7) scales as O (1 /N ) and becomes irrelevant, as was shown in [71], leaving only the contribution fromthe linking L ij between different vortex lines which isconserved in Euler flows:lim N →∞ H c ( N Γ) = lim N →∞ N (cid:88) i (cid:88) j (cid:54) = i L ij = H Euler Γ (8)Hence the rescaled centerline helicity of an infinite collec-tion of vortex lines is conserved in Euler flows. However,for a finite number of singular vortex lines, the writheterm remains relevant albeit O (1 /N ) and the rescaledcenterline helicity is not conserved. The case of a super-fluid is interesting in the context of this discussion, sincequantization imposes a fundamental granularity in thevorticity field.Since the above calculation is independent of the dy-namics of the vortices, it leaves unanswered the questionof what the dynamics of the rescaled centerline helicityof collections of superfluid vortex lines will be. In par-ticular, will the centerline helicity remain unchanged asin Euler flows, follow the dynamics observed in viscousflows, or have entirely different dynamics?In the case of Euler flows, the helicity dynamics aresimple: H c remains constant (in the limit of an infinitenumber of vortex lines). In the case of viscous flows, thedynamics are more subtle. For a freely evolving helicalvortex, as shown in a recent study [72], the total helicityconverges to the writhe over time. This can be rational-ized by separating the helicity into contributions from (a)the linking between bundles, (b) the writhing (coiling) ofbundles and (c) the local twisting of vortex lines, withthe total twist being the difference between the total he-licity and the former two. Since the twist is the only localcomponent of helicity, it is the only one acted upon byviscosity and thus the only one that dissipates.The special role of twist can be understood by comput-ing the instantaneous rate of helicity dissipation: ∂ t H = − ν (cid:82) d x ωωω · ∇ × ωωω = − ν (cid:82) d x | ωωω | ˆ ωωω · ∇ × ˆ ωωω , whereˆ ωωω · ∇ × ˆ ωωω captures the local twisting of vortex lines [73],and vanishes identically for a twist-free thin-core vortex[72]. While the role of the twist-free state as the zero-dissipation state is clear, the dynamics of the approachto such a state are more challenging to study because oftheir dependence on the local details of the vortex core[72].Thus for a collection of superfluid vortices, a constantrescaled centerline helicity would suggest Euler-flow likebehavior, while the convergence of the rescaled center-line helicity to the writhe would suggest viscous flow-likebehavior. FIG. 6. The ratio h ( T ) /h (0) approaches the ratio (cid:104) Wr(0) (cid:105) /h (0) of the average initial writhe to the initialrescaled helicity for a variety of helical vortex bundles (1209simulations) in the shape of 2 (aspect ratio:0.35),3 (aspectratio: 0.25), and 4-fold (aspect ratios: 0.16, 0.18, 0.2) heliceswith N = 5 and N = 6 vortex filaments where T is a proxy forthe time at which the vortex bundle disintegrates. To divideby the initial helicity h (0), we only consider vortex bundleswhose initial helicity satisfies: | h (0) | > .
25. Vortex bundleswith initial helicity | h (0) | < .
25 also display similar behaviorwith h ( T ) → (cid:104) Wr(0) (cid:105) as shown in Fig. 5 and the SI.
VI. CENTERLINE HELICITY OF SUPERFLUIDVORTEX BUNDLES
Superfluid vortex bundles which approximate thestructure of a classical thin-core vortex tube, have beenshown to be robust coherent structures [23, 24]. We con-struct thin bundles of equally spaced vortex lines windingaround a central vortex loop as shown in Fig. 3(a), whoseshape controls the writhe (coiling) of the vortex bundle.These superfluid vortex bundles evolve coherently overdistances of the order of their size (see Figs. 3,4, sup-plementary movies) before becoming unstable and dis-integrating, as observed in previous work [23, 24]. Thecoherent portion of the evolution of these bundles resem-bles the dynamics of single vortex loops in superfluidsand the evolution of vortices in classical fluids, and hasbeen studied for ring bundles [24] and reconnecting linebundles [23]. When the vortex bundles become unstable,the number of individual vortices quickly proliferates asshown in the bottom panel of Fig. 5, with the number ofvortex strands acting as a natural indicator of whetherthe bundle has disintegrated. We use the earliest time T at which the number of vortex filaments N ( T ) exceedstheir initial number N by 50% as the time until whichthe bundle evolves coherently. Figure 5 shows that thetransition between the coherent phase and the disinte-gration phase of the vortex bundle is sharp.In order to inject different amounts of centerline helic-ity in the bundle, we twist the lines of the bundle aroundthe central vortex, thus varying the centerline helicityindependently of the writhe of the bundle. An initialcomplex order parameter ψ for these vortex bundles isconstructed following the methods outlined in [9, 34, 68],and evolved by numerically solving the Gross-Pitaevskiiequation (Eq. (1)) using a split-step method. Simulationsof vortex bundles in the shape of helices and trefoil knotsshow that their coherent evolution is much like their clas-sical vortex tube counterparts [9, 74]. Helical vortex bun-dles propagate coherently without a significant change inshape (see Fig. 3) for longer times, while knotted vortexbundles stretch and reconnect (see Fig. 4) to form dis-connected loop bundles which quickly become unstable.Vortex bundles which evolve coherently over long timesallow us to study the dynamics of their rescaled center-line helicity h = H c / ( N Γ) . We focus on helical vortexbundles which evolve coherently over distances of 6¯ r orgreater, and in particular study bundles in which the cen-tral vortex is a toroidal helix (see Figs. 5,6) winding 2,3,4times in the poloidal direction around tori of aspect ra-tios 0 .
35 (2-fold), 0 .
25 (3-fold), 0 . , . , . N = 5 and N = 6 The twisting of vortex lines mentioned here describes the windingof one vortex line around another, and is distinct from the twistTw ∗ in Eq. (5) of the ribbon formed by a phase isosurface endingon a vortex line. vortex lines each having a circulation Γ = 2 π , an initialinter-vortex spacing of d ∼ ξ (see Fig. 3) and an overallr.m.s. radius ¯ r ∼ ξ . To avoid the possibility that sym-metry stabilizes the vortices, we add a small amount ofGaussian noise to each vortex line in the transverse di-rection. To obtain sufficient statistics, we simulated theevolution of a total of 1,156 vortex bundles with a volumeof (256 ξ ) and a grid spacing of 1 ξ . A small number ofsimulations at double resolution (but the same physicalvolume) yield identical observations.Unlike in Euler flows, where the rescaled centerline he-licity h of a bundle of singular vortex lines emerges as aconserved quantity in the limit of large N , the rescaledcenterline helicity h of superfluid vortex bundles appearsto change with time. Assuming these superfluid vortexbundles approximate thin-cored vortex tubes, we can fur-ther decompose their rescaled centerline helicity (Eq. (7))into contributions from the twisting of the vortex linesaround each other, and their individual writhes. Using L ij = Tw ij + Wr i ( j ) , the rescaled centerline helicity be-comes: H c ( t )( N Γ) = 1 N (cid:88) i (cid:88) j (cid:54) = i (Tw ij ( t ) + Wr i ( t )) + 1 N (cid:88) i Wr i ( t )= 1 N (cid:88) i (cid:88) j (cid:54) = i Tw ij ( t ) + 1 N (cid:88) i Wr i ( t )= 1 N (cid:88) i (cid:88) j (cid:54) = i Tw ij ( t ) + (cid:104) Wr( t ) (cid:105) (9)where the average writhe (cid:104) Wr( t ) (cid:105) = (cid:80) i Wr i ( t ) /N in-cludes contributions from the writhe term in Eq.(7), aswell as from the linking term by decomposing it intowrithe and twist contributions.Our numerical simulations show that the rescaled cen-terline helicity of long-lived superfluid vortex bundlestends towards their average initial writhe (cid:104) Wr(0) (cid:105) , as inFig.s 5, 6, suggesting that the twist term in Eq. (9) de-cays over time. The dynamics of the rescaled centerlinehelicity h are thus classical. The role of writhe in the dynamics of centerline helic-ity of superfluid vortex bundles in our simulations hasa striking resemblance to the role of writhe in the he-licity dynamics of vortices in viscous flows [72]. Thispoints to a “classical limit” in which classical behavior isrecovered from quantized vortex filaments geometrically by replacing single vortex filaments with vortex bundles.However, owing to reconnections, the classical behaviorthat is recovered is not that of Euler flows, but that of theNavier-Stokes equations in which viscosity acts to dissi-pate twist. Our results corroborate the role of writhe asan attractor for the helicity at long times, adding a ge-ometric lens to previous work [75, 76] on the dissipativeeffects of vortex reconnections in superfluids. VII. CONCLUSION
We have addressed the existence of an additional con-servation law in superfluids—conservation of helicity—bygeneralizing to superfluids the particle relabeling symme-try, which underlies helicity conservation in Euler flows.The application of Noether’s second theorem to the parti-cle relabeling symmetry [42, 50] yields the conservation ofhelicity and circulation in Euler flows, however for super-fluid flows it yields a trivially vanishing conserved quan-tity. This is owing to the appearance of an additionalterm that comes from the phase of the superfluid orderparameter, not present in Euler flows. This additionalterm has a well-known geometric interpretation for thevanishing of “superfluid helicity” in terms of a relationbetween the linking and writhing of vortex lines, and thetwisting of phase isosurfaces near vortex lines.On replacing superfluid vortices with superfluid vortex bundles , their centerline helicity becomes the classical he-licity in the limit of an infinite collection of vortices. Westudy the dynamics of the centerline helicity of super-fluid vortex bundles via numerical simulations and findbehavior akin to that of classical helicity in a viscousfluid, with the writhe acting as an attractor for the finalvalue of helicity. [1] L. D. Landau and E. M. Lifshitz,
Fluid Mechanics , 2nded., Landau and Lifshitz course of theoretical physics,Vol. 6 (Elsevier, 1987).[2] D. Christodoulou, Bull. Amer. Math. Soc. , 581 (2007).[3] P. Constantin, Bull. Amer. Math. Soc. , 603 (2007).[4] T. Dombre, U. Frisch, J. M. Greene, M. Hnon, A. Mehr,and A. M. Soward, J. Fluid Mech. , 353 (1986).[5] J. T. Beale, T. Kato, and A. Majda, Commun. Math.Phys. , 61 (1984). the difficulty of calculating the average writhe at later timesstems from the small-wavelength fluctuations in the vortex lineswhich contribute to large fluctuations in their writhe. [6] L. Woltjer, PNAS , 489 (1958).[7] J. J. Moreau, C.R. Acad. Sci. Paris , 2810 (1961).[8] H. K. Moffatt, J. Fluid Mech. , 117 (1969).[9] M. W. Scheeler, D. Kleckner, D. Proment, G. L. Kindl-mann, and W. T. M. Irvine, PNAS , 15350 (2014).[10] Y. Kimura and H. K. Moffatt, J. Fluid Mech. , 329(2014).[11] E. Levich and A. Tsinober, Phys. Lett. A , 293 (1983).[12] A. K. M. F. Hussain, J. Fluid Mech. , 303 (1986).[13] N. Yokoi and A. Yoshizawa, Phys. Fluids A: Fluid Dyn. , 464 (1993).[14] C. F. Barenghi, R. J. Donnelly, and W. F. Vinen, eds., Quantized Vortex Dynamics and Superfluid Turbulence (Springer Berlin Heidelberg, 2001).[15] M. S. Paoletti, M. E. Fisher, K. R. Sreenivasan, and
D. P. Lathrop, Phys. Rev. Lett. , 154501 (2008).[16] W. F. Vinen, Phys. Rev. B , 1410 (2000).[17] W. F. Vinen and J. J. Niemela, J. Low Temp. Phys. ,167 (2002).[18] J. Yepez, G. Vahala, L. Vahala, and M. Soe, Phys. Rev.Lett. , 084501 (2009).[19] P. Clark di Leoni, P. D. Mininni, and M. E. Brachet,Phys. Rev. A , 043605 (2016).[20] P. Akhmetev and A. Ruzmaikin, in Topological Aspects ofthe Dynamics of Fluids and Plasmas (Springer Nether-lands, 1992) pp. 249–264.[21] R. Hnninen, N. Hietala, and H. Salman, Sci. Rep. ,37571 (2016).[22] H. Salman, Proc. R. Soc. A , 20160853 (2017).[23] S. Z. Alamri, A. J. Youd, and C. F. Barenghi, Phys.Rev. Lett. , 215302 (2008).[24] D. H. Wacks, A. W. Baggaley, and C. F. Barenghi,Physics of Fluids , 027102 (2014).[25] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari,Rev. Mod. Phys. , 463 (1999).[26] E. P. Gross, J. Math. Phys. , 195 (1963).[27] L. P. Pitaevskii, Sov. Phys. JETP , 451 (1961).[28] C. F. Barenghi and N. G. Parker, A Primer on QuantumFluids , SpringerBriefs in Physics (Springer InternationalPublishing, 2016).[29] R. J. Donnelly,
Quantized Vortices in Helium II (Cam-bridge University Press, 1991).[30] E. Madelung, Naturwissenschaften , 1004 (1926).[31] E. Madelung, Z. Physik , 322 (1927).[32] D. DeTurck and H. Gluck, Journal of Differential Geom-etry , 87 (2013).[33] J. Koplik and H. Levine, Phys. Rev. Lett. , 1375(1993).[34] D. Proment, M. Onorato, and C. F. Barenghi, Phys.Rev. E , 036306 (2012).[35] G. P. Bewley, M. S. Paoletti, K. R. Sreenivasan, andD. P. Lathrop, PNAS , 13707 (2008).[36] J. D. Bekenstein, Phys. Lett. B , 44 (1992).[37] T. R. Mendelson, arXiv:hep-th/9908194 (1999).[38] J. E. Marsden and T. S. Ratiu, Introduction to Mechanicsand Symmetry (Springer New York, 1999).[39] J. E. Marsden, T. S. Ratiu, and J. Scheurle, J. Math.Phys. , 3379 (2000).[40] P. J. Morrison, Rev. Mod. Phys. , 467 (1998).[41] N. Padhye and P. J. Morrison, Phys. Lett. A , 287(1996).[42] N. Padhye and P. J. Morrison, Plasma Phys. Rep. ,869 (1996).[43] Y. Kuroda, Fluid Dyn. Res. , 273 (1990).[44] Y. Fukumoto, Topologica , 003 (2008).[45] R. Salmon, Ann. Rev. Fluid Mech. , 225 (1988).[46] C. J. Cotter and D. D. Holm, Found. Comput. Math. ,457 (2012).[47] T. Tao, “Noether’s theorem, and the conservation lawsfor the Euler equations,” (2014).[48] P. J. Olver, Applications of Lie Groups to DifferentialEquations , Graduate Texts in Mathematics (Springer-Verlag, New York, 1986).[49] Y. Kosmann-Schwarzbach,
The Noether Theorems: In-variance and Conservation Laws in the Twentieth Cen-tury , Sources and Studies in the History of Mathematicsand Physical Sciences (Springer-Verlag, New York, 2011).[50] G. Webb,
Magnetohydrodynamics and Fluid Dynamics:Action Principles and Conservation Laws , Lecture Notes in Physics (Springer International Publishing, 2018).[51] F. P. Bretherton, J. Fluid Mech. , 19 (1970).[52] C. Miniatura, L.-C. Kwek, M. Ducloy, B. Grmaud, B.-G. Englert, L. Cugliandolo, A. Ekert, and K. K. Phua, Ultracold Gases and Quantum Information (Oxford Uni-versity Press, 2011).[53] T. Rindler-Daller and P. R. Shapiro, MNRAS , 135(2012).[54] P. H. Roberts and N. G. Berloff, in
Quantized Vortex Dy-namics and Superfluid Turbulence (Springer Berlin Hei-delberg, 2001) pp. 235–257.[55] L. P. Pitaevskii and S. Stringari,
Bose-Einstein Conden-sation (Clarendon Press, 2003).[56] R. L. Jerrard, Ann. Scuola Norm-Sci. , 733 (2002).[57] F. Lund, Phys. Lett. A , 245 (1991).[58] C. Sulem and P.-L. Sulem, eds., The NonlinearSchrdinger Equation: Self-Focusing and Wave Collapse ,Applied Mathematical Sciences, Vol. 139 (Springer NewYork, 2004).[59] T. Kambe,
Elementary Fluid Mechanics (World Scien-tific, 2007).[60] R. L. Ricca and B. Nipoti, J. Knot Theory Ramifications , 1325 (2011).[61] G. Calugareanu, Rev. Math. pures appl (1959).[62] G. Calugareanu, Czechoslovak Mathematical Journal ,588 (1961).[63] J. H. White, Am. J. Math. , 693 (1969).[64] F. B. Fuller, PNAS , 815 (1971).[65] H. Seifert, Math. Ann. , 571 (1935).[66] A. Ruzmaikin and P. Akhmetiev, Phys. Plasmas , 331(1994).[67] J. J. v. Wijk and A. M. Cohen, IEEE Trans. Vis. Comput.Graphics , 485 (2006).[68] D. Kleckner, L. H. Kauffman, and W. T. M. Irvine, Nat.Phys. , 650 (2016).[69] V. I. Arnold and B. A. Khesin, Topological Methodsin Hydrodynamics , Applied Mathematical Sciences, Vol.125 (Springer New York, New York, NY, 1998).[70] M. A. Berger, Plasma Phys. Control. Fusion , B167(1999).[71] M. A. Berger and G. B. Field, J. Fluid Mech. , 133(1984), bibtex: Berger1984.[72] M. W. Scheeler, W. M. van Rees, H. Kedia, D. Kleckner,and W. T. M. Irvine, Science , 487 (2017).[73] H. Kedia, On the Construction and Dynamics of KnottedFields , Ph.D. thesis, University of Chicago (2017).[74] D. Kleckner and W. T. M. Irvine, Nat. Phys. , 253(2013).[75] C. F. Barenghi, Physica D: Nonlinear Phenomena EulerEquations: 250 Years OnProceedings of an internationalconference, , 2195 (2008).[76] P. Clark di Leoni, P. D. Mininni, and M. E. Brachet,Physical Review A , 053636 (2017).[77] R. Kaiser, C. Westbrook, and F. David, Coherent atomicmatter waves - Ondes de matiere coherentes (Springer-Verlag Berlin Heidelberg, 2001).[78] X. Z. Wang, Phys. Rev. D , 124009 (2001). Appendix A: Helicity—as a Casimir invariant
In Euler flows, helicity emerges a special constant ofmotion: a Casimir invariant [40, 43], i.e. it has a vanish-ing Poisson bracket with any function of the phase spacevariables: {H , F ( u , ρ ) } = 0 ∀ F , where the density ρ and the fluid velocity u are the phase space variables,and { · , · } denotes the Poisson bracket.Solving for the Casimir invariants in Euler flow, i.e.solving {C , F ( u , ρ ) } = 0 ∀ F gives rise to helicity as anadditional conserved quantity. We seek an analogous con-served quantity in superfluids by solving for the Casimirinvariants for the Gross-Pitaevskii equation.The Hamiltonian corresponding to the Gross-Pitaevskii equation is: H = (cid:90) d x (cid:20) (cid:126) m |∇ ψ | + V | ψ | (cid:21) (A1)with the canonical Poisson bracket: { F , G } = − i (cid:126) (cid:90) d x (cid:18) δFδψ δGδψ ∗ − δGδψ δFδψ ∗ (cid:19) (A2)Solving for the Casimir invariants {C , F ( ψ, ψ ∗ ) } = 0 ∀ F reduces to the equations: δ C δψ = 0 , δ C δψ ∗ = 0 (A3)which gives only trivial constants as Casimir invariants.Since Casimir invariants of the Gross-Pitaevskii super-fluid should yield a conserved quantity analogous to he-licity in Euler flows, the above calculation suggests thatthe conserved quantity analogous to helicity in super-fluids is a trivial constant. This is consistent with ourcalculation based on the relabeling symmetry which sug-gests that the conserved quantity analogous to helicity insuperfluids vanishes identically.We note that an alternative path to seeking Casimirinvariants, by taking the phase space variables tobe { j , ρ } = { ( ψ ∗ ∇ ψ − ψ ∇ ψ ∗ ) / (2 i ) , ψ ∗ ψ } instead of { ψ, ψ ∗ } runs into difficulties because of the singular na-ture of vorticity: ∇ × ( j /ρ ). This difficulty manifests interms of an erroneous equation of motion for a vortex,as the Poisson bracket for the new phase space variablesdenoted by {· , ·} j ,ρ incorrectly gives: ∂ t ( ∇ × ( j /ρ )) = {∇ × ( j /ρ ) , H } j ,ρ = 0, suggesting that vortex lines arestationary.We now briefly review the underlying symmetry—therelabeling symmetry—that gives rise to helicity as a con-served charge via Noether’s theorem, and calculate theanalogous conserved charge in superfluids. Appendix B: Helicity as a Noether charge
We now consider a classical fluid which obeys the sameequation of motion as the Gross-Pitaevskii superfluid, ex-cept for the quantum pressure term, and show that therelabeling symmetry which gives rise to helicity conser-vation via Noether’s theorem in Euler fluids gives a van-ishing conserved charge in such a classical fluid. +1+1+1+1+1 +1 +1 +1 +4-1-1 -1 -4-1-1-1-1 -1
FIG. 7. An illustration of the linking of a vortex tangle witha path that is phase-offset from the original vortex. The or-ange (light gray) lines indicate a vortex path, while the teal(dark gray) lines indicate a path offset from each vortex pathalong a direction of constant phase. (The phase fields arecomputed by numerical integration of the Biot-Savart law.)Each signed crossing of an offset line with the original vortexpath is indicated; in each case the total linking between theoffset path and all vortex paths is 0. From left to right, thevortex topologies correspond to a trefoil knot, Hopf link, and6 from the Rolfsen table of links.
1. Superfluid equations of motion
On setting (cid:126) = m = 1, the Gross-Pitaevskii equationof motion for a superfluid is: i∂ t ψ = − ∇ ψ + V | ψ | ψ (B1)On substituting ψ = √ ρ exp( iφ ), the above complexequation gives two real equations for the evolution of ρ and φ as follows: ∂ t ρ + ∇ · ( ρ ∇ φ ) = 0 (B2) ∂ t φ + 12 ( ∇ φ ) + V ρ − (cid:18) ∇ √ ρ √ ρ (cid:19) = 0 (B3)On applying a spatial gradient operator ∇ to Eq. (B3),and substituting the expression for the superfluid velocity u = ∇ φ , we find: ∂ t u + ∇ (cid:18) u + V ρ − ∇ √ ρ √ ρ (cid:19) = 0 (B4)Note that the above equation contains the quantum pres-sure term ∇ (cid:0) ∇ √ ρ/ √ ρ (cid:1) containing spatial derivativesof the density, is dominant only near the vortex core.Such a term is not present in classical hydrodynamics,since the pressure is assumed to depend only on the localdensity, and not on the spatial derivatives of the den-sity. We now make the Thomas-Fermi approximation[25, 55, 77, 78] and neglect the quantum pressure term inthe above equation, thereby considering a hypotheticalclassical fluid which obeys the above equation of motionwithout the quantum pressure term, i.e. ∂ t u + ∇ (cid:18) u + V ρ (cid:19) = 0 (B5)The above equation describes the superfluid well in theregion excluding the vortex core. Note that the above0
FIG. 8. Left-handed helical vortex bundles with positive initial writhe display helicity dynamics similiar to the right-handedhelical vortex bundles shown in Fig. 5 in the main text. The rescaled helicity h for superfluid vortex bundles constructed withvarying degrees of twist, i.e. having different initial helicity, trends towards their initial average writhe (horizontal grey band)as long as the bundle-like structure is preserved, before eventually decaying towards zero (as indicated by the grey dottedlines) for (a) 2-fold helical vortex bundles, (b) 3-fold helical vortex bundles, and (c) 4-fold helical vortex bundles. For eachhelical vortex bundle configuration corresponding to a given initial rescaled helicity h (0), multiple simulations are performedwith random Gaussian noise (r.m.s is 2% of the r.m.s. radius) added to the initial bundle. The mean rescaled helicity isindicated by the solid lines, and the width of the shaded band around the solid line indicates the standard deviation (2 σ ).After the vortex bundle disintegrates, its rescaled helicity is shown by a grey dotted line. The bottom row shows the ratio ofthe number of vortex filaments at time t (cid:48) to the initial number of vortex filaments: N ( t (cid:48) ) /N (0). The time at which a vortexbundle disintegrates is measured as the earliest time at which the number of vortex filaments N ( t (cid:48) ) exceeds the initial numberof vortex filaments N (0) by more than 50%. equation is similar to the equation of motion for an irro-tational Euler fluid: ∂ t u + ∇ (cid:18) u + e (cid:19) = 0 (B6)where e : de = dp/ρ is the enthalpy per unit mass and pis the pressure.
2. Relabeling symmetry in a classical Euler fluid
The action for a classical (isentropic) Euler fluid is: S Euler = (cid:90) d a dτ (cid:32) (cid:18) ∂ x ( a , τ ) ∂τ (cid:19) − E ( ρ ) (cid:33) (B7)where x ( a , τ ) is the position of the fluid element labeledby a at time τ , and the fluid velocity u ( a , τ ) = ∂ τ x ( a , τ ).The label co-ordinates a are chosen such that ρ d x = d a ⇒ ∂ ( x ) ∂ ( a ) = ρ − . It is easily verified [40, 41, 43] thatextremizing the action with respect to variations in theposition field x ( a , τ ), gives the Euler equations of motion.Mass conservation follows from: ∂∂τ ρ − = ∂∂τ (cid:16) ∂ ( x ) ∂ ( a ) (cid:17) . As shown in [40, 41, 43, 44] and can be easily verified,the transformation a i → ˜ a i = a i + (cid:15) η i , such that ∂∂τ η i =0 , ∂∂a i η i = 0 is a symmetry of the action and gives thecorresponding conserved Noether charge: Q = (cid:90) d a u i ∂x i ∂a j η j (B8)When the fluid labels are displaced infinitesimally alonga closed material curve, the conserved charge Q simpli-fies to the circulation around the material loop Γ C , thusgiving Kelvin’s circulation theorem. This can be veri-fied by substituting η j = (cid:72) C : a ( s ) ds δ (3) ( a − a ( s )) ∂a j ( s ) ∂s inEq. (B8). When the fluid labels are displaced infinites-imally along vortex lines, the conserved charge Q is thehelicity of the fluid: Q = H = (cid:82) d x u · ∇ × u . Thiscan be verified by substituting η j = (cid:15) jkl ∂∂a k u p ∂∂a l x p inEq. (B8).1 FIG. 9. The rescaled helicity h ( T ) trends towards the averageinitial writhe (cid:104) Wr(0) (cid:105) for a variety of helical vortex bundlesin the shape of 2,3, and 4-fold helices with N = 5 and N = 6vortex filaments. Here T is the time at which the vortex bun-dle disintegrates, i.e. the earliest time at which the numberfilaments N ( t (cid:48) ) exceeds the initial number of filaments N (0)by more than 50%. The large spread in values of h ( T ) comesfrom vortex bundles whose initial rescaled helicity h (0) is farfrom their average initial writhe (cid:104) Wr(0) (cid:105) , and is removed onrescaling both the axes by h (0), as shown in Fig. 6 in themain text. The final rescaled helicity h ( T ) trends towardsthe average initial writhe as shown in Figs. 5,8 but such vor-tex bundles often disintegrate before the final rescaled helicity h ( T ) becomes equal to the average initial writhe (cid:104) Wr(0) (cid:105) , giv-ing rise to the large observed spread in h ( T ).
3. Relabeling symmetry in a superfluid
The action corresponding to the Gross-Pitaevskii equa-tion is: S gpe = (cid:90) dt d x (cid:18) iψ ∗ ∂ t ψ − |∇ ψ | − V | ψ | (cid:19) (B9)which can be written in terms of ρ, φ as follows: S gpe = − (cid:90) dt d x (cid:32) ρ ∂ t φ + 12 ρ ( ∇ φ ) + V ρ + 12 ( ∇√ ρ ) (cid:33) (B10)It is easy to verify that extremizing the above action inEq. (B10) with respect to ρ, φ gives the desired equationsof motion: Eq.s (B2),(B3), and that the last term in theaction: ( ∇√ ρ ) corresponds to the quantum pressureterm in Eq.s (B3),(B4).We now model the superfluid in the region excludingvortex cores as a classical fluid which carries with it aphase φ ( x , t ). We neglect the quantum pressure term(making the Thomas-Fermi approximation), and use the FIG. 10. Fig. 6 of main text including vortex bundles withlower initial helicity, i.e. | h (0) | < .
25. The larger spreadcomes from dividing by a small number i.e. h (0). The ratioof the rescaled helicity h ( T ) to the initial rescaled helicity h (0) approaches the ratio of the average initial writhe (cid:104) Wr(0) (cid:105) to the initial rescaled helicity for a variety of helical vortexbundles in the shape of 2,3, and 4-fold helices with N = 5and N = 6 vortex filaments. Here T is the time at which thevortex bundle disintegrates, i.e. the earliest time at which thenumber filaments N ( t (cid:48) ) exceeds the initial number of filaments N (0) by more than 50%. relation u = ∇ φ to get the following new action:˜ S gpe = − (cid:90) dt d x (cid:18) ρ u + ρ ∂ t φ + V ρ (cid:19) (B11)In region excluding the vortex cores, we assume thatwe can label the fluid particles with labels a where d a = ρ d x , and track the positions of these particles x ( a , τ )over time τ . We now rewrite the above action in termsof label co-ordinates a , τ using ∂ τ = ∂ t + u · ∇ :˜ S gpe = (cid:90) dτ d a (cid:18) u − ∂ τ φ − V ρ (cid:19) (B12)It is easy to verify that extremizing the above actionwith respect to x ( a , τ ) gives the desired hydrodynamicequation of motion: Eq. (B5), suggesting that trans-forming S gpe in Eq. (B10) to the above action ˜ S gpe inEq. (B12) is akin to performing the Madelung transfor-mation.We now perform the same relabeling transformationthat gives the circulation theorem and helicity conserva-tion in Euler fluids, to seek analogous conservation laws.It is easily verified that the relabeling transformation: a i → ˜ a i = a i + (cid:15) η i , such that ∂η i /∂τ = 0 , ∂η i /∂a i = 0, isa symmetry of the above action ˜ S gpe . The correspondingNoether charge is found to vanish identically, indepen-2dent of η i , as shown below: Q gpe = (cid:90) d a η j (cid:18) ∂x i ∂τ ∂x i ∂a j − ∂φ∂a j (cid:19) = (cid:90) d a η j (cid:18) u i ∂x i ∂a j − ∂φ∂a j (cid:19) = (cid:90) d a η j (cid:18) ∂φ∂x i ∂x i ∂a j − ∂φ∂a j (cid:19) ( ∵ u = ∇ φ )= (cid:90) d a η j (cid:18) ∂φ∂a j − ∂φ∂a j (cid:19) = 0 (B13)The above calculation suggests that the conservedcharges analogous to helicity, and circulation triviallyvanish for superfluids.Note that the presence of an additional phase term ( − ∂ τ φ ) in addition to the terms present in the Euler ac-tion S Euler , is necessary to ensure Galilean invariance (asdefined in [58]) of the modified action ˜ S gpe , much like theconstant term ( − c ) [59] is necessary to ensure Galileaninvariance of the classical fluid action. The presence ofthe additional phase term gives rise to mass conserva-tion in the original Gross-Pitaevskii action S gpe , which ismanifestly Galilean invariant. However, mass conserva-tion is inherent to the description of the superfluid whenexpressed in terms of the particle label co-ordinate frame( a , τ, τ