Bathtub vortex in superfluid 4 He
aa r X i v : . [ c ond - m a t . o t h e r] D ec Bathtub vortex in superfluid He Sosuke Inui, Tomo Nakagawa, and Makoto Tsubota Department of Physics, Osaka City University, 3-3-138 Sugimoto, 558-8585 Osaka, Japan Department of Physics & Nambu Yoichiro Institute of Theoretical and Experimental Physics(NITEP) & The OCU Advanced Research Institute for Natural Science and Technology (OCARINA),Osaka City University, 3-3-138 Sugimoto, 558-8585 Osaka, Japan (Dated: December 24, 2020)We have investigated the structure of macroscopic suction flows in superfluid He. In this study, weprimarily analyze the structure of the quantized vortex bundle that appears to play an importantrole in such systems. Our study is motivated by a series of recent experiments conducted by aresearch group in Osaka City University [Yano et al . , J. Phys. Conf. Ser. , 012002 (2018)];they created a suction vortex using a rotor in superfluid He. They also reported that up to 10 quantized vortices accumulated in the central region of the rotating flow. The quantized vortices insuch macroscopic flows are assumed to form a bundle structure; however, the mechanism has notyet been fully investigated. Therefore, we prescribe a macroscopic suction flow to the normal fluidand discuss the evolution of a giant vortex ( i . e . , one with a circulation quantum number exceedingunity) and a bundle of singly quantized vortices from a small number of seed vortices. Then, usingnumerical simulations, we discuss several possible characteristic structures of the bundle in such aflow, and we suggest that the actual steady-state bundle structure in the experiment can be verifiedby measuring the diffusion constant of the vortex bundle after the macroscopic normal flow has beenswitched off. By applying extensive knowledge of the superfluid He system, we elucidate a newtype of macroscopic superfluid flow and identify a novel structure of quantized vortices.
I. INTRODUCTION
We often encounter “vortices” of various length scales:the dropping of milk into coffee, whirlpools, the GreatRed Spot on the surface of Jupiter, and so on. The suc-tion vortex, also referred to as the “bathtub vortex,” isone of the most familiar classical vortices; it can be eas-ily produced by unplugging a bathtub filled with water.However, this vortex’s simple generation procedure doesnot entail that its structure can be easily understood. In-deed, despite several attempts, no theory of the vortexhas yet been completed [1–3]. In this paper, we elucidatethe bathtub vortex from a different perspective: that ofa bathtub vortex in superfluid He.Liquid He, at a saturated evaporation pressure belowthe lambda point T λ ≈ .
17 K, exhibits superfluidity; inthis state, its sheer viscosity vanishes and a number ofeccentric phenomena (e.g., fountain and capillary effects)can be observed. These effects are often explained using aphenomenological model (the so-called “two-fluid model”[4–6]), in which the superfluid He at 0 < T < T λ fea-tures two fluid components: an inviscid superfluid withdensity ρ s ( T ) and a viscous normal-fluid with density ρ n ( T ). One of the most notable properties of superfluidsis that their circulation κ ≡ H L v · d l can be quantized as κ = hm n, (1)where n is an integer, h is Planck’s constant, and m is themass of a He atom. This quantization assumes that thepath L encloses a filamentary topological defect in thesuperfluid. The topological defects with a quantized cir-culation always form closed loops or terminate their endsat boundaries, and thus they are called quantized vortex loops or lines [7]. In a bulk superfluid, the kinetic energyper unit length of the vortex line ǫ is proportional to n ;thus, it is more energetically stable to have two vorticeswith n = 1 than one vortex with n = 2. The superfluidsystem is very clean and offers an ideal experimental en-vironment for many fields of physics; thus, it has beenextensively studied over the decades by researchers hop-ing to understand various physical phenomena, includingturbulence [8–11], the Kibble–Zurek mechanism [12–14],and pulsar glitches in neutron stars [15, 16].In the experiments conducted by a research group atOsaka City University (OCU), Yano et al . created amacroscopic bathtub vortex by sucking superfluid He(temperature: T = 1 . r . The nor-mal fluid has a viscosity; thus, it can be reasonably as-sumed that its steady-state flow profile resembles the pro-file discussed in Refs. [1, 2]; that is, the down-flow is nar-rowly confined in the central region, forming a flow tubeabove the drain hole. It is classically understood that anup-flow surrounds the down-flow, owing to the vorticitygenerated near the central region [2]. However, the vor-ticity of the superfluid is only carried by quantized vor-tices; therefore, this might not apply in the non-classical Region IRegion IIRegion III
Drain holeRotor (a) (b) (c)
FIG. 1. (a) – Schematic overview of the “bathtub vortex”[17, 18]. The entire length scale of the system (from the sur-face to the bottom of the fluid) is approximately 20–30 cm.The system can be roughly separated into three regions: Re-gion I, in which the surface of the superfluid He dimples anda large vortex with circulation quantum number n > case, and the formation of classical-like macroscopic suc-tion flows is not trivial. Moreover, from observations ofsecond sound attenuation, the vortex line density L L atthe core region (radius ∼ . × m − [18].These vortex lines are thought to be attracted towardthe axis of rotation, thereby forming a vortex bundle [25]through the particular macroscopic flow geometry of thesystem of two fluids. In the presence of the downflow ofnormal fluid, we argue that such a highly accumulatedvortex line density in the core region can be developedwith a structural pattern inherited from the flow geom-etry. Throughout this analysis, we prescribe the profileof the normal fluid velocity and perform a series of nu-merical simulations to follow the dynamics of individualvortices, rather than a coarse-grained vortex line densityfield, to investigate the large-scale structure of the vortexbundle.The generation mechanism of a macroscopic bathtubvortex in a superfluid is not trivial. To understand suchnovel macroscopic flows in superfluid He, it is neces-sary to construct models that do not contradict the ex-perimental results; for this, we apply extensive back-ground knowledge on superfluidity and computational techniques developed over several decades. The objectiveof this study is to qualitatively understand the structureof quantized vortices in such a macroscopic suction flow.In this article, we argue that 1) the deformed superfluidsurface is identified as a giant vortex, 2) a strongly polar-ized vortex bundle is developed along the rotational axisbeneath the dimpled surface, and 3) the polarization ofthe bundle may be assessed experimentally by measur-ing the diffusion constant of the bundle. We divide thesystem into three regions, as shown in Fig. 1, based onthe boundary conditions. Region I is where the surfaceboundary cannot be neglected. The dimpled surface cre-ated in Region I may be identified as a giant vortex fromthe 1 /r velocity profile around it (see Sec. III for thediscussion). Region II is a bulk, where there is presum-ably no giant vortex, but a bundle of singly quantizedvortices that would resemble the configuration in Fig. 6( b ) or ( c ). Region III is a region in which the bottomboundary condition is not negligible. The flow geome-try near the bottom layer, known as an Ekman layer inclassical hydrodynamics, is not trivial, and the discus-sion on the vortex dynamics in this region is beyond thescope of our current work. In Sec. II we briefly reviewthe numerical model used to simulate vortex dynamics, i . e .
3D vortex filament model (VFM). In Sec. III, theprocess of giant vortex production is discussed. Then,we discuss how vortices are transported from Region I toRegion II, using VFM simulations. In Sec. IV, we showthat, depending on the geometry of the normal fluid flow,two characteristic vortex bundle structures are possiblein Region II: a linear-vortex structure and a cylindricalvortex-layer-like structure. In Sec. V, we argue that thelarge scale vortex bundle structure may be determined bythe experimental observation of the diffusion constant ofthe vortex bundle. We qualitatively estimate the charac-teristic diffusion time scale of the bundle from the VFMsimulations. Finally, in Sec. VI, we summarize theoverall structure of a bathtub vortex.
II. EQUATION OF MOTION FOR VORTICES
The core radius of a quantized vortex in superfluid Heis of the order of ˚ A , and a vortex segment carries a po-tential flow of velocity v s ∝ /r around it, where r is theradial distance from the vortex core. Thus, quantizedvortices are often treated as having a delta-function-likevorticity at position s ( ξ ), using the arc length param-eterization ξ . Thus, the motion of a quantized vortexobeys the Helmholtz’s theorems and follows the local su-perfluid flow v s ( ξ ). However, at finite temperatures, thetemperature-dependent mutual friction terms α and α ′ become significant, and the equation of motion is [26]d s ( ξ, t )d t = v s + α s ′ ( ξ ) × ( v n − v s ) − α ′ s ′ ( ξ ) × [ s ′ ( ξ ) × ( v n − v s )] , (2)where v s and v n are the velocity fields of super- andnormal-fluids, respectively; the prime symbol ′ denotesthe derivative with respect to arc length ξ . Therefore,we can calculate the time-evolution of a vortex, once weobtain the velocities, v s and v n , at s ( ξ ).In Region II, we consider a symmetrically rotating flowof normal fluid along the z -axis that resembles a Rankinevortex velocity profile of the form v n ( r, φ, z ) = Γ n π rR v z ( r ) for r < R (3) v n ( r, φ, z ) = Γ n π r v z ( r ) for r > R , (4)where R is the radius of the down-flow tube (which isthe same size as the drain hole at the bottom of the con-tainer), and Γ n is the circulation of the normal fluid. Thevertical velocity profile v z ( r ) is not known, experimen-tally nor theoretically. Since we qualitatively investigatethe macroscopic structure that is imprinted on a vortexbundle by such a flow in Region II, we assume that thestructure is not highly dependent on the detail of the flowprofile v z ( r ). Thus, for simplicity, we take it as constantif r < R , and 0 otherwise, and To identify the velocity v s at s ( ξ ) we apply the vortex filament model (VFM),which is briefly explained in the following subsection. A. Numerical Method: Vortex Filament Model
First, we consider a 3D vortex line configuration, dis-cretizing it into segments of length dξ . A vortex segmentat s ( ξ ) tends to move with velocity v s ( s ( ξ )). The term v s ( s ( ξ )) can be decomposed into three contributions: thevelocity v s, , which is induced by all vortices in the sys-tem; the velocity v s, ext , which is imposed externally; andthe velocity v s, b , which is induced by the boundaries.The superfluid velocity v s, at ξ is obtained by calculat-ing the following Biot–Savart integral: v s, ( ξ ) = κ π Z L s ′ ( ξ ) × ( s ( ξ ) − s ( ξ )) | s ( ξ ) − s ( ξ ) | d ξ = v s, loc + v s, non-loc . (5)The integral (5) diverges as ξ → ξ , because we ne-glect the core radius a of the vortex. Computation-ally, we avoid the divergence by separating out the localterm from the total integration path L , to obtain v s, loc and v s, non-loc . Applying the local induction approxima-tion, v s, loc can be evaluated as v s, loc ≈ β s ′ × s ′′ , where β = ( κ/ π ) ln( R/a ). To solve Eq. (2) and perform thesimulation, the path L is divided into segments of ∆ ξ ,and the integration in Eq. (5) is calculated for eachsegment and at every time step ∆ t in the fourth-orderRunge–Kutta scheme. III. GIANT VORTEX AND VORTEXTRANSPORT IN REGION I
The steady bathtub vortex in He features a deep cav-ity in the central region. The shape of the cavity in-dicates that the azimuthal velocities of both the super-and normal-fluids are inversely proportional to the ra-dial distance r around it. This implies that, for a fullydeveloped bathtub vortex, the cavity behaves like a gi-ant vortex; that is, a quantized vortex with a circulationquantum number n > z -axis, and the quantizedvortices are also transported toward the central regionfrom the surrounding bulk fluid. As these gather, theystart to exhibit a collective rotational motion, formingsome type of lattice structure; this is analogous to thetriangular-lattice formation observed in solid-body rotat-ing superfluid helium [27], BEC [21–24], and supercon-ducting currents [28]. Then, the surface of the superfluid He gradually starts to deform in the central region, dueto the pressure difference and down-flow. The surfacebecomes increasingly deformed and generates a cavity ofdepth h ∞ − h (as measured from the height of the sta-tionary surface h ∞ at r → ∞ ) as the vorticity of thenormal fluid accumulates and vertical vortices enter thevicinity; we can identify this as a giant vortex of circu-lation quantum number n giant >
1. Taking the cavitydepth h as a function of radial distance r , h (0) = h andlim r →∞ h ( r ) = h ∞ ; thus, the quantum number n giant at h ( r ) can be identified as the number of singly quantizedvortex lines attached below the surface, as shown in Fig.2 (c). In a steady state, the macroscopic flow profiles of (a) (b) (c) n=1n=3n=5n=6n=2n=4 h ∞ − h “Initial stage” “Intermediate stage” “Late stage” FIG. 2. (a) – (c) Snapshots of the three stages of giant vor-tex production (color online). The blue lines in each panelrepresent the singly quantized vortices, and the shaded re-gion around the vertical axis ( z -axis) represents the regionin which the vorticity of the normal fluid accumulates andforms a strong down-flow. Each stage is briefly described asfollows: (a) – Initial stage: vortex lines gather and tend toform a lattice. (b) – Intermediate stage: the surface of thecentral region dimples owing to the azimuthal velocity, whichis inversely proportional to the radial distance r and pressuredifference. (c) – Late stage: the dimple grows to become acavity by “absorbing” singly quantized vortices. At this stage,the normal fluid circulation Γ n is not necessarily equal to thatof the giant vortex, κn giant . v z S i d e v i e w T op v i e w .
25 cm
FIG. 3. Series of snapshots of VFM simulation at t = 0 .
0, 1 .
4, 2 .
8, 4 .
2, 5 .
6, and 7 . .
25 cm) in each panel representsa giant vortex, around which the circulation of both fluids are non-zero. In the system, the external normal fluid velocity v z isapplied downward. The top and bottom surfaces of the box are subject to the periodic boundary condition. the super- and normal-fluids coincide with each other, tominimize the mutual friction; this means that the circula-tion of each fluid around the entire system is equal; thatis, Γ n = Γ s + κN vor . Here, Γ s = κn giant with κ = h/m ,and N vor is the number of freely floating vortex lines.If the system is ideally clean ( i . e . , no remnant vortexrings exist), then after a sufficiently long time, Γ n = Γ s and κN vor = 0 are satisfied, because all the singly quan-tized vortices are “absorbed” into the giant one. How-ever, because of the geometry of the experimental setup,vortex rings can be constantly transported to the cen-tral regions from the side, under the macroscopic flowgenerated by the rotor (see Fig. 1 (c) ). We conductednumerical simulations to qualitatively assess the vortexline distributions in the presence of flows proportional to1 /r ; that is, the azimuthal velocity profiles for normal-and super-fluids were v n = Γ n / πr and v s = Γ s / πr , re-spectively, for an r outside the giant vortex (radius: 0 . n > Γ s . Figure 3 shows a seriesof snapshots of the simulation, conducted with the pa-rameters Γ n = 5 . × − m /s and Γ s = 4 . × − m /s; the prescribed vertical normal velocity component v z = − . He, where the giantvortex (with circulation Γ s ) is assumed to exist. Initially,three vortex lines exhibiting a Kelvin wave excitation areplaced around the giant vortex. The vortex lines and gi-ant vortex are aligned mutually parallel, hence they tendto repel each other. However, because Γ n > Γ s , thesingly quantized vortex lines are pulled toward the cylin-der under mutual friction. In the presence of externalflows proportional to 1 /r , the vortices are stretched andspiraled in toward the cylindrical surface, as shown in Fig. 3. Locally, the orientation of the vortex line nearthe wall is almost parallel to that of the wall; eventually,the tip of the vortex reaches the surface.In this simulation, special attention must be paid whenhandling the reconnection events between the singlyquantized vortices and the giant vortex. When a vortexline approaches and hits the surface of the hollow cylinderof the giant vortex, a reconnection event is highly likely;this is thought to be a crucial mechanism that sustainsthe growth of the circulation Γ s when Γ n > Γ s . How-ever, the conventional method of managing these eventsalgorithmically [26] may not be valid in this system, be-cause the boundary condition at the surface of the giantvortex is unknown. We can assume that the singly quan-tized vortex lines must intersect the surface of the giantvortex perpendicularly, so that the superfluid does notflow out of the fluid through the boundary. The perpen-dicularity of the vortices at the reconnecting points isapproximately attained by introducing an “effective fric-tion” to the ends of the vortex lines where they meet thewall of the giant vortex. In the numerical simulation,we simply set the azimuthal and vertical velocity com-ponents of the vortex segment to be zero when it entersthe cylinder through the wall. The reconnected segmentscircles around the giant vortex, and the remaining vor-tex lines are wound around the cylinder; this can be ob-served in the panels in Fig 4 and in the video found inRef. [29]. However, in the presence of the vertical nor-mal flow, only the vortex segments whose orientationsare such that they induces a superfluid flow along thenormal flow grow selectively; meanwhile, those with theopposite orientation tend to diminish gradually throughmutual friction. This means that spiral-shaped vortex fil-aments with the same helical orientation are tend to beformed, which is be similar to the vortex mill discussedby Schwarz in Ref. [30].Through the processes discussed in this section, quan- FIG. 4. Series of magnified snapshots of VFM simulation at t = 6 .
1, 6 .
2, 6 .
3, and 6 . tized vortices with a specific orientation were selectivelyproduced in Region I; they then travelled to Region II.As vortex lines continue to wind around the giant vortex,the value of the circulation Γ s increases. When the valueof Γ s becomes sufficiently close to that of Γ n , the giantvortex no longer attracts the free vortices, and the vor-tices enter a quasi-stable equilibrium state. The vorticessteadily produced in Region I can behave as a “vortexbath,” which is essential to bundle formation in RegionII; we discuss this in Sec. IV. IV. BUNDLE FORMATION IN REGION II
In the presence of a steady down-flow and anazimuthal-flow of normal fluid in Region II, some char-acteristic structural patterns/ polarization may be im-printed on the vortices that are densely produced in Re-gion I and transported to Region II. We consider the nor-mal fluid velocity profile given by Eqs. (3) and (4), andwe neglect the flow profile perturbation attributable tothe quantized vortices generated through mutual friction.Microscopically, this assumption does not hold. Recentstudies [31–34] have shown that the normal fluid profileis non-trivially modulated by the presence of quantizedvortices, through mutual friction on the scale of the inter-vortex distance. However, in the analysis below, we onlyconsider the macroscopic vortex bundle structure thatdevelops in the macroscopic steady normal flow; a studyof the characteristic small-scale structures that emergedue to coupled dynamics remains a future work to bedealt with.One factor that characterizes the macroscopic vortexbundle structure is the ratio of the vertical velocity v z tothe azimuthal velocity v φ of the normal fluid. To observethe effects of this factor, we consider a helical vortex line s ( ξ ) with arc length parametrization ξ ∈ R : s ( ξ ) ≡ x ( ξ ) y ( ξ ) z ( ξ ) = X cos k ξY sin k ξξ . (6)On the right-hand side of Eq. (2), we neglect all terms ex- cept the one proportional to v n (the second term); then,the equation of motion for r < R simplifies to ˙ s ( ξ, t ) ≈ α s ′ × v n = A (cid:16) k v z X Y − Γ n πR (cid:17) x (cid:16) k v z Y X − Γ n πR (cid:17) y Γ n k πR (cid:16) Y X − X Y (cid:17) , (7)where A = α/ p k ( X + Y ) + 1. When X = Y , theequation of motion for the helix amplitude r ≡ p x + y is simply ˙ r = (cid:18) k v z − Γ2 πR (cid:19) r. (8)Equation (8) indicates that when Γ n πR > k v z , the right-hand side of Eq. (8) becomes negative, and the amplitude r diminishes. Assuming that the maximum wavelengthof a vortex line in such a rotating normal fluid tube (ra-dius: R ) is at most λ max ≡ π/k , min ∼ R , then thecriterion for the helical excitation on the vortex line todiminish becomes v φ v z & π, (9)where v φ ≡ Γ n / πR is the azimuthal velocity at ra-dial distance r = R . The validity of the criterion isconfirmed through numerical simulations of the VFM inSec. IV A. A. VFM simulations for Region II
We consider the dynamics of six seed vortex rings ran-domly placed near the central region of radius R (whichis shown as the shaded region in Fig. 1(b) schemati-cally) and see how the flow ratio modifies the polariza-tion of the growing vortices. In numerical simulations,we set the radius R = 2 . v z = −
10 mm/s, andwe adjust the circulation of normal fluid Γ n such that v φ = 10 × π and 1 × π mm/s. Figure 5 (a) and the videoin Ref. [35] show the case in which v φ /v z = π . Smallexcitations/Kelvin waves in the horizontal direction onthe vortex lines are visibly damped, and straight vortexlines tend to align themselves and lengthen in the cen-tral region along the z -axis, as the rough estimate in Eq.(9) indicates. However, when the ratio was sufficientlysmall, the amplitudes of the Kelvin waves are amplified;this can be seen in Fig. 5 (b) and in the video found inRef. [35]. A helical excitation is amplified in the flowcylinder of radius R . However, when the radius of theexcitation exceeds R , it ceases to grow because of theabsence of normal flow that transfers energy through mu-tual friction. As more helical excitations are generated,a helically polarized vortex bundle is formed. Now, theindividual vortices are repelled from the central region S i d e v i e w T op v i e w (a) (b) FIG. 5. (a) Snapshots of VFM simulation with v z = 10 mm/s and v φ = 10 × π mm/s. The ratio v z /v φ is chosen to satisfythe relation in Eq. (9). The formation of a bundle of vertical vortex lines can be observed in the central region within thecylindrical shell of radius R = 2 . v z = 10 mm/s and v φ = 1 × π mm/s. Becausethe relation is not satisfied, the amplitudes of the excitations are significant, and eventually the cylinder of radius R is coveredby helical vortex lines. and form a “vortex layer” surrounding the cylinder ofradius R . The vortex layer induces a superfluid flow in-side cylinder of the vortex layer, analogous to a magneticfield generated by a current passed through a coil.In both cases, the growth of the vortices along the z -axis appears to be indefinite while the steady normal flowprofile is prescribed; however, in simulations, the maxi-mum vortex line density is limited by the computationalresolution ∆ ξ . Furthermore, in reality, the dense vortexbundle would significantly deform the normal profile, andour method will eventually break down. We note that ourabove analysis only applies to the initially growing stateof the bundle; however, it is crucial for understanding thestructure of the bundle.The growth of the vortex line density in such an ex-ternal flow may be obtained in the numerical simulationsin the framework of the HVBK hydrodynamics as well.However, in the HVBK framework the quantized vorticesare treated as a coarse-grained vortex-line density fieldin which the microscopic information of vortices, such aslocal curvature, is lost. Therefore, such a method maynot be suitable to investigate the vortex bundle structuredirectly ascribed to individual vortex dynamics. V. ESTIMATION OF DIFFUSION TIME SCALE
We have discussed the structure of the vortex bundlein Region II. However, the direct determination of thebundle in experiments is difficult. Therefore, we proposethat the structure (polarization/helicity) of the bundlemay be assessed qualitatively by the determination ofthe diffusion constant of the bundle.The bundle in the steady state is energetically sus-tained by the normal fluid; thus, when the rotor isstopped, the normal flow slows down and the bundle dif-fuses. The diffusion constant D of a homogeneous vortextangle is reported to be of the order of the circulationquantum number κ = h/m [36–38]. However, in our case, the bundle is assumed to possess an ordered struc-ture; this would allow the system to have a structure-dependent diffusion constant, which is an experimentallymeasurable quantity.We consider a system of N vertical, mutually parallelquantized vortices distributed evenly within a cylindricalregion of radius R = 0 .
25 mm. The height of the systemis set to 2 . z -axis ( n twist = 0) (as shown in Fig.6 ( b )), the scenario is relatively simple: The vortices forma triangular lattice as the radius R of the occupied cross-sectional area grows from its initial value R . Then, thesuperfluid velocity within the radius R mimics a rigid ro-tation. However, when the bundle is “twisted” such thatall the vortices are helically deformed (as in Fig. 6 ( c )),the situation becomes more complex.First, to qualitatively understand the diffusion processin this system, we consider the kinetic energy E R of abundle of N vertical vortex lines confined in a region ofradius R . For simplicity, we assume that the vorticesare not twisted ( i . e . , n twist = 0) and that the superfluidvelocity profile induced by the vortices is given (in cylin-drical polar coordinates) as v s ( r, φ, z ) = Γ s π rR for r < R , (10)and v s ( r, φ, z ) = Γ s π r for r > R, (11)where Γ s = κN . Then, the kinetic energy per unit heightcan be calculated as E R /L z = ( ρ s / π R R max drrv s .Substituting Eqs. (10) and (11) into the integral, theenergy is expressed as E R L z = Γ ρ s π (cid:20)
14 + ln R max − ln R (cid:21) , (12)where R max is the radius of the cylindrical container. Interms of the area A ≡ πR , the time derivative of Eq.(12) is ddt E R L z = − Γ ρ s π ˙ AA . (13)We can also estimate the energy dissipation rate ε fromthe mutual friction per unit length between the restingnormal fluid and the vortex lines. In the first-order ap-proximation, the frictional force f per unit length of avortex segment is known to be proportional to its veloc-ity, and the proportionality constant γ depends on thetemperature T [39]. Therefore, we obtain ε = f · v s = γ Γ s π N X i =1 r i R ≈ γ Γ s N πA . (14)The sum P Ni =1 r i in the second line is approximatelyevaluated as N R /
2, assuming an even distribution. Theonly major factor determining energy loss in the systemis the mutual friction; thus, we equate Eq. (13) and Eq.(14) to finally obtain A ( t ) = A + γ Nρ s t. (15)Figure 6 ( a ) plots the computationally obtained valuesfor the properly normalized areas of the bundle cross-section ( i . e . , ( A ( t ) − A ) ρ s /γ N t ) as functions of time t . It can be clearly seen that when n = 0, the valuesagree with Eq. (15). However, they start to diverge astime elapses; thus, higher-order estimates are needed fora more precise discussion. Interestingly, when n twist > n twist and the reduction from unity in Fig. 6 ( a ) has notyet been established, the significant suppression of vor-tex bundle diffusion can be expected in the experimentsif the bundle is twisted.The expression in Eq. (15) relates to the diffusionconstant D in conventional 2-dimensional diffusion prob-lems; that is, ˙ n = D ∇ n . A solution to the partial dif-ferential equation, using an instantaneous delta function-like source at time t = 0, takes the form n ( r, t ) = N πDt exp (cid:18) − r Dt (cid:19) , (16)where n ( r, t ) is the vortex number density such that N = 2 π R ∞ rn ( r, t ) dr is the total number of vortices. ( A ( t ) − A ) ρ s / γ N t [ a r b . un i t ] t [s] n twist = 0 n twist = 1 n twist = 2 n twist = 3 : N = 60: N = 40: N = 20 : N = 100 n twist = 0 n twist = 1 t = 0 . t = 4 . t = 0 . t = 4 . ( a )( b ) ( c ) FIG. 6. (a) Normalized cross-sectional areas of the bundles asfunctions of time for various numbers of vortices N and twists n twist . The values of the functions are proportional to thediffusion constant D . The proportionality constant is foundin Eq. (17). (b) – (c) Snapshots of VFM simulations with N = 60 vortices for n twist = 0 and 1, respectively. The topand bottom boundaries are subject to the periodic boundarycondition. The disks in each panel are of radii 0 . The radius R of the cross-sectional area of the bundle ischaracterized by the exponential function in Eq. (16),and R ∼ √ Dt . Combining this result with Eq. (15), weobtain the final expression: D ≈ γ N πρ s . (17)In the experiment at OCU, because the temperature T was 1 . N was of order 10 ,the diffusion constant was approximately D ≈ /s.The values of the temperature-dependent quantities γ and ρ s can be found in Ref. [41].If we linearly extrapolate our computational results forthe simplified system, then the diffusion constant mea-surable in the experiment is ∼ /s if the vortexbundle is twisted. In our above analysis, the normal fluidis assumed to be at rest for the sake of simplicity; how-ever, in the case of an experiment where 10 vortices arepresent, this assumption may not be valid. If the bundleof vortices and the normal flow co-rotate about the z -axis, then the energy loss via mutual friction in Eq. (14)is reduced; this would lead to further reduction of thediffusion constant, at least initially. Therefore, we wouldneed to wait for sufficiently long time after the rotor isstopped (so that the vortex line density becomes smalland normal fluid comes to rest) to observe the predictingdecay behavior. VI. CONCLUSIONS AND DISCUSSION
Motivated by an experimental report on the “bath-tub” vortex of superfluid He, we discussed the structureof the quantized vortex bundle that can be formed insuch a macroscopic flow, based on numerical simulationsusing the VFM. The superfluid bathtub vortex systemwas investigated by separating it into three regions. Thetop region (Region I) is assumed to contain a giant vor-tex with multiply quantized circulation. By analogy withrotating superfluid He, we illustrated the developmentprocess of the giant vortex (or the surface dimple). InRegion I, a vortex bundle can develop alongside the giantvortex. The bundle that forms around the giant vortexappears to act as a major source of the vortices that aretransferred to Region II; thus, it can be considered as a“vortex-line bath.” Region II is the region in which theboundary effect of the vessel bottom is negligible and avortex-line bath is present at the top. Because the nor-mal fluid has an intrinsic viscosity, we assume that it es-tablishes a macroscopic steady flow. The steady normalflow “stirs” the transferred vortex loops; this presumablydeforms the bundle structurally, reflecting the geometryof the normal flow. Then, the bundle settles in a steady state such that the mutual friction between the two fluidsis minimized.The velocity profile of the normal fluid in our analysisis that of a Rankine-vortex-like flow, containing a verticalflow within a radius R along the z -axis, as described inEqs. (3) and (4). In such environments, the vortices thatconstitute a bundle either (a) align themselves parallellyalong the z -axis or (b) wind around the down-flow regionof radius R and form a cylindrical vortex layer. Whetherthe bundle takes the structure (a) or (b) depends on theratio of the vertical velocity v z to the azimuthal veloc-ity v φ of the normal fluid. Because of the complexity ofthe experimental setup, no direct experimental data arecurrently available to indicate size of the ratio. Insteadof measuring the ratio, we proposed that the structurecould be elucidated indirectly, by measuring the decay ofthe vortex bundle. In the OCU experiment, the expectedvortex diffusion constant D was approximately 8 mm /s,if the bundle was not twisted along the z -axis. A seriesof VFM simulations indicate that the diffusion constantis significantly reduced if the bundle is twisted. By ex-perimentally measuring the extent to which the diffusionconstant diverges from its expected value, we can furtherour understanding of the structures of vortex bundles inmacroscopic bathtub vortices. ACKNOWLEDGMENTS
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