Classical and quantum vortex leapfrogging in two-dimensional channels
Luca Galantucci, Michele Sciacca, Nick G Parker, Andrew W Baggaley, Carlo F Barenghi
CClassical and quantum vortex leapfrogging in two-dimensional channels
Luca Galantucci, Michele Sciacca, Nick G. Parker, Andrew W. Baggaley, and Carlo F. Barenghi
Joint Quantum Centre Durham–Newcastle, School of Mathematics, Statistics and Physics,Newcastle University, Newcastle upon Tyne, NE1 7RU, United Kingdom
The leapfrogging of coaxial vortex rings is a famous effect which has been noticed since the timesof Helmholtz. Recent advances in ultra-cold atomic gases show that the effect can now be studiedin quantum fluids. The strong confinement which characterizes these systems motivates the studyof leapfrogging of vortices within narrow channels. Using the two-dimensional point vortex model,we show that in the constrained geometry of a two-dimensional channel the dynamics is richer thanin an unbounded domain: alongsize the known regimes of standard leapfrogging and the absenceof it, we identify new regimes of backward leapfrogging and periodic orbits. Moreover, by solvingthe Gross-Pitaevskii equation for a Bose-Einstein condensate, we show that all four regimes existfor quantum vortices too. Finally, we discuss the differences between classical and quantum vortexleapfrogging which appear when the quantum healing length becomes significant compared to thevortex separation or the channel size, and when, due to high velocity, compressibility effects in thecondensate becomes significant. a r X i v : . [ c ond - m a t . qu a n t - g a s ] J un I. INTRODUCTION
The leapfrogging of two co-axial vortex rings (in three dimensions) or of two vortex-antivortex pairs (in two di-mensions) is a benchmark problem of vortex interaction [41] which dates back to [24]. The time evolution of thisvortex configuration is striking: the vortex ring (or pair) which is ahead widens and slows down, while the ring behindcontracts, speeds up, catches up with the first ring and goes ahead through it; this ‘leapfrogging’ game is then repeatedover and over again, unless instabilities disrupt it. A number of papers have been written on different aspects of thisproblem, ranging from the stability [1, 26, 36, 64] to the deformation of the vortex cores and to the effects of viscosity[60] using numerical [10, 13, 54] as well as experimental methods [37, 40, 51, 73]. The most recent developmentsconcern leapfrogging of vortex bundles [68] and helical waves [27, 52, 57].Our work is motivated by recent experiments with atomic Bose-Einstein condensates, which constitute a dilutequantum fluid and provide an idealised platform to study fundamental vortex dynamics [70]. In these experiments,atomic gases are confined by suitable magnetic-optical traps and cooled to nano-Kelvin temperatures. If the atomsof the gas are bosons (i.e. have integer spin), a phase transition occurs upon cooling below a critical temperature T c , and the gas forms a macroscopic coherent quantum state [4] called a Bose-Einstein condensate (BEC). From thepoint of view of the hydrodynamics, a BEC has three key properties: it is superfluid (i.e. it suffers no viscous losses ofkinetic energy when it flows), it is compressible, and its vorticity is concentrated to thin hollow vortex lines with fixedwidth a and fixed circulation ± h/m where h is Planck’s constant and m is the mass of a boson (while vortices withlarger quanta of circulation, ± h/m, ± h/m, · · · , are possible, they are unstable to decay into multiple singly-chargedvortices). Thus, in BECs, vortices are well-defined and identical objects, evolving in an inviscid compressible fluid.There are several additional characteristics of atomic BECs that make them attractive for probing vortex dynamics.Firstly, the physical parameters of the fluid (including the width and speed of the vortices) are tunable, for example,through the density of the gas and the strength of the atom-atom interaction (which can be modified by means ofFeshbach resonances [29]); this should be contrasted with superfluid liquid helium - historically the most studiedquantum fluid - whose physical parameters are fixed by nature. Secondly, the potential experienced by the gas can becontrolled through magnetic and optical fields. Such trapping is essential, on one hand, to contain the gas, and givesrise to the boundary effects which are central to this work. However, the potential can also be exploited to engineerthe dimensionality of the gas - particularly, quasi-two-dimensional geometries in which vortex lines effectively becomepoint-like vortices - and to stir and shake the condensate. Finally, recent techniques have enabled the observation ofvortex lines [59] and vortex points [58] in real-time, including inference of their individual circulations.Atomic BECs have been employed as a context to study a range of fundamental vortex phenomena, including vortexnucleation from moving obstacles [16, 33, 43, 46, 62] and flow constriction [11, 66, 72], von K´arm´an vortex streets[34, 56], vortex-antivortex annihilations [58], vortex line reconnections [17, 59], vortex chaos [42], vortex scattering[3, 12, 23], quantum turbulence [19, 25, 32, 44, 63, 65, 70], and self-organisation and clustering of vortices [6, 21, 30, 61].With regards to vortex leapfrogging, this has been considered theoretically in idealised unconfined condensates [28],including spinor condensates [31].Atomic BECs however are characterized by their small dimensions, typically from 10 to 100 times the vortex coresize, for which the motion of vortices can be significantly affected by the presence of boundaries. This drawbackis mainly due to the loss of atoms in the final evaporative stage of cooling the gas. There are even experimentsin which, by design, the most interesting physics occurs in the most restricted region of the system, for examplevortex rings nucleated in the weak link of the Josephson junction between two condensates [66, 72]. The aim ofthe present work is to provide insight in the interpretation of current and future experimental studies of vortexdynamics in confined condensates (rather than idealised open domains), where leapfrogging dynamics, which canbe established if the vortex nucleation frequency is sufficiently high, is affected by the presence of boundaries. Thecharacteristics of leapfrogging motion in such confined systems is likely to show significant dissimilarities comparedto the corresponding dynamics in unbounded systems stemming from the role played by image vortices arising fromthe presence of boundaries. Despite the expected impact of geometrical confinement, to the best of our knowledgethe role of boundaries in leapfrogging dynamics has never been investigated in literature neither for classical nor forquantum fluids ([31] and [28] indeed studied leapfrogging in homogeneous condensates, without boundaries). In orderto assess the impact of the boundaries and disentangle the latter from other concurrent physical effects existing inquantum fluids ( e.g. compressibility), in this research we compare the leapfrogging of vortices in plane channels in (i)ideal incompressible classical fluids and (ii) box-trapped Bose-Einstein condensates. In order to simplify the systemunder investigation, our theoretical and numerical analysis is performed in two-dimensions, employing the point vortexmodel for classical fluids and the Gross-Pitaevskii equation for BECs. We stress that the Gross-Pitaevskii equationhas proved an excellent quantitative model of experiments with Bose-Einstein condensates at temperatures T (cid:28) T c ; atrelatively high values of temperature, the condensate exchanges energy and particles with the thermal cloud, and theGross-Pitaevskii equation requires modifications [5, 7, 8, 50]. We also remark that on one hand the two-dimensionalnature of the system that we consider is an idealisation (the aim is to get insight in the motion of three-dimensionalvortex rings), but, on the other hand, where atomic Bose-Einstein condensates are tightly confined in one directionthe system becomes effectively two-dimensional and our two-dimensional approach becomes realistic.The article is organised as follows. In Section II, we illustrate the two theoretical models employed, namelythe classical point vortex model and the Gross-Pitaevskii equation describing the dynamics of BECs in the zero-temperature limit. In Section III, we report the results obtained in both classical and quantum fluids, focusing on therole of boundaries and on the differences between classical and quantum systems. Finally, in the last Section IV, wesummarise our findings and illustrate their importance in the future of quantum vortex experiments. II. MODELSA. Point vortex model
The simplest model of our system is the classical point vortex model: a two-dimensional inviscid incompressibleirrotational fluid in an infinite channel of width 2 D containing two vortex-antivortex pairs (the two-dimensional analogof three-dimensional coaxial vortex rings), each of circulation ± Γ. In view of comparing the results obtained withthis classical model to quantum vortices in confined BECs, the hypotheses behind the point vortex model must becarefully considered.The classical model describes a fluid with constant density. In the bulk of the condensate, i.e. sufficiently far fromboundaries or vortices, this assumption is realistic: indeed, although in past experiments condensates were usuallyconfined by harmonic trapping potentials resulting in density gradients [14], current experimental techniques [20] allowbox-like trapping potentials which lead to uniform density profiles in the bulk of the condensate as in the classicalpoint vortex model. In particular, in the vicinity of a vortex, the classical model assumes constant density at anyradial distance r to the vortex axis, including the vortex axis r = 0 itself. In Bose-Einstein condensates, a vortex isa topological defect of the phase of the governing complex wavefunction (or order parameter), as we shall describewith more details in Section II B 1. Therefore the vortex core is a thin tubular region around the vortex axis which isdepleted of atoms: as r →
0, the velocity tends to infinity, as in the point vortex model, but the fluid density tends tozero. The radius of this tube is of the order of the quantum mechanical healing length ξ (see Section II B 1). A similardifference between the classical point vortex model and Bose-Einstein condensates occurs near a hard boundary: theclassical model assumes that the fluid’s density is constant up to the boundary; in a Bose-Einstein condensate athin boundary region (again of the order of ξ ) forms near the boundary where, in the case of box-like traps, thecondensate’s density rapidly drops from the bulk value to zero. We conclude that, from a geometrical point of view,the classical point vortex model can be used to model Bose-Einstein condensates provided that vortex-vortex andvortex-boundary distances are larger than the healing length ξ .From a dynamical point of view, the assumption of constant density implies that the classical point vortex modelneglects sound waves which are radiated away by quantum vortices when they accelerate [3]. The point vortexmodel, in fact, is based on the classical ideal Euler equation which conserves energy. In the low temperature limit T /T c (cid:28) et al. [61]. It must also be noticed that Mason et al. [38] have shown that the motion of a realisticvortex at distance d to a boundary can be described in terms of a classical image vortex even if ξ is comparableto d (although a small correction is needed to account for the density depletion in the boundary region). In thesuitable physical limits, we hence expect the point vortex model to correctly describe the impact of boundaries onthe leapfrogging of quantised vortices.
1. Equations of motion
Our physical domain under investigation is a two-dimensional infinite strip C R defined as C R = { ( x, y ) ∈ R :( x, y ) ∈ ( −∞ , ∞ ) × (0 , D ) } , which hereafter we will refer to as the channel . We assume the flow to be two-dimensional, i.e. the velocity vertical component v z = 0 and the horizontal components v x and v y only depend onhorizontal coordinates x and y and time t . The incompressibility assumption implies that the continuity equation canbe written as follows ∇ · v = 0 . (1)The velocity field v can hence be expressed as the curl of vector field Ψ which, given the two-dimensionality of theflow, has non-vanishing components only in the z direction, Ψ = (0 , , ψ ( x, y, t )). The velocity components havehence the following expressions in terms of the function ψ which is often denominated streamfunction : v x = ∂ y ψ and v y = − ∂ x ψ , where ∂ i indicates spatial derivatives in the i direction.The irrotationality of the flow implies that the velocity field can be expressed via a potential function ϕ , i.e. v = ∇ ϕ , (2)leading to the following relations for the components v i = ∂ i ϕ . Equations (1) and (2) imply that both ϕ and ψ satisfyLaplace equation, ∆ ϕ = ∆ ψ = 0, and the following equalities between their spatial derivatives: ∂ x ϕ = ∂ y ψ , (3) ∂ y ϕ = − ∂ x ψ. (4)Equations (3) and (4) coincide with the well known Cauchy-Riemann relations for the complex function Ω( z ) := ϕ +i ψ ,where z = x + iy . Hence, following basic complex analysis, the function Ω( z ), denominated complex potential , is ananalytical complex function on the simply connected open domain C = { z ∈ C : 0 < (cid:61) m z < } (cid:40) C . As aconsequence, Ω( z ) is differentiable and its derivative w ( z ) := d Ω dz = v x − iv y (5)is the so-called complex velocity . In the framework of complex potentials, the impermeable boundary conditions forideal fluids correspond in our channel C to the following constraint: (cid:61) m Ω( z ) | z ∈ ∂ C = α ( t ), with α ( t ) ∈ R dependingonly on time t .The description of incompressible and irrotational flows of ideal fluids via the complex potential-based formulation isparticularly useful in the present work as it allows the employment of conformal mapping techniques for the derivationof the analytical expression of the complex potential Ω( z ) describing the velocity field induced by a point vortex inour channel C . The essential steps for this derivation are as follows. The necessary ingredients are mainly two: (a) theknowledge of the complex potential Θ( ζ ) describing the flow induced by a point vortex in a simply connected opensubset D of the complex plane, with ζ ∈ D (cid:40) C ; and (b) the construction of a conformal map ζ = f ( z ) transformingour channel C onto the domain D .Conformal maps f are transformations defined on the complex plane which preserve angles. Such maps are per-formed by analytical complex functions with non-vanishing derivative, i.e. , in the present case, f (cid:48) ( z ) (cid:54) = 0 for all z ∈ C .The requirement D not coinciding with the entire complex plane C , is fundamental in order to exploit the RiemannMapping Theorem which ensures the existence of the conformal map f mapping C onto D . Once Θ( ζ ) and f ( z ) aredetermined, the complex potential Ω( z ) for a vortex flow in C is obtained by transforming the potential Θ( ζ ) via theconformal map f − ( ζ ), i.e. Ω( z ) = Θ( f ( z )) . (6)The reasons why the derived complex function Ω( z ) via Eq. (6) is the seeked complex potential are the following.First, Ω( z ) is analytic on C (as it is obtained via the composition of two analytic functions, f and Θ), implying that thereal and imaginary parts of Ω( z ) are related to each other via the Cauchy-Riemann equations and are both harmonicfunctions. Hence, they do satisfy all the necessary conditions for corresponding respectively to a potential functionand a streamfunction of an incompressible and irrotational flow of an inviscid fluid. Second, the correspondence of ∂ C and ∂ D under the conformal mapping performed by f transposes the boundary conditions enforced by Θ( ζ ) on ∂ D to the boundary ∂ C [35]. Finally, via conformal mappings, the flow induced by a vortex of circulation κ is indeedmapped to a vortex flow with the same circulation [45].In the present work, we choose D to coincide with the upper half complex plane, i.e. D = { ζ ∈ C : (cid:61) m ζ > } . Inthis domain, the complex potential Θ( ζ ) describing the flow induced by a vortex placed in ζ ∈ D is obtained by themethod of images, namely Θ( ζ, ζ ) = − sgn( ζ ) iκ π log (cid:18) ζ − ζ ζ − ζ ∗ (cid:19) , (7) FIG. 1. Schematic illustration of the conformal map ζ = f ( z ) = e πz D transforming C onto D and a vortex placed in z into avortex in ζ , ζ = f ( z ). where sgn( ζ ) is the sign of the vortex placed in ζ (positive for anti-clockwise induced flow, negative for clockwise), ζ ∗ is the complex conjugate of ζ where a vortex of opposite sign is placed (the image-vortex of ζ ) and κ is thecirculation of the flow generated by the vortex. The analytical function f transforming conformally the channel C = { z ∈ C : 0 < (cid:61) m z < D } onto D is as follows (see Fig. 1 for a schematic illustration) ζ = f ( z ) = e πz D . (8)The conformal map f transforms ∂ C onto ∂ D , with f ( { z ∈ C : (cid:61) m z = 0 } ) = R + and f ( { z ∈ C : (cid:61) m z = 2 D } ) = R − .Employing Eq. (6), the determination of the complex potential Ω( z ) is straightforward, namelyΩ( z, z ) = − sgn( z ) iκ π log (cid:32) − e − π D ( z − z − e − π D ( z − z ∗ (cid:33) , z = f − ( ζ ) (9)leading to the following complex velocity w ( z, z ) = − sgn( z ) iπ κ πD (cid:110) coth (cid:104) π D ( z − z ) (cid:105) − coth (cid:104) π D ( z − z ∗ ) (cid:105)(cid:111) = χ ( z, z ) + χ ( z, z ∗ ) , (10)where χ ( z, z ) = − sgn( z ) iπ κ πD coth (cid:104) π D ( z − z ) (cid:105) and sgn( z ∗ ) = − sgn( z ).The complex function χ ( z, z ) (and, correspondingly, χ ( z, z ∗ )) can be physically interpreted as the complex velocitygenerated by an isolated vortex placed in z (whose complex potential would be Ω( z, z ) = − sgn( z ) iκ log( z − z ) / (2 π ))and its infinite images with respect to the walls of the channel, (cid:61) m z = 0 and (cid:61) m z = 2 D . The expression (10) forthe complex potential w ( z, z ) can indeed be derived by considering two sets of infinite images of a vortex placed in z and an anti-vortex in z ∗ [22].If the channel is characterised by the presence of N vortices, the complex velocity w ( z, z k { k =1 ,...,N } ) generated bythe the set of N vortices is obtained via the superposition principle, i.e. w ( z, z k { k =1 ,...,N } ) = N (cid:88) k =1 w ( z, z k ) = N (cid:88) k =1 [ χ ( z, z k ) + χ ( z, z ∗ k )] . (11)A crucial role in this N -vortex problem is played by the equations of motion of a generic j -th vortex. In order to derivesuch equations of motions, we define the position z j ( t ) := x j ( t ) + iy j ( t ) occupied by the vortex at time t in the channel C . Indicating with the superscript ‘ ˙ ’ derivation with respect to time, we define the quantity ˙ z j ( t ) := ˙ x j ( t ) + i ˙ y j ( t ),where the real and imaginary part correspond to the x and y components of the j -th vortex velocity. As vortices areadvected by the local fluid velocity, i.e. ˙ x j ( t ) = v ( x j ( t ) , t ), the following relation holds˙ z j = w ∗ ( z j , z k { k =1 ,...,N } ) , (12)where we have omitted the time dependence of z j and z k to ease notation and the complex conjugation on the r.h.s.arises from the definition (5) of complex velocity. In order to determine the complex velocity w ( z j , z k { k =1 ,...,N } ), weemploy Eq. (11) subtracting the term corresponding to the vortex placed in z j , obtaining the following relation˙ z j = w ∗ ( z j , z k { k =1 ,...,N ; k (cid:54) = j } ) + χ ∗ ( z j , z ∗ j )= (cid:88) k (cid:54) = j w ∗ ( z j , z k ) + χ ∗ ( z j , z ∗ j )= (cid:88) k (cid:54) = j [ χ ∗ ( z j , z k ) + χ ∗ ( z j , z ∗ k )] + χ ∗ ( z j , z ∗ j ) , (13)which coincides with the equations of motion of the j -th vortex. The equations of motion of the whole N -vortexproblem are hence a set of 2 N coupled ordinary differential equations. B. Gross-Pitaevskii equation model
The Gross-Pitaevskii model is a well-established theoretical framework for the investigation of the dynamics ofBECs at temperatures much smaller than the critical transition temperature. The Gross-Pitaevskii (GP) equationdescribes the temporal evolution of the complex order parameter Ψ = Ψ( x , t ) of the system, and reads as follows, i (cid:126) ˙Ψ = − (cid:126) m ∆Ψ + V Ψ + g | Ψ | Ψ , (14)where the dot is the time derivative, (cid:126) = h/ (2 π ) is the reduced Planck’s constant, m is the boson mass, V = V ( x , t )is an externally applied potential, and g = 4 π (cid:126) a s /m models the two-body contact-like boson interaction, where a s is the s-wave scattering length for the collision of two bosons. The order parameter Ψ can be written in terms of itsamplitude and its phase as Ψ = √ ne iθ , (15)where n = n ( x , t ) = | Ψ | is the particle number density (number of bosons per unit volume) and θ = θ ( x , t ) is thephase. Without loss of generality, the order parameter Ψ can be written as Ψ( x , t ) = e iµt/ (cid:126) Φ( x , t ) where µ is calledthe chemical potential and Φ( x , t ) obeys i (cid:126) ˙Φ = − (cid:126) m ∆Φ + V Φ + g | Φ | Φ − µ Φ . (16)
1. Quantum vortices
In the context of BECs described by the Gross-Pitaevskii equation, quantum vortices are topological defects ofthe phase θ of the order parameter, at which Ψ = 0 (hence θ is undefined) and around which θ wraps by 2 qπ with q ∈ Z \ { } . In three dimensions, vortices take the form of one-dimensional curves which may form a vortex tangle,as observed both in BECs [69] and superfluid helium [67]. In two dimensions, vortices coincide with vortex pointswhich have been observed extensively in oblate (pancake-like) BECs [39]. For the purpose of the present work, wewill restrict our discussion to two dimensional systems.The velocity field v ( x , t ) associated to a BECs whose dynamics is described by the order parameter Ψ, is definedfrom the phase θ via the relation v ( x , t ) = (cid:126) m ∇ θ. (17)Employing the definition (17) of the velocity and the 2 qπ phase wrapping existing around a vortex, it is straightforwardto verify that the circulation Γ of the velocity field on any closed curve γ enclosing a vortex point is quantised interms of the quantum of circulation κ = h/m , i.e. Γ = (cid:73) γ v · dl = qκ , q ∈ Z \ { } . (18)Choosing γ to be a circle of radius r and assuming the flow around a vortex to be axisymmetric, the azimuthalcomponent of the flow velocity around a vortex is given by the relation v φ = qκ/ (2 πr ), coinciding with the expressionfor a classical point vortex. Hence, from a velocity point of view, quantum and classical vortices are identical.The important and dynamically significant distinction between classical and quantum vortices is that the latter arecharacterised by a finite core whose size is of the order of the so-called healing length ξ = (cid:126) / √ mgn . As we will verybriefly illustrate in the next section, quantum fluids are indeed compressible fluids.
2. Fluid dynamical equations for a BEC
The Gross-Pitaevskii equation (14) may be rewritten via the
Madelung transformation consisting in expressing Ψ inpolar form (15) and separating the real and imaginary parts of (14). This procedure leads to the following equations˙ n + ∇ · ( n v ) = 0 , (19) mn [ ˙ v + ( v · ∇ ) v ] = −∇ ( p + p (cid:48) ) − n ∇ V, (20)where p and p (cid:48) are respectively pressure and quantum pressure p = gn , (21) p (cid:48) = − (cid:126) m n ∆ (ln ( n )) . (22)Equation (19) coincides formally with the continuity equation of a classical fluid, while equation (20), exception madefor the presence of the quantum pressure p (cid:48) , is formally identical to the momentum balance equation for a barotropic,compressible classical Euler (ideal) fluid. At length scales (cid:96) much larger than the healing length ξ (which is the typicallength scale for density variations, associated e.g. to the presence of vortices or boundaries) p (cid:48) /p (cid:28)
1, implying thatin this limit the BEC can indeed be considered as a barotropic, compressible classical inviscid fluid. Hence, at lengthscales (cid:96) (cid:29) ξ , the dynamics of quantum and classical point vortices only differ on the basis of compressible phenomenawhich may arise in BECs. In the other limit of (cid:96) ∼ ξ , the physics may be significantly different. For instance, if therelative distance between quantum vortices of opposite sign is of the order of ξ , the quantum pressure term wouldtrigger the annihilation of the vortex pair, while in the classical point vortex model no loss of circulation is included inthe model. Moreover, the behaviour of a co-rotating pair of quantum vortices of same sign also shows dissimilaritieswith respect to the classical case, in particular for the finite value of the rotation frequency ω τ as the distance (cid:96) tendsto zero (in the classical model, the frequency diverges, ω τ ∼ /(cid:96) ). III. RESULTSA. Classical fluids
To make progress in understanding the impact of boundaries on the leapfrogging behaviour of classical pointvortices in a two-dimensional channel, we consider the motion of four vortices, half with positive circulation κ , halfwith negative − κ . In Fig. 2 we show this initial condition. If we interpret our two-dimensional configuration as amodel of a three-dimensional configuration of vortex rings, point vortices of same colour in the figure correspondto cross-sections of the same ring. Initially, the four vortices are vertically aligned on the y axis, i.e. x j (0) = 0 for j = 1 , . . . , y = D ,namely y j (0) = D ± R for the first pair j = (1 ,
2) and y j (0) = D ± r for the second pair j = (3 , R/D < r/R <
1. In order to characterise the dependence of vortex trajectories on the two non-dimensionalparameters r/R and
R/D which determine the flow, we numerically integrate the equations of motion (13) for thefour vortices, j = 1 , . . . ,
4, varying r/R and
R/D . In particular, we choose r/R = n/
10 and
R/D = m/
10, with m, n = 1 , . . . ,
9. The time-advancement scheme employed in the numerical simulations is a second-order Adams-Bashforth method with a time step ∆ t = T / T = 2 π δ /κ is the rotation period of a pair of vortices ofthe same polarity placed at distance δ . In our numerical simulations δ is set to 10 − D .For classical unbounded fluids, since the study performed by Love over a century ago [36], it is well known thatvortices undergo leapfrogging motion only if r/R is larger than a critical value α c = 3 − √ ≈ . r/R < α c ,leapfrogging does not occur: the smaller, faster pair moves “too fast” for the larger ring to influence its dynamics in FIG. 2. Initial vortex configuration for the classical point vortices numerical simulations: filled (open) circles correspond tovortices with positive (negative) circulation. Numerical labels close to vortices indicate the vortex numeration employed. a significant way, and the vortices separate. More recently, Acheson [1] extended numerically the study performed byLove and established that leapfrogging motion is unstable when α c < r/R < α (cid:48) c , with α (cid:48) c = 0 . backward leapfrogging and periodic orbits . The phase diagram of the system resulting from the numericalsimulations is illustrated in Fig. 3.For values of R/D ≤ /
2, the dynamics is very similar to what is observed in an unbounded fluid, the role of theboundaries being only marginal. For a given value of
R/D ≤ /
2, in fact, as we increase r/R , we first observe nonleapfrogging motion (in black in Fig. 3), defined as the dynamics characterised by ˙ y j ( t ) = 0 for all j at late times; thenwe notice unstable leapfrogging motion (open red squares), and finally stable leapfrogging (filled red squares). Thesedynamical regimes therefore coincide with the scenario outlined by Acheson [1], the only significant and importantdifference being the dependence of α c on R/D : for small values of
R/D , α c is very close to the constant value 0 . e.g. for R/D = 0 . α c = 0 . R/D ( e.g. α c = 0 .
216 for
R/D = 0 . α c on R/D stems from the interaction of the outer vortices 1and 2 in Fig. 2) with their corresponding images with respect to the closest channel wall; essentially, the interactionwith image vortices is stronger compared to the interaction of the inner pair with the corresponding images. Theseimages, of opposite sign, slow down the outer vortex pair, allowing the inner pair to escape towards infinity for valuesof r/R which would produce leapfrogging motion in an unbounded fluid; in order to recover leapfrogging, r/R wouldhave to increase. As
R/D increases, this effect is amplified as the outer pair is closer to the channel walls.This increasing monotonous behaviour of α c with respect to R/D extends also for
R/D > /
2, where the role playedby boundaries becomes significant, triggering a much richer dynamics. As
R/D is larger than 1 /
2, for large values of r/R , we observe backward-leapfrogging , indicated by blue diamonds in Fig. 3. This dynamics, again, originates from theinteraction of vortices with their images with respect to the closest channel wall. In particular, each vortex, paired toits image of opposite sign, forms a virtual vortex-anti vortex pair on its own. As a consequence, we observe two distinctleapfrogging motions, each involving two virtual vortex-anti vortex pairs. Due to the vortex polarity, the leapfroggingmotion induces a net translation in the opposite direction with respect to standard (forward) leapfrogging. In the(
R/D, r/R ) plane, the forward-leapfrogging to backward-leapfrogging transition occurs via an intermediate regime inwhich vortices follow periodic orbits, indicated by green stars in Fig. 3. As shown in detail in the next section and inthe analytical derivation presented in the Appendix, periodic orbits are observed when R + r = D , corresponding to thegreen dashed line in Fig. 3. For large values of R/D ( R/D (cid:38) /
1. Derivation of periodic orbits
In this section we derive theoretically the existence of periodic orbits in the leapfrogging motion of four vortices ina channel using the classical point-vortex model. We show that under suitable conditions, namely when R + r = D ,each pair of same signed vortices moves around a fixed point. Some analytic details are discussed in the Appendix.With reference to Fig. 2, we consider the pair of vortices P = ( x ( t ) , D − R ( t )), with negative circulation − κ ,and P = ( x ( t ) , D + R ( t )), with positive circulation κ , and the pair of vortices P = ( x ( t ) , D − r ( t )), with negative FIG. 3. Phase diagram of the classical motion of two vortex-anti vortex pairs in a two-dimensional plane channel. All symbolsrefer to performed numerical simulations. Black circles indicate no leapfrogging motion; red filled (open) squares stand for stable(unstable) forward, standard leapfrogging; blue filled (open) diamonds correspond to stable (unstable) backward leapfrogging;green stars stand for periodic orbits. The dashed green line indicates the analytical solution for periodic orbits (see sectionIII A 1 and Appendix). The dashed violet curve is the numerically computed α c dependence on ( R/D ). circulation − κ , and P = ( x ( t ) , D + r ( t )), with positive circulation κ , where t is time. In the complex domain, omittingthe time dependence to ease notation, these vortices are located in z = x + i ( D − R ) for P , z = x + i ( D + R ) for P , z = x + i ( D − r ) for P and z = x + i ( D + r ) for P , and they generate the following complex velocity in thepoint z , as given by Eq. (11), w ( z ) = w ( z, z ) + w ( z, z ) + w ( z, z ) + w ( z, z ) . (23)We now consider the midpoint M between the vortex points P and P , namely z M ( t ) = x ( t ) + x ( t )2 + i (cid:18) D − r ( t ) + R ( t )2 (cid:19) and the complex velocity generated by vortices in z M which we indicate with w ( z M )0 FIG. 4. Examples of dynamical regimes and trajectories for classical 4-vortex motion in a two-dimensional channel. Filled(open) symbols indicate positive (negative) vortices. Top Left:
R/D = 5 / r/R = 1 /
10, no-leapfrogging (vortices moving tothe right). Top Right:
R/D = 4 / r/R = 4 /
10, forward (standard) leapfrogging (vortices moving to the right). Bottom Left:
R/D = 8 / r/R = 6 /
10, backward-leapfrogging (vortices moving to the left); Bottom Right:
R/D = 72 / r/R = 39 / w ( z M ) = iκ D (cid:16) − e iπ ( r + R ) D (cid:17) e π (x0+x1)2 D (cid:16) e π (4 ir +4 iR +x0+x1)2 D − e π (2x0+ i ( r +3 R ))2 D − e π (2x1+ i (3 r + R ))2 D + e π (x0+x1)2 D (cid:17) . (24)If we look for the conditions such that the velocity w ( z M ) of the midpoint M is zero, we have w ( z M ) = 0 ⇐⇒ e iπ ( r + R ) D − ⇐⇒ π ( r + R ) D = 2 kπ , k ∈ Z . (25)Note that the same result Eq. (25) is found for the midpoint N between the two vortex points P and P .Since r , R , and D are positive real parameters, the only admissible values of k in (25) are k ∈ Z + . Moreover, weknow that r < R < D , leading to r + R < D , which implies that the only admissible value for k is k = 1, i.e. r ( t ) + R ( t ) = D . (26)This is the most interesting result: it states that when the four vortices satisfy the condition (26) then the midpoints M and N are at rest: the two pairs of vortices ( P , P ) and ( P , P ) move hence symmetrically with respect to theircorrespondent midpoints, i.e. ˙ x ( t ) = − ˙ x ( t ) and ˙ R ( t ) = − ˙ r ( t ). The last equality is fundamental as it expresses thatif condition (26) is satisfied at a given t = t , it will be satisfied for every t > t . Thus, if the initial condition isprepared such that x (0) = x (0) = 0 and r (0) + R (0) = D , vortices will always move symmetrically with respect totheir midpoints z M = i D z N = i D M and N as, in principle more general trajectories with therestriction ˙ R ( t ) = − ˙ r ( t ) (for instance, ˙ R ( t ) = ˙ r ( t ) = 0) could be possible, not leading to periodic orbits. We tacklethis issue in the Appendix, to ease the readability of the manuscript. B. Quantum fluids
The next step is to numerically probe the dynamical regimes of two quantum vortex-antivortex pairs interactingin a two-dimensional channel. We shall compare the results with the corresponding classical results outlined in theprevious Section (III A).We consider a two-dimensional BEC in a channel geometry, imprinting quantum vortices in the positions initiallyoccupied by classical vortices. Note that, in addition to the parameters R , r and D already present in the classicalpoint vortex formulation, in the Gross-Pitaevskii formulation of the problem we have an extra length scale - thehealing length ξ - which plays a fundamental role in the dynamics. To assess the relevance of this extra lengthscale, we present numerical simulations of leapfrogging quantum vortices employing two distinct values of the channelhalf-width D : D = 40 ξ and D = 20 ξ . In order to model the channel confinement, we use the following potential V : V = V ( y ) = < y < D µ if y ≤ y ≥ D, (27)corresponding to a channel of half-width D , where the density | Φ | is constant everywhere with the exception of thinlayer whose width is of the order of the healing length at the channel boundaries y = 0 and y = 2 D .The trajectories of the quantum vortices are calculated as a function of time by numerically solving the equationof motion of the order parameter Φ, the dimensionless Gross-Pitaevskii equation i ˙Φ = −
12 ∆Φ + Vµ Φ + | Φ | Φ − Φ . (28)Equation (28) is obtained from Eq. (16) after introducing characteristic units of length, time and energy: ξ = (cid:126) / √ mµ (the healing length), τ = ξ/c (where c = (cid:112) µ/m is the speed of sound), and µ (the chemical potential) respectively,and normalising the order parameter with respect to the unperturbed homogeneous solution Φ = (cid:112) µ/g of Eq. (16).In these units the healing length and the bulk density in the channel are unity.The numerical integration of Eq. (28) is performed employing a fourth-order Runge-Kutta time advancementscheme and second-order finite differences to approximate spatial derivative operators. Time step ∆ t/τ is set to1 . × − and spatial discretization ∆ x/ξ = ∆ y/ξ is chosen to be equal to 0 .
25. In the set of simulations where D = D = 40 ξ , the numbers of grid-points in the x and y directions are N x = 6400 and N y = 400 respectively,leading to the computational box − ξ ≤ x ≤ ξ and − ξ ≤ y ≤ ξ . On the other hand, when D = D = 20 ξ , N x = 3200 and N y = 240 respectively, leading to the computational box − ξ ≤ x ≤ ξ and − ξ ≤ y ≤ ξ .The initial imprinting of vortices is made by enforcing a uniform 2 π phase wrapping around the positions employedas initial condition for the classical point vortex simulations and letting the system relax in imaginary time beforestarting the integration of Eq. (28) for t ∈ R . In Fig. 5 we report the density | Φ | ( x, y ) (left) and the phase θ ( x, y )(right) of the initial condition employed for R/D = 0 . r/R = 0 . D = D = 40 ξ . It can be easily observedthat the density | Φ | rapidly drops to zero at the vortex positions and outside the channel. Correspondingly, the four2 π phase wrappings can be distinguished in Fig. 5 (right).To verify the existence in a BEC of all distinct regimes observed in the classical point vortex model (Section III A),we perform numerical simulations of quantum vortex leapfrogging along the vertical line R/D = 0 . R/D because along this line, as r/R varies from 1 /
10 to9 /
10, all regimes which we have identified using the classical point vortex model are present.The results are schematically outlined in Fig. 6, where classical vortex dynamics (left) is compared to quantum vortexdynamics at D = D = 40 ξ (middle) and D = D = 20 ξ (right). When D = D , the boundaries of the phase diagramat R/D = 0 . r/R in the classical and in the quantum case. When D = D we observe twodifferences: first, periodic motion now occurs at ( R/D, r/R ) = (0 . , .
7) instead of (
R/D, r/R ) = (0 . , . R/D, r/R ) = (0 . , .
1) the internal vortex-anti vortex pair annihilates as their initial distance is only 2 . ξ . Thesedifferences at the smaller value of channel size are expected, as the healing length scale starts playing a role: only if D/ξ is sufficiently large we can expect classical and quantum dynamics to be the same.2
FIG. 5. Initial condition for numerical simulation of leapfrogging of quantum vortices in a two dimensional channel for
R/D = 0 . r/R = 0 . D = D = 40 ξ . (Left) the density of the BEC | Φ( x, y ) | (presented as a ratio of the bulk density | Φ | ) is displayed: it is unity (yellow) in the bulk of the channel and vanishes (blue) in the vortex cores and at the channel’sboundaries; (right) the phase θ ( x, y ) of the BEC is illustrated in the range [ − π, π ). The exact matching of the observed dynamical regimes when comparing classical and quantum leapfrogging in atwo-dimensional channel if D ≥ ξ is confirmed in Fig. 7, which shows the trajectories of quantum vortices for( R/D , r/R ) pairs selected as for the classical trajectories illustrated in Fig. 4.It is worth noting some minor differences between the quantum vortex trajectories and their classical counterpartsreported in Fig. 4. Since the initial condition is not stationary with respect to any frame of reference, when we startintegrating in time Eq. (28) for t ∈ R there is a sudden emission of sound waves, and as a result the entire vortexconfiguration is translated towards the positive x direction. The effect (which has been reported in the literature [16]),is visible in the top right, bottom left and bottom right panels of Fig. 7 when compared Fig. 4. In particular, thishorizontal shift affects the periodic orbits reported in Fig. 7 (bottom right) whose center is slightly shifted towardspositive x values. In addition, the number of periods observed in the x range [ − D, D ] is different from the classicalcounterpart, possibly due to the compressible nature of a quantum Bose gas, in which incompressible kinetic energymay be transformed into compressible kinetic energy (sound) when vortices change their velocities (accelerate), asshown by Parker et al. [48], exactly as the accelerated motion of charged particles emit electro-magnetic radiation.The role played by this effective dissipation of kinetic energy into sound will be assessed in a future study. IV. CONCLUSIONS
In conclusion, we have demonstrated that, in the confined space of a two dimensional channel, the classical problemof vortex leapfrogging acquires new aspects. Using the point vortex model we have found that, besides the knownregimes of standard leapfrogging and absence of leapfrogging, there are two new regimes: backward leapfrogging andperiodic motion. Using the Gross-Pitaevskii equation to model an atomic Bose-Einstein condensate (a compressiblequantum fluid) confined within a channel, we have verified that all four regime also exist for quantum vortices. In largechannels, the boundaries between these regimes are the same for classical and quantum vortices. Some differencesappear if the channel size is reduced, and the finite-size nature of the quantum vortex core starts playing a role, orif the vortices are very close and sound radiation becomes important. The determination of a richer dynamics forthe leapfrogging of vortices occuring in confined geometries will be particularly important for the interpretation andplanning of ongoing and future experiments with atomic Bose-Einstein Condensates, where the dynamical regimesreported in the present work can be potentially observed.Future work will address the problem in three dimensions, paying attention to the excitation of Kelvin waves alongthe vortex rings and the departure from axisymmetry.
V. ACKNOWLEDGEMENTS
LG, NGP and CFB acknowledge the support of the Engineering and Physical Sciences Research Council (grant No.EP/R005192/1). M.S. acknowledges the support of MIUR-Italy through the project PRIN “Multiscale phenomenain Continuum Mechanics: singular limits, off-equilibrium and transitions” (grant No. PRIN2017 2017YBKNCE).3
FIG. 6. Cuts in the dynamical regimes phase diagram corresponding to
R/D = 0 . D = D = 40 ξ (middle) and D = D = 20 ξ (right). Symbols as in Fig 3 except for the newlyintroduced up-pointing orange triangle corresponding to the annihilation of the inner vortex-anti vortex pair. Declaration of Interests. The authors report no conflict of interest. [1]
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In order to show the existence of periodic orbits, we have to prove that if condition (26) is satisfied, the trajectoriesof the vortex points are closed curves with vortices rotating around the two midpoints M and N defined in sectionIII A 1.For the sake of simplicity, and with reference to section III A 1, we prove the closedness of the trajectory only forthe vortex point P , as the proof for the other vortex points is an iterative procedure. We consider equation (12)for the vortex point z with the complex velocity given by the expression (23) evaluated on the vortex point z .Since the middle point z M is at rest for r + R = D , we rewrite the dynamic equation of z , namely ˙ z = w ∗ ( z ),in the polar coordinate system ( ρ, θ ) centered on z M . The middle point z M , under the condition r + R = D ,becomes z M = x + x i D ρ < D P and P are in { z ∈ C : 0 < (cid:61) m z < D } .Thus, in the new reference system the vortex points correspond to z = z M − ρ cos( θ ) − iρ sin( θ ) ,z = z M + iD − ρ cos( θ ) + iρ sin( θ ) ,z = z M + ρ cos( θ ) + iρ sin( θ ) ,z = z M + iD + ρ cos( θ ) − iρ sin( θ ) , (A1)where z M is now the origin of the new frame of reference, which can be set z M = 0 + 0 i . Note that the condition (26)is automatically satisfied by construction; indeed, z − z i (cid:18) D ρ sin( θ ) (cid:19) and z − z i (cid:18) D − ρ sin( θ ) (cid:19) ,implying R ≡ D ρ sin( θ ) and r ≡ D − ρ sin( θ ) and, hence, condition (26). We now substitute the coordinates7(A1) into the equation ˙ z = w ∗ ( z , z k { k =1 ,..., } ) , (A2)according to (12), and change the vectorial basis from (ˆ x , ˆ y ) to (ˆ u ρ , ˆ u θ ) by means of the following rotation:ˆ u ρ = cos( θ )ˆ x + sin( θ )ˆ y , ˆ u θ = − sin( θ )ˆ x + cos( θ )ˆ y . By writing ˙ z = − ˙ ρ ˆ u ρ − ρ ˙ θ ˆ u θ , we then find the following equations for ˙ ρ and ˙ θ :˙ ρ = f ( ρ, θ ) = k D csch (cid:16) πe − iθ ρD (cid:17) csch (cid:16) πe iθ ρD (cid:17) (cid:104) cos( θ ) tan (cid:16) πρ sin( θ ) D (cid:17) cosh (cid:16) πρ cos( θ ) D (cid:17) − sin( θ ) cos (cid:16) πρ sin( θ ) D (cid:17) tanh (cid:16) πρ cos( θ ) D (cid:17)(cid:105) (A3)˙ θ = f ( ρ, θ ) = − k Dρ csch (cid:16) πe − iθ ρD (cid:17) csch (cid:16) πe iθ ρD (cid:17) (cid:104) cos( θ ) cos (cid:16) πρ sin( θ ) D (cid:17) tanh (cid:16) πρ cos( θ ) D (cid:17) ++ sin( θ ) tan (cid:16) πρ sin( θ ) D (cid:17) cosh (cid:16) πρ cos( θ ) D (cid:17)(cid:105) (A4)From equations (A3) and (A4), we finally derive the equation for ρ (cid:48) = d ρ/ d θ as follows: ρ (cid:48) = ˙ ρ ˙ θ = ρ sin( θ ) cos (cid:16) πρ sin( θ ) D (cid:17) tanh (cid:16) πρ cos( θ ) D (cid:17) − ρ cos( θ ) tan (cid:16) πρ sin( θ ) D (cid:17) cosh (cid:16) πρ cos( θ ) D (cid:17) cos( θ ) cos (cid:16) πρ sin( θ ) D (cid:17) tanh (cid:16) πρ cos( θ ) D (cid:17) + sin( θ ) tan (cid:16) πρ sin( θ ) D (cid:17) cosh (cid:16) πρ cos( θ ) D (cid:17) , (A5)which is well-defined in A = { ( ρ, θ ) ∈ R + × R : 0 < ρ < D/ } because: a) all the elementary functions are well-defined(included the function tan( ... ) through the condition 0 < ρ < D/ θ ) · tanh (cid:16) πρ cos( θ ) D (cid:17) ≥ θ ) · tan (cid:16) πρ sin( θ ) D (cid:17) ≥
0) and never zero (both terms are neverzero in A ).In order to prove that the trajectory of vortex P is a closed curve, we need to show that the function ρ ( θ ) is acontinuos and periodic function. However, the integration of equation (A5) is a hard task to achieve. Therefore, wechoose to prove that ρ ( θ ) is a continuos and periodic function without finding the exact integral of (A5). In order toachieve this goal, we first need to recall a result from mathematical analysis, which states: Theorem 1
Given a continuos and periodic function f : R → R with period T such that (cid:90) T f ( x ) d x = 0 , then theprimitive function of f ( x ) is periodic with period T . Having recalled Theorem 1, we now need to prove the following theorem:
Theorem 2
The primitive function ρ ( θ ) of ρ (cid:48) ( θ ) (as defined in (A5) ) is C ( R ) and periodic with period at least π . The proof consists in three steps:a) ρ ( θ ) is C ( R ) function;b) ρ (cid:48) ( θ ) is a periodic function, at least of period T = 2 π ;c) (cid:90) π ρ (cid:48) ( θ ) d θ = 0.Below the proof of each step:8a) As stated in the previous sections, the complex velocity w ( z ) is an analytic function, and hence the curvedescribing the trajectory of the vortex point P . This implies that the function ρ ( θ ) is C ( R ). Moreover, we canassert that the denominator of ρ (cid:48) ( θ ) is (cid:54) = 0, or, better, it is easy to show that it is always positive for ( ρ, θ ) ∈ A .Indeed, the two terms in the denominator in (A5) are always positive (both for sin θ and cos θ positive, negativeor null).b) ρ (cid:48) ( θ ) is a periodic function: in fact it follows directly from (A5) that ρ (cid:48) ( θ + 2 π ) = ρ (cid:48) ( θ ) . (A6)c) A sufficient condition to prove the last step is that the function ρ (cid:48) ( θ ) is an odd function in R . The proof followsdirectly from (A5) after substituting θ by − θ obtaining: ρ (cid:48) ( − θ ) = − ρ (cid:48) ( θ ) (A7)Finally, we apply Theorem 1 to our function ρ (cid:48) ( θ ) and the theorem is proved. Theorem 2 leads hence to the conclusionthat ρ ( θ + 2 π ) = ρ and thus that the trajectory of vortex point P1