Decay of homogeneous two dimensional quantum turbulence
DDecay of homogeneous two-dimensional quantum turbulence
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 (Dated: October 2, 2018)We numerically simulate the free decay of two-dimensional quantum turbulence in a large, ho-mogeneous Bose-Einstein condensate. The large number of vortices, the uniformity of the densityprofile and the absence of boundaries (where vortices can drift out of the condensate) isolate theannihilation of vortex-antivortex pairs as the only mechanism which reduces the number of vortices, N v , during the turbulence decay. The results clearly reveal that vortex annihilations is a four-vortexprocess, confirming the decay law N v ∼ t − / where t is time, which was inferred from experimentswith relatively few vortices in small harmonically trapped condensates. I. MOTIVATION
Quantum turbulence (the chaotic motion of quan-tum vortices in superfluid helium[1] and cold gases[2])has become a prototype problem of nonlinear statisti-cal physics. The absence of viscosity and the natureof vorticity distinguish quantum turbulence from ordi-nary turbulence: in quantum fluids infact, vorticity isnot a continuous field of arbitrary shape and strength(as in ordinary fluids), but is concentrated on the nodalpoints (in 2D) or lines (in 3D) of a complex wavefunc-tion ψ . Around these points or lines where ψ = 0, thephase of ψ changes[3] by 2 π . The large scale proper-ties of quantum turbulence thus depend on the inter-actions of discrete vortices, which induce effects suchas Kelvin waves[4–7], vortex reconnections[8–10] andphonon emission[11, 12]. At temperatures sufficientlyclose to the critical temperatures, the interaction of vor-tices with thermal excitations[14, 15] induces friction ef-fects [13].In turbulence, the study of the free decay is fruit-ful because it removes the arbitrariness of the forcingwhich is necessary to sustain a statistical steady state.In 3D, experiments[16, 17] and numerical simulations [18]of the decay of quantum turbulence in superfluid heliumhave revealed the existence of two turbulent regimes: aquasi-classical (or Kolmogorov) regime, which decays as L ( t ) ∼ t − / , and an ultra-quantum (or Vinen) regime,which decays as L ( t ) ∼ t − , where the vortex line density L (defined as the length of vortex lines per unit volume)measures the turbulence’s intensity. Physically, the Kol-mogorov regime is characterized by a cascade of kineticenergy from large to small eddies (similar to what hap-pens in ordinary turbulence), whereas the Vinen regimelacks a cascade and is more akin to a random flow [19].Recent studies of 3D turbulence in atomic condensateshave identified these two regimes [20], despite uncertain-ties due to the small number of vortices in the systemcompared to liquid helium experiments.In 2D, quantum turbulence takes the form of a chaoticconfiguration of quantized point vortices. Since no di-rect vortex visualization is available in superfluid heliumfilms, all relevant 2D experiments have been performed in trapped atomic Bose-Einstein condensates where vor-tices can be easily imaged. The 2D context has uniquefeatures (absent in 3D) associated to the possibility ofnonthermal fixed points[21], an inverse energy cascade[22] and the emergence of vortex clusters[23, 24]. In thiswork we are concerned with a simpler question: in anal-ogy with 3D, what is the law governing the free decayof a random vortex configuration consisting of an equalnumber of positive and negative vortices ? This questionwas experimentally addressed in a harmonically trappedcondensate by Kwon et al. [25]: they found that the timeevolution of the number of vortices, N v ( t ) (the 2D equiv-alent of the vortex line density L ( t )), is fairly well de-scribed by the logistic equation dN v dt = − Γ N v − Γ N , (1)In analogy with the kinetic theory of gases, Kwon etal. argued that the rate coefficients Γ and Γ repre-sent one-vortex and two-vortex processes respectively:the drift of vortices out of the condensate, and annihi-lations of vortex-antivortex pairs (the 2D analog of 3Dreconnections). Stagg et al. [26] modelled numericallythe experiment of Kwon et al. , analyzed the results us-ing Eq. (1), and determined that annihilations increasewith temperature (see also [15]). Cidrim et al. [28] at-tempted to generalize Eq. (1) to the case of net polariza-tion P = ( N +v − N − v ) / ( N +v + N − v ) (cid:54) = 0 (where N +v and N − v are the numbers of positive and negative vortices re-spectively and N v = N +v + N − v ). They noticed that theoriginal interpretation of Γ and Γ as one-vortex andtwo-vortex processes cannot be correct, as negative val-ues of Γ were required to fit decays during which novortices visibly entered the condensate. Using differentmodel equations for N +v and N − v , they obtained a betterfit to the observed decay. In the case P = 0 (correspond-ing to the experiment of Kwon et al. ), the model ofCidrim et al. reduces to dN v dt = − Γ N / − Γ N , (2)where the N / and N dependence of the drift and a r X i v : . [ phy s i c s . f l u - dyn ] F e b the annihilation terms were derived using physical ar-guments. In particular, the quartic nature of the anni-hilation term in Eq. (2) agrees with the observation ofGroszek et al. [27] that the annihilation of a vortex anti-vortex pair is a four-vortex process, not a two-vortex pro-cess (hence N rather than N ). Briefly, the argumentis the following. Without dissipation, a vortex and anti-vortex alone would be a stable configuration which trav-els at constant velocity. A third vortex is necessary tobring the two vortices together, destroying the circulationand creating a stable nonlinear wave; this wave, called‘crescent-shaped’ by Kwon et al. and ‘vortexonium’ byGroszek et al. , was identified as a soliton by Nazarenkoand Onorato [29, 30]. The fourth vortex is necessaryto destroy the nonlinear wave upon collision, radiatingphonons away. Groszek et al. [27] also highlighted therole played by the trapping potential; in particular, theyfound that vortex clustering is energetically less likely inharmonically trapped condensates compared to recentlydeveloped box-traps [31, 32].In contrast, in the presence of dissipation, vortices ofopposite circulation move towards one another and an-nihilate directly. Hence, it is natural to expect that inthe presence of dissipation the decay of two-dimensionalquantum turbulence follows a two-vortex process. In-deed, our results will verify that this is the case.Unfortunately the number N v ( t ) of point vortices inthe cited studies is relatively small due to the constraintsof current experimentally available condensates. The de-cay curves N v ( t ) are therefore noisy, and it is difficult todetermine with precision the exponents of the two effects- vortex drift and vortex annihilation. Moreover, the driftof vortices out of the condensate is likely to depend onthe steepness of the trapping potential, which now is notnecessarily harmonic [31, 32].To make progress towards understanding the law of 2Dturbulence decay, we concentrate on the annihilation pro-cess which here we study in the absence of vortex driftby performing numerical simulations in a uniform con-densate without boundaries. In other words, we want todetermine accurately the exponent k of the rate equation dN v dt = − Γ N k v , (3)when the only mechanism responsible for decreasing N v ( t ) is annihilations of vortex-antivortex pairs. Forlarge times, the solution of Eq. (3) scales as N v ∼ t / (1 − k ) , if k >
1. A precise measurement of the ex-ponent k will help future works to determine the decayin finite-sized, non-uniform condensates, where the decaydepends also on vortices drifting out of the boundaries. II. MODEL
Our model is the 2D Gross-Pitaevskii equation (GPE)for an atomic condensate ( i − γ ) (cid:126) ∂ψ∂t = − (cid:126) m (cid:18) ∂ ψ∂x + ∂ ψ∂y (cid:19) + g | ψ | ψ − µψ, (4)where ψ ( x, y, t ) is the wavefunction, m is the boson mass, g is the interaction strength, µ is the chemical poten-tial, (cid:126) = h/ (2 π ) and h is Planck’s constant. The phe-nomenogical dissipation coefficient γ [33] is used in someour numerical simulations to mimic the interaction of thecondensate with the thermal cloud, in particular the lossof energy (i.e. the reduction in size) of vortex-antivortexpairs.Eq. (4) is made dimensionless using the length scale ξ = (cid:126) / √ mµ , the time scale (cid:126) /µ and the density scale | ψ | = µ/g , and solved in the (dimensionless) periodicdomain − D ≤ x, y ≤ D with D = 512 ξ . The large sizeof the domain (compared to the vortex core size whichis of the order of ξ ) and the absence of boundaries allowus to track the evolution and annihilations of thousandsof vortices, a number which is larger than in the typicalexperiments and previous numerical simulations. Spaceis discretized onto a N = 2048 uniform cartesian mesh,spacial derivatives are approximated by a 6 th –order finitedifference scheme and a 3 rd –order Runge-Kutta schemeis used for time evolution.The initial conditions of our simulations consist ofa large number N v of vortices with approximately netzero polarization ( N +v ≈ N − v ). To create this conditionmodelling an experimentally feasible manner, we initial-ize the system with the non-equilibrium state [34, 35], ψ ( x ,
0) = (cid:80) k a k exp( i k · x ), where k = ( k x , k y ) is thewavevector, and the coefficients a k are uniform and thephases are distributed randomly. By taking k x , k y ∈ Z we ensure our initial configuration satisfies the periodicboundaries we impose. We perform three sets of simu-lations; without dissipation ( γ = 0) and at two differentlevels of dissipation, γ = 0 .
01 & 0 . π winding of the phase, and an associated density de-pletion, see Fig. 1. III. RESULTS
Figure 2 displays the evolution of the condensate den-sity on the x, y plane at different times t for the non-dissipative( γ = 0, left column) case and a dissipative( γ = 0 .
01, right column) case. As vortices move chaoti-cally (accelerate) in each others’ velocity fields, they radi-ate sound waves [11], turning part of their kinetic energyinto acoustic energy (phonons). Two vortices of oppo-site signs which collide annihilate, radiating more soundenergy[8, 12]. It is apparent from the figure that dissi-pation damps out density oscillations and removes vor-tices more quickly. The number of vortices N v vs time t (ensemble-averaged over 10 simulations) is displayed inFig. 3 for both non-dissipative and dissipative cases. Asexpected, the decay of vortices is much faster in the pres-ence of (larger) dissipation.Figure 4 analyses the decay in a quantitative way. Theleft panel of Fig. 4 shows that, in the absence of dissipa-tion, the vortex number decays as N v ( t ) ∼ t − . in agree-ment with the k = 4 scaling in Eq. (3) of a four-vortexprocess[27, 28] which would yield N v ∼ t − / (red dashedline). The blue dot-dashed line of this panel shows thatthe exponent k = 2 of the two-vortex process would notbe a good fit.The central and right panels of Fig. 4 show that, with γ = 0 . .
01, the final part of the decay is steeper( N v ∼ t − ) and more similar (particularly for t > × )to the prediction N v ∼ t − (red dashed line) of the two-vortex process. Clearly, the N v ∼ t − . decay associatedwith the four-vortex process would not be a good fit atlarge times.We also observe that dissipation introduces a tran-sient N v ∼ t − / regime (clearly visible in the centraland right panels) before the final N v ∼ t − regime isachieved. This transient regime is the predicted outcomeof a three-vortex process. It seems reasonable to assumethat early in the simulations, when the vortex density islarge, the annihilation of two vortices is predominantlyinduced by vortex dynamics (i.e. the presence of a thirdvortex), and not by dissipation. The four-vortex scalingis not seen if the soliton that emerges from the annihila-tion is strongly damped by the dissipation. However oncethe vortex density becomes sufficiently small ( N v < IV. CONCLUSION
We have performed numerical simulations of the freedecay of 2D vortex configurations, which initially con-tain thousands of vortices. The very large homoge-neous condensate and the absence of boundary effectshas clearly confirmed that vortex annihilation is a four-vortex process which is described by the rate equation dN v /dt = − Γ N proposed by Cidrim et al. [28] andGroszek et al. [27]. The presence of dissipation adds ad-ditional complexity. Initially the decay follows the three-vortex rate equation dN v /dt = − Γ N , as dissipationeliminates the need for a fourth vortex to dissipation theresulting soliton. However at small vortex densities dis-sipation brings the vortex and the closest antivortex to-gether without the need of the presence of other vortices,and the late-time decay is faster, in qualitative agreee-ment with the rate equation dN v /dt = − Γ N first pro-posed by Kwon et al. [25].Having established the contribution to the turbulencedecay arising from the annihilation of vortices with an-tivortices, it will be easier in future experiments to findthe contribution from vortices drifting out of the conden-sate (an effect which likely depends on the steepness ofthe confining potential). V. ACKNOWLEDGMENTS
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01, right) at times a) t = 7 . × ; c) t = 2 . × ; e) t = 1 × , b) t = 1 × ; d) t = 1 × ; f) t = 5 × . The small holes in these density plots are the vortices. t × N v FIG. 3. (Color online). The total vortex number, N v , plottedvs time, t . The solid black curve, dashed blue curve anddot-dashed red curve correspond to ensemble-averaging 10simulations without dissipation ( γ = 0) and with dissipation( γ = 0 . γ = 0 .
01) respectively. Notice the more rapiddecay induced by increasing dissipation. t N v N v ∼ t − . N v ∼ t − t N v N v ∼ t − N v ∼ t − / N v ∼ t − / t N v N v ∼ t − N v ∼ t − / N v ∼ t − / FIG. 4. (Color online) Log log plots of the data presented in Fig. 3, with corresponding best fits plotted as red dashed lines(position adjusted for clarity). The left panel corresponds to the case γ = 0, the central panel to γ = 0 . γ = 0 ..