T.-L. Horng et al- Two-dimensional quantum turbulence in a nonuniform Bose-Einstein condensate

8/3/2019 T.-L. Horng et al- Two-dimensional quantum turbulence in a nonuniform Bose-Einstein condensate

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Two-dimensional quantum turbulence in a nonuniform Bose-Einstein condensate

T.-L. Horng, 1 C.-H. Hsueh, 2 S.-W. Su, 3 Y.-M. Kao, 2 and S.-C. Gou 2

1 Department of Applied Mathematics, Feng Chia University, Taichung 40724, Taiwan2 Department of Physics, National Changhua University of Education, Changhua 50058, Taiwan

3 Department of Physics, National Tsing Hua University, Hsinchu 30047, TaiwanReceived 26 May 2009; published 24 August 2009

We investigate the dynamics of turbulent ow in a two-dimensional trapped Bose-Einstein condensate bysolving the Gross-Pitaevskii equation numerically. The development of the quantum turbulence is activated bythe disruption of an initially embedded vortex quadrupole. By calculating the incompressible kinetic-energyspectrum of the superow, we conclude that this quantum turbulent state is characterized by the Kolmogorov-Saffman scaling law in the wave-number space. Our study predicts the coexistence of two subinertial rangesresponsible for the energy cascade and enstrophy cascade in this prototype of two-dimensional quantumturbulence.

DOI: 10.1103/PhysRevA.80.023618 PACS number s : 03.75.Kk, 47.32.C , 47.37. q, 67.25.dk

I. INTRODUCTION

The turbulent ow in superuid is an important and long-standing subject in low-temperature physics 1 . It was al-ready known in the early 1950s that the ow of superuidcomponent of liquid helium can become turbulent in thepresence of a counterow of the two uids driven by asteady-injected heat current 1 . Feynman speculated thatthis superuid turbulence can be described as a chaotictangle of quantized vortex laments 2 . Later, Vinen experi-mentally conrmed this picture by showing that superuidturbulence is maintained by the mutual friction between vor-tices and the normal ow 3 . Subsequently, superuid tur-bulence in the thermal counterow was intensively investi-gated and many aspects of this type of turbulence are nowwell understood. Nevertheless, due to the complexity of thetwo-uid structure, the counterow turbulence has no classi-cal analog and thus attracted little attention of the investiga-tors working on classical hydrodynamics. More recently, su-peruid turbulence of noncounterow types have beenachieved experimentally in liquid helium, in analogy withsome rather simple cases of classical turbulence CT . Forexample, Maurer and Tabeling demonstrated the generationof turbulence by two counter-rotating disks 4 and Stalp et al. observed the decay of grid turbulence 5 .

Superuid turbulence is sometimes termed as quantumturbulence QT to highlight the key role played by quan-tized vortices in the dynamics which differs from that of the

chaotic but continuous vorticity of CT. Despite the funda-mental difference between QT and CT, experimental evi-dences 4,5 and numerical simulations 6–8 have revealedthat, even at very low temperatures, QT is characterized byan energy spectrum following the Kolmogorov’s −5 / 3 powerlaw, a sign that also hallmarks the occurrence of CT 9 .These ndings have aroused considerable interest in explor-ing the possible connection between CT and QT occurring atzero temperature. In particular, due to the close relationshipbetween liquid helium and atomic Bose-Einstein condensateBEC in superuid behavior and quantized vorticity 10 ,

QT in the trapped BEC has been well studied by severalgroups in the last few years. In these investigations, QT is

predicted to occur in the following cases: in the evolvingstage prior to the nucleation of vortex lattice in a rotatingcondensate 11,12 , in the collision of two condensates 13 ,in the combined rotations around different axes of a conden-sate 14 , in the disruption of a single vortex ring throughbending wave instability 15 , and in a phase-imprinted two-dimensional 2D BEC 16 .

From the perspective of classical statistical physics, thefully developed turbulence can be explained by means of thetransport of inviscid invariants between different scales. Withthis reasoning, Kolmogorov assumed that CT is statisticallyself-similar in an inertial range of wave number k , whereenergy can be transferred from large scales to small scales ata constant rate without being dissipated by viscosity, aprocess called cascade 9 . Using simple dimensional analy-sis, Kolmogorov obtained a scaling law for the energy spec-trum E k 2 / 3k −5 / 3, which imposes a general criterion forcharacterizing three-dimensional 3D CT. Here the energyspectrum E k is dened as E = dk E k , where E is the totalincompressible kinetic energy and = d E / dt . Kraichnan 17and Batchelor 18 hereafter referred to as KB extendedKolmogorov’s theory to 2D turbulence by considering en-strophy half of the integral of squared vorticity as a secondinviscid invariant. According to KB theory, the dual cascadesof energy and enstrophy give rise to two distinct scalingregimes for 2D turbulence in the inviscid limit. For forced2D turbulence, specically, where energy and enstrophy areinjected into the ow at a given wave number k f , enstrophytends to cascade from k f toward higher k , leading the energy

spectrum to fall off like k −3

in this regime. On the other hand,energy tends to cascade upward, i.e., from k f toward lower k ,with the kinetic-energy spectrum scaling like k −5 / 3, which isalso known as the inverse cascade. As for the decaying 2Dturbulence, KB theory also predicts a k −3 spectrum in theinertial range. More detailed review of the theory and phe-nomenology of 2D CT can be found in Ref. 19 and allreferences therein.

Although the picture of enstrophy cascade proposed byKB theory is also supported by other approaches, such as theclosure theory and closure calculations, however, direct nu-merical simulations do not very strongly support the k −3 law19 . In fact, depending on the numerical schemes employed,

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the power law of the energy spectrum in the enstrophy cas-cade regime appears in divergent disagreements with k −3 law19 , leaving 2D CT open to question. For all that, it is still

signicant to ask whether there exists any analog between2D CT and QT just as the Kolmogorov’s −5 / 3 power lawbeing common in the 3D cases for CT and QT. In this work,we study the turbulent dynamics of the 2D BEC conned ina harmonic trap V = 1 / 2 m 2 x2 + y2 y2 by numericallysolving the time-dependent GP equation. We shall carry outthe computations in the oscillator units for mathematicalconvenience, in which the length, time, and energy are, re-spectively, scaled in units of / m

1 / 2, −1 and , and the

time-dependent GP equation takes the following dimension-less form:

i , t

t = −

2

2+

x2 + y2 y2

2+ g , t 2

, t . 1

Here, = x, y and the condensate wave function , t isnormalized by , t 2d =1. Correspondingly, the cou-

pling strength is expressed by g =4 a˜ N , where N is the totalnumber of atoms in the condensate and a is the effective

s-wave scattering length in 2D space.The organization of this paper is as follows. In Sec. II, the

time evolution of the wave function , t is solved bynumerically integrating Eq. 1 . Hydrodynamic propertiesthat are relevant in examining the occurrence of turbulentow in superuid BEC, such as the kinetic energy, enstrophyare determined from the condensate wave function. In Sec.III , we discuss the scaling laws for the incompressiblekinetic-energy spectrum. In particular, a scenario for dualcascade in 2D trapped BEC is presented. Finally, some con-cluding remarks are given in Sec. IV.

II. NUMERICAL INTEGRATIONSOF THE GROSS-PITAEVSKII EQUATION

To numerically integrate Eq. 1 , we use the method of lines with spatial discretization by highly accurate Fourierpseudospectral method and time integration by adaptiveRunge-Kutta method of orders 2 and 3 RK23 . Furthermore,to stir the condensate to turbulence, we assume that a vortexquadrupole consisting of two singly quantized vortices lo-cated at 4 ,0 and two singly quantized antivortices lo-cated at 0 , 4 is initially placed in an isotropic condensate

y=1 with a Thomas-Fermi TF radius RTF = 2 =6,

where is the chemical potential of the condensate. Such aconguration can be constructed following the method de-scribed in Ref. 20 . The time evolution of the above vortexconguration is shown in Fig. 1 by plotting , t 2 at vari-ous times. In Figs. 1 b –1 d , we see that the vortices rstpair up as two vortex dipoles under the inuence of trappotential Fig. 1 b and start to traverse in the condensate.Since vortex is a phase singularity and can rotate the masssurrounding it, the traveling vortices that pass through theouter dilute area of the trapped BEC can generate surfacewaves. Consequently, the interplay between surface wavescreates mass blobs of opposite phases in the outer dilute area,which may collide with each other and nucleate vortices via

snake instability. Now as time goes on, the two vortex di-poles move toward each other along the line y = x under mu-tual interaction and eventually come to a head-on collision,which annihilates all initial vortices and forms a radially ex-panding ringlike dark soliton preceded by a shock wave.Owing to the trapping potential, the outgoing shock wave isreected at the border area and subsequently pulsates in thecondensate for a while. This back-and-forth pulsation givesrise to more violent surface waves which would also gener-ate vortices as described above. Apart from the above mecha-

nism, the shock wave itself could also generate vorticeswhen traveling in the outer dilute area. Such a mechanism isdiscussed by Hau in Ref. 21 , where the shock wave causesstripes of dark or gray soliton vortex sheet , which evolveslater into vortices via snake instability.

The pulsation of the shock wave fades until most of itsenergy disperses and causes signicant density uctuationsinside the condensate. After this stage, the sound waves den-sity uctuations fully develop in the interior of the conden-sate, whereas all newly generated quantized vortices aredriven to the outer region of the condensate. Finally, after asufciently long period, say t =100, the density prole of thecondensate is plotted in Fig. 1 f , where the cumuluslike pat-tern of the spatial distribution suggests that the superowmight have become turbulent there.

In applying the spectral scaling approach to identify theoccurrence of turbulence in the trapped BEC, it is more ap-propriate to express the condensate wave function in theform of Madelung transformation, namely, , t = n , t exp i , t . Substituting the above form into Eq.1 , we obtain the continuity equation and Euler equation,

respectively,

t n + · nu = 0, 2

FIG. 1. Color online Density prole of the trapped BEC atvarious times: a t =0, two singly quantized vortices and two singlyquantized antivortices are located at 4 ,0 and 0, 4 , respec-tively, in a condensate with RTF =6 and =1; b t =3.5; c t =5.4,annihilation of the four energetic vortices in the central region; dt =5.8, an outwardly propagating shock wave is formed; e t =9.5;and f t =100, achieving the stationary turbulent state.

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t u + u · u = − Q , 3

where u , t = , t is the velocity eld of the superowand Q = gn + V − 2 n / 2 n . The total-energy functional iscalculated in terms of n and u , that can be expressed as the

sum of four terms

E tot =12

ne −i 2 + x2 + y

2 y2

2n +

12

gn 2 d

= E kin + E q + E tr + E int , 4

where

E kin =12

nu 2d ,

E q =

1

2 n2

d ,

E tr =12

n x2 + y2 y2 d ,

E int =12

gn 2d ,

represent the superuid kinetic energy, the quantum pressureenergy, the trap energy, and the interaction energy, respec-tively. Following Ref. 7 , we decompose the vector eld

nu into the solenoidal and irrotational parts or, correspond-ingly, the incompressible and compressible parts, i.e., nu= nu i + nu c, where · nu i =0 and nu c =0. Toaccomplish such decomposition, we may let nu i = Aand nu c = , where A and are the vector and scalarpotentials for the eld nu , such that nu = A + . Tak-ing divergence on both sides of the last expression, we getthe Poisson equation for the scalar potential

2 = · nu , 5

which can be numerically solved by Fourier pseudospectralmethod. Once the scalar potential is numerically solved,

the vector potential A can be determined and, hence, the eldcomponents nu i and nu c. Thus, the incompressible andcompressible kinetic energies are dened by E kin

i,c

= 1 / 2 d nu i,c 2. Moreover, since nu i and nu c aremutually orthogonal, it follows that E kin = E kin

i + E kinc . Physi-

cally, E kini and E kin

c correspond to the kinetic energies of swirls and sound waves in the superow, respectively. Apartfrom the kinetic energy, we also need to calculate the enstro-phy of the superow for comparison with the results of 2DCT. Classically, the enstrophy Z is dened by Z = 1 / 2 2d , where is the vorticity vector. In this paper,in analogy with the replacement u → nu , for nonuniformcondensate density, we use the modied vorticity vector,

= nu = n u + j

2 s j n − j z , 6

instead of the usual form = u to evaluate the density of enstrophy in a trapped BEC. Here 2 s j s j = 1 is the cir-culation quantum carried by the jth vortex and the delta

functions entering in the right-hand side indicate the loca-tions of singularity of the enstrophy distribution. It should beclaried that, although Eq. 6 introduces anomalous genera-tion of vorticity due to the gradient of density in addition tothe usual contribution made by quantized vortices, the rstterm in the right-hand side of Eq. 6 only contribute slightlyto the total vorticity as we shall see in the following demon-strations in Fig. 4. As a matter of fact, the denition of vor-ticity is not crucial for the current study since is not in-volved in the calculations of kinetic-energy spectrum.

The proles of nu i,c 2 / 2 and 2 / 2, at t =0 and 100are shown separately in Figs. 2–4. Initially, E kin

i and Z arecompactly localized around the core of each individual quan-

tized vortex whose singular nature is revealed by the granu-lar distribution of Z . The snapshot of the spatial distributionsfor E kin

i,c and Z at a much later instant of time t =100 areshown in Figs. 2 b , 3 b , and 4 b for comparison. We seethat the incompressible eld is almost distributed in the outerannular area of the condensate with 6 8, whereas thecompressible eld is largely distributed in the inner regionwith 6. Thus, the incompressible and compressiblecomponents of the kinetic energy are basically spatially sepa-rated despite that they somewhat overlap around the borderarea about 6. This result is consistent with the scenarioof vortex-sound separation in our previous study for the for-

FIG. 2. Color online Spatial distribution of the incompressiblekinetic energy E kin

i at a t =0; b t =100. The circle of white dashedline indicates the Thomas-Fermi radius of the condensate RTF =6 .

FIG. 3. Color online Spatial distribution of the compressiblekinetic energy E kin

c at a t =0; b t =100. The circle of white dashedline indicates the Thomas-Fermi radius of the condensate RTF =6 .

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mation of QT in a 3D BEC 15 , and Fig. 4 b indicates veryclearly that all vortices are randomly localized in the outerregion.

The time evolutions of kinetic energies and enstrophy inthe time interval 0 t 100 are illustrated in Fig. 5. By vir-tue of the initial vortex conguration, E k

c comes up at a muchsmaller value than E k

i initially. However, the subsequent mo-tion of the vortices causes E k

c and Z to increase but E k i to

decrease shortly after the evolution sets out. In particular, themovement of the vortices toward the high-density centralregion leads to a rapid rise in the enstrophy as shown in Fig.5 b . Nevertheless, both E kin

i and Z have an abrupt drop in

the period 5.5 t 6.0, when the four initial vortices aredriven to annihilate each other at the central region, which isconsistent with our previous observation.

From Fig. 5, we see that E kini and E kin

c stay almost station-ary when the system reaches fully turbulent, yet the uctua-tions about their mean values reveal the occurrence of inten-sive energy interchange between all components of the totalenergy. To gain more insight into this observation, here wederive an equation which relates the change rate of E kin

i tothat of E kin

c . In doing so, we rst note that Eqs. 2 and 3

can be combined to yield the continuity equation for thecurrent J = nu , i.e.,

J t

+ · Ju = − n Q . 7

Adding up the inner product of u and Eq. 7 , together with

that of J and Eq. 3 , we obtain12

t n u 2 = −

12

· J u 2 − J · Q . 8

Consequently, the change rate of the total kinetic energy canbe obtained by taking volume integral over the whole spacein both sides of the last equation and this leads to the follow-ing identity:

d

dt E kin

i + E kinc = − J · Qd , 9

where the integral of the rst term in the right-hand side of Eq. 8 vanishes as a consequence of Stokes theorem. Equa-tion 9 then accounts for the balance of energy, in the sensethat the change in the total kinetic energy is compensated bythe work done by the eld − Q on the trapped condensateparticles. Accordingly, the power in the right-hand side of Eq. 9 represents the sum of conversions of E int , E tr , and E q,into the total kinetic energy . Note that it is not possible togure out exactly the fractions of power converting, respec-tively, into E kin

i and E kinc from further derivations of Eq. 9 .

However, from Fig. 5, we see that E kini and E kin

c do constantlyuctuate around their mean values, suggesting that energyinterchange between incompressible and compressible eldsmediated by the eld − Q occurs from time to time in thecondensate. Now, since we consider the scaling behavior of the spectrum of E kin

i as the sole criterion for identifying theformation of QT, it is expected that E kin

c , which is inseparablefrom E kin

i , may play a signicant role in the dynamics of QTin the trapped BEC, and this will be addressed in the follow-ing section.

III. SCALING LAWS OF THE INCOMPRESSIBLEKINETIC-ENERGY SPECTRUM

The angle-averaged kinetic-energy spectrum E kini,c k , as a

function of wave number k , is dened by 7

E kin

i,c

k =

k

2 d F

nui,c 2

, 10

such that E kini,c = 0 E kin

i,c k dk . Here F . . . denotes the Fouriertransform of a given function or vector eld in 2D space. InEq. 10 , the integral over the k shell in the momentum spaceis accomplished by numerically summing over the gridpoints with k x

2 + k y2 1 / 2 = k , where k x and k y are the Cartesian

components of the wave vector k . In analogy with Eq. 10 ,the angle-averaged enstrophy spectrum is dened by Z k = k / 2 d F 2 such that the total enstrophy is given bythe integral Z = 0 Z k dk . From Eq. 6 , it follows that Z k = k 2 E kin

i k , which is well known in the classical uid dynam-ics.

FIG. 4. Color online Spatial distribution of the enstrophy Z ata t =0; b t =100. The circle of red dashed line indicates the

Thomas-Fermi radius of the condensate RTF =6 . Comparing thepresent gures with Fig. 2, we see that the distribution of Z basi-cally coincides with that of E kin

i . Obviously, every brightest speck indicates the core of a quantized vortex. On the other hand, in theinterior region of the condensate that is free from vortices, the dis-tribution of vorticity is scarcely detectable, indicating that the con-tribution of the term n u to the total enstrophy can beneglected.

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

t

E n e r g y

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

t

E n s t r o p h y

E k

i

E k

c

FIG. 5. Color online Time developments of theincompressible/compressible kinetic energy upper and the enstro-phy lower .


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With Eq. 10 , we are now able to compute the spectra inquestion. In doing so, we have observed that although thevelocity eld of the superow may change all the time, theenergy spectrum E kin

i k evolves into a stationary form astime gets large. To be specic, let us consider the incom-pressible energy spectrum of the ow at t =100. Accordingly,we nd that there are two ranges of spectral scaling, in which E kin

i k takes a power law as shown in Fig. 6. In the rstrange 2 k 4, E kin

i k k −5 / 3, which is consistent with theKolmogorov spectrum and, in the second range k 10, E kin

i k follows closely with the k −4 law. Between these tworanges, there is a transitional zone 4 k 10, where E kin

i k exhibits no scaling behavior. To double check the validity of the scaling laws, we calculate Z k directly by numerical

integration according to the denition for the angle-averagedspectrum. As expected, Z k indeed exhibits the power-lawbehavior of k −1 / 3 and k −2 correspondingly. Furthermore, toverify whether the spectral scaling behavior for E kin

i k de-pends on the geometry of the trapped BEC, we proceed tocalculate E kin

i k for BEC with various values of RTF and y.Remarkably, all our numerical results indicate that when thequantum turbulent ow is well developed, only the k −5 / 3 andk −4 ranges persist in E kin

i k , whereas the k −3 spectrum aspredicted by KB theory fails to develop in our system.

The emergence of k −5 / 3 law in our numerical calculationsfor E kin

i k is well expected, as we have seen in various ear-lier studies concerning the turbulent motion of superow in

the 3D atomic condensate. In this work, we are more inter-ested in the k −4 spectrum with QT in trapped BEC. As amatter of fact, the k −4 spectrum had been reported by somenumerical calculations in 2D CT and Saffman conjecturedthat this scaling behavior is due to the dissipation of ran-domly distributed discontinuities of vorticity in the classical2D turbulent ow 22 . In comparison with those classicalcases, our results are consistent with Saffman’s theory for thefact that there are indeed randomly distributed discontinuitiesof vorticity in the quantum turbulent ow as demonstrated inFig. 4 b . On the other hand, the derivation of k −4 spectrumas a dissipation spectrum in Saffman’s theory is inconsistentwith our presumption on the nondissipative dynamics gov-

erned by Eq. 1 . Such a discrepancy might be explained interms of energy transfer between E kin

i and E kinc as we shall see

in the following passages. Furthermore, we note that thespectrum shown in Fig. 5 shares some similarities with thoseof forcing 2D CT, although both dissipation and externalforcing are not taken into account in the present problem.The coexistence of k −5 / 3 and k −4 spectra draws forth the fol-lowing question: does a quantum analog for dual cascades in2D CT exist in the 2D turbulent superow of a trapped BEC?

A straightforward way for probing into the above questionis to evaluate the cascade rates of E kin

i and the associatedenergy ux through the wave number k by solving the scale-by-scale energy budget equations derived from GP equation23 . However, as we have shown in Fig. 1, the temporal and

spatial variations in the modulus of the order parametern , t can no longer be ignored in the turbulent regime

and, consequently, some extra terms mixing up the energy

contents of two different kinematic categories incompress-ible and compressible would enter the desired equation.This makes the scale-by-scale calculations for the cascaderate and energy ux through wave number k for E kin

i ex-tremely complicated by using the method of Ref. 23 .Therefore, to circumvent this difculty, we shall give an em-pirical interpretation of the cascading process in the currentproblem based on the numerical results so far presented. Indoing so, let us survey some subtle ingredients of the model.The rst subtlety is the meaning of the “size” for an energy-carrying eddy in a trapped BEC. From the denition of thevorticity vector Eq. 6 , we note that the singular andregular parts of are modulated by n and its gradient,

respectively. As a result, the magnitude of vorticity of avortex in high-density area is greater than that of a vortex inlow-density area and, with such reasoning, we say that aneddy is energetically “larger” if it contains more circumuentparticles. In Fig. 7, the streamlines or the level curves of

= C for various values of the constant C are plotted forthe ow at t =0 and t =100, respectively, where is thestream function 24 associated with the velocity eld andthe false colors indicate the corresponding values of . Tobe conspicuous, part of Fig. 7 b is enlarged and shown inFig. 8, so that the locations and structures of eddies can beeasily recognized by families of closed streamlines. Note thatthe superow acquires the largest velocity where the contour

100

10110

−13

10−12

10 −11

10−10

10 −9

10−8

10 −7

k

e n e r g y

s p e c t r u m

enstrophy cascade

energy cascade

slope=−4

slope=−5/3

energy input from E kc

energy output to Ekc

energy output to E kc

FIG. 6. Color online The incompressible kinetic-energy spec-trum E kin

i k at t =100. This spectrum follows Kolmogorov’s k −5 / 3

law in the interval of 2 k 4 and Saffman’s k −4 law for k 10.The energy input regime is particularly marked with a circle.

(a)

x

y

−10 −5 0 5 10

−10

−5

0

5

10

−0.2

−0.1

0

0.1

0.2

(b)

x

y

−10 −5 0 5 10

−10

−5

0

5

10

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

FIG. 7. Color online The streamlines or the level contours of the stream function associated with the superow at a t =0;b t =100. The circle of red dashed line indicates the Thomas-Fermi

radius of the condensate RTF =6 . The false colors indicate thevalues of .


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lines are closest together. Furthermore, by comparing thesizes of regions occupied by a same color i.e., the regionscharacterized by the same = C , we conclude that themore energetic the eddy is, the larger spatial structure it pos-sesses. Thus, considering the turbulent state represented byFig. 2 b , the inhomogeneity of the system gives rise to agradational spatial distribution of eddies over the area en-closing the active region of compressible eld, with the larg-est also the most energetic eddies located along the borderat 6 and the smaller also the less energetic ones lo-cated more outwardly from the border.

Another subtlety yet to be addressed is the role of thecompressible eld in the dynamics of QT. In 2D CT, nonlin-

ear interactions transfer energy primarily from the forcingscales to larger scales where it is ultimately dissipated byexternal friction while the enstrophy goes to small scales tobe removed by viscosity. Recall the scaling laws in the spec-trum E kin

i k and the corresponding subinertial ranges asshown in Fig. 5. In comparing our results with the cases of forcing 2D CT, we note that the transitional zone 4 k

10 sandwiched in between the k −5 / 3 and k −4 ranges servesas the energy input regime. This is very intriguing since thereare no external forcing terms in our theoretical setting. Wereason that the emergence of this energy input regime is dueto the activity of sound eld, as it is evident that E kin

c sur-passes E kin

i considerably see Fig. 5 and, above all, cannot

be insulated from interchanging with the latter. Following thearguments given at the end of Sec. II, we may conceive thatsound eld acts as the energy source and sink simultaneouslyfor the vortices provided that the scaling behavior of E kin

i k is held as the top concern for the current problem. Nowconsidering that the sound waves generate vortices mostly inthe outer region of BEC, we may conclude that the energy istransferred in the following two types. The rst type is viathe enstrophy cascade manifested by the migration of vorti-ces toward the low-density region. This is clearly demon-strated in Fig. 4 b , where the majority of vortices generatedby the input energy diffuse to the more and more dilute areaand so the incompressible kinetic energy is transferred into

smaller scales. Surely, this cascading is only effective withina certain range, beyond which the density is too low to sus-tain the formation of vortices and, hence, the energy is even-tually “dissipated” by the dilution of density. Since the totalenergy is conserved, the decrease in the incompressible ki-netic energy is balanced by the replenishment of compress-ible kinetic through the sound waves emitted by the vorticeswhen they are driven to move outwardly. This explains theemergence of Saffman’s k −4 spectrum at large- k side despitethat no authentic dissipations are introduced 25 .

On the other hand, the k −5 / 3 spectrum at the small- k side2 k 4 demonstrates a distinct way to cascade energy. In

analogy to 2D CT, we may interpret this part of spectrum asa consequence of inverse energy cascade from the forcingscales 4 k 10 up to the larger scales with a cutoff at k c=2. In 2D CT, the reciprocal of the smallest wave numberspecies the size for the largest vortices available, which isnormally dependent on the mechanism assumed for remov-ing the energy. By the same token, we presume in our casethat k c

−1 indicates a length scale comparable to the sizes of

those few largest quantized vortices clumped around the rimof the condensate, where the active regions of compressibleand incompressible elds overlap and the energy exchangetakes place persistently and expeditiously. Considering thatthe overwhelming compressible eld acts simultaneously asthe sink and source for the incompressible eld, inverse cas-cade occurs with energy being injected from the forcing,conveyed among the few largest vortices, and eventuallydumped to the sound eld all around. In fact, as the soundwaves keep pulling and dragging the ambient vortices, thisincoherent action prevents the accumulating of incompress-ible kinetic energy at large scales, so the vortices are neverenergetically large enough to re-enter the inner part of the

condensate.

IV. CONCLUDING REMARKS

In this paper, we have investigated a 2D QT in a harmoni-cally trapped BEC. By solving Gross-Pitaevskii equation nu-merically, we calculate the incompressible kinetic-energyspectrum E kin

i k for the fully turbulent superow in the con-densate. We conclude that the energy spectrum is character-ized by dual cascades following the Kolmogorov-Saffmanscaling laws, with E kin

i k k −5 / 3 at small- k regime and E kin

i k k −4 at large- k regime in the wave-number space.Although our results of 2D QT in a trapped BEC possess

some similarities with those of 2D CT determined by Navier-Stokes equations in classical uid mechanics, there are somefundamental differences yet to be pointed out. First, our sys-tem described by Euler equation after Madelung transforma-tion is compressible with the relationship of pressure anddensity being polytropic; but the Navier-Stokes equations arefor incompressible ow with density being constant. Second,viscosity in Navier-Stokes equation serves as an energy sink and that means the classical uid is a dissipative system. InBEC, there is no viscosity and, therefore, the system is non-dissipative. However, the trap potential and binary interac-tion between particles serves as a medium for energy con-versions such that the compressible and incompressible

x

y

−6 −4 −2 0 2 4

3

4

5

6

7

8

9

10

11

12

13

14

FIG. 8. Color online Detailed structure for part of the eddies inFig. 7 b are shown by the enlarged streamlines plot.


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kinetic energies can be converted into each other through thismedium with compressible kinetic energy acting like sink and source for incompressible kinetic energy comparable to2D CT. Third, our Euler equation additionally has a disper-sive quantum pressure term though small , which Navier-Stokes equations do not have. Finally, for incompressible andinviscid ow in classical uid mechanics, a pair of point

vortices with same circulation but in opposite signs are im-possible to annihilate each other due to Kelvin’s theorem.However, it is possible in BEC when this kind of vorticesbump into each other. This is probably because the rotationalincompressible kinetic energy can be almost totally turned tocompressible but irrotational kinetic one.

We have veried in the present investigation that the scal-ing behavior of the spectrum is independent of the trapping

geometry. To see whether the Kolmogorov-Saffman scalinglaws are universal for all cases of 2D QT with trapped BEC,it is desirable to study the dynamics of turbulent superowsfor various condensate systems, such as the dipolar BEC,two-component BEC, spinor BEC, and so forth. Results of these studies will be reported elsewhere in the near future.

ACKNOWLEDGMENTS

This work is supported by Taida Institute for Mathemati-cal Science TIMS , Taipei, Taiwan, and National Center forTheoretical Sciences NCTS , Hsinchu, Taiwan. T.-L. Horngand S.-C. Gou thank Professor M. Tsubota for helpful com-ments on this work.

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T.-L. Horng et al- Two-dimensional quantum turbulence in a nonuniform Bose-Einstein condensate