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Energy spectra of vortex distributions in two-dimensionalquantum turbulence

Ashton S. BradleyJack Dodd Center for Quantum Technology, Department of Physics, University of Otago, Dunedin 9016, New Zealand.

Brian P. AndersonCollege of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA.

We theoretically explore key concepts of two-dimensional turbulence in a homogeneous compressible super-fluid described by a dissipative two-dimensional Gross-Pitaeveskii equation. Such a fluid supports quantizedvortices that have a size characterized by the healing length ξ. We show that for the divergence-free portionof the superfluid velocity field, the kinetic energy spectrumover wavenumberk may be decomposed into anultraviolet regime (k ≫ ξ−1) having a universalk−3 scaling arising from the vortex core structure, and an in-frared regime (k≪ ξ−1) with a spectrum that arises purely from the configuration ofthe vortices. The Novikovpower-law distribution of intervortex distances with exponent−1/3 for vortices of the same sign of circulationleads to an infrared kinetic energy spectrum with a Kolmogorov k−5/3 power law, consistent with the existenceof an inertial range. The presence of thesek−3 andk−5/3 power laws, together with the constraint of continuity atthe smallest configurational scalek ≈ ξ−1, allows us to derive a new analytical expression for the Kolmogorovconstant that we test against a numerical simulation of a forced homogeneous compressible two-dimensionalsuperfluid. The numerical simulation corroborates our analysis of the spectral features of the kinetic energydistribution, once we introduce the concept of aclustered fractionconsisting of the fraction of vortices thathave the same sign of circulation as their nearest neighboring vortices. Our analysis presents a new approach tounderstanding two-dimensional quantum turbulence and interpreting similarities and differences with classicaltwo-dimensional turbulence, and suggests new methods to characterize vortex turbulence in two-dimensionalquantum fluids via vortex position and circulation measurements.

I. INTRODUCTION

Turbulence in three-dimensional (3D) classical fluids is as-sociated with a cascade of energy from large length scales de-fined by the details of an energy-forcing mechanism, to smalllength scales where viscous damping removes kinetic energyfrom the fluid. This range of length scales, and the range ofassociated wavenumbersk, define theinertial rangeof energyflux [1]. As shown by Kolmogorov in 1941 [2], the energycascade corresponds to a kinetic energy spectrum that is pro-portional tok−5/3 in the inertial range. Turbulence in a 3Dfluid is also often associated with the decay of large patchesof vorticity into ever smaller regions of vorticity; thisRichard-son cascadeof vorticity provides an important visual pictureof the fluid dynamics involved in 3D turbulence [3].

Remarkably, two-dimensional (2D) incompressible classi-cal fluids exhibit very different turbulent flow characteristicsdue to the existence of an additional inviscid invariant: intheabsence of forcing and dissipation, the totalenstrophy[4] ofa 2D fluid is conserved in addition to the fluid’s kinetic en-ergy [5–9]. The fluid dynamics during forced 2D turbulenceare highly distinctive when compared with 3D flows: ratherthan decaying into smaller patches, vorticity aggregates intolarger coherent rotating structures [10] (see [11] for a moredetailed picture in terms of turbulent stress imposed on small-scale vortices). Accompanying these 2D fluid dynamics is aninverseenergy cascade in which energy moves from a smallforcing scale to progressively larger length scales, defining aninertial range for energy transport with energy flux in a di-rection opposite that of 3D turbulence. Eventually, energyis transported into flows characterized by length scales thatare on the order the system size [12], for which dissipationmay occur. Additionally, there is an enstrophy cascade, in

which enstrophy is transported from the forcing scale to pro-gressively smaller scales [13]. Thus in 2D turbulence, the ki-netic energy distribution contains at least these two distinctlydifferent spectral regimes.

Quantumturbulence [14] involves chaotic flow in a super-fluid [15–19] and is often associated with a random vortextangle in 3D [15]. In general, the quantization of circula-tion strongly constrains the velocity fields allowed in quan-tum turbulence, which must be irrotational everywhere withinthe fluid, yet inertial ranges withk−5/3 spectral dependenceare still found in 3D quantum turbulence [20]. In anincom-pressiblesuperfluid (such as HeII), the vortex core diametercan be neglected for all practical purposes, inspiring the studyof point-vortex models of superfluid dynamics. Such a modelwas used by Onsager to first predict the aggregation of vor-tices within inviscid 2D fluids, and was the context for hisprediction of the quantization of vortex circulation in a super-fluid [21]. Despite the historical importance of this approachin stimulating advances in 2D classical turbulence [22], char-acteristics of 2DQT remain little known, due in part to thedifficulty of achieving the necessary 2D confinement for in-compressible superfluids. The increasing relevance of 3Dturbulence concepts to dilute-gas Bose-Einstein condensate(BEC) experiments [23–25] and recent theoretical work on2DQT [19, 26–31] have highlighted the need for a treatmentof turbulence in 2D superfluid systems that incorporates theconcept of compressibility from the outset. Motivated by re-cent experimental demonstrations of the confinement neededto study 2DQT in a dilute-gas BECs [32, 33] our aim in thepresent paper is to present a new approach to solving some ofthe open problems of 2DQT in the context of such a system.

In a BEC, the vortex core size is non-negligible, and stemsfrom the healing lengthξ, a scale of fundamental importance

2

in BEC dynamics that is typically about two orders of magni-tude smaller than the system size [34]. Compressibility alsoallows for a rich array of physical phenomena in these super-fluids; in particular, a vortex dipole [32] (comprised of twovortices of opposite sign of circulation) can recombine, re-leasing vortex energy as a burst of acoustic waves. The oppo-site process of vortex dipole generation from sound may alsoefficiently occur.

Recent theoretical studies of decaying quantum turbulencein 2D BECs have shown that when the vortex dipole annihi-lation process is dominant it sets up a direct cascade of en-ergy over the scales associated with the dipole decay, sug-gesting that this annihilation mechanism could prohibit anin-verse energy cascade from occurring in a compressible super-fluid [29, 30]. Moreover, enstrophy in a quantum fluid is asso-ciated with the number of vortex cores; if vortices annihilate,then enstrophy may not be conserved, bringing into questionthe existence of energy and enstrophy cascades in 2DQT, andthe universal nature of 2D turbulence and its correspondencewith 2DQT.

The general characteristics of 2D quantum turbulence incompressible quantum fluids, including the capacity for thesesystems to show an inverse energy cascade, enstrophy con-servation, and vortex aggregation have thus remained largelyunknown. However, a recent study of the formation of vortexdipoles during the breakdown of superfluid flow around an ob-stacle in a highly oblate BEC experimentally and numericallyobserved aggregation of like-sign vortices into larger-scale co-herent structures [32], and found time scales over which vor-tex number and hence enstrophy may remain constant. Thevortexclusteringeffect inhibits the dipole-decay mechanismby keeping vortices distant from antivortices (vortices ofop-posite circulation), and suggests that an inverse cascade mightbe observed under the right conditions of forcing. System dy-namics consistent with the existence of an inverse energy cas-cade were indeed found in a recent study of forced 2DQT in aBEC [33].

In this article we address 2D quantum turbulence in a com-pressible quantum fluid from an analytical perspective. Wedetermine the kinetic energy spectra of vortex distributionsin a homogeneous compressible superfluid in a quasi-exactmanner via an analytical treatment of the physics of the vor-tex core. We are thus able to study the properties of vortexconfigurations and their resulting spectra in BEC. We developa technique to sample spatially localized vortex distributionswith power-law behavior over a well defined scale range. Weare thus able to identify the conditions for an inertial rangein fully polarized and neutral systems. A polarized clusterissampled using a specific exponent for the vortex locations rel-ative to the cluster center, which is size and scale dependent.The specific radial exponent is shown to determine the veloc-ity distribution in the classical limit and we thus identifyanexpanding inertial range with a steepening velocity distribu-tion.

Making use of the universality of the spectral region gen-erated by the vortex core, we identify an analytical formof the Kolmogorov constant that we test against dynamicalsimulations of the damped GPE. The derivation of the Kol-

mogorov constant occurs for a highly idealized vortex distri-bution. Thus the complex flows generated by real forcing re-quire that we introduce a new parameter called theclusteredfraction, and modify our ansatz to account for imperfect clus-tering, based on the universality of the Kolmogorov constant.The modified ansatz agrees well with the numerical simula-tions of grid turbulence, supporting our analytical identifica-tion of the Kolmogorov constant.

II. BACKGROUND

The starting point for much of BEC theory is the Gross-Pitaevskii equation (GPE), which provides a capable descrip-tion of trapped Bose-Einstein condensates at zero temperature[34]. Our model, outlined below, consists of a damped GPE(dGPE) description of a finite-temperature BEC which can bederived from the stochastic GPE theory [35, 36]. In this sec-tion we develop a link between the dGPE and the classicalNavier-Stokes equation, identifying aquantum viscosityaris-ing from the damping. The corresponding Reynolds number isdefined in direct analogy with classical fluids. We then statesome key properties of a single quantum vortex, and definethe decomposition of kinetic energy into its compressible andincompressible components.

A. Damped Gross-Pitaevskii theory

The damped Gross-Pitaevskii equation of motion for thequantum fluid wave functionψ(r , t) has been obtained phe-nomenologically [37], within ZNG theory [38], and via amicroscopic reservoir theory [35, 39, 40], and we will con-sider it within the context of the latter framework, for whichthe full equation of motion is the Stochastic Projected Gross-Pitaevskii equation (SPGPE). The SPGPE is derived by treat-ing all atoms with energies above an appropriately cho-sen energy cutoff ǫcut as thermalized, leading to a grand-canonical description of the atoms belowǫcut. A dimension-less temperature-dependent rateγ describes Bose-enhancedcollisions between thermal reservoir atoms and atoms in theBEC. Neglecting the noise, we obtain the equation of motionfor the condensate wave function (in the frame rotating withthe chemical potentialµ)

i~∂ψ(r , t)∂t

= (iγ − 1)(µ − L)ψ(r , t). (1)

For atoms of massm in an external potentialV(r , t), the oper-atorL gives the GPE evolution:

Lψ(r , t) ≡(

− ~2

2m∇2 + V(r , t) + g|ψ(r , t)|2

)

ψ(r , t), (2)

where the interaction parameter isg = 4π~2a/m, for s-wavescattering lengtha. This equation of motion has been used ex-tensively in previous studies of vortex dynamics [38, 39, 41,42] and provides a capable description of dynamical BEC phe-nomena. In general the damping parameter is small (γ ≪ 1),

3

and it is typically much smaller than any other rates charac-terizing the evolution.

Defining the Gross-Pitaevskii Hamiltonian

HC =

∫

d3r

~2

2m|∇ψ(r , t)|2 + V(r , t)|ψ(r , t)|2 + g

2|ψ(r , t)|4

,

(3)and condensate atom number

NC =

∫

d3r |ψ(r , t)|2, (4)

the equation of motion (1) evolves the grand-canonical Hamil-tonianKC = HC − µNC according to

dKC

dt= −2γ~

∫

d3r |(µ − L)ψ(r , t)|2. (5)

The stationary solution minimizingKC is the ground state sat-isfying µψ0(r ) ≡ Lψ0(r ). This is a consequence of the nonlin-ear form of the damping in (1). The damping term arises fromcollisions between high energy atoms that lead to a Bose-enhanced growth of the matter wave field, with instantaneousenergy determined byL. The equation of motion thus de-scribes a system coupled to a thermal reservoir in the chosenframe of reference.

The SPGPE provides a rigorous framework for the dGPEderivation, originating from a microscopic treatment of thereservoir interaction. In particular,γ can be calculated explic-itly [36] for a system with well-defined reservoir parametersµ,T, andǫcut, i.e. a system close to thermal equilibrium. Inessence it is computed via a reduced Boltzmann collision in-tegral that accounts for all irreversibles-wave interactions thatcan change the condensate population by interacting with thethermal cloud. If the thermal cloud is 3D (i.e.β−1 ≡ kBT isgreater than the potential well mode spacing in each spatialdimension) the damping takes the explicit form

γ = γ0

∞∑

j=1

eβµ( j+1)

e2β jǫcutΦ

[

eβµ

eβ jǫcut, 1, j

]2

, (6)

whereΦ[z, s, α] is the Lerch transcendent, and

γ0 = 8a2/λ2dB, (7)

with λdB ≡√

2π~2/mkBT the thermal deBroglie wavelength.The dimensionless rateγ0 provides a useful estimate of thefull damping strength when the cutoff ǫcut is unknown. Equa-tion (6) is independent of position, and valid over the regionV(r , t) ≤ 2ǫcut/3, provided the potential can be treated semi-classically [36]. The summation gives Bose-enhancement cor-rections due to the Bose-Einstein distributed reservoir atoms,and is typically of order 1-20 in SPGPE simulations with aconsistently determined energy cutoff [43]. Typicaly γ ∼5× 10−4 in 87Rb experiments [42, 44].

B. Heuristic derivation of a quantum Reynolds number

In this section we consider the role of dissipation withinthe dGPE description, and show how to recover the celebrated

Navier-Stokes equation (NSE). In doing so we find an explicitexpression for the viscosity which has a microscopicquan-tum origin, stemming from s-wave scattering of incoherentreservoir particles with a coherent superfluid. While not of-fering a practical reformulation (the GPE and its generaliza-tions are capable numerical workhorses), this indicates a con-nection between the dGPE and the NSE in the hydrodynamicregime, allowing the identification of a parameter analogousto the kinematic viscosity of classical fluids.

The fluid dynamics interpretation of the Gross-Pitaevskiiequation is based on the Madelung transformation, whichwe now apply to the damped GPE (1), writingψ(r , t) =√

ρ(r , t) exp [iΘ(r , t)], whereρ(r , t) is the number density ofthe superfluid (number of atoms per unit volume), andΘ(r , t)is the macroscopic phase of the quantum fluid. The velocity isthen given byv(r , t) = ~∇Θ(r , t)/m. The resulting equationsof motion (with implicit t andr dependence) for density andvelocity are then given by

∂ρ

∂t+ ∇ · (ρv) =

2ργ~

(µ − Ueff), (8)

m∂v∂t= −∇

(

Ueff −~γ

2ρ∇ · (ρv)

)

, (9)

where an effective potentialUeff is defined as

Ueff(r , t) =mv2

2+ V + gρ − ~

2

2m

∇2√ρ√ρ. (10)

The last term is called thequantum pressure, which is verysmall except whereρ changes sharply, such as near vortexcores. By neglecting this term in the absence of dissipationwe are considering the so-calledhydrodynamic regime.

We now consider theγ term in (9)

~γ

2∇

(

1ρ∇ · (ρv)

)

=~γ

2

(

∇(∇ · v) + ∇v · ∇ρρ

)

. (11)

Note thatv · ∇ρ ≡ 0 for an isolated quantum vortex. In the ab-sence of acoustic energy this will also be a good approxima-tion for a system of vortices provided their cores are well sep-arated, since the density gradient of each vortex is localized toa region where the velocity is dominated by the single-vortexvelocity field. It should thus be a reasonable approximationtoneglect the second term in (11). In a superfluid the curl termin the expansion∇(∇ · v) = ∇ × (∇ × v) + ∇2v may also beconsistently neglected away from vortex cores; similarly weneglect the curl term in∇(v · v) = 2 (v · ∇)v + 2v × (∇ × v)when taking the gradient of (10). We then find that (9) reducesto a quantum Navier-Stokes equation for the velocity field:

∂v∂t+ (v · ∇)v = − 1

m∇ (V + gρ) + νq∇2v (12)

where thekinematic quantum viscosityis

νq ≡~γ

2m, (13)

in analogy with classical fluids. In this regime, (12) is coupledto the continuity equation

∂ρ

∂t+ ∇ · (ρv) =

2ργ~

(µ − UH) (14)

4

with hydrodynamic potential

UH(r ) ≡ mv2

2+ V + gρ. (15)

The source term in (14) drives the system towards particle-number equilibrium with the reservoir. In the Thomas-Fermiregimeµ−UH(r ) ≈ 0, restoring approximate particle-numberconservation.

Making use of (7), we can give an order-of-magnitude esti-mate for the viscosity

ν0q ≡~γ0

2m=

2a2kBTπ~

. (16)

We can also estimate a quantum Reynolds number as

Re0q ≡

UL

ν0q=π~

2a2

ULkBT

(17)

for BEC flow with characteristic speedU and length scaleL.We can write the quantum Reynolds number as

Re0q =

λ2dB

a2

mUL4~

. (18)

We note that temperature only enters the expression throughthe deBroglie wavelength of the matter wave field, in the ra-tio λdB/a, which is typically very large for a BEC. Note thatstrong scattering corresponds to strong damping, and hencelow Reynolds number. A large deBroglie wavelength corre-sponds to a relatively cold system, which is hence expected tobe weakly damped and have a high Reynolds number.

We thus have a dimensionless ratio in the form

Re0q ∼

[deBroglie wavelength]2

[scattering length]2· [flow momentum]

[quantum momentum](19)

where~/L is interpreted as the quantum momentum associ-ated with the transverse length scale of the flow.

A concrete example is provided by a recent experimen-tal study of 2DQT generated by stirring a highly oblate,toroidally confined BEC [33]. The initial system consists of∼ 2.6× 106 atoms of87Rb at a temperature of∼ 100nK. Us-ing these numbers in our analysis the dimensionless dampingparameterγ0 ∼ 6 × 10−4 gives a kinematic quantum viscos-ity ν0

q ∼ 6× 10−2µm2 s−1. The trapping potential confines theflow to an annular channel of widthL ∼ 30µm. The nomi-nal flow speed can be estimated from numerical simulationsof the dGPE [33], giving a valueU ∼ 5µm s−1 as the peakvalue occurring in the bulk flow during the stirring sequence.These values give an estimateRe0

q ∼ 600. We can alterna-tively estimate a Reynolds number at the scale of the forcingin the experiment, which is of order of the size of the vortexdipoles nucleated,d ∼ 10ξ, with healing lengthξ ∼ 0.5µm.Such dipoles have a characteristic speed vd ∼ 146µms−1, forwhich we estimate the Reynolds number of the forcing scaleas∼ vdd/ν0

q = 6.2×103. These large values suggest that turbu-lent flow in a finite-temperature BEC may exist across a widerange of length scales if the analogy is made with the classicalReynolds numbers that correspond to turbulent flow [1]. This

interpretation is broadly consistent with the experimental andnumerical observations of chaotic vortex dynamics [33].

We emphasize that the quantum Reynolds number estimateproposed here is applicable to a finite-temperature weakly in-teracting superfluid and may provide a general condition inanalogy with classical fluids that is independent of dimension.However, taking the zero-temperature limit gives an infinitevalue for Re0

q, and in this regime the superfluid dissipationstems from vortex reconnections (or annihilation in two di-mensions) and coupling to the sound field [45–47]. The de-tailed description of criteria for superfluidity in the cross-overfrom the zero-temperature to high-temperature regimes is anopen problem [48]. Furthermore, due to the reversed directionof energy transfer in 2DQT, the scales of interest have to bereexamined; we do not pursue this here. Our aim is to estab-lish a conceptual link between the dGPE and the NSE, givenby Eq. (12). In doing so we have shown how to identify theequivalent viscosity in a finite temperature BEC.

C. Two-dimensional vortex wavefunction

In the remainder of this work we limit our analysis to ho-mogeneous compressible quantum fluids in two dimensions,and redefine our spatial and velocity coordinates accordingly:r = (x, y) = r (cosθ, sinθ) andv = (vx , vy). We thus confineour attention to the regime of an effective 2D GPE, with modi-fied interaction parameter. While 2D BEC systems can be cre-ated through extremely tight confinement in one dimension, aregime of effective 2D vortex dynamics can also be accessedin less oblate systems, giving a 2D analysis wider applicabil-ity. For example, although the BECs of References [32] and[33] were three dimensional, the confinement along one di-mension was strong enough to limit vortex motion to a planeand suppress vortex bending and tilting away from the tight-trapping direction. Aspects of BEC dimensionality in regardsto vortices and Kelvin waves were analyzed in [44], further in-dicating that sufficiently oblate 3D BECs may be considered2D as far as vortex dynamics and turbulence are concerned.At the same time, such systems can remain far enough awayfrom the quasi-2D limits in which a Berezhkinski-Kosterlitz-Thouless (BKT) transition has been observed [49], and BKTphysics may thus be neglected.

For our analysis of kinetic energy spectra, we require cer-tain properties of a quantized vortex, namely the asymp-totic character of the wavefunction for large and small lengthscales. The Gross-Pitaevskii equation describing the homo-geneous (V = 0) 2D Bose gas is obtained from (1) by takingγ = 0 and using an interaction parameterg2 = g/l wherel isthe characteristic thickness of the 3D system [30]:

i~∂ψ(r , t)∂t

=

(

−~2∇2⊥

2m+ g2|ψ(r , t)|2

)

ψ(r , t). (20)

For example, in a system with harmonic trapping in thez-direction characterized by trapping frequencyωz, the lengthscale isl =

√2πlz wherelz =

√

~/mωz is thez-axis harmonicoscillator length, and the confinement is assumed sufficient

5

to put the wavefunction into thez-direction single-particleground state.

For solutions with chemical potentialµ containing a singlevortex at the origin (with circulation normal to the plane ofthequantum fluid) we can write [50]

ψ1(r , t) =√

n0e−iµt/~χ(r/ξ) e±iθ (21)

whereξ = ~/mc is the healing length for speed of soundc =√

µ/m, andn0 = µ/g2 is the 2D particle density forr ≫ ξ andis taken to be a constant. The vortex radial amplitude functionχ(σ), whereσ = r/ξ is a scaled radial coordinate, is a solutionof

(

−σ−1∂σ σ∂σ + σ−2

)

χ = 2(χ − χ3). (22)

The boundary conditions areχ(0) = 0, and the derivativeχ′ ≡ dχ/dσ evaluated atσ = 0 must be chosen such thatit is consistent withχ(∞) = 1 andχ′(∞) = 0. The value

Λ ≡ χ′(0) = limr→0

ξ√

n0

∣

∣

∣

∣

∣

dψ1

dr

∣

∣

∣

∣

∣

(23)

is determined numerically to beΛ = 0.8249. . . . The state(21) has the velocity field of a quantum vortex

v(r ) =~

mr(∓ sinθ,± cosθ). (24)

D. Kinetic energy decomposition

We make use of the decomposition of the kinetic energyinto compressible and incompressible parts [28, 51]. The 2Dcase of the Gross-Pitaevskii energy functional (3) can be de-composed asE = EK + EV + EI + EQ, where

EK =m2

∫

d2r ρ(r , t)|v(r , t)|2, (25)

EV =

∫

d2r ρ(r , t)V(r , t), (26)

EI =g2

2

∫

d2r ρ(r , t)2, (27)

EQ =~

2

2m

∫

d2r |∇√

ρ(r , t)|2. (28)

Respectively, these define the components of energy thatcan be attributed to kinetic energy, potential energy, inter-action energy, and quantum pressure. We are interested inthe kinetic energy,EK . We define a density-weighted ve-locity field u(r , t) ≡

√

ρ(r , t)v(r , t), then decompose this intou(r , t) = ui(r , t) + uc(r , t), where the incompressible fieldui

satisfies∇ · ui = 0, and the compressible fielduc satisfies∇ × uc = 0. We can further decompose the kinetic energy asEK = Ei +Ec, where the portion ofEK attributed to compress-ible or incompressible kinetic energy is defined as

Ec,i =m2

∫

d2r |uc,i(r , t)|2. (29)

The compressible component is attributed to the kinetic en-ergy contained in the sound field, while the incompressible

part gives the contribution from quantum vortices. Our analy-sis below only involvesEi . Because we focus on vortex con-figurations at instants in time, we drop the explicit time de-pendence from the remainder of our expressions.

In k−space, the total incompressible kinetic energyEi isgiven by

Ei =m2

∑

j=x,y

∫

d2k |F j(k)|2 (30)

where

F j(k) =12π

∫

d2r e−ik·r uij(r ). (31)

The one-dimensionalspectral densityin k-space is given inpolar coordinates by integrating over the azimuthal angle togive

Ei(k) =mk2

∑

j=x,y

∫ 2π

0dφk|F j(k)|2, (32)

which, when integrated over allk, gives the total incompress-ible kinetic energyEi =

∫ ∞0

dk Ei(k).

III. INCOMPRESSIBLE KINETIC ENERGY SPECTRA

A. Incompressible kinetic energy spectrum of a vortex

We now consider the kinetic energy spectrum of a singlequantum vortex in a 2D BEC. For an arbitrary wavefunctionthe decomposition into compressible and incompressible partsmust be performed prior to carrying out the transformation tothe spectral representation. However, for a quantum state con-taining a single vortex and no acoustic energy [i.e. the singlevortex wavefunctionψ1 (21)] we note that the wavefunctionis automatically incompressible, i.e. the compressible part isidentically zero:

∇ · (√

ρ(r )v(r )) = v · ∇√

ρ(r ) +√

ρ(r )∇ · v ≡ 0. (33)

The first term vanishes due to the orthogonality of the densitygradient and velocity of a vortex, and the second due to theform of (24). Thus the incompressible spectrum is the entirespectrum for a single quantum vortex.

For a single vortex we can thus ignore the incompress-ible decomposition and cast the kinetic energy spectrum interms of the properties of the radial amplitude functionχ(σ) =√

ρ(σξ)/n0 obtained from (22). We have

Fx(k) = − ~2πm

∫

d2r e−ik·r√

ρ(r )

rsinθ

=i~m

ddk

∫ ∞

0dr

√

ρ(r)

rJ0(kr)

=−i~√

n0ξ

m1kξ

∫ ∞

0dσ χ′(σ)J0(kξσ), (34)

6

whereJ0 is the zeroth-order Bessel function of the first kind.Similar analysis givesFy(k) = −Fx(k). We can thus find theone-vortex spectrum [see (32)]

E1i (k) = Ωξ3F(kξ), (35)

where we define the dimensionlesscore spectral function

F(kξ) ≡ 1kξ

(∫ ∞

0dσ χ′(σ)J0(kξσ)

)2

, (36)

and we have introduced the unit of enstrophy

Ω ≡ 2π~2n0

mξ2, (37)

giving Ωξ3 as the natural unit for the kinetic energy density.The core spectral function has the small-kξ asymptotic form

F(kξ)∣

∣

∣

∣

kξ≪1=

1kξ

(∫ ∞

0dσ χ′(σ)

)2

=1kξ. (38)

For kξ ≫ 1, J0(kξσ) is highly oscillatory except atσ = 0where it is unity, and the Taylor expansion ofχ′(σ) can betruncated at zeroth order to give

F(kξ)∣

∣

∣

∣

kξ≫1=Λ2

kξ

(∫ ∞

0dσ J0(kξσ)

)2

=Λ2

(kξ)3. (39)

We thus have the asymptotic spectra for a single vortex

E1i (k)

∣

∣

∣

∣

kξ≪1=Ωξ3

kξ, (40)

E1i (k)

∣

∣

∣

∣

kξ≫1= Λ2 Ωξ

3

(kξ)3. (41)

Thekξ ≪ 1 regime arises purely from the irrotational veloc-ity field of a quantum vortex, while thekξ ≫ 1 regime is aproperty of the core of a compressible quantum vortex. Thekξ ≫ 1 regime explicitly depends on the slope of the wave-function at the core of a vortex. The cross-over between theseregions occurs in the vicinity ofkξ ≈ 1, hence we takekξ = 1as distinguishing the infrared (kξ < 1) and ultraviolet (kξ > 1)regimes in the remainder of our analysis. The scalekξ = 1thus serves to define an important length scale of the problem,namelylv ≡ 2πξ. In Fig. 1 we see that at this distance fromthe vortex core the deviation of the amplitude from the back-ground value is very small. This is the scale beyond which thedetails of the core structure are no longer important in charac-terizing the wavefunction, or equivalently that the fluid den-sity has approximately reached its bulk value. The irrotationalvelocity field in Eq. (24) is the only remaining signature of avortex at this range from its center and beyond. We note thatour derivation of thek−3 power-law stemming from the quan-tum vortex core structure is consistent with recent analysis ofthe Kelvin-wave cascade in 3D [52].

Now that we have identified the properties of a single vor-tex, it is natural to ask whether a unit of enstrophy can beattributed to a single quantum vortex, and to compare thiswith the quantity defined in (37). The point-vortex model sug-gests that this can be done, but gives a singular result, which

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

σ

χ(σ

)

FIG. 1. Amplitude of the wavefunction for a single vortex solution ofthe Gross-Pitaevskii equation. The numerical solution of (22) (solidline) is compared with the ansatz (44) (dashed line). The inclineddashed line shows the slopeΛ of the exact solution at the origin. Thevertical line is the pointσ = 2π.

is nevertheless known to be proportional to the number of vor-tices [30]. The problem is also evident if we attempt to eval-uate the enstrophy of a single vortex from the spectrum (35).Multiplying by k2 to produce an enstrophy spectral densityand integrating this over the ultraviolet regimekξ > 1, we arefaced with the singular integral

∫ ∞1

dk/k.In this work we therefore define a new asymptotic quantity

with units of enstrophy as

ζ ≡ limk→∞

k3Ei(k). (42)

This quantity plays a fundamental role for a compressiblesuperfluid because it completely specifies the large-k regionof the incompressible kinetic energy spectrum. Because thespectrum in this region ofk-space is determined by the corestructure of quantized vortices, we call this unit theonstrophyto both recall Onsager’s contribution to our understandingofquantized vorticity in a superfluid and emphasize the differ-ence between enstrophy in classical and quantum fluids. Fora single quantum vortex we find, using (35), the onstrophy

ζ1 = Λ2Ω, (43)

which differs from (37) by the factorΛ2 = 0.6805. . . , a prop-erty of the vortex core in a compressible superfluid.

B. Vortex wavefunction ansatz and kinetic energy spectrum

To study 2D kinetic energy spectra we will make extensiveuse of an algebraic ansatz for the wavefunction of a singlevortex in a homogeneous superfluid. Numerical evaluationof the exact core function (36) is not straightforward due tothe highly oscillatory integrand, and the need to determinethevortex amplitudeχ(σ) extremely accurately over a large rangeof length scales. In order to accurately represent the spectrumit will be crucial that our ansatz have the correct asymptoticproperties for small and large length scales described imme-diately above (23). Making use of the slope at the origin com-

7

puted for the exact solution in (23), we use the ansatz wave-function:

φv(r ) =√

n0re±iθ

√

r2 + (Λ−1ξ)2. (44)

The general form of this ansatz has been previously used todescribe the shape of a vortex core [50], but here we usea length scaleΛ−1ξ that enforces matching the slope of theansatz density distribution to the exact value at the centerofthe core. The state (44) has the irrotational velocity field of aquantum vortex specified in (24) and reproduces the asymp-totic slope of the exact solution near the origin, as shown inFigure 1.

We now compute the kinetic energy spectrum for a singlevortex by evaluating (31) using the form (44). TakingΛ−1ξ =

b for brevity, we have

Fx(k) =i~√

n0

mddk

∫ ∞

0

dr J0(kr)√

r2 + b2

= i~√

n0b

2m

[

I1

(

kb2

)

K0

(

kb2

)

− I0

(

kb2

)

K1

(

kb2

)]

(45)

whereI j andK j are modified Bessel functions of the first andsecond kind, respectively, of orderj. Since|Fy|2=|Fx|2, wefind the incompressible energy spectrum of a single vortex

E1i (k) = Ωξ3FΛ(kξ), (46)

where

FΛ(kξ) ≡ Λ−1 f (kξΛ−1), (47)

and where we define

f (z) ≡ (z/4)[I1(z/2)K0(z/2)− I0(z/2)K1(z/2)]2. (48)

The functionf (z) has the following asymptotics: forz≪ 1

f (z) =1z+

(

γ + ln( z4

)) z2+ . . . , (49)

where γ = 0.57721... is the Euler-Masceroni constant; forz≫1

f (z) =1z3+

3z5+ . . . . (50)

The functionFΛ(kξ) thus has the asymtotics

FΛ(kξ)∣

∣

∣

∣

kξ≪1=

1kξ

(51)

FΛ(kξ)∣

∣

∣

∣

kξ≫1=Λ2

(kξ)3(52)

which are identical to those ofF(kξ). The two functions arevery similar, with only small differences evident in the cross-over regionkξ ∼ 1, as seen in Figure 2. We useFΛ(kξ) insteadof F(kξ) for describing the kinetic energy spectrum for a vor-tex core in the remainder of this work as it is numerically ex-pedient and does not alter any of the physical consequences ofour analysis. Towards the end of this paper we will compare

!""

!"!#

!""

kξ

Ei(k

)/Ω

ξ3

FIG. 2. Comparison of the numerically computed kinetic energyspectrum [(35), blue solid line] and that obtained from the core func-tion FΛ(kξ) [(46), red dashed line] in the cross-over regimekξ ∼ 1.The asymptotic expressions (38) and (39) are shown by the black andgreen lines respectively.

the asymptotic results of our analysis with spectra determinednumerically from forced dGPE dynamics.

The spectrum of a single vortex is shown in Figure 3, andcompared with the spectrum of a vortex-antivortex pair (a vor-tex dipole), and that for two vortices of the same sign (a vortexpair). These two-vortex spectra are analyzed in the followingsection.

C. Two-vortex spectra

Extension of the discussion in Section III A leads us to con-clude that a wavefunction thatonly contains vortices (i.e., nosound field) separated by more than a few healing lengthswill thus be approximately incompressible according to thedecomposition. The approximation breaks down through thenon-orthogonality ofv and∇

√

ρ(r ) near a vortex core due tothe velocity field induced by the other vortices. However, inthe close vicinity of a vortex core, where∇

√

ρ(r ) is signif-icant, the velocity is dominated by the velocity field of thatvortex core. An arrangement of vortices separated by morethan a few healing lengths will thus be approximately incom-pressible. In the following analytical treatment we will ne-glect any compressible part that arises from an assembly ofvortices described by the ansatz (44).

A two-vortex state in a homogeneous system with noboundaries has density-weighted velocity field

√

ρ(r )[v1(r ) +v2(r )], wherev j is the velocity field around vortexj = 1, 2taken separately. If the vortex cores are separated byd ≫ ξ

it will also be a very good approximation to writeρ(r ) =ρ1(r )ρ2(r )/n0, with

ρ j(r ) =n0 |r − r j |2

|r − r j |2 + (Λξ−1)2. (53)

The density weighted two-vortex velocity field can then be

8

!"!#

!"!!

!""

!"!

!"!#

!"!!

!""

!"!

!"#

kξ

Ei(k

)/Ω

ξ3

FIG. 3. Incompressible kinetic energy spectra for a single vortex(chain line), a vortex dipole (solid line), and a vortex pair(dashedline). The vortex dipole and pair are both shown for vortex separationd = 20ξ, and the wavenumberkd ≡ 2π/d is shown as a verticaldashed line. The cross-over scalekξ = 1 is given by the solid verticalline.

written as√

ρ1(r )ρ2(r )n0

[v1(r ) + v2(r )] =√

ρ1(r )v1(r ) +√

ρ2(r )v2(r )

+K12(r ). (54)

The final term is

K12(r ) =

√

ρ1(r )ρ2(r )n0

(

v1(r )

[

1−√

n0

ρ2(r )

]

+v2(r )

[

1−√

n0

ρ1(r )

] )

(55)

which is only significant when considering the velocity fieldof one vortex in the close vicinity of the other vortex core.K12

is therefore a negligible correction to the spectrum for widelyseparated vortices. This approximation may be trivially gen-eralized to arbitrary numbers of vortices provided the cores donot overlap appreciably. This approximation is central to ourtreatment, as it allows recovery of familiar point-like vortexphysics in the infrared regimekξ ≪ 1. Strict validity is lim-ited to the regime where the intervortex spacingd is boundedbelow by∼ lv = 2πξ, the scale at which the core structurebecomes evident (see Fig. 1).

We require the transform

F dj (k) =

12π

∫

d2r e−ik·r[ √

ρ1(r )v1(r )]i

j

+12π

∫

d2r e−ik·r[ √

ρ2(r )v2(r )]i

j(56)

where the superscriptd denotes the case of a vortex dipole,and subscriptj = x, y indicates thex andy components of thedensity-weighted velocity fields of each vortex. As above, thesubscripts 1 and 2 denote vortices 1 and 2, and the superscripti denotes that it is the incompressible, or divergence-free por-tion of the density-weighted field that is of interest here. Toaccount for the opposite signs of circulation for the two vor-tices, without loss of generality we choose vortex 1 as posi-tively charged, located atr0 = (d/2)x, so thatv1(r ) = v(r − r0)

wherev(r ) is the central vortex velocity field (24). Vortex 2has velocity fieldv2(r ) = −v(r + r0). We then have

F dj (k) =

12π

∫

d2r e−ik·(r+r0)[√

ρ(r )v(r )]i

j

− 12π

∫

d2r e−ik·(r−r0)[ √

ρ(r )v(r )]i

j. (57)

For a vortex dipole we can then write

F dj (k) = F j(k)e−ik·r0 − F j(k)eik·r0 (58)

whereF is the spectrum of a single vortex. Using (45), andthe fact thatFy(k) = −Fx(k), we find for a vortex dipole

Edi (k) = 2Ωξ3FΛ(kξ)(1− J0(kd)). (59)

The spectrum of a pair of vortices of the same circulation sep-arated byd is calculated similarly to be

Epi (k) = 2Ωξ3FΛ(kξ)(1+ J0(kd)). (60)

The spectra (59) and (60) are shown in Figure 3. It is clearthat for scales less than the vortex separation distanced thereis interference ink-space, leading to oscillations in the spec-trum. The difference between the dipole and pair is that the in-terference fringes are offset, and the infrared asymptotics aredifferent, a feature we discuss further below. The spectrum ofthe vortex pair is clearly similar to that of the single vortex inthe far infrared region, but the additional kinetic energy of thevortex pair state is observed throughout the spectrum.

D. Kinetic energy spectrum ofN-vortex configurations

Extending the above analysis, for a general system ofNsingly quantized vortices with circulation signsκp = ±1 lo-cated atr p the kinetic energy spectrum is given by

ENi (k) = Ωξ3FΛ(kξ)

N∑

p=1,q=1

κpκqJ0(k|r p − rq|). (61)

We note the resemblance to point-vortex results which alsohave the Bessel function dependence [29, 53]. The functionFΛ(kξ) gives the incompressible limit for smallk [J0(kr) ap-proaches unity for smallk and finite r], and introduces thephysics of compressible superfluids for 1. kξ.

We can write (61) as

ENi (k) = NΩξ3FΛ(kξ)GN(k) (62)

where

GN(k) ≡ 1+2N

N−1∑

p=1

N∑

q=p+1

κpκqJ0(k|r p − rq|) (63)

is a purely configurational function involving summation ofM = N(N − 1)/2 distinct intervortex distances. This functionhas the limits

limk→∞

GN(k) = 1, (64)

limk→0

GN(k) = 1+2N

N−1∑

p=1

N∑

q=p+1

κpκq =Γ2

N, (65)

9

where the total dimensionless circulation is defined by

Γ ≡ m~

∮

Cv · dl =

N∑

p=1

κp = N+ − N− (66)

for any contourC enclosing allN+ positive andN− = N − N+negative vortices.

We then find that the onstrophy for theN-vortex system is

ζN = limk→∞

k3ENi (k) = NΛ2Ω = Nζ1, (67)

and consequently

ENi (k)

∣

∣

∣

∣

kξ≫1=ζN

k3=

NΛ2Ωξ3

(kξ)3. (68)

This is one of our central results: the ultraviolet regimekξ ≫1has a universal asymptotic form that is independent of the vor-tex configuration, and that resembles the ultraviolet spectrumof classical 2D turbulence that is identified with a direct cas-cade of enstrophy. If we try to evaluate the classical definitionof enstrophy, the result is singular, yet the onstrophy definition(42) gives a well-defined additive quantity that is singularityfree and depends only on the total number of vortices in thesystem.

E. Infrared behavior

WhenΓ , 0 the far infrared limit (65) gives

ENi (k)

∣

∣

∣

∣

kξ≪1=Ωξ3Γ2

kξ. (69)

This configuration-independentk−1 power law arises from thefar-field velocity distribution of a collection of point vortices,which becomes equivalent to that of a single vortex of chargeΓ at sufficiently large scales.

WhenΓ ≡ 0 we use the small-argument expansionJ0(z) ≃1 − z2/4, and the asymptotic form (49), to find thekξ ≪ 1behavior determined by the configurational information con-tained in the intervortex distances|r p − rq|:

ENi (k)

∣

∣

∣

∣

kξ≪1= −Ωξ

2k2

N−1∑

p=1

N∑

q=p+1

κpκq|r p − rq|2. (70)

The simplest case involves a single vortex dipole and has onlyone length scale, namely the vortex separation, and the low-kform Ed

i (k) ≃ Ωξ2d2k/2, as shown in Fig. 3. In general, whenΓ = 0 the infrared region of the spectrum is sensitive to thevortex configuration, but approaches a power-law for low-kthat has a configuration-independent exponent. The linear de-cay of kinetic energy ask→ 0 stems from the cancellation ofthe far-field velocity profiles for length scales greatly exceed-ing the largest intervortex separation in any neutral configura-tion of vortices.

IV. KOLMOGOROV SPECTRUM

In the previous section we obtained an explicit expression(62) for the incompressible kinetic energy spectrum that in-corporates the compressible nature of individual vortex coresthrough the functionFΛ(kξ) (derived via an ansatz for the vor-tex core profile), which captures the essential physics of thecorresponding exact solutionF(kξ) defined in (36). For bothfunctions, point-vortex physics is recovered at large length-scales (kξ ≪ 1). If the dynamical evolution is such that aninertial range associated with an inverse energy cascade devel-ops, we should expect a Kolmogorov power lawEi(k) ∝ k−5/3

over the inertial range. It is clear from the form of (62) thatthis law can only depend on the spatial configuration of thevortices. We now seek to understand the simplest situationsthat may show evidence for the existence of such an inertialrange. We consider forcing occurring via vortex and energyinjection at a forcing scalekF ∼ ξ−1, and describe vortex con-figurations that do and do not lead to a Kolmogorov law fork < ξ−1.

We now assume an idealized case in which the infraredspectrum is continuous with the universalk−3 law of the ultra-violet spectrum at the scalekξ ≈ 1. This constraint imposes astrong restriction on the infrared spectrum, completely deter-mining its form in the case that it satisfies a power law. In thisrespect the universal ultraviolet region has significant physicalconsequences. The power-law approximation to the universalultraviolet region based on (67) has the form

ENi,U(k) = Λ2 NΩξ3

(kξ)3= ζNk−3. (71)

The number of vortices determines theN-vortex onstrophyζN

(67), from which the power-law approximation to the ultravi-olet energy spectrum is completely determined. This powerlaw is a very good approximation, as will be seen by sam-pling different vortex configurations below. The infra-red orconfigurational regime is then given by thekξ ≪ 1 regime ofFΛ(kξ):

ENi,C(k)

∣

∣

∣

∣

kξ≪1=

NΩξ3

kξGN(k). (72)

At this point we consider the consequences of assuming that aturbulent system will have ak−5/3 law in the configurationalregime, and that this power law is continuous with (71) atkξ = 1. We suppose thatEN

i,C(k) ∝ k−5/3. Continuity atk = 1/ξ then requiresEN

i,C(1/ξ) = ENi,U(1/ξ), and gives the

infrared spectrum

ENi,C(k) = Λ2 NΩξ3

(kξ)5/3= ζN ξ

4/3 k−5/3. (73)

Thus the constraint that the universal regime is continuousatthe cross-over scalek = ξ−1 completely constrains the form ofthe configurational spectrum. Physically this may correspondto an inertial range that extends upwards from the smallestscale of the infrared region given forcing at a wavenumberkF ∼ ξ−1. Note that this expression (73) has no reference tothe signs of the vortex circulations, the degree of circulation

10

polarization, or vortex clustering. By assuming continuity atkξ = 1 with a UV spectrum that has a universalN-vortex form,we have implicitly assumed that allN vortices are involved indetermining the spectrum of the inertial range.

We might expect that this will give a very good descriptionfor a completely polarized system exhibiting fully developedturbulence. When there is clustering in mixtures of differentsign vortices, the spectrum may well still approach a Kol-mogorov law, but there is no reason to expect that it will crossover so smoothly. We return to this problem when we compareour analysis with numerical simulations of forced turbulencein Sec. V C.

It is useful at this point to give a simplified reiteration ofNovikov’s argument for the power law for the vortex distri-bution being−1/3 in Kolmogorov turbulence [53]. To obtainpower-law behavior we must consider the spectrum for a vor-tex distribution involving many length scales. For simplic-ity we assume all vortices have the same sign of circulation,κp ≡ κ. The configuration function (63) hasM = N(N − 1)/2terms in the summation, and can be written as

GN(k) = 1+2N

M∑

p=1

J0(ksp), (74)

in terms of an average over distinct vortex separationssi . Weintroduce the intervortex distance distributionP(s) such thatP(s)ds is the fraction of intervortex distances in the range[s, s+ ds). In the continuum limit

GN(k) ∝∫

P(s)J0(ks)ds. (75)

We seek a distributionP(s) that will generate a Kolmogorovlaw from theN-vortex spectrum (62) for scales larger than thevortex core,k ≪ ξ−1, in the large-N regime. We then findfrom (72) that

ENi,C(k)

∣

∣

∣

∣

kξ≪1∼ 1

k

∫ ∞

ξ

P(s)J0(ks)ds (76)

≃ 1k

∫ ∞

0P(s)J0(ks)ds. (77)

The scale invariance of turbulence naturally leads to theassumption that the intervortex separation distribution is apower-lawP(s) ∼ s−α over the scale range of interest. Therequirement of a power law in the kinetic energy then givesthe scaling relation

ENi,C(k) ∼ 1

kβ∼ 1

k

∫ ∞

0s−αJ0(ks)ds

=1

k2−α

∫ ∞

0τ−αJ0(τ)dτ. (78)

The integral is convergent for−1/2 < α < 1, allowing

1 < β < 5/2. (79)

In particular, the universal Kolmogorov lawβ = 5/3 occursfor

P(s) ∼ s−1/3 (80)

as obtained by Novikov [53] for point-vortices. We will testthis scaling argument for the exponent in numerical samplingof localizedvortex configurations in the following sections.Testing if this vortex separation power-law holds in simula-tions and experiments may give a quantitative measure of fullydeveloped 2D turbulence in a compressible superfluid, and away to identify the inertial range as the scale range over whichthis power-law can be identified.

In 2D classical turbulence ak−3 region of the kinetic energyspectrum is often associated with a direct enstrophy cascade.We note that this exponentβ = 3 is ruled out by (79). Hence,within this continuum analysis thek−3 power-law spectrumcannot occur in the configurational region for 2D quantumturbulence, as long as the vortex distribution follows a sim-ple power law. This result suggests that if a direct enstro-phy cascade were to occur in the configurational region ofthe spectrum, a different type of vortex distribution would benecessary. This makes intuitive sense, since direct enstrophycascades may be associated with the stretching of patches ofvorticity in the 2D plane. Furthermore, direct enstrophy cas-cades and energy spectra proportional tok−3 have been notedin simulations of superfluid helium thin films [54, 55]. Nev-ertheless, since we have also shown that thekξ > 1 range isdetermined entirely by the core structure, this gives a strongindication that a direct enstrophy cascade cannot occur in thisregion in compressible 2DQT.

It is clear that configurations containing one or a few char-acteristic length scales, such as a vortex dipole or a vortexlattice, cannot lead to a power law spectrum forEN

i,C(k). In thecase of a vortex lattice the intervortex distance distribution hasmany discrete peaks [36]. The vortex dipole and a vortex paireach have a single length scale and this leads to characteristicinterference fringes in the energy spectrum seen in Fig. 3.

A. Sampling spatial vortex distributions

We now test our analysis of the spectrum by numericallysampling several vortex distributionsr p, κpNp=1 and evaluat-ing (62). A straightforward test of the statistical argumentfor the Kolmogorov power law to occur involves samplingthe power law (80) and evaluating (74). Indeed, it is easilyverified that this generates ak−5/3 spectrum in the configura-tional region. However, the connection of such a sampling toparticular spatial vortex distributions is not clear, and in factthe mapping is not unique. To make this connection concretewe require a way of sampling finite, localized, spatial vortexdistributions for which the vortex separations are power-lawdistributed.

In an ideal, infinitely extended vortex configuration exhibit-ing the power law (80), the system is translationally invariantand the coordinate origin can be placed at any particular vor-tex, yielding the same power law for the radial distributionofvortices from the origin. In a finite system the scale invariancecan only persist up to scales of order the largest vortex sepa-ration. Furthermore, the vortices must be separated by a min-imum distance to satisfy the assumptions used in deriving theenergy spectrum from the point-vortex model. In practice it

11

is necessary to use a self-consistent sampling scheme in orderto generate the correct power-law distributions for localizedfinite configurations.

Our sampling scheme for a configuration ofN vortices is:

1. Sample the radial distancerp of each vortex from thecoordinate origin according to a power law probabilitydistribution∝ r−αp . The exponent ¯α is distinct fromα

due to the finite system size and localization of the dis-tribution.

2. Assign each vortex a randomly chosen, uniformly dis-tributed angleθp ∈ [0, 2π). The cartesian coordinatesfor vortexp are then

(xp, yp) = rp(cosθp, sinθp). (81)

3. For a givenN, compute the kinetic energy spectrum,averaging overns = 100 samples of vortex position datafound via the foregoing routine. Iterate until the spectralpower law of interest is found.

We will sample a power-law distribution of vortex distancesfrom the origin. In practice there is a lower (rmin ∼ ξ) andupper (rmin ∼ R = system size) cutoff for power-law scaling.We thus wish to sample the distribution

Pα(r) =1− α

r1−αmax− r1−α

min

r−α, (82)

that is normalized on the intervalrmin ≤ r < rmax. We sampler values from this distribution using uniform random variatesx ∈ [0, 1) via the transformation [56]:

r =[

r1−αmaxx+ r1−α

min (1− x)]

11−α. (83)

Note that when ¯α < 1, as is always the case in this work,some kind of ultraviolet cutoff is required for the distributionto be normalizable. Here we have made a choice that givesa power-law distribution over a well-defined scale range (Seee.g. Ref. [56] for other common choices).

B. Classical velocity distribution of a large cluster

A fully polarized configuration of vortices with a given ra-dial power law forms a quantum analogue of the coherentvortices of forced 2D turbulence in classical fluids. As an ar-rangement of many vortices, the velocity distribution mustap-proach a classical limit, according to Bohr’s correspondenceprinciple, in much the same way that a rotating Abrikosov lat-tice generates a velocity field that approaches that of a rotatingrigid body [50]. In what follows we find the classical veloc-ity field, and identify the physical significance of the radialexponent ¯α.

To determine the velocity field we first compute the fractionof vortices enclosed by a circular contour around the origin,

with radiusr, for the distribution (82). Takingrmin = ξ, rmax =

R, and considering scalesξ ≪ r ≪ R we have

fr =∫ r

ξ

Pα(u)du= N∫ r

ξ

duuα=N

1− α(

r1−α − ξ1−α)

≃( rR

)1−α, (84)

where we used the fact thatN−1 =∫ R

ξs−αds = (1 −

α)−1(R1−α − ξ1−α) ≃ R1−α/(1− α) normalizes the distributionup to the largest scaleR. Considering the average azimuthalvelocity component vφ(r), the circulation is

∮

v · dl =hm

n = vφ(r)2πr (85)

wheren = fr N is the number of vortices enclosed by the con-tour of radiusr. Using (84), we obtain the velocity profile

vφ(r) ≃cN

(R/ξ)1−α(r/ξ)α. (86)

This inertial cluster has a power-law velocity profile deter-mined by the specific radial exponent ¯α. In contrast, the veloc-ity profile of an Abrikosov vortex lattice rotating at frequencyω approaches that of a rigid body vφ(r) = ωr, and a singlequantum vortex has profile vφ(r) = ~/mr. The inertial clustervelocity profile (86) is compared with sampled distributions(see below) in Figure 4.

C. Scale expansion of a large cluster

We now illustrate the role of the radial power law expo-nent α and the classical velocity distribution by sampling alarge cluster. We vary our choice of scale range for the vor-tices (rmax in Eq. 82), and investigate how the range ofk−5/3

changes.A characteristic wave number measuring cluster size for a

sample involvingnc vortices in a given cluster is given by

k ≡ 2π/r (87)

Where ¯r = 1nc

∑nc

p=1 rp. In the figure we plot the scalek to

give an indication of the range ofk−5/3 scaling. We also plotkmax corresponding to the largest vortex separation scale in thesystem

kmax = 2π/max|r p − rq|. (88)

For k < kmax the velocity field approaches that of a singlevortex.

In Figure 4 vortex distributions are sampled forN = 100vortices of the same sign using the sampling scheme (81),(83), forrmin = ξ and different values ofrmax. Individual sam-ples are shown to indicate the spread of vortices, and the ve-locity profiles and kinetic energy spectra are computed by av-eraging overns = 100 samples. The mean azimuthal velocitycompares well with Equation (86), showing that the specificradial exponent ¯α in (82) also determines the power law ofthe azimuthal velocity field, as seen in (86). For scales larger

12

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y/ξ

!"!#

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!"%

Ei(k

)/N

Ωξ3

!!"" " !""

!!""

"

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x/ξ

y/ξ

!!"" " !""

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x/ξ

y/ξ

(a)

(b)

(c)

!"#"$

!"#"%

!"#"&

!"#"'

!"#"(

!"#")

!"#"*

Eq. (71)

Eq. (73)

kξ = 1

kmax

k

Eq. (69)

Eq. (61)

!!""" !#""" " #""" !""""

"$!

"$%

"$&

"$'

#

#$!

x/ξ

|vφ(x

,0)/

c|

!!"" " !"""

"#!

"#$

"#%

"#&

'

'#!

x/ξ

|vφ(x

,0)/

c|

!!"""" " !"""""

"#"$

"#!

"#!$

"#%

"#%$

"#&

x/ξ

|vφ(x

,0)/

c|

!"!#

!"!$

!""

!""

!"%

Ei(k

)/N

Ωξ3

!"!#

!"!$

!""

!""

!"%

kξ

Ei(k

)/N

Ωξ3

FIG. 4. Locations ofN = 100 vortices for particular samples (left), with velocity distributions [middle, with the analytical result (86) (dashedline)], and kinetic energy spectra (right, blue curves), averaged overns = 100 samples distributed according to (81) and (82), withrmin = ξ,and (a)rmax = 60πξ, α = 0.4, (b) rmax = 400πξ, α = 0.65, and (c)rmax = 4000πξ, α = 0.83. The increasing scale range of power-law behaviorgives an inertial range in the corresponding kinetic energyspectrum of∼ 1, 1.5, and 2 decades respectively.

thanrmax, vφ(r) returns to ther−1 scaling for a chargeN vortex[clearly seen in Fig. 4 (a)]. N = 100 vortices distributed up tormax = 60πξ gives an inertial range in the kinetic energy spec-trum of approximately one decade, for ¯α = 0.4 (as the numberof vortices in a given scale range increases, ¯α → 1/3). In-creasing the upper scale cutoff to rmax = 400πξ, 4000πξ givesα = 0.65, 0.83, with 1.5 and 2 decades of inertial range re-spectively. Thus expanding the scale range of power-law be-havior expands the inertial range, but requires the azimuthalvelocity profile to steepen. In the UV region of the spectrumthe k−3 law always holds [Eq. (71)], while the inertial rangeholds fork . k . ξ−1, and single vortex behavior is apparentfor k < kmax.

D. Clustering in a neutral distribution

We now consider the role of increasing clustering in ex-panding the inertial range of the kinetic energy spectrum.

First, in ns = 100 samples, we distributeN+ = N− = 100vortices over a uniform periodic domain; one such sampleis shown in the inset of Figure 5 (a). The corresponding ki-netic energy spectrum shows the correct UV-region spectrumgiven by Eq. (71), and also approaches theEi(k) ∝ k form fork < kmax, given by Eq. (70). Forkmax . k . ξ−1 the spec-trum is less steep thank−5/3 and the system lacks an inertialrange. In Figure 5 (b) the vortices are sampled as 40 clustersof nc = 5 vortices of the same sign according to (81) and (83),with α = 0.8, andrmin = ξ, rmax = 10πξ. The 10+ve and 10-ve cluster centers are uniformly distributed as in Fig. 5 (a),as seen in the sample (inset). This distribution yields ak−5/3

power-law kinetic energy spectrum over∼ 1 decade of wavenumbers. By further expanding the scale of clustering whilereducing the number of clusters to preserveN+ and N−, wefind the inertial range can be extended. In Figure 5 (c) sam-ples consists ofnc = 20 vortices in each cluster, distributedbetweenrmin = ξ−1 and rmax = 100πξ, with α = 0.75, andgiving ∼ 2 decades of inertial range. We note that, compared

13

!"!#

!"!$

!""

!""

!"%

kξ

Ei(k

)/N

Ωξ3

!"!#

!"!$

!""

!""

!"%

Ei(k

)/N

Ωξ3

!"!#

!"!$

!""

!""

!"%

Ei(k

)/N

Ωξ3

(b)

(c)

(a)

!"#"$

!"#"%

!"#"&

!"#"'

!"#"(

!"#")

!"#"*

Eq. (70)

Eq. (71)

Eq. (73)

kξ = 1

kmax

k

Eq. (61)

!!"" " !""

!""

"

!""

!!"" " !""

!""

"

!""

!!"" " !""

!""

"

!""

FIG. 5. Kinetic energy spectra forN+ = 100, andN− = 100 vortices,averaged overns = 100 samples (blue curves). The vortices are dis-tributed (a) uniformly over the (600ξ)2 square domain (inset), andclusters are sampled according to (81) and (82), with (b)nc = 5 vor-tices in each cluster,rmin = ξ, rmax = 10πξ, and (c)nc = 20 vorticesin each cluster andrmax = 100πξ, with each cluster center uniformlydistributed over the periodic domain as in (a). The power-law sam-pling in figures (b) and (c) requires a radial exponent of ¯α = 0.8,and 0.75, and gives a Kolmogorovk−5/3 region in the correspondingkinetic energy spectrum of∼ 1 and 2 decades respectively.

with Fig. 4,k, shown in Fig. 5 (b) and (c) does not correspondso well with the lower bound on the inertial range, presum-ably because of the significant space between clusters in theneutral system.

V. FORCED 2D QUANTUM TURBULENCE

A. Scenario of forced turbulence

kF k

k = ξ−1k ∼ L

−1

E(k)

k−5/3

k−3

k

FIG. 6. Illustration of an inertial range (non-shaded region) forthe incompressible portion of kinetic energy in forced compressible2DQT. TheE(k) ∝ k−3 region arises from the structure of the vortexcore and thus is not a signature of vortex configurations and vortexturbulence. This ultraviolet region can thus not support energy cas-cades, nor does this region correspond to enstrophy cascades. Netenergy injected atkF ∼ ξ−1 in the form of vortices can only move to-wards the infrared. The Kolmogorov lawE(k) ∝ k−5/3 occurs in theinertial range of fully developed turbulence. The far-infrared regionis given byE(k) ∝ k for a system with no net vorticity, and is evidentfor k≪ L−1 whereL is the largest intervortex distance. For forcing atsmaller wavenumbers, the spectrum may be more complex, possiblyinvolving other forms of energy and enstrophy flux.

in a compressible 2D superfluid

The canonical model of 2D classical turbulence consists ofa velocity field described by the 2D Navier-Stokes equation

∂v∂t+ (v · ∇)v = −1

ρ∇p+ ν∇2v − λv + fv. (89)

The densityρ of the incompressible fluid is held constantby the pressure fieldp, ν is the kinematic viscosity,fv is aforcing term andλ represents linear frictional damping aris-ing from irreducible 3D aspects of the system in which the2D flow resides. If the fluid is subjected to suitable forcingit will develop an inverse energy cascade and a direct enstro-phy cascade, with associatedk−5/3 andk−3 power laws respec-tively [13]. An inverse energy cascade induced by small-scaleforcing can be steady because the−λv term damps energy atlarge length scales [9]. For a homogeneous compressible su-perfluid subject to forcing from an external potential, (12)canbe written as

∂v∂t+ (v · ∇)v = − g

m∇ρ + νq∇2v + fv, (90)

where the forcingfv ≡ −∇V(r , t)/m is assumed to be spa-tially localized. The lack of a−λv frictional damping termmeans, in the classical case, that if an inverse energy cascadedevelops as a result of steady forcing, it is not expected to bestationary. In the superfluid case, the compressibility of thefluid allows vortex-antivortex annihilation, which couples en-ergy into the sound field. This interaction between the sound

14

and vorticity fields renders the calculation of energy fluxesin compressible superfluid systems particularly difficult andsomewhat ambiguous [30].

As we have shown above, in contrast with the classicalKraichnan scenario utilizing a 2D Navier-Stokes analysis,ak−3 spectrum fork > ξ−1 for a 2D quantum fluid is not causedby a direct enstrophy cascade but is rather a consequence ofthe vortex core structure, and thus should not be interpretedin terms of vortex configuration dynamics (note that vortexcore shape excitations can be neglected, since they constitutea component of the sound field). Given forcing at a wavenum-berkF ∼ ξ−1, and minimal vortex-antivortex annihilation, theincompressible kinetic energy can only move toward the in-frared. This scenario is shown schematically in Figure 6.

It has been shown that dipole recombination provides aroute for a direct energy cascade to develop in 2D GPE dy-namics [30]. This mechanism provides a means for opposite-sign vortices to approach zero distance, coupling vortex en-ergy to the sound field during vortex annihilation. However,ifthe forcing leads to significant clustering of like-sign vorticesfaster than recombination occurs, or prior to recombinationoccurring, dipole-decay will be strongly inhibited. This sug-gests that under the right conditions of forcing an inverse en-ergy cascade can become the dominant mechanism of energytransport between distinct length scales.

B. Kolmogorov constant and clustered fraction

By making use of the universal onstrophy and the conditionof continuity atkξ ≈ 1 we have found that thek−5/3 power-lawgiven by (73) describes the spectrum of numerically sampledvortex configurations that exhibit as−1/3 power law for thevortex separation data. While individual spectra and configu-rations do not give information about dynamics, in particular,the direction ink-space of any energy cascades, the powerlaw suggests the existence of an inertial range comprised ofvortices. In a cascade, such a configuration will transfer in-compressible energy between scales while conserving energy.Assuming the infrared portion of our double-power-law anal-ysis, namely (73), will also describe such a cascade, we cancast it as a statement about the Kolmogorov constant in termsof the one-vortex onstrophy and the slope of the radial wave-function at the vortex core.

To write (73) in standard form, we introduce the uniqueN-vortex quantity with dimensions energy/mass/time that can beconstructed fromΩξ2, m, and~/Ωξ2:

ǫN ≡(Ωξ2)2

m~N3/2. (91)

We then find

ENi,C(k)

m= C2Dǫ

2/3N k−5/3 (92)

where the remaining quantities have been absorbed into thedimensionless Kolmogorov constant:

C2D ≡ Λ2

(

µ

Ωξ2

)1/3

, (93)

µ = ~2/mξ2 in the homogeneous system, and the bar notationdistinguishes the quantum system. In the dilute Bose gas the2D interaction parameter isµ/n0 = g2 = 4π~2a/ml, wherel is the characteristic thickness of the three-dimensional sys-tem [30, 57]. In terms of this length we find

C2D = Λ2( mg2

2π~2

)1/3= Λ2

(

2al

)1/3

. (94)

We emphasize that the physical input needed to arrive atthis form of the Kolmogorov constant is (i) accounting forthe structure of a compressible quantum vortex in determin-ing the ultraviolet spectrum, and (ii) imposing continuityofthe ultraviolet spectrum atkξ = 1 to a Kolmogorov power-law in the infrared. In classical turbulence,C2D ≃ 7 [58]. Togive an example of howC2D may be evaluated for a compress-ible superfluid exhibiting 2DQT, we consider a87Rb BEC thatis homogeneous in thex − y plane, and that is harmonicallytrapped in thez-dimension with trap frequencyωz = 2π×5000Hz. We usem = 1.44 × 10−25kg, a = 5.8nm, for whichl =

√

2π~/mωz = 0.38 µm, andg2 = 0.197~2/m. For thesevalues,C2D = 0.212. Note also that by defining the configura-tional rate constant (91), we have confined this discussion to ascale invariant distribution involvingN vortices. This expres-sion suggests that such a configuration can support an inverseenergy cascade at the rateǫN.

The foregoing discussion involves an ideal distribution ofN vortices configured with theα = 1/3 power law. It is clearfrom Fig. 5 that the vortices do not have to all have the samesign of circulation, but they must be configured into clustersof vortices with the same sign. The universality ofC2D inclassical turbulence in incompressible fluids leads us to postu-late that the condition that allN vortices are power-law clus-tered can be further relaxed. For fully developed quantumturbulence involvingN vortices, we interpretN as thepar-ticipation number, representing the number of vortices in ascale-free turbulent configuration, which in the case of a fullypolarized cluster is maximal. Imperfect clustering involvesfewer vortices in power-law cluster configurations, and an ef-fective participation number that is the number of clusteredvorticesNc < N, namely, the number with nearest-neighborsof the same sign. Making the replacementN → Nc in (73),we propose the ansatz spectrum

ENi,C(k) = Λ2 NcΩξ

3

(kξ)5/3, (95)

as a more general definition for systems that have incompleteclustering in the inertial scale range. We test this hypothesis inthe next section in dynamical simulations of the forced dGPE.

It is important to note that the condition of continuity atkξ = 1 is no longer exactly met, since only the valueNc ≡ Nwill produce an infrared spectrum withk−5/3 that is continuouswith the ultraviolet power law approximation atk = 1/ξ. Wealso note that a more general measure of clustering, namelythe polarization index(P) was introduced in Ref. [59] mea-suring the degree and type of spatial clustering of like-signvortices in 3DCT. The Kolmogorovk−5/3 spectrum was foundto correspond to the partially polarized valueP = 1/3, while

15

FIG. 7. Time evolution of grid turbulence in damped GPE. Left: particle density (rescaled to the peak density). Center: vortices colored bycharge, with total (N) and clustered (Nc) numbers of vortices. The field of view is (1024ξ)2. Right: Incompressible energy spectra (circles),with the Kolmogorov ansatz [red line, Eq. (95)], the ansatz for a polarized cluster ofN vortices [dashed line, Eq. (73)], and the universalk−3

region [blue line, Eq. (71)].

other scaling laws yield differing polarizations. The clusteredfraction used in the present work is a simpler (global) mea-sure of polarization as it does not contain information aboutthe spatial distribution of vortices, but it is only relevant for2DQT.

We can also write down an expression forC2D for a gen-eral energy spectrum that may be computed numerically fromsimulation data, by making use of the ansatz (95). This isequivalent to using (92) withN → Nc, from which we candefine the function

C∗2D(k) = ENi,C(k)

(kξ)5/3

NcΩξ3

(

µ

Ωξ2

)1/3

. (96)

In a region where the spectrum is approximatelyk−5/3, C∗2D(k)

will be approximately constant, and it may be compared withthe prediction (93). We test this numerically in the next sec-tion.

We note that in 3DQT an energy bottleneck has been pre-dicted for the direct energy cascade [59, 60], and also ob-served in GPE simulations [61]. It occurs due a mismatchbetween the rates of energy transport at large length scales(hydrodynamic regime) and small length scales (Kelvin wavecascade). The mismatch causes energy to pile up at the lengthscale where the two cascades meet. This raises the possibilityof a bottleneck in 2D, although Kelvin waves are disallowedin 2DQT, so this particular mechanism would not be relevant.However, for a given forcing mechanism, it is possible that therate of transporting energy to large length scales may not be

16

!"!!

!""

!"!#

!"!!

!""

kξ

C∗ 2D(k

)

t = 600ξ/c

t = 1040ξ/c

t = 1450ξ/c

C2D = 0.212

FIG. 8. Plot ofC∗2D(k) [see Eq. (96)] computed from the grid turbu-lence kinetic energy spectra of Figure 7. The horizontal dashed linegivesC2D from Eq. (94).

high enough to remove all of the vortex energy introduced atthe forcing scale. Thus a bottleneck could still occur, and ourassumption of continuity of the spectrum atkξ ≃ 1 may nothold in general. We return to this question in the next subsec-tion, where we find some indication of an energy bottleneckat the forcing scale in numerical simulations.

C. Damped Gross-Pitaevskii Dynamics

We now consider a simulation of the forced dGPE that gen-erates significant clustering of vortices of the same sign. Thesystem consists of a homogeneous superfluid with periodicboundary conditions, stirred by dragging four Gaussian ob-stacle beams through it at a constant speed [32], thus model-ing grid turbulence in a BEC. When an obstacle is draggedthrough a superfluid sufficiently rapidly, superfluidity cannotbe maintained. For slowly moving obstacles, the superfluidwill adapt to the forcing, and vortices are not formed. Aboveacritical velocity vc [62] vortex dipoles are periodically formedin the wake of the obstacle, injecting linear momentum intothe superfluid. Sufficiently rapid motion (v≫ vc) causesmany vortices to be nucleated behind the obstacles in a chaoticfashion [63], involving clustering of like sign vortices. Ourchoice of obstacle speed puts the system dynamics in the lat-ter category.

We work in units ofµ, ξ, andξ/c for energy, length, andtime, respectively. In these units the specific parameters wechoose (see the previous subsection) areg2 = 0.197µξ2, cor-responding to homogeneous densityn0 = µ/g2 = 5.26ξ−2,andNtot = 5.5 × 106 particles in a homogeneous 2D systemof side lengthL = 1024ξ. The Gaussian potentials each havefixed 1/e2 width of w0 =

√8ξ, and heightV0 = 100µ, and are

initially located atx = −L/2+ 8ξ, y = ±L/8,±3L/8. Numer-ically, we proceed by first finding a ground state of (1), for ahomogeneous system with periodic boundary conditions, sub-ject to the localized obstacle beams. We then transform intoa frame translating at v0 = 0.8c, and maintain the obstaclelocations relative to this frame, creating a dragging grid of

stirring beams. A small amount of initial noise is added to thewave function to break the reflection symmetries of the sys-tem. We thus evolve the system according to (1) for the samepotential, but with the Galilean transformed nonlinear oper-atorL → L + i~v0∂x. During evolution the dimensionlessdamping rate is set toγ = 0.003.

The time evolution of the system is shown in Figure 7 atthree times, at approximately (1/3, 2/3, 1)L/v0, so as to avoidany periodic flow effects in thex-direction. The four obstaclebeams generate many vortices (up toN ∼ 103), and significantclustering (Nc/N & 0.6). The incompressible kinetic energyspectrum shows a wide region that is well described by thek−5/3 form of Equation (95). For later times [Figure 7 (b), (c)]the spectrum shows a significant pile up around the forcingscalekF ∼ ξ−1, suggesting a mismatch between the rates ofinjection and transport of incompressible kinetic energy.

In Figure 8 we compare the functionC∗2D(k) [Eq. (96)], asnumerically computed from our simulation data, with the ana-lytical prediction of the Kolmogorov constantC2D [Eq. (94)].The region ofk−5/3 appears as a broad flat region that is inclose agreement withC2D = 0.212 pertaining to our simula-tion parameters.

VI. CONCLUSIONS

To summarize, we have investigated relationships betweenthe concepts of 2D turbulence in classical fluids and theemerging topic of 2D quantum turbulence of vortices, specif-ically as it relates to Bose-Einstein condensates. We estab-lished a link between the hydrodynamic limit of the dampedGPE and the Navier-Stokes equations, providing an estimateof a quantum Reynolds number for superfluid flows in BECs.We have given a theoretical treatment of the incompressiblekinetic energy spectrum that explicitly incorporates the vor-tex core structure in a compressible superfluid. The incom-pressible kinetic energy spectrum for a compressible super-fluid is deconstructed in terms of single-vortex contributionsdetermining a unique ultraviolet power-law where the energyspectrum scales ask−3, and a contribution that depends onthe configuration of vortices within the fluid that determinesthe infrared region of the spectrum. For the configurationalregime we find:

1. The spectrum only depends on the distribution of vortexseparations and the sign of the circulation of each quan-tum vortex. If the distribution of vortex separations fora system of vortices of thesame signis a power-law∝ s−α with exponentα = 1/3, the kinetic energy spec-trum will take the universal Kolmogorov form∝ k−5/3,as shown for point vortices [53].Localizedclusters ofN vortices of the same circulation with this power lawdistribution can be constructed by sampling using a spe-cific radial exponent ¯α that depends on the number ofvortices and the scale range over which they are dis-tributed.

2. The azimuthal velocity field of a large cluster is deter-mined byα. By inflating the scale range of a cluster we

17

find thatα increases, the velocity field is steepened andthe inertial range expands to larger scales. In a neutralsystem the inertial range can be extended by increasingthe size of clusters while decreasing their number.

3. The universal form of the UV region of the kinetic en-ergy spectrum imposes a strong constraint. If the Kol-mogorov power law occurs in the infrared region, thenthe postulate of continuity between the infrared and ul-traviolet regions completely determines the spectrumwhen the ultraviolet and infrared regions are approxi-mated as power laws. Physically, this corresponds tothe inertial range extending down to the smallest con-figurational scale of the system∼ ξ. We note that thepostulate of continuity may not be relevant for all sys-tems or forcing mechanisms.

4. We infer an analytical value for the Kolmogorov con-stant [Eq. (93)] under the conditions of spectral conti-nuity at the cross-over scale for a system of vortices ofthe same sign. To assess the validity of this inferencefor dynamical situations we compare our analytical re-sults with spectra from a numerical simulation of theforced dGPE for the specific case of a dragging a grid ofobstacles through an otherwise homogeneous BEC. Wefind reasonable agreement provided we introduce theconcept of aclustered fraction Nc/N ≤ 1, which is thefraction of vortices that have same-sign nearest neigh-bors. This measure discounts all vortex dipoles from theconfigurational analysis. We then observe good agree-ment between our Kolmogorov ansatz [Eq. (95)], andthe spectrum calculated from the dGPE data. We alsofind that the predicted value of the Kolmogorov con-stant is in close agreement with the numerical simula-tions [Figure 8].

We note that while our analysis indicates that vortex positionsand circulations are enough to determine an approximate in-compressible kinetic energy spectrum, the reverse is not nec-essarily true: a Kolmogorov spectrum does not carry infor-mation about specific vortex distributions. Nevertheless,ouranalysis does indicate that the number of vortices in a quantumfluid can in principle be directly determined from the ultravi-olet energy spectrum. Moreover, the concept of a cascade inturbulence implies system dynamics and energy transport, yetaside from our numerical simulation example, our analyticalapproach is an instantaneous measure. Importantly, one mustdetermine means of characterizing vortex motion and relate

such dynamics to the cascade concept.The field of 2D quantum vortex turbulence is relatively new,

compared with the much longer histories of 3D superfluid tur-bulence, 2D classical turbulence, and even dilute-gas Bose-Einstein condensation. Point-vortex models have been ex-tensively used in descriptions of superfluid dynamics as wellas in 2D classical turbulence, although point-vortex distribu-tions can only serve as approximate models of real 2D classi-cal flows. Our approach merges concepts from each of theabove subjects in order to develop a new understanding of2D quantum turbulence. By considering the compressibilityof a dilute-gas BEC, we find an analytical expression for theultraviolet incompressible kinetic energy spectrum and anN-vortex equivalent of enstrophy in a quantum fluid, which weterm the onstrophy. For the infrared region, point vortex mod-els are sufficient, and vortex configurations serve to identifyspectra as summarized above. Taken together, the primarynew outcome of our analysis is a link between vortex distri-butions, vortex core structure, and power-law spectra for 2Dcompressible quantum fluids.

Future work on 2D quantum vortex turbulence will involvenumerical simulations and comparisons with our analyticalre-sults, extension of this analysis to confined and inhomoge-neous density distributions, characterization of vortex dynam-ics and the time-dependence of energy spectra particularlyinrelation to clustering [64], inverse-energy cascades, andthenonthermal fixed point [31, 65], and investigation of connec-tions with weak-wave turbulence in BEC [16, 18, 19, 66–69].We also believe that observing vortex distributions such astheα = 1/3 power-law for localized clusters may provide a newmeans of quantitatively characterizing 2D quantum vortex tur-bulence through direct experimental observations of vortex lo-cations in a forced 2D superfluid, and we are working towardsrealizing such experimental observations.

ACKNOWLEDGMENTS

We thank Matt Reeves for assistance with numerical simu-lations, and Aaron Clauset for providing code for samplingpower-law distributions. We also thank Thomas Gasenzer,Gary Williams, Matt Reeves, Sam Rooney, Blair Blakie, Mur-ray Holland, Giorgio Krstulovic, Michikazu Kobayashi, andEwan Wright for useful discussions.

We are supported by the Marsden Fund of New Zealand(contract UOO162) and The Royal Society of New Zealand(contract UOO004) (AB), and the US National Science Foun-dation grant PHY-0855467 (BA).

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