Turbulent flux and the diffusion of passive tracers in electrostatic turbulence

PHYSICS OF PLASMAS VOLUME 10, NUMBER 7 JULY 2003

Turbulent flux and the diffusion of passive tracersin electrostatic turbulence

Ronni Basu, Thomas Jessen, Volker Naulin,a) and Jens Juul RasmussenAssociation EURATOM—Riso” National Laboratory, Optics and Fluid Dynamics Department,OFD-128 Riso”, 4000 Roskilde, Denmark

~Received 26 February 2003; accepted 28 March 2003!

The connection between the diffusion of passive tracer particles and the anomalous turbulent flux inelectrostatic drift-wave turbulence is investigated by direct numerical solutions of the 2DHasegawa–Wakatani equations. The probability density functions for the point-wise and fluxsurface averaged turbulent particle flux are measured and compare well to a folded Gaussian,respectively a log-normal distribution. By following a large number of passive tracer particles weevaluate the diffusion coefficient based on the particle dispersion. It is found that the particlediffusion coefficient is in good agreement with the one derived from the turbulentE3B-flux byusing Fick’s law. Employing the Lagrangian conservation of the ‘‘Potential Vorticity’’ in theHasegawa–Wakatani equations, the analytical support for this result is obtained. The transportestimated by passive tracer dispersion and turbulent plasma flux are found to coincide. ©2003American Institute of Physics.@DOI: 10.1063/1.1578075#

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I. INTRODUCTION

The understanding of particle dispersion and anomaltransport in turbulence is of great importance in plasmaswell as in neutral fluids. The high levels of energy acharged particle transport across magnetic field linesserved in tokamaks and stellarators, known as anomatransport, are generally agreed to be due to low-frequeelectrostatic, micro-turbulence, with theE3B-drift as thedominating velocity, in particular in the edge region~see,e.g., Ref. 1!. While the turbulence observed in a wide ranof different experiments shows universal behavior andfluctuations are found to have a self-similar character ovebroad range of spatial scales~see, for example, Ref. 2!, adetailed understanding of the underlying mechanisms drivthe fluctuations and the associated transport is still nohand. A good candidate to understand and explainanomalous transport from first principles is drift-waturbulence.3–5

Here we use a simple model for plasma turbulenceinvestigate the transport properties of a turbulent plasflow-field. We would like to point out, that there are twways to determine the plasma particle transport. One wato look directly at the convected density, leading to the

miliar expressionG5nvE3B for the local turbulent particle

flux. Density fluctuations are denoted byn andvE3B is thefluctuatingE3B-velocity. The radial component ofG is thecross-field transport as usually measured in experimentssimulations of electrostatic turbulence, see, e.g., Refs. 6Another approach to determine the particle transport isderive a diffusion coefficient by following passive tracer paticles convected by the turbulent flow~see, e.g., Refs. 10–12! and references therein. This method has it roots in

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classical theory of particle dispersion in turbulent flows.13,14

It is not from the outset at all evident that these twapproaches should yield the same result. The flux is deging the plasma confinement while the tracer particle diffusrather reflects the mixing properties of the turbulent flothan giving information of the direct cross-field transport.transport of mass is not connected to the tracer particlefusion, as the center of mass of the considered test partstays fixed for situations with purely fluctuating velocifields, i.e.,^v&50. However, the passive tracer particle dfusion coefficient can—if we are able to use Fick’s law—related to a flux in the presence of a gradient, and therebused to predict the confinement time of the plasma.

In the present work we investigate in detail the relatibetween test particle diffusion and plasma transport, derifrom the turbulent flux across the magnetic field for driwave turbulence. We employ the Hasegawa–Wakatani eqtions ~HWE!,15 that describe the self-consistent developmof drift-wave turbulence driven by unstable resistive driwaves. The HWE are solved numerically for a large paraeter regime and we have obtained the turbulentE3B-fluxfor various parameters. The turbulent density flux is motored directly and analyzed in terms of the probability desity function~PDF!, for the point-wise flux as well as for thepoloidally averaged flux. While the point-wise flux PDFapproximated by the folding of two Gaussian PDFs for desity and velocity fluctuations, the PDF of the averaged fluxwell described by a log-normal distribution. Tracing a larnumber of passive particles which are uniformly distributin the developed turbulence, we determine a diffusion coficient for the particle dispersion. It is clearly demonstratthat the diffusion coefficient derived from the total plasmflux G0 by using Fick’s law is in good agreement with thdiffusion coefficient obtained from the absolute dispersionthe tracer particles. This result is supported analytically

6 © 2003 American Institute of Physics

license or copyright, see http://pop.aip.org/pop/copyright.jsp

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2697Phys. Plasmas, Vol. 10, No. 7, July 2003 Turbulent flux and the diffusion of passive tracers . . .

employing the Lagrangian invariance of the so-called ‘‘Ptential Vorticity.’’ We should stress, however, that this clecorrespondence between the flux and the particle dispermay be restricted to cases with an exponential density grent, whereLn[(“n0 /n0)21 is constant, that is for a locadescription. Note that additionally also radial boundary coditions as well as long range correlations could destroyapplicability of Fick’s law.

The paper is organized as follows: In Sec. II we descrthe model equations and the associated invariants. Sectiocontains the numerical results for the particle dispersionthe turbulent flux including a discussion of the transpPDF. In Sec. IV we describe the relation between the fland the particle dispersion. Finally, in Sec. V we discussresults.

II. MODEL EQUATIONS

The investigations are based on the Hasegawa–Wakaequations~HWE!15 for resistive drift-wave turbulence at thplasma edge:

] tn1]yw1$w,n%52C~n2w!1mn¹2n, ~1!

] t¹2w1$w,¹2w%52C~n2w!1mw¹4w, ~2!

where n and w are the fluctuating part of the density anpotential, respectively. 1/C51/ki

2L i2 is the adiabaticity pa-

rameter and contains via the parallel scale lengthL i5(Ln Te /me cs nei)

1/2 the parallel resistivity.It is assumed that only one mode, parallel to the hom

geneous outer magnetic fieldBW 5B0z, is excited. All differ-ential operators work in the (x,y)-plane, and the Poissobracket is $w,c%[ z3“w•“c5(]w/]x) (]c/]y)2(]c/]x) (]w/]y). We identify thex-coordinate with theradial and they-coordinate with the poloidal direction. Thradial component of the fluctuating velocity is designatedu52]w/]y and the poloidal componentv5]w/]x. Normal-ization scales arers5cs /V i for lengths perpendicular to thmagnetic field andLn /cs for the time, wherecs5ATe /mi isthe sound-speed andLn5„¹n0(x)/n0(x)…21 is the lengthscale of the radial background density gradient. We(Te /e) (rs /Ln) to normalize the potential-fluctuations, ann0rs /Ln for the density fluctuations. The electrontemperature is constant and we assume cold ions.mn is des-ignating a diffusivity andmw is the viscosity, without loss ogenerality we takemn5mw5m.

In the limit of weak nonadiabaticityC@1, we obtain anapproximate linear dispersion relation:

n5ky

11k21 i

1

C

ky2k2

~11k2!32 im

k4

11k2, ~3!

wheren is the frequency andk5kxx1kyy is the wavenum-ber. The deviation from adiabaticity, given by the parame1/C, leads to an instability with a maximum growth rateg'1/8C at k'1.

The HWE ~1!, ~2! contain two limits:~i! C→0, wherethe density and potential decouples, thus the systemscribes standard 2D Navier–Stokes turbulence withn pas-sively advected.~ii ! C→`, which corresponds to the adia


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batic density responsew'n and the HWE reduce to theHasegawa–Mima equation~HME!.16 By varying the cou-pling parameterC in Eqs. ~1!, ~2! we may thus span theregion from pure hydrodynamic 2D turbulence to fully aisotropic drift-wave turbulence.

The energetics of the drift-wave turbulence are governby

dE

dt5E 2n]ywd2r 2CE ~n2w!2d2r

2mE „v21~¹n!2…d2r , ~4!

dW

dt5E 2n]ywd2r 2mE „¹~n2v!…2d2r , ~5!

where

E[ 12 „~“w!21n2

…, W[ 12 ~v2n!2

are the energy and the generalized enstrophy, respectiandv5(“Ãv)• z5¹2f is the vorticity. Only the first termon the right side, the integratedE3B-flux,

G05E Gd2r 5E nu d2r 5E 2n]yw d2r ,

may become positive. TheE3B-flux mediates the instabilitywhen nÞw, i.e., in the nonadiabatic case (1/CÞ0). Theterm 2C*(n2f)2d2r is the parallel current damping anthe two last terms are the viscous damping terms.

In the inviscid limit the HWE has furthermore a Lagrangian conserved quantity,

P5~v2n1x!, ~6!

which we term the Potential Vorticity~PV! in analogy withthe notation employed in geophysical fluid dynamics.17 P isonly convected and is kept constant on any fluid, resplasma element, thusdP/dt50, where the Lagrangian derivative d/dt[]/]t1v•“5]/]t1$w,•%. The conserved PVrestricts the motion of the fluid elements in the radial diretion as we will discuss in Sec. IV.

III. TURBULENCE DYNAMICS AND TRANSPORT

The HWE ~1!, ~2! are solved numerically on a periodidomain by means of a de-aliased pseudo-spectral cod12

Typically we have used at least 2563256 modes on a squarof length L540, but confirmed some of the results by caculations with a higher resolution~up to 102431024modes!. The system is initialized with low amplitude randopotential fluctuations with Gaussian distribution ink-space,centered around zero. After the initial linear growth of tmost unstable modes leading to amplitudes of the orunity, the system ultimately enters a saturated turbulent swhere the global statistical properties show little or no timdependence, nor any dependence on the initializamethod.

We solved the equations for a wide range ofC-valuesspanning from the 2D hydrodynamic limit~C50! to the lim-


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2698 Phys. Plasmas, Vol. 10, No. 7, July 2003 Basu et al.

its of adiabatic electron response (C→`), where the HW-system reduce to the Hasegawa–Mima equation.

The energy spectra are characterized by power [email protected].,E(k);ka] in the inertial range, with an exponenof a;23.5. It is interesting to note that thekx- andky-spectra are almost identical. This might at first sight sesurprising since the linear instability primarily pumps enerinto the kx50 andky;1 modes, and, moreover, an anisoropy is explicitly introduced into the system through thedial density gradient. However, the isotropy in the specmerely demonstrates that in the saturated turbulent statenonlinear convection terms dominate, which are isotropWe are in a regime of strong turbulence. The spectral inonly varies slightly withC, with a tendency of decreasinwith decreasingC. Also, the rms value of the radial velocitfluctuations is roughly independent ofC: It decreases slightlywith C from 1.43 for C50.5 until 1.35 forC55. Visualinspection of the developing turbulent fields reveals the pence of transient vortical structures moving predominantlythe poloidal direction, with a typical lifetime oft510220and a spatial scale ofl;224, much smaller than the domain size. The properties of these structures have beenvestigated in detail by Naulinet al.9,12 for moderate values oC.

A. Dispersion of passive tracer particles

We investigated the dispersion of passive particles infully developed turbulence by tracing a large number of pticles, typically 5000, that are passively convected byflow. The particles are initialized at random positions uformly over the domain, when the turbulence has reachedquasi-steady regime. The trajectories of the particlesfound by integrating along their paths:

x~ t !5x01E0

t

v„x~ t8!,t8…dt8, ~7!

wherex0 is the initial position of the particle andv is theE3B velocity evaluated at the position of the particle frothe electrostatic potentialf,

v5~u,v !5~2]yf,]xf!. ~8!

We have employed a bi-cubic interpolation scheme to pvide the velocity at the exact particle position, which is usally not at the grid nodes. This method uses the values offunction and its derivatives at the grid nodes forming tcorners of the rectangle that encloses the particle. Therivatives of the function at the grid are found using a specscheme based on fast Fourier transform. We have compthe bi-cubic interpolation method with the most accurate oavailable on a periodic grid based on a full spectral interlation of the velocity field,12 and the agreement of the resuwas excellent. The full spectral interpolation is very timconsuming as fast Fourier routines cannot be used andtherefore not practically applicable for tracing a large nuber of particles in a high resolution simulation.

We further tested the accuracy of the particle tracing aif the tracer trajectories cover the whole domain by comping the probability distribution function—PDF—of theEule-


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rian andLagrangianvelocity fluctuations. TheEulerianPDFis obtained from velocities measured at fixed positions ongrid nodes over a given time interval, while theLagrangianPDF is found by tracing a sample of particles and measuthe distribution of their velocities. In the limit of having botan ‘‘infinite’’ amount of grid-points and an ‘‘infinite’’ amountof particles, the Lagrangian and Eulerian PDFs shouldidentical for the case of homogeneous turbulence.13 In Fig. 1the Lagrangian velocity PDFs found by tracing 5000 pticles is compared with the corresponding Eulerian PDwhere the measurements have been performed on all3256 grid points. Here only the radial component of tvelocity is shown. We observe a negligible difference btween the two PDFs~a similar agreement is observed for thpoloidal component of the velocity!. This indicates that theensemble of 5000 particles fully represents the flow, the tring routines are correct and that the turbulence may welconsidered to be homogeneous. Additionally the PDFsvery close to a Gaussian distribution with the varianû2&1/251.35.

We verified that the center of mass of an evenly sprelarge sample of particles stays constant, i.e.,d^x&/dt50 andd^y&/dt50, which indeed is the case for an incompressiflow.

We emphasize that the Lagrangian PDF, as discusabove, should not be confused with the PDF of the Lagraian velocity fluctuations, which is obtained by monitoring thvelocity of individual particles along their trajectory. ThPDF, which will resemble the step size PDF of the dispersparticles, may deviate from the Eulerian PDF, and it wonly approach a Gaussian distribution in the diffusive lime.g., Ref. 18.

To describe the radial diffusion of a large ensembleparticles we introduce the running diffusion coefficient dfined as

Dx~ t !5X2~ t !

2t5

^@x~ t !2x~ t50!#2&2t

. ~9!

FIG. 1. The Eulerian PDF of the radial velocity compared to the PDF ofLagrangian velocity sampled over all particles uniformly distributed in tflow for C55. The distributions are very close to Gaussians withsv51.35.


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The brackets denote an average over all particles andX2(t)is the mean square displacement of particles. For a diffuprocess the running diffusion coefficient,D(t) reaches as-ymptotically a time independent valueD, sinceX2(t);t fora diffusion process. The classical Taylor result for partidispersion in turbulent flows predicts that the absolute difsion coefficient is given by~see, e.g., Ref. 14!

Dx5û2&tL , ~10!

where

tL5E0

`

CL~t!dt ~11!

is the Lagrangian integral time scale andCL(t) is the La-grangian autocorrelation function, andu2& is the meansquare radial velocity fluctuations averaged over time. Nthat we have already seen that the turbulence is homneous, and, thus, we are able to replace time averagesspatial averages. Generalizations of the Taylor descrip@Eq. ~10!# of particle dispersion were discussed by, e.g., Vet al.,11 who used of a stochastic model for electrostatic tbulence of a plasma. They considered the case characteby a high value of the so-called Kubo numberK.1, whereK5^v2&1/2Tc /l. The Kubo number measures the averagdistance covered by a particle during the correlation timTc , relative to the typical spatial scale of the fluctuations,l.Their results are summarized by the dependence of thefusion coefficient on the Kubo number:D}Ka, with a52for K!1 anda;0.7 for K@1. At high Kubo numbers particle trapping effects partly hinder the diffusion.

It is obvious that the expression@Eq. ~10!# requires afinite integral time scaletL , which implies a sufficientlyrapid decrease of the correlation functionCL(t). For longrange correlations one could expect ‘‘strange’’ dynamics lsuper-diffusive behavior, see, e.g., Ref. 19. Actually, intermdiate super-diffusive behavior is often observed for partidispersion in 2D fluid turbulence.18,20

In Fig. 2 we show the evolution of the running diffusiocoefficient defined in~9! for different values of the coupling

FIG. 2. Running diffusion coefficient in the radial directionDx versus timefor various values ofC.


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coefficient,C. After an initial ballistic phase,Dx}t, the co-efficient decreases and finally tends to flatten out at a lethat decreases withC. Thus, it appears that the evolutioafter a period of sub-diffusive behavior, characterizedX2(t);tb with b,1, approaches asymptotically a diffusivregime, with a diffusion coefficient that is strongly decreaing with C, although it was found thatu2& only weaklydecreased withC. Additionally, the value of the asymptotidiffusion coefficient is much lower than the ‘‘classical’’ valugiven by Eq.~10!, which in the present case amounts toDx

;425, since the Lagrangian integral time scale (tL;223) was also found to be only weakly depending onC.Using tL as a typical correlation time and the characterissize of the structures (l;224) as the spatial scale, we obtain a rough approximation for the Kubo number:K;223. Thus, the observed strong variation ofDx is not reflect-ing a strong variation in the Kubo number, and the Kunumber scaling discussed above cannot be conclusitested.

The sub-diffusive regime was found to be related to traping of particles in vortical structures.12 Note that the poloi-dal diffusion increases withC with the tendency for superdiffusive behavior for largeC values. This behavior and, inparticular, the strong anisotropic effect for largeC is similarto the dispersion of passive particles in the framework ofHME ~see, e.g., Refs. 21, 22!. In that situation the formationof strong poloidal flows due to finite boundary conditioplays an important role as well. In our case these flows wlimited due to the periodic boundary conditions; indeedobserved that the rms value of the flow component (ky50)was only a fraction of the total velocity fluctuation rms.detailed study of the anisotropic dispersion for varying prameters in our model will be presented elsewhere; hereconcentrate on the radial diffusion and the relation todensity flux.

Before discussing the density flux we examine the intmediate sub-diffusive behavior in more detail and verify tha diffusive regime is reached asymptotically even for tcase of largeC values. We have plottedX2(t) in Fig. 3 forthe caseC55. It is observed that after the initial ballisti

FIG. 3. The mean square radial particle displacementX2(t) versus time forC55. The full curve is a least mean square fit of the formAtb1B to thedispersion fort.400, whereb51, A50.184,B568.75. The dashed linehas the slope 1 and the dot–dashed line has the slope 0.73.


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behavior a clear sub-diffusive regime@X2(t);tb with b;0.5] is entered for 10,t,500. This regime appears asslow transition to the asymptotic diffusive behavior. The fuline curve is the best fit of a function of the formAtb1B fort.500, which resulted in a fitting valueb51. We note thata ‘‘naive’’ fit in the regime 200,t,2000, the dot–dasheline, would suggest a sub-diffusive behavior:X2(t);tb withb.0.73 and clearly belowb.1, the dashed line.

B. Density flux

The point-wise radial densityE3B-flux, G5nu, ismonitored in the simulations at the grid positions. It is geerally found that this flux is strongly intermittent, showinlarge bursts in agreement with standard experimeobservations.6–8 The averaged total flux,G05^nu&, is foundto decrease with increasingC; see also Ref. 9 and the folowing section. In Fig. 4 we have depicted the PDF of tflux for an intermediate value ofC52. This PDF corre-sponds to the single point transport PDF measured by ua probe array in plasma experiments. It is worthwhile to nthat the long tail in the PDF for positive values ofG indicatesthe bursty nature of the flux. A significant part of the poinwise flux is actually carried by flux events that are larcompared to the mean flux. The point-wise flux PDF cancompared with the PDF estimated under the assumptionthe fluctuations inu (w) and n both are Gaussian, but arcorrelated to each other with an averaged phase shift relto the coupling coefficientC. Note that a finite correlation isnecessary to make the average of the flux nonzero. The fing of the correlated Gaussian PDFs foru andn results in8

pG51

p

A12g2

snsuK0S uGu

snsuDexpS 2g

G

snsuD , ~12!

whereg is the correlation:

g52ûn&

û2&1/2^n2&1/25cosanu . ~13!

anu is the relative phase betweenn and u, su[û2&1/2(12g2)1/2 and sn[^n2&1/2(12g2)1/2. The averaged flux isthen obtained as

FIG. 4. PDF of theE3B flux for C52.0 and a mean fluxG&50.38~dashed curve!. For comparison the result due to Eq.~12! using the corre-lation g520.28 is shown~full curve!.


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G052gû2&1/2^n2&1/2. ~14!

Indeed, we verified that the fluctuations inu andn are veryclose to Gaussians and we have used the measured rmsues and correlation when obtainingpG , which is plotted inFig. 4 as the full curve. The agreement with the measuPDF is very good for the bulk of the PDF, while we obsersmall deviations in the tails. The presence of large flevents is at least partly just due to the finite correlationtween density and potential fluctuations, and in the prescase not necessarily a sign of ‘‘strange’’ or anomalous kinics. However, the contribution of coherent structures totails of the transport PDF has been show to lead to deviatfrom PG .12

As we noted above the PDF in Fig. 4 is based on a sinpoint measurement. Long range correlations, which wolead to a deviation of the flux PDF from a Gaussian disbution and imply strange kinetics should, however, be mumore visible in their contribution to transport PDF, when wtake theaveraged fluxas the statistical quantity. Thus, walso consider the PDF of themagnetic flux surface averageflux: GFS5^G&y5*Gdy, which in the present case corresponds to the point-wise flux averaged along the poloidaly)direction. The PDF ofGFS is shown in Fig. 5. The distribu-tion is shifted along theGFS-axis with the center close toG0 ,which of course is the mean value ofGFS; it is skewed withan exponential tail extending to largeGFS-values. A deviationof the transport PDF from a Gaussian is sometimes inpreted as an indication of long range correlations and aciated non-Gaussian or anomalous behavior. As the fluxface averaged particle flux appears to take only non-negavalues, we expect that the normal distributed fluctuatquantity is the logarithm of the flux. The fluxGFS shouldtherefore have a log-normal distribution of the form

Pln~G!51

A2psexpS 2

1

2 S ln G2 ln GM

s D 2D . ~15!

GM is the value ofG wherePln is maximum. This functionfits the observed PDF extremely well, as depicted in Fig

FIG. 5. PDF of the flux surface averagedE3B flux ^G&y for C52.0. Thedashed curve is a Gaussian with mean 0.21 and widths50.08 and thedashed–dotted curve overlying the data is a fitted log-normal distributio


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One should note that if the average flux is large comparethe level of the fluctuations the log-normal distribution bcomes very close to a Gaussian. Normal~Gaussian! and log-normal distributions are direct consequences of the cenlimit theorem, for PDFs with finite variance. Otherwise,infinitely large transport events were possible and the vance of the distribution function for the point-wise transpwould not exist, we would expect a Le´vy-Pareto type ofPDF19 for the flux surface averaged transport. Only in thcase, we expect to observe nondiffusive behavior in a podic domain. For the considered situation, this is obviounot the case.

IV. DIFFUSION COEFFICIENT AND DENSITY FLUX

Here we discuss the relation between the diffusion coficient obtained from the dispersion of passive particlesfrom the turbulent flux. We may define an effective diffusiocoefficient related to the flux by using Fick’s law:DG

5G0 /“n0 , which with our normalizations reduce toDG

5G0 . First, we note that bothDx and the normalized fluxwas found to be independent of the box size at least foL.10, which corresponds to the normalized flux being indpendent of the magnetic field.9 It is seen that the unnormaized diffusion coefficient will read as

Dun5Dxcsrs2/Ln .

Here Dx will depend onC, but apparently not on box sizei.e., the magnetic field as argued above. Thus,Dun will fol-low the gyro-Bohm scaling.

We derive an expression for the flux in relation to tdispersion of fluid elements~passive tracer particles! by us-ing the inviscid Lagrangian conserved ‘‘Potential VorticityP5(v2n1x), Eq. ~6!. The conserved PV restricts the mtion of the fluid elements in the radialx-direction and sofrom Eq. ~6! we have

x2x052~z2z0!,

wherez[v2n is the fluid part of the PV. The index ‘‘0’’designates the initial position/z-value of a fluid element, thusthe motion of the fluid element is restricted by the ‘‘allowedchange ofz. From the relation above we obtain

^~x2x0!2&p5^z2&p1^z02&p22^z0z&p , ~16!

where the average•&p is taken over all fluid elementsparticles. If we assume that the fluid elements are uniformdistributed over the domain, and that the turbulence is hogeneous, we may replace the averaging over the fluidments with averaging over the fields13 (^•&p5^•& f). In thetime asymptotic limit, we assume that the last term on thevanishes~this designates the correlation between the fluidand the initial fluid PV!. Taking the time derivative and usinEq. ~5! we obtain


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d^~x2x0!2&p

dt5

d^z2& f

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dW

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It is readily observed that the radial component of the difsion coefficient can, using Eq.~5! once again, be evaluateas

Dx[1

2

d^~x2x0!2&p

dt5G05DG . ~18!

Thus, we observe for the presented case that theE3B fluxand the radial particle diffusion are strongly related; thprovide the same diffusion coefficient. In the Appendix, wprovide a further discussion of the relation between thefusivity, the flux of particles and vorticity. The argumenabove are based on the inviscid conservation of theHowever, viscosity plays a major role in the simulationssaturated turbulence. In the saturated state we have forensemble or time average of energy and vorticity t^dE/dt&50 and^dW/dt&50 in Eqs.~4!–~5!. The flux as thedriving term will then be balanced by the dissipative termleading one to the assumption that any measured transpofinally a consequence of viscosity. However, the viscosmainly acts on the small scales and the contribution tointegral in the last term in Eq.~5! mainly arises from thesesmall scales. The flux integral on the other handmainly hascontributions originating from the larger or intermediascales. Therefore, we are confident that we may applyinviscid arguments leading to Eq.~18! on these large scaleswhich are also the scales that mediate the diffusivity. Indethe relation between particle diffusion and flux (Dx5DG) asgiven by Eq.~18! is confirmed by the direct numerical simulations.

In Fig. 6, we compare the normalized fluxG0 or DG withthe measured particle diffusion coefficientDx . We observethat bothDG andDx decrease with increasingC, additionallythey depend only weakly on the viscosity, as well the vaof the viscosity as the type of viscosity. The results in F6~b! are obtained by using instead of the normal viscosithyper-viscosity of the type¹6. Note that the agreement betweenDG andDx is particularly good for low values of theviscosity, as to be expected. Note also that for extremely hvalues of the viscosities, the Lagrangian invariant PV wobe destroyed, but that would happen outside the turburegime.

The relation, Eq.~18!, is far from being trivial as wemay easily imagine a system having a finite diffusion coficient, but zero related flux, even in the presence of a gdient. The diffusion is then merely a measure of the mixicapability of the turbulent flow. One example is providedthe driven Hasegawa–Mima system~the C→` limit of theHWE!. In the driven case we have a finite diffusion coefcient, while the flux isidentically zero. Using argumentssimilar to those leading to the relation, Eq.~18!, it is easilyseen that the radial diffusion vanishes for thenondrivenHME case, and the spread of the particles in the radial dirtion is controlled by the variance of the fluid potential voticity @Eq. ~16!, herez[v2w]. This is important for con-siderations involving the HME for discussing drift-wavtransport.21 When the turbulence is sustained by an exter


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drive, a finite diffusion coefficient will arise, however, it wibe directly related to the driving forces and the obtaindiffusivities may indeed be of little relevance for the undestanding and scaling properties of the anomalous diffusioself-sustained drift-wave turbulence.

FIG. 6. A comparison of radial diffusion coefficientDx denoted by3 andthe diffusion coefficient obtained from the particle fluxDG (s), for variousvalues ofC. ~a! Newtonian viscositym50.001.~b! Hyper-viscosity with theoperator¹6. ~c! Different values of the Newtonian viscosity andC52.


d-in

V. DISCUSSION

We investigated the transport properties of resistive drwave turbulence. We considered two approaches; one wwe measured the dispersion of passive tracer particlesanother, where we measured the direct density fluxG5nu.

For the particle dispersion we found sub-diffusive bhavior for intermediate time intervals. However, asymptocally the dispersion approaches a normal diffusive behawith the variance increasing linearly with time, in accodance with the classical Taylor expression for correlatfunctions having finite integrals, viz., a finite integral timscale@Eq. ~11!#. This seems to be a standard observationdeveloping turbulent flows on a periodic domain~see, e.g.,Ref. 20!. Indeed, it should be expected for systems on podic domains, where a finite correlation scale of the flucttions is naturally connected to the system size. An excepwould be for cases with dominating zonal flows~character-ized byky50), where this argument is not relevant, and tdispersion may be ballistic, as observed for the HME instrongly anisotropic limit.22 The actual value of theasymptotic diffusion coefficient was significantly lower thapredicted by the Taylor expression, Eq.~10!, and the cleardecrease with increasing adiabaticity parameterC could notbe conclusively related to a scaling with an estimated Kunumber, which appeared to be roughly independent ofC.

For the density flux we observed that point-wise mesurements gave a strongly intermittent flux with a PDF wa long tail towards positive flux events. This PDF, howevwere found to agree well with the PDF obtained from foldiGaussian PDFs for correlated velocity and density flucttions. We also examined the poloidally~magnetic flux sur-face! averaged flux,GFS, which was found to have a PDF iclose agreement with a log-normal distribution, which chacterizes random variables that can only take non-negavalues. Thus, no signatures of long range correlations wfound in the present model. A more detailed study of tissue of transport PDFs and correlations is beyond the scof the present paper and will be subject to further investitions.

Finally, we found that the diffusion coefficient,DG , de-rived from the density flux, was in close agreement with tdiffusion coefficient obtained for passive particles,Dx . Thisstresses the fact that the transport in this model systemdiffusive like properties. It should be emphasized, howevthat our result was obtained for the particular case wheredensity gradient scale length,Ln[(“n0 /n0)21 is constant.Clearly, only in this case will it be possible to directly relathe particle diffusion to the density flux, as described in SIV. For more complicated density profiles a constant flthrough the system will inevitably lead to a profile in theffective diffusion coefficient if we apply Fick’s law locallyStill, it will be possible in a system with periodic boundarieto obtain one diffusion coefficient for the particle dispersiobut it will not be directly related to the flux. In this contexwe note that Linet al.23 recently provided evidence for thcomparison between the heat flux and tracer particle dission in a gyro-kinetic simulation. In the local regime thefound that the PDF of the particle displacement is Gauss


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which is closely related to the Eulerian velocity PDF,discussed in Sec. III A. Their results confirm that the hflux is ‘‘carried’’ by the radial diffusion of particles and thaintermittent, bursty transport events appear to be absent

Additionally, we remark that for the case of driven sytems, i.e., systems driven at one boundary, the concepparticle dispersion should be reformulated. It does not remake sense to talk about a diffusion coefficient in the clsical sense fort→`, rather one may invoke the conceptexit time statistics as have been outlined by Carreraset al.24

In that respect one could also speculate that the anomadispersion effects would be much more dominant, say ifcharacteristic domain size corresponds to a time scale wthe intermediate regime of anomalous dispersion. Investtions of particle dispersion in drift-wave turbulence onbounded domain is in progress.

ACKNOWLEDGMENTS

Discussions with Anders H. Nielsen and Poul K. Micelsen on various aspects on particle tracing and drift-wturbulence are gratefully acknowledged.

This work was partly supported by the Danish NatuScience Foundation~SNF Grant No. 9903273!.

APPENDIX: RELATION BETWEEN THE PARTICLEDISPERSION AND FLUXES

We detail the discussion of the implications of the coserved PV@Eq. ~6!# for the radial movement of fluid elements and thereby the transport of both density and vorticBy following the ideas used by Rhines17 in the context ofgeophysical fluid transport, we write the conservation of PdP/dt50, in the form of the vorticity equation:

d

dt~v2n!52u. ~A1!

This may formally be solved by following a plasma eleme

v2n52E0

t

u~ t8!dt81v02n0 , ~A2!

where u takes the role as the radial (x) component of theLagrangian velocity, and the quantities labeled by 0 areinitial values att50. We defineh5*0

t u(t8)dt8, which is theradial displacement of the plasma element„h5(x2x0)….Multiplying Eq. ~A2! with u, averaging over the poloidal (y)direction ( •&y[*•dy) we obtain for the local radial fluxG(x)5ûn&y in the positionx

G~x!51

2

d^h2&y

dt1]xûv&y2û~v02n0!&y , ~A3!

where the first term on the right hand side is the local radturbulent diffusivity—for homogeneous turbulence and pticles uniformly distributed over the whole domain the avaging over the fields may be replaced by averaging over fl


t

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useina-

e

l

-

y.

,

:

e

l--id

elements; see, Sec. IV. The second term is the derivativthe Reynolds stress, which is the local radial flux of vortic(]xûv&y5ûv&y), while the last term designates the corrlation between the velocity at the positionx and the initialfluid PV ~at x0), which we may assume vanishes at lartimes. Thus, Eq.~A3! demonstrates that the local radial difusivity is mediating both the flux of particles, resultingthe evolution of the density profile and the flux of vorticitresulting in setting up a radial potential profile, which wdrive theE3B zonal flows. Note, however, that the correltion û(v02n0)&y does not vanish, but diverges in the caof long range correlations in the flow or in the presencefinite boundary conditions.25

If we average Eq.~A3! over the radial (x) direction, weobserve that the total flux,G0 , is given by the radial component of the turbulent diffusivity as in Sec. IV@Eq. ~18!#, i.e.,the diffusion coefficients of plasma elements, which will albe the diffusion coefficient experienced by passive partic

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Plasmas4, 628 ~1997!.22S. V. Annibaldi, G. Manfredi, R. O. Dendy, and L. O’C. Drury, Plasm

Phys. Controlled Fusion42, L13 ~2000!; S. V. Annibaldi, G. Manfredi, andR. O. Dendy, Phys. Plasmas9, 791 ~2002!.

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Turbulent flux and the diffusion of passive tracers in electrostatic turbulence