Confining turbulence in plasmasVincenzo Carbone, Fabio Lepreti, and Pierluigi Veltri Citation: Physics of Plasmas (1994-present) 11, 103 (2004); doi: 10.1063/1.1632905 View online: http://dx.doi.org/10.1063/1.1632905 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/11/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Turbulent excitation of plasma oscillations in the acoustic frequency range Phys. Plasmas 14, 082304 (2007); 10.1063/1.2755944 Magnetic presheath in a weakly turbulent multicomponent plasma Phys. Plasmas 14, 013504 (2007); 10.1063/1.2428278 Turbulence and intermittent transport at the boundary of magnetized plasmas Phys. Plasmas 12, 062309 (2005); 10.1063/1.1925617 Transition in multiple-scale-lengths turbulence in plasmas Phys. Plasmas 9, 1947 (2002); 10.1063/1.1455005 Comparing simulation of plasma turbulence with experiment Phys. Plasmas 9, 177 (2002); 10.1063/1.1424925

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Confining turbulence in plasmasVincenzo CarboneDipartimento di Fisica and Istituto Nazionale per la Fisica della Materia, Universita` della Calabria,Ponte P. Bucci Cubo 31C, 87036 Rende (CS), Italy

Fabio LepretiDepartment of Physics, University of Thessaloniki, 54124 Thessaloniki, Greece

Pierluigi VeltriDipartimento di Fisica and Istituto Nazionale per la Fisica della Materia, Universita` della Calabria,Ponte P. Bucci Cubo 31C, 87036 Rende (CS), Italy

~Received 28 May 2003; accepted 9 October 2003!

The transport properties of electrostatic turbulence in plasmas are investigated by using test-particlesimulations. In particular, the possibility of control of the transport in a given synthetic turbulentfield, which evolves both in space and time, is explored. The fluctuations are built up taking intoaccount observations of real turbulence in laboratory plasmas, that is, by allowing the field tocontain structures lying on all dynamically interesting scales. It is shown that, inside a given regionof space, the transport can be reduced when phases of the field are randomized, that is, whencorrelations of the field, which are responsible for the generation of structures, are annihilated. Thismeans that a barrier for the transport can be achieved in a plasma even without actually suppressingturbulence. When the barrier is active, a flux of particles toward the center of the simulation box ispresent inside the region where the barrier has been located. ©2004 American Institute of Physics.@DOI: 10.1063/1.1632905#

I. INTRODUCTION

High-amplitude fluctuations in plasmas represent themain cause for the enhancement of transport.1,2 This kind oftransport is often called anomalous, with respect to the clas-sical transport, and in general its effects are catastrophic. Themain example of this is the disruption of magnetic confine-ment in fusion plasmas~see, e.g., Ref. 3!. Also in nonfusionplasmas,4 the presence of fluctuations leads to an enhance-ment of transport towards the edge of the device. In otherdevices, for example, magneto-plasma-dynamic engines de-veloped for space flights, turbulent induced transport causesa limitation of the thrust.5 Even if we cannot completelyeliminate transport, the mitigation of its effect remains by farthe most crucial challenge to improving performances in dif-ferent plasma devices. Roughly speaking, since the enhance-ment of transport is due to the presence of turbulent fluctua-tions in a plasma, the effects of the enhanced transport can bereduced when turbulence disappears. This point of view,which invokes the disappearance of turbulence to achieve awell confined state for plasma, is quite in contrast with as-trophysical plasmas where, for example, the presence of tur-bulence is strongly invoked to confine cosmic rays withingalaxies for a long time.6

A barrier for transport is defined as a region where trans-port is effectively reduced, and its generation can beachieved in different ways. In laboratory plasma a barrier hasbeen generated mainly through an externally imposed strongshear flow,2 or through the injection of pellets in the plasma.7

These operations generate a transition between a low con-finement mode to a high confinement mode~say the L→Htransition!, that is, a kind of control for transport can be

achieved. In some cases, zonal flows spontaneously generatethe L→H transition.8–10What happens in the plasma when ashear flow is applied has been interpreted as follows.11,12

When the mean streaming rate of the shear flow is greaterthan the decorrelation rate of turbulence, the eddies’ lifetimeis shortened and the decrease of correlation time should im-ply a decrease in turbulent intensity. The effect of the appli-cation of a shear flow seems to be a transient decay of theturbulent energy.2

However, it has been observed that transport can be sen-sitively reduced also without an effective reduction of turbu-lent energy.2 For example, the transport due to theEÃB drift~E and B are the electric and magnetic field! depends notonly on the intensity of fluctuations, but also on the cross-phase of quantities involved in the average flux.13 Whetherthe reduction of transport is due to cross-phase decorrelationsor the reduction of turbulent amplitudes is a matter of recentinvestigations.1,2,13

A different way to reduce and to control transport shouldbe due to spontaneous self-organized properties of turbu-lence in plasmas. In fact, it has been observed that in somecases a turbulent state spontaneously bifurcates to a newstate where a magnetic mode at large scale is present,14 andthis mode dominates the spectrum for some time. This iscalled a quasi-single helicity~QSH! state. After some time anew turbulent state is spontaneously re-established into theplasma. This effect has been investigated in detail in reversedfield pinch configurations, but a detailed explanation of whatphysical effect generates the QSH states is far from beingfully understood.15 Of course, the role played by the QSHstate of plasma in connection to transport properties is not

PHYSICS OF PLASMAS VOLUME 11, NUMBER 1 JANUARY 2004

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yet clear and some investigations are required. The underly-ing idea might be trivially simple: since the QSH state ismade by few interacting modes, it is quasi-laminar. Then,when this state is established in the plasma a barrier for thetransport could set up automatically.

It is now well established that statistical properties ofturbulence in laboratory plasmas are not really different fromstatistics of fluctuations measured in the solar wind plasmaor in generic fluid flows.16–18As an example, the same kindof intermittency19 observed in fluid flows is a phenomenonwhich characterizes both magnetic and electrostatic fluctua-tions in laboratory plasmas.20,21The probability density func-tion of fluctuations at small scales is non-Gaussian and dis-plays algebraic tails, which means that high-amplitudefluctuations have a high probability of occurrence. From aphenomenological point of view, this means that coherentstructures within turbulence, that is, regions of fluid wherephase correlations exist, accumulate at small scales. Struc-tures continuously appear and disappear, at some randomlocations in the fluid, and they carry a large quantity of en-ergy of the flow.22As shown, for example, in some numericalsimulations,23,24 the presence of structures on all scalesseems to play a relevant role in the properties of transport.

In the present paper we try to investigate, through asimple model describing both space and time evolution of aturbulent field, a mechanism which could lead to the genera-tion of a barrier in order to reduce transport, without actuallyreducing the amplitudes of turbulent fluctuations. Throughtest-particle simulations, we will show how fully developedturbulencecan actuallyact as a confining medium. The un-derlying idea is relatively simple, being based on the obser-vation that correlations of the velocity fields are responsiblefor the enhancement of transport. After a brief presentationof test-particle simulations in electrostatic fields~Sec. II!, weintroduce a simple model describing realistic turbulent fluc-tuations with correlations on all dynamically interestingscales~Sec. III! and we present results of simulations. In Sec.IV we analyze the motion of test-particles in the turbulentfield when the transport barrier is active. In Sec. V we brieflyaddress the problem of anomalous transport due to Lagrang-ian chaos in nonturbulent fields, and finally in Sec. VI wesummarize and discuss the results of the paper.

II. TEST-PARTICLE SIMULATIONS IN ELECTROSTATICFIELDS

The passive evolution of test particles in assigned turbu-lent fields is the subject of a huge amount of numerical simu-lations ~see, e.g., Ref. 25 and references therein!. Here weinvestigate the motion of test particles in a given electrostaticturbulent field which evolves both in space and time. In thisframework the particle is allowed to move with the driftvelocity udr5EÃB/B2. Since we want to investigate thegross features of test particles, we neglect the polarizationdrift, this latter being a secondary effect. The dynamics ofparticles in electrostatic fields has been studied by some au-thors in different situations~among others cf., e.g., Refs. 26–28!.

We consider a slab geometry with a strong confining

magnetic field directed along thez-axis of a reference systemB5B0ez . The dynamics then lies on the (x,y) plane, a situ-ation which is usual in laboratory plasmas.1 By introducingthe electrostatic potential2“c(r ,t)5E(r ,t) the Lagrangianevolution of the particle is given by a Langevin equation ofthe charge in the guiding-center approximation,

dr

dt52

“c~r ,t !ÃB0

B02 . ~1!

We assume thatc(r ,t) evolves according to a turbulent pat-tern both in space and time, whose statistical properties aresimilar to that observed in plasmas. This pattern is obtainedfrom a model for synthetic turbulence. Previous investiga-tions used both assigned spatially turbulent fields~withouttime evolution!, or independent numerical simulations~withan expensive computational time!.

The system~1! in general is nonintegrable, so that cha-otic particle trajectories may exist.29,30A test particle is thensubject to Lagrangian chaos, and diffusive motion, due to aturbulent eddy-diffusivity, can take place without invokingstochastic collisions due to molecular diffusion~cf., e.g., Ref.31 for a discussion on anomalous diffusion in that limit!. Thebasic properties of transport can be investigated when thescaling properties of the time evolution of the mean squaredisplacement,25 say ^@r (t)2r (0)#2& ~brackets being en-semble averages!, are determined. From~1! we have

r i~ t !5r i~0!1e i jm~B0 j/B0

2!E0

t ]c~r ,t8!

]xmdt8, ~2!

and this, assuming statistical homogeneity, allows us to de-fine the eddy-diffusivityDi j

E25 through the correlation tensorfor the electrostatic field,

Ci j ~t!5 K F]c„r ~t!…

]xiGF]c„r ~0!…

]xjG L . ~3!

In practice we get

^@r ~ t !2r ~0!#2&.2DiiEt, ~4!

which corresponds to standard diffusion, i.e., particles makea Brownian-like motion. Nevertheless there exist caseswhere diffusion is non-Gaussian,31,32 in the sense that themean square displacement has a long-time dependence assome power of time. In these cases we can define both adiffusion coefficientDe and a scaling exponentnÞ1/2, bythe asymptotic slope of the curve given by the relation

^@r ~ t !2r ~0!#2&.2Det2n. ~5!

Superdiffusion is obtained whenn.1/2, while n,1/2 cor-responds to subdiffusion.32 Very interesting and general re-sults on anomalous transport of particles by velocity fields,where stochastic molecular diffusion is absent, have beenobtained in the past.33,34

III. A MODEL FOR ELECTROSTATIC TURBULENCE

Measurements in laboratory plasmas20,21showed that in-termittency characterizes the statistics of turbulent fluctua-tions. This means that turbulent structures~‘‘coherent’’ struc-tures in some sense! are present on all dynamically

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interesting scales. Then, in order to build up a syntheticmodel for turbulent fluctuations which mimics real observa-tions, we must take into account the presence of these struc-tures. A field with this feature can be built up in differentways, for example, from the celebratedp-model,35 or usingdifferent approaches.36,37 Here we use a model which can beeasily integrated, recently proposed in Ref. 38 to reproduceturbulent pair diffusion in fluid flows.

As a first step, we consider a coherent mode to repro-duce structures both in space and time, for example,the function c(r ,t)5c0@a sin(k"r2vt)2b cos(k"r2vt)#.Then, in order to reproduce scaling properties of turbulence,we introduce the electrostatic field as the superposition ofdifferent wave vectors, frequencies and coupling coefficients,

“c~r ,t !5~c0 /L !(n

ane'

(an) cos~kn(a)•r2vnt !

1bne'

(bn) sin~kn(b)•r2vnt !, ~6!

whereL is a characteristic length. Both wave vectorskn(a,b)

5kne'

(an ,bn) lie in the (x,y) plane (e'

(an ,bn) represents unitvectors in the direction perpendicular to thez-axis!. In orderto reproduce the stochastic properties of real turbulent flows,both directions of wave vectorse

'

(an ,bn) are chosen randomlyfor each moden, with the same probability.

One of the main problems encountered in the investiga-tion of the statistics of test-particles in turbulent fields is theextension of the scaling range of turbulence. Ranges that aretoo small imply that small scales of turbulence are not welldescribed, while by introducing a large number of modes thenumber of particles which can be used in the simulations issmall. In order to reproduce small scale turbulence with asufficient approximation, but at the same time to be able torun the equation of motion for a large number of test-particles, we divided the wave vector space in logarithmi-cally spaced shells, each characterized by a discrete wavevectorkn52nk0 , wherek052p/L andn50,1,. . . ,N. Shellmodels, which mimic the gross features of turbulent flows,are usually built up using the same approach.39 Turbulentproperties of the field are specified by making the amplitudesrelated to the energy spectrum, that isan

25bn25(kn11

2kn21)E(kn), and using a given spectrumE(kn);kn2a .

The frequency of each mode as a function of the wave vec-tors is not given by a dispersion relation, because here weconsider strong turbulence. Rather, in a turbulent environ-

ment, we assume that the frequency at a given scale is relatedto the eddy-turnover timetn

21;vn5Akn3E(kn). After these

steps we obtain a two-dimensional turbulent field with statis-tical features similar to those observed in turbulent plasmas,where coherent structures are present at all scalesl n;kn

21 .We investigate the motion of test particles in the above

field by using a second order Runge–Kutta scheme for thetime integration of Eq.~1!, with a time stepDt5531023.Here we report results obtained forN510 shells andk051that is the integration region is the square2p<(x,y)<p.Equation~1! can be rewritten in dimensionless form by in-troducing the characteristic drift speedvdr5c0 /LB0 , and acharacteristic timeL/vdr . In Fig. 1 we report the meansquare displacement^@r (t)2r (0)#2& vs time~averages havebeen made on 104 different initial realizations of the turbu-lent field! for different values of the spectral index in therange 1<a<3. For long times, we found a linear range anda fit gives the value of the diffusion coefficientDe ~in unitsof L2/t) and the scaling exponentn. In Fig. 2, we report thevalues of bothDe andn versusa. The value ofn is alwaysslightly greater than 0.5, meaning that a weak super-diffusion, due to the presence of small scale correlations inthe field, is present. The diffusion coefficient significantly

FIG. 1. Test-particle mean square displacements as a function of time, fordifferent values of the spectral index, namelya51,1.5,2,2.5,3. The curvesare ordered from the bottom to the top asa increases and offsets have beenintroduced for clarity. As a reference, the dashed line corresponds to thelinear function^@r (t)2r (0)#2&;t.

FIG. 2. Values of both the diffusion coefficientDe andthe diffusion exponentn as functions of the spectralindex a.

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increases witha, because the large scale structures becomesmore and more efficient in the diffusion process.

Let us consider now what happens when the phases ofthe fieldc(r ,t) are shuffled, in such a way to destroy struc-tures on all scales. We add random numbers in the argumentsof trigonometric functions of~6!, namely (kn

(a)•r2vnt

1fn), uniformly distributed in the intervalfn5@0,2p#. InFig. 3 we report the mean square displacements as a functionof time both when phases are not randomized and for ran-dom phases, fora51.5. Since correlations of velocity fieldsintroduced through trigonometric functions are destroyed,diffusion becomes normal, so that we foundn50.50. Whatis, however, interesting is the fact that the diffusion coeffi-cient in that case is lowered fromDe.0.23 to the valueDe.2.531023. This means that diffusive properties are en-hanced in presence of coherent structures. As a consequencethe randomization of phases should be able to generate abarrier for the transport.

IV. A BARRIER FOR TRANSPORT

In this section, we investigate the possibility of generat-ing a barrier for transport in a typical plasma device. To thisaim, we consider an experiment in which the electrostaticfield is continuously perturbed within a given region at theedge of the physical domain. In real experiments, this situa-tion could be easily obtained in different ways, for example,by using a perturbing electric field at the edge of the plasmacolumn. Perhaps the perturbation could be also small in am-plitude, since, in order to just randomize the phases, there isno need to change the amplitudes of the fluctuations insidethe plasma.

To simulate the generation of a barrier for transport inthis situation, we introduce in our model a randomization ofthe phases of the electrostatic fieldonly within a region at theedge of the integration domain. This region is a strip of size(12g)p, that is the region2p<(x,y)<2gp and gp<(x,y)<p. A visualization of what happens when the bar-

rier is active is given in Fig. 4, where we show a contour plotof the so, called Weiss fieldQ(x,y).40 This field, defined asQ(x,y)5strain22vorticity2, represents the negative curva-ture of the stream function. Then, we can distinguish be-tween elliptical regions whereQ,0 ~rotation dominatesover deformation!, and hyperbolic regions whereQ.0 ~de-formation dominates over rotation!. Figure 4 was obtainedfor a typical realization of the electrostatic field att50, withthe barrier starting atg50.8. A look at Fig. 4 makes clearthat the large scale structures, visible in the core of the inte-gration domain, are destroyed at the boundary due to theappearance of the barrier.

As a consequence, the particle motion, which is super-diffusive inside the core of the simulation domain, becomesBrownian in the region where the barrier is active. This canbe easily seen in Fig. 5, where the trajectory of a particle inthe plane (x,y) is shown.

FIG. 3. Mean square displacements as a function of time, for the casea51.5, both when phases are not randomized~full line! and for randomphases~dotted line!. As a reference, the dashed line corresponds to the linearfunction ^@r (t)2r (0)#2&;t.

FIG. 4. A contour plot of the Weiss fieldQ(x,y), obtained through a real-ization of the electrostatic field att50. The barrier at the edge of the do-main, obtained withg50.8 ~see the text!, is clearly visible. Black corre-sponds to negative regions ofQ, white corresponds to positive regions ofQ.

FIG. 5. Trajectory of a particle in the plane (x,y), obtained in our simula-tion when the barrier is active, withg50.8 ~see the text!.

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In order to investigate the effect of the barrier, we per-formed some simulations with 103 particles and calculatedthe cumulative number of particlesG(t), as a function oftime, which are lost from the simulation region, both in thecase when the barrier is activated and when it is not. In thesesimulations the barrier is confined to a strip withg varying inthe range 0.5<g<0.9. The results of these simulations areshown in Fig. 6, where it can be clearly seen that in thepresence of the barrier the number of particles which, at afixed time, reach the edge of the integration region is signifi-cantly reduced with respect to the no-barrier case. It is alsoobserved that the transport reduction becomes more andmore efficient as the barrier width increases.

As a further description of the barrier behavior we alsoadopted the following procedure. We calculated the numberof times a particle crosses the boundaries of a square whosecorners are located at coordinatesx* 5y* 560.95p, for thecaseg50.9. In more detail, we determined the number oftimesm1 a particle crosses the side of the square going fromthe center to the edge~outcoming flux!, and the number oftimes m2 it makes the opposite way, that is it crosses theside of the square going from the edge towards the center~incoming flux!. We then calculated the cumulative numbersN6(t) of crossings for all particles, as a function of time.The coefficientC(t)5(N12N2)/(N11N2) represents anindex which gives information on the time evolution of therelative sign of fluxes of particles which crossed the bound-aries of the square at times smaller thant. The time evolu-tion of C(t) is shown in Fig. 7 in both cases~the presenceand absence of the barrier!. When the barrier is not appliedthe coefficientC(t) rapidly reaches a valueC(t).0.6, indi-cating that a large number of particles cross the square goingfrom the center toward the edge, and, at long times, only fewparticles cross the square in the opposite direction. When thebarrier is applied we found thatC(t) is very small at shorttimes, and it rapidly tends to a valueC(t).0.01, thus indi-cating that in this case the incoming flux becomes rapidly

comparable to the outcoming flux. This represents evidencefor the fact that the application of a barrier for the transportthrough a randomization of the phases of the electrostaticfield at the edge of the device, could give rise to a reversedflux of particles from the edge towards the center. It is inter-esting to note that this feature has been observed in recentexperiments on the CASTOR Tokamak device,2,41 where ex-periments with perturbative electrostatic fields at the borderof the devices have been performed.

V. TEST-PARTICLES IN NONTURBULENT FIELDS

It is worthwhile to remind the reader here of the casewhere anomalously enhanced diffusion is observed alsowhen only few modes are present. The field is nonturbulent,that is, we consider a very simple case wherec(r ,t) is madeby few time-dependent vortices, that is,

c~r ,t !5c0 (kx ,ky ,v

k21 sin@kx~x1m sinvt !#sin~kyy!,

~7!

wherekx5pn andky5pm with integersn,m. In the hydro-dynamical case this form forc(r ,t) has been used to repre-sent the stream functions, which describe the velocity field ofRayleigh–Be´nard convection. The termm sinvt takes intoaccount a lateral oscillation of rolls. The model, in the aboveframework, has been extensively studied in Refs. 30, 31when a single moden5m52 is present. Despite its simplic-ity, the dynamics of particles in that field displays interestingdynamical properties. In fact, it has been shown that diffu-sion is anomalous when there exists a synchronization be-tween the lateral frequencyv and the frequency of the fieldc0 /L2.31

In our framework, we use2“c(r ,t) as a realization ofthe electric field. As an example, we investigate the casem50.1, c050.1, andv51.0. We report here two situations.The first one includes a single mode on small scale, namely

FIG. 6. The cumulative number of particlesG(t) as a function of time,which escape off from the simulation region. The solid line refers to the casein which the barrier for transport is not active, while in the other cases thebarrier is active, theg parameter being 0.9~dots!, 0.8 ~dashes!, 0.7 ~dashes–dots!, 0.6 ~dashes–3 dots!, and 0.5~long dashes!, respectively.

FIG. 7. A cumulative coefficient defined through (N12N2)/(N11N2) asa function of time~see the text! for test-particles in the turbulent field, bothwhen the barrier for transport is not active~full line! and when the barrierfor transport is active~dot line!. The barrier has been introduced usingg50.9 ~see the text!.

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n5m58, while in the second case a larger scale moden5m52 is also present. In Fig. 8 we report the motion of atest particle in the plane (x,y) for both cases, showing thatthe particle is subject to Lagrangian chaos near the separa-trix. To get information on the statistics of diffusion, wefollow the time behavior of 104 test particles and we calcu-late the scaling exponentn and the diffusion coefficientDe .In both cases we found that, even if the diffusion is Gauss-ian, that isn50.5 ~in the present paper we are not interestedin investigating the anomalous diffusion induced by the reso-nance, as in Ref. 31!, the diffusion coefficient is significantlydifferent in the two cases. In fact, we foundDe52.8931024 when only the small scale mode is present, whileDe56.4531023 when also the large scale mode is present.This means that the presence of a large scale mode enhancesthe diffusion of particles. As a further confirmation, we re-port in Fig. 9 the cumulative number of particlesG(t) whichescape off from the simulation box as a function of time. It isevident that when a large scale structure is present, the es-caping of particles is favored, thus enhancing diffusion. Inthe perspective of reducing transport, a single mode at thelargest available scale could be dangerous.

VI. CONCLUSION

The starting point of the paper is the surprising remarkthat turbulent fluctuations are invoked in literature both as aconfining environment, when people deal with cosmic raysin astrophysics, and something to be avoided to confine labo-ratory plasmas. We used test-particle simulations in a turbu-lent environment, showing that diffusive properties are en-

hanced due to the presence of regions where correlationsexist ~coherent structures!. These structures, being dynami-cally generated by the nonlinear energy cascade, are presentin plasmas on all dynamical scales. Then, in order to reducetransport, it is sufficient to eliminate structures. This can beeasily made by randomizing the phases of the field, and thisis able to generate a barrier for the transport. It is worthwhilepointing out that the randomization of phases does not cor-respond to cancel out turbulence, rather it generates, in somesense, a more turbulent~stochastic! situation. In our modelthe amplitudes and the spectral properties of turbulence areeven the same in the whole physical domain. Then in orderto generate a barrier for the transport there is no need to killturbulence, it suffices that structures on all scales are de-stroyed. The annihilation of large scale structures has beenreported, for example, in tokamak experiments where a bar-rier of transport is built up through a shear flow.42 From anexperimental point of view, the barrier of transport due to theannihilation of structures through a randomization of phasescould be generated in some more efficient way rather thanthrough the application of a shear flow. Finally, we showedthat Lagrangian chaos is also responsible for the transport ofparticles in nonturbulent plasmas. This should be very im-portant as far as a single large scale mode state is invoked toeliminate transport in laboratory plasmas. Further investiga-tions in this perspective will be reported in a forthcomingpaper.

ACKNOWLEDGMENTS

We acknowledge discussions with P. Devynck, P. Bu-ratti, F. Paganucci, V. Antoni, and C. Riccardi.

This paper is a product of activities within the projectCoBiChao, PAIS 2002-2003 of Istituto Nazionale di Fisicadella Materia, and the projectMHD turbulence phenomenaand their control in thrusters, of the Italian Space Agency.F.L. is supported by the European Commission, through the

FIG. 8. The trajectory of a test-particle in the plane (x,y), obtained by usingthe field described by Eq.~7!. The upper panel refers to the case in which asingle small scale mode (n5m58) is present. The lower panel shows theresult obtained by introducing also a larger scale mode (n5m52).

FIG. 9. The cumulative number of particlesG(t) as a function of time,which escape off from the simulation region, obtained by using the fielddescribed by Eq.~7!. The dashed line refers to the case in which a singlesmall scale mode (n5m58) is present, while the solid line shows the resultobtained by introducing also a larger scale mode (n5m52).

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Research Training Network ‘‘Theory, observation and simu-lation of turbulence in space plasmas’’~Contract No. HPRN-CT-2001-00310!.

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109Phys. Plasmas, Vol. 11, No. 1, January 2004 Confining turbulence in plasmas

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