Simulation of plasma transport by coherent structures in scrape-off-layer tokamakplasmasNirmal Bisai, Amita Das, Shishir Deshpande, Ratneshwar Jha, Predhiman Kaw, Abhijit Sen, and RaghvendraSingh Citation: Physics of Plasmas (1994-present) 11, 4018 (2004); doi: 10.1063/1.1771658 View online: http://dx.doi.org/10.1063/1.1771658 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/11/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Electrostatic transport in L-mode scrape-off layer plasmas in the Tore Supra tokamak. I. Particle balance Phys. Plasmas 19, 072313 (2012); 10.1063/1.4739058 Scrape-off layer tokamak plasma turbulence Phys. Plasmas 19, 052509 (2012); 10.1063/1.4718714 Reduced model simulations of the scrape-off-layer heat-flux width and comparison with experiment Phys. Plasmas 18, 012305 (2011); 10.1063/1.3526676 Effect of discrete coherent structures on plasma-wall interactions in the scrape-off-layer Phys. Plasmas 15, 082316 (2008); 10.1063/1.2974802 Edge and scrape-off layer tokamak plasma turbulence simulation using two-field fluid model Phys. Plasmas 12, 072520 (2005); 10.1063/1.1942427

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.114.34.22 On: Thu, 27 Nov 2014 07:20:24

Simulation of plasma transport by coherent structures in scrape-off-layertokamak plasmas

Nirmal Bisai, Amita Das, Shishir Deshpande, Ratneshwar Jha, Predhiman Kaw,Abhijit Sen, and Raghvendra SinghInstitute for Plasma Research, Bhat, Gandhinagar-382 428, India

~Received 13 January 2004; accepted 17 May 2004; published online 16 July 2004!

The formation of coherent structures by two-dimensional interchange turbulence in the scrape-offlayer ~SOL! of tokamak plasmas and their subsequent contribution to anomalous plasma transporthas been studied in recent years using electron continuity and current balance equations. In thispaper, it is demonstrated that the inclusion of electron energy equation in the simulations changesthe nature of coherent structures in a significant manner and gives results which are in betteragreement with experiments. Specifically, it is observed that radial potential gradients areestablished which give a poloidal elongation and movement to the structures. Only during the radialtransport events do the structures get significantly extended in the radial direction giving radialvelocities of order 1 km/s. Sometimes detachment of density structures from the main plasma isobserved. These detached structures either decay into the background plasma or are transported outfrom the SOL. The simulated particle flux and its statistical properties also are discussed. ©2004American Institute of Physics.@DOI: 10.1063/1.1771658#

I. INTRODUCTION

Investigation of turbulent transport and coherent struc-tures in edge/scrape-off layer~SOL! region of tokamaks hasbecome an active area of experimental and theoretical re-searches. Experimentally, using an array of probes and im-aging diagnostics, vortex-like structures extended in themagnetic field direction and localized perpendicular to ithave been observed.1–15These structures have been shown tobe responsible for intermittent turbulent transport events inSOL region of the tokamak. Detailed measurements showthat the structures are typically poloidally elongated but dur-ing turbulent transport events they become radially elongatedand extend deep into the SOL.1 In the simulation studies, anumber of authors have investigated the problem using non-linear codes16–21 and found similar structures and intermit-tent turbulent transport. Sarazinet al.16,17have simulated theSOL turbulence using a two-dimensional~2D! nonlinearsimulation code which is based on a model of flux driveninterchange turbulence. The authors have assumed a constantelectron temperatureTe and solved the electron continuityequation and current balance equation. They have simulatedthe intermittent nature of particle transport and its triggeringmechanism. In order to study the dynamics of the structuresD’Ippolito et al.20,21 have also solved electron conservationand current balance equations. Here too, the authors haveassumed a constantTe and have additionally neglected ioninertia in the current balance equation. The assumption ofconstantTe generally simplifies the problem. However, theradial gradient ofTe can contribute to the generation of aradial electric field. If such a field exists, then the differentmagnetic field lines in the radial direction will be at a differ-ent potential. This extra contribution to the total radial elec-

tric field contributes to the self-consistent poloidal velocityshear and plays an important role in the nature and dynamicsof the turbulence.

In this paper we have carried out a simulation of the fluxdriven 2D interchange turbulence where we have includedthe dynamics ofTe by additionally solving the electron en-ergy equation. In our model, however, the ion temperatureTi

is still assumed to be negligibly small (Te@Ti). We presentresults from two simulations to show the effect of electrontemperature in the SOL. In the first simulation~named as theFDET simulation—flux driven simulation with electron tem-perature dynamics! we solve electron continuity equation,current balance equation, and the electron energy equation.In the second simulation~named as the FDUT simulation—flux driven simulation with uniform temperature! we solveonly electron continuity equation and current balance equa-tion and assumeTe to be uniform in the SOL.16 We firstcompare the potential structures observed in the FDET andFDUT simulations. In the FDET simulation, we find that thepotential structures are mainly poloidally elongated but be-come radially extended during intermittent transport events.In the FDUT simulation potential structures are always radi-ally elongated. This is consistent with the observation thatthe magnitude of the poloidal flows in the FDET simulationis considerably larger than those in the FDUT simulation. Wealso present results of the time evolution of coherent densityand temperature structures from the FDET simulations. Thedensity structures originate from the main source plasma lo-cated at the last closed flux surfaces and move both in theradial ~0.5–1.2 km/s! and poloidal~0.8–2.5 km/s! directions.Sometimes the radially extended density structures get de-tached from the main structures and either decay or get trans-ported out from the main plasma. We have also simulated theflux time series from the FDET simulation. The statistical

PHYSICS OF PLASMAS VOLUME 11, NUMBER 8 AUGUST 2004

40181070-664X/2004/11(8)/4018/7/$22.00 © 2004 American Institute of Physics

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.114.34.22 On: Thu, 27 Nov 2014 07:20:24

properties of the flux show that the probability distributionfunction ~PDF! deviates from the Gaussian and it has posi-tive skewness and large kurtosis. Overall the FDET simula-tion results on the dynamics of the density and potentialstructures and the statistical properties of the flux time seriesare in excellent agreement with the experimental results.

This paper is organized as follows. The model equationsare presented in Sec. II and linear instabilities are studied inSec. III. In Sec. IV, we present the numerical simulationresults and also compare simulation results with experi-ments. General discussion of the results is given in Sec. V.

II. MODEL EQUATIONS AND NUMERICALSIMULATION

Our model equations for the FDET simulations are elec-tron conservation equation, current balance equation, and en-ergy equation. The assumptions made are quasineutralplasma (ni5ne5n), zero electron inertia, and negligiblysmall Ti . We have also neglected the particle source due tothe ionization in the SOL. The SOL model geometry isshown in Fig. 1. The model equations are

dn

dt1gS n

]Te

]y1Te

]n

]y2n

]f

]y D2D¹'2 n

52snATeeL2f/Te1Sn , ~1!

d¹'2 f

dt1gS Te

n

]n

]y1

]Te

]y D2n¹'4 f

5sATe~12eL2f/Te!, ~2!

dTe

dt1

2

3gS 7

2Te

]Te

]y1

Te2

n

]n

]y2Te

]f

]y D 2ke¹'2 Te

524sTe3/2eL2f/Te1ST , ~3!

whered

dt5

]

]t1VE .¹' , ¹'5 x

]

]x1 y

]

]y,

and VE5E3B/B252¹'f3B/B2.

Here, the electron densityn and theTe are normalized by aconstant arbitrary densityn0 and temperatureT0 . The elec-tric potentialf is normalized byT0 /e and time is normal-ized by Vs

21 , where Vs is ion cyclotron frequency. Thetransverse scalesx andy are normalized by ion Larmor ra-diusrs (rs5cs /Vs5ATe /mi /Vs) wheremi is the ion mass.The normalized magnetic curvature induced gravity is de-noted byg'rs /R0 , whereR0 is the tokamak major radius.The normalized sheath conductivity is represented bys[rs /Lc (Lc is the connection length!. The value ofLc , andhences, depends on the detailed limiter geometry but istypically of orderR0

21. The last terms in Eqs.~1! and~3! aresource terms, which drive the instability. We assumed thatthe source terms have the formsSn5S0 exp(2x2/ls

2) andST52Sn , wherels is thee-folding length of the source. Theperpendicular diffusion coefficientD and viscosityn are nor-malized byT0 /eB0 . The floating potential is denoted byL50.5 ln@4/p(mi /me)#.

For the FDUT simulations we assume that theTe is uni-form (Te51) in the SOL, and consequently Eqs.~1!–~3!reduce to following equations:

dn

dt1gS ]n

]y2n

]f

]y D2D¹'2 n52sneL2f1Sn , ~4!

d¹'2 f

dt1

g

n

]n

]y2n¹'

4 f5s~12eL2f!. ~5!

Equations~4! and~5! are similar to the Sarazin equations inRef. 16 with the difference that we include the curvatureinduced divergence terms in the electron continuity equation~4!; however we have found that for parameters of interestthese terms do not change the results in a significant manner.

The input parameters for the simulation correspond totypical tokamak SOL parameters. We have useds5231024, g5831024, D5n5ke50.01, andL53.9 for thesimulation. The parametersSn andls are estimated from thetotal particle flux coming from the core plasma into the SOL.The estimated values areS05531024 andls510. We havetakenST52Sn as the energy confinement time is about halfof the particle confinement time. The sources are shown inthe Fig. 2. The size of 2Ly is estimated from the experimen-tally measured correlation lengths, which is typically about128rs . The radial sizeLx is taken same asLy which is aboutmore than three times the experimentally measured SOLwidth.

We solve Eqs.~1!–~3! using a pseudospectral code with128 Fourier modes each in thex andy directions. We havealso separately solved Eqs.~4!–~5! for comparison. In bothcases the time integration is performed using fourth-orderaccurate Adams–Bashforth predictor-corrector method. Thealiasing instability due to the nonlinear terms is suppressedby 2/3 antialiasing techniques, which is the same as was usedin the Rayleigh–Taylor code in Ref. 22. We have used ran-dom initial condition ofn andTe which were superimposed

FIG. 1. The modeling of a typical tokamak SOL~HFS and LFS representhigh field side and low field side!. Both hatched portions are joined togetherand considered for simulations.

4019Phys. Plasmas, Vol. 11, No. 8, August 2004 Simulation of plasma transport by coherent structures . . .

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.114.34.22 On: Thu, 27 Nov 2014 07:20:24

over a constant ‘‘floor’’n andTe . The initial condition offis random. The boundary conditions are periodic both in theradial and poloidal directions. The periodicity inx leads tounphysical cross talk16 between the positive and negativexregions, which can be minimized by increasing 2Lx . Wehave takenLx of about three times of experimental SOL sizeto minimize the cross talk. The integration time stepdt isequal toVs

21 . The total number of time steps;Vst;105

which typically corresponds to a few milliseconds of realtime.

III. LINEAR INSTABILITY

We first carry out a linearized instability calculation toverify how the instability due tog is altered by the inclusionof electron energy equation. We have linearized the source-less equations~1!–~3! using expansion ofn(x,y,t)5n(x,t)1n(x,y,t), Te(x,y,t)5Te(x,t)1Te(x,y,t) and f(x,y,t)5f(x,t)1f(x,y,t). Here,n(x,t), f(x,t), andTe(x,t) areequilibrium density, potential, and electron temperaturewhich change slowly in time compared to the fluctuatingn(x,y,t), Te(x,y,t), and f(x,y,t). The equilibrium poten-tial can be represented byf(x,y,t)5LTe(x,t) and all thefluctuating terms n, Te , and f can be expressed by;e2 ivt1 ikyy where all radialx variations are neglected com-pared to the poloidaly variations. The linearized form ofEqs.~1!–~3! can be expressed by

~2 iv1a1!n1b1f1c1Te50, ~6!

a2n1~b21 iky2v!f1c2Te50, ~7!

a3n1b3f1~2 iv1c3!Te50, ~8!

where a15Dky21s2 (L/ l T) iky1 igky , b15( iky / l n) 2s

2giky , c15giky1s(L11/2), a25giky , b25(2nky2

1 iky L/ l T)ky22s1 iky L/ l T

3 , c25Ls1giky , a35 23giky ,

b35 iky / l T 24s2 23ikyg, and c352 (L/ l T) iky1keky

2

1g 73iky14s(L13/2); l n and l T are the density and tem-

perature scale lengths in equilibrium which are assumed tobe fixed and given in this linearized calculation. The disper-sion relation is

U2 iv1a1 b1 c1

a2 b21 iky2v c2

a3 b3 2 iv1c3

U50. ~9!

The cubic dispersion relation has one unstable and two stableroots. We plot the unstable root ofv with respect toky forl n5 l T510.0 as shown in the Fig. 3~curve I!. Figure 3 showsthat growth rate is negative for smallky and largeky . Thesmall ky and largeky are stable due to the effect of parallelconductivity s and dissipative termsD,n and ke , respec-tively, as is indicated by Eqs.~1!–~3!. The growth rate ismaximum atky;0.17.

We have also studied the linear instability from the Eqs.~4! and ~5!. The dispersion relation obtained from Eqs.~4!and ~5! is

v2ky21 iv~a18ky

22b28!1a18b282g2ky21sgiky1

gky2

l n50,

~10!

where a185Dky21s1giky and b2852nky

42s. The disper-sion relation, Eq.~10!, has one unstable root and one stableroot, where the stable root is almost identical with one of thestable roots in Eq.~9!. The unstable root is plotted in Fig. 3~dashed curve II!, where maximum growth rate is obtainedfor ky;0.3.

It is to be noted that the maximum growth rate obtainedfrom Eq. ~9! is about two times higher than the growth rateobtained from Eq.~10!. The vertical dashed line shows thevalue ofky (;0.2) where the modulus of Fourier transformof a fluctuating quantityn or f ~obtained from the FDUTsimulation! is maximum in the fully developed turbulentstate. The vertical solid line shows the value ofky (;0.09)where the modulus~obtained from the FDET simulation! ismaximum. Therefore both simulations show evidence of in-verse energy cascades where the time averaged spectral en-ergy transfer function23–25 f (k8→k) is positive (k8.k).

FIG. 2. Particle (Sn) and temperature (ST) source profiles in normalizedunits. FIG. 3. The linear growth ratesg vs ky . The curve I represents growth rate

obtained from dispersion relation, Eq.~9!, for l n5 l T510.0 and the verticalsolid line represents value ofky (;0.1) where the modulus of the poloidalFourier spectrum of a fluctuating quantity is maximum during turbulentphase of the FDET simulation. The curve II represents growth rates obtainedfrom dispersion relation, Eq.~10!, for l n510.0 and the dashed vertical lineindicates value ofky (;0.2) where modulus of the poloidal Fourier trans-form is maximum during turbulent phase of the FDUT simulation.

4020 Phys. Plasmas, Vol. 11, No. 8, August 2004 Bisai et al.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.114.34.22 On: Thu, 27 Nov 2014 07:20:24

IV. SIMULATION RESULTS

Our main aim in this paper is to study the coherent struc-tures and related intermittent transport using the FDET@Eqs.~1!–~3!# and FDUT@Eqs.~4! and~5!# simulations. A numberof authors have already studied the SOL turbulence using theFDUT simulations. Therefore, we are not giving in detail theresults of the FDUT simulations, but a comparison betweenthe two will be given to show the importance of the role ofelectron temperature in the SOL turbulence. Unless it is men-tioned, all quantities are expressed in normalized units andalso unless specified we shall be presenting only the FDETsimulation results.

The simulated turbulence data are characterized by cor-relation time and length scales. The decorrelation timetd ofall signals~i.e., density, potential, and temperature! are in therange of 400–600 normalized time. The poloidal and radialcorrelation lengths are in the ranges (40– 50)rs and(20– 25)rs , respectively. The instantaneous plot~snapshot!of the potential, density, and temperature show localizedstructures on 1283128 grid points. The lifetime of the struc-tures can be determined by taking the time averages of thedata for different durations. We observe that the time aver-ages of the data also show localized structures as long as theaveraging time is less than or equal to 3td . When the aver-aging time is increased further, the structures gradually fadeand finally vanish altogether after 4td . Thus the lifetime ofthe structures is about 3td and hence the structures are co-herent. The dynamics of these coherent structures is pre-sented in the following.

A. Dynamics of coherent structures

The dynamics of coherent structures over a 1283128grid points is followed by taking snapshots at an interval of200 normalized times. The typical coherent potential struc-

FIG. 4. Time evolution of the potential structures~from FDET simulations!is shown using time breakup plots~a!–~f!. Note the dynamics of structureswhich are marked by the arrow. The time difference between two successiveplots ~a!–~f! is 200 normalized time steps.

FIG. 5. Potential structures obtained from the FDUT simulations. The timedifference between two successive plots~a!–~f! is 200 normalized timesteps. Note that the potential structures are always radially elongated.

FIG. 6. Time evolution of the density structures~from the FDET simulation!is shown using time breakup plots~a!–~f!. Note the dynamics of structuresindicated by arrows. The time difference between two successive plots~a!–~f! is 200 normalized time steps.

4021Phys. Plasmas, Vol. 11, No. 8, August 2004 Simulation of plasma transport by coherent structures . . .

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.114.34.22 On: Thu, 27 Nov 2014 07:20:24

tures are shown in Figs. 4~a!–4~f! using gray plots. Thewhite color indicates the highest value of the total potentialand black color indicates the lowest value of the potential.The time difference between two figures is 200 normalizedunits which is of the order of a few microseconds. Figures4~a!–4~f! show that the potential structures are mainly poloi-dally elongated and that a radial elongation is seen only atthe position where transport is taking place~shown by ar-rows!. We have compared the potential structures betweenthe FDET and FDUT simulations. The potential fluctuationsin FDUT simulation has decorrelation time of 1200 normal-ized time. At long delay time, the correlation function showssignificant coherent oscillation. The averaging of snapshotsshows that structures survive upto 4td and decay slowly atlonger averaging time. Figures 5~a!–5~f! show the time evo-lution of the potential structures obtained from the FDUTsimulations. It is seen that the potential structures are mainlyradially elongated and are remarkably different from those inFigs. 4~a!–4~f!. The poloidal elongation of the potentialstructures as seen in the FDET simulations@Figs. 4~a!–4~f!#is consistent with the experimental observations in mosttokamaks.1–15 In the case of FDET simulations, the electrontemperature plays an important role as the radial profile ofthe electron temperature is connected with the generation ofradial electric field. If a radial gradient of electron tempera-ture (]Te /]x) exists, the different magnetic field lines alongthe radial direction will be at different potential. This radialgradient of potential (]f/]x) generates a radial electric fieldEx52]f/]x which is responsible for a generation of poloi-dal velocity (Vy5Ex3B). The radial dependence ofVy gen-erates a velocity shear]Vy /]x, which is responsible for thepoloidal elongation of the coherent structures.

A complete time evolution of the coherent structure inthe density is shown in Figs. 6~a!–6~f!. The presence of theparticle source is clearly seen in the dominance ofhigh den-sity at x50. The density structure which is studied here isshown by an arrow. The time difference between two succes-sive figures in Figs. 6~a!–6~f! is also 200 normalized units,i.e., a few microseconds. In Fig. 6~a!, the density structureoriginates from the position~46,46!. The density structure inFig. 6~b! extends along the radial direction further and apoloidal movement is also clearly seen. Figures 6~c! and 6~d!show that the density structure has detached from the mainplasma. Figures 6~e! and 6~f! show that the detached struc-ture is moving radially and as well as poloidally and finally apart of it is transported out from the SOL. The coherentstructure also diffuses into the background plasma which is

shown in Fig. 6~d! by the decrease of density as compared toFigs. 6~b! and 6~c!. We have measured the radial velocity ofthe coherent density structures. It is found that they extendradially with speed.0.5– 1.2 km/s, which is consistent withtokamak experiments. Figures 6~a!–6~f! also show that thedensity structures move in a curved path because of nonuni-form velocities in (x,y) plane. The velocities depend on theradial (Ex) and poloidal (Ey) electric fields, and numericallyit is found that these fields are functions of (x,y) coordi-nates. These fields are shown in Figs. 7~a! and 7~b!. The dataof Figs. 7~a! and 7~b! are taken at the same instant of time asthat of Fig. 6~c!. It may be noted that as compared to thebackground valuesEx is lower andEy is higher at the posi-tion of the elongated density structures.

We have also studied the detachment of density struc-tures or ‘‘blobs’’ by the FDET simulations. A typical bloblikedensity detachment is shown in Figs. 8~a! and 8~b!. The timedifference between Figs. 8~a! and 8~b! is 200 normalizedtime units. Figures 8~a! and 8~b! show that the detachedstructure moves radially as well as poloidally. The poloidal

FIG. 7. Radial (Ex) and poloidal (Ey)fields obtained from the FDET simula-tions. The data are taken at the sameinstances of time as in Fig. 6~c!. Theradial elongation of the density struc-tures is taking place as theEx is lowerandEy is higher.

FIG. 8. Detachment of density structure and its radial and poloidal motions.The detached structure forms like a blob and moves in the radial and as wellas poloidal direction. The movement is shown by horizontal and verticallines. The time difference between upper and lower figures is about 200normalized time.

4022 Phys. Plasmas, Vol. 11, No. 8, August 2004 Bisai et al.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.114.34.22 On: Thu, 27 Nov 2014 07:20:24

velocity is about 2.5 km/s and the radial velocity is about 0.5km/s. It is observed that the poloidal velocity is alwayshigher than the radial velocity.

We have also simulated coherent structures in the tem-perature field. The simulation shows that the coherent tem-perature structure dynamics is similar to that of density struc-tures. Only the levels of the temperatures are different fromthe density structures as the temperature source and the par-allel thermal conductivity are different from the particlesource and parallel particle transport. A typical temperaturestructure is shown in Fig. 9. It is to be noted that Fig. 9 istaken at the same instant of time as Fig. 6~f!, which showsthat both density and temperature structures are similar.

We now compare the radial electric fields from the twosimulations. The results are shown in Figs. 10~a! and 10~b!.These figures show poloidally averaged values of radial elec-tric fields which are obtained from the FDET and FDUTsimulations, respectively. Figures 10~a! and 10~b! are takenat an instant of time when the poloidally averaged electricfield is a maximum. The radial electric field from FDETsimulations is about four to five times higher than that in theFDUT simulations. In the FDUT simulations, the major con-tribution to generation of radial electric field comes from thenonlinear Reynold stress terms which drive zonal flows. Wemay thus estimate the radial electric field for FDUT simula-tions from the zonal flow~ZF! equation, which is obtainedfrom integration and poloidal averaging of Eq.~5!:

]

]t^vy&52

]

]x^vyvx&1sE

0

x

^12e(L2f)&dx

1n]2

]x2 ^vy&, ~11!

where ^Ex&5^vy&52( ]/]x )^f&, the Reynold stress term^vyvx& is obtained fromz“f3“¹2f term in Eq.~5!, and^ f & indicates poloidal average off . Neglecting the smallcontribution from viscosity term the maximum radial electricfield obtained from Eq.~11! is

^Ex&.U ky4

skxl n2U. ~12!

For s5231024, ky;0.2, l n;20 ~estimated from simula-tion!, andkx;0.6 ~estimated from the smallest structure ob-tained from FDUT simulation which is about 10rs), thevalue of Ex is about;431022, which is consistent withFig. 10~b!. In the FDET simulation the equilibrium radial~poloidally averaged! temperature profile generates radialelectric field, which dominates over the nonlinearly gener-ated zonal flow field; this equilibrium field is given by

^Ex&5LdTe~x!

dx52

L

l TTe~x!. ~13!

For l T520.0, Te(x);1 the value of Ex& is about 0.2, whichis consistent with the order of magnitude of simulation resultin Fig. 10~a!. The exact calculation of electric field from theZF and equilibrium effects shall be presented elsewhere.

B. Particle transport

The particle transport is examined from the FDET simu-lations only. The intermittent transport is due to the radialpropagation of the coherent structures. We have measuredthe flux G(x,y,t)52n]f/]y time series at the location~33,0!. Figure 11 shows the plot of flux with respect to nor-malized time (tVs). The total length of the time series is1.23106 normalized time units which corresponds to a fewmilliseconds in real time units. It is shown that the timeseries has both negative~inward! and positive~outward!fluxes. But the time average flux is positive and outward. Theoccasional large amplitude of the flux indicates that thetransport is intermittent or bursty. The PDF of the flux is

FIG. 9. A typical temperature structure. The data of this figure are taken atthe same instant of time as Fig. 6~f!. Note that the temperature structure isalmost similar with that of the density structure as in the Fig. 6~f!.

FIG. 10. The poloidally averaged radial electric field^Ex& as a function ofx. The plot ~a! is obtained from the FDET simulations and the plot~b! isobtained from the FDUT simulations.

4023Phys. Plasmas, Vol. 11, No. 8, August 2004 Simulation of plasma transport by coherent structures . . .

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.114.34.22 On: Thu, 27 Nov 2014 07:20:24

shown in Fig. 12. The PDF is not Gaussian, it is asymmetricand positively skewed. The flux has been analyzed usingstatistics to determine skewness and kurtosis. The values ofthe skewness and the kurtosis are 1.4 and 5.8, respectively.The positive skewness along with finite mean value indicatesa predominance of large positive flux in the time series. Thenon-Gaussian kurtosis (K.3) indicates bursty nature of theflux. However, it should be noted that the PDF as well as thekurtosis is only mildly non-Gaussian compared to the experi-mental results. The experimental data show strongly non-Gaussian PDF~Ref. 6! and kurtosis.26

V. SUMMARY AND DISCUSSIONS

We have simulated the SOL turbulence using two sets ofmodel equations. The FDET simulations@Eqs. ~1!–~3!# in-clude the electron energy equation whereas the FDUT simu-lations @Eqs.~4! and ~5!# assume uniform electron tempera-ture in the SOL. In the case of FDET simulations theinstability is driven by a constant particle flux and heat~tem-perature! flux and in the case of the FDUT simulation theinstability is driven only by a constant particle flux. Thesedriving sources are assumed poloidally constant and dependon radial coordinates only. Although in reality these sources

may have spatial and temporal variations, the poloidally con-stant nature is taken to simplify the problem and so that itcan be easily treated numerically. The results of the twosimulations are significantly different from each other. Thisdifference indicates that the electron temperature plays animportant role in the SOL. Due to the inclusion of electrontemperature dynamics the simulation results are in betteragreement with the experimental results. The potential struc-tures are poloidally elongated most of the time but duringturbulent transport events the structures become radiallyelongated. This observation is consistent with probe mea-surements in most tokamaks. We have also simulated thetime evolution of density structures. The radial and poloidalmotions of the structures are consistent with the experimen-tal measurements. A detachment of density structures fromthe main plasma is observed. The detached structures eitherdecay or get transported out from the SOL region. We havealso simulated particle transport, which shows intermittency.The statistical properties of the particle transport are found tobe close to the experimental measurements.

1B. K. Joseph, R. Jha, P. K. Kaw, S. K. Mattoo, C. V. S. Rao, Y. C. Saxena,and the Aditya Team, Phys. Plasmas4, 4292~1997!.

2R. Jha, S. K. Mattoo, and Y. C. Saxena, Phys. Plasmas4, 2982~1997!.3R. Jha, P. K. Kaw, S. K. Mattoo, C. V. S. Rao, Y. C. Saxena, and the AdityaTeam, Phys. Rev. Lett.69, 1375~1992!.

4S. Benkadda, T. Dudok de Wit, A. Verga, A. Sen, ASDEX team, and X.Garbet, Phys. Rev. Lett.73, 3403~1994!.

5S. J. Zweben, Phys. Fluids28, 974 ~1985!.6M. Endler, H. Niedermeyer, L. Giannone, E. Holzhauer, A. Rudyj, G.Theimer, N. Tsois, and Asdex team, Nucl. Fusion35, 1307~1995!.

7R. A. Moyer, R. D. Lehmer, T. E. Evans, R. W. Conn, and L. Schmitz,Plasma Phys. Controlled Fusion38, 1273~1996!.

8J. Boedo, D. Rudakov, R. Moyer, S. Krashninnikov, G. Tynan, S. Allen, P.Stangeby, P. West, and D. Whyte, Phys. Plasmas8, 4826~2001!.

9S. Zweben, R. Maqueda, J. Terry, B. LaBombard, C. S. Pitcher, M. Green-wald, B. Rogers, and W. Davis, Bull. Am. Phys. Soc.45, 188 ~2000!.

10R. Maqueda, S. Kaye, S. Sabbagh, and R. Maingi, Rev. Sci. Instrum.72,931 ~2001!.

11J. L. Terry, B. LaBombard, C. S. Pitcher, M. J. Greenwald, and S. Zweben,Bull. Am. Phys. Soc.45, 320 ~2000!.

12J. Boedo, D. L. Rudakov, R. A. Moyeret al., Phys. Plasmas10, 1670~2003!.

13S. J. Zweben, D. P. Stotler, J. L. Terryet al., Phys. Plasmas9, 1981~2002!.

14J. L. Terry, S. J. Zweben, K. Hallatscheket al., Phys. Plasmas10, 1739~2003!.

15A. W. Leonard, P. H. Osborne, M. E. Fenstermacheret al., Phys. Plasmas10, 1765~2003!.

16Y. Sarazin and P. H. Ghendrih, Phys. Plasmas5, 4214~1998!.17Y. Sarazin, P. H. Ghendrih, G. Attuelet al., J. Nucl. Mater.313–316, 796

~2003!.18P. Beyer, Y. Sarazin, X. Garbet, P. Ghendrih, and S. Benkadda, Plasma

Phys. Controlled Fusion41, A757 ~1999!.19N. Bian, S. Benkadda, J. V. Paulsen, and O. E. Garcia, Phys. Plasmas10,

671 ~2003!.20D. A. D’Ippolito, J. R. Myra, and S. I. Krasheninnikov, Phys. Plasmas9,

222 ~2002!.21D. A. D’Ippolito and J. R. Myra, Phys. Plasmas10, 4029~2003!.22A. Das, S. Mahajan, P. Kaw, A. Sen, S. Benkadda, and A. Verga, Phys.

Plasmas4, 1018~1997!.23M. Lesieur, inTurbulence in Fluids, in Fluid Mechanics and its Applica-

tions, edited by R. Moreau~Kluwer Academic, Dordrecht, 1993!, p. 272.24B. D. Scott, H. Biglari, P. W. Terry, and P. H. Diamond, Phys. Fluids B3,

51 ~1991!.25C. P. Ritz, E. J. Powers, and R. D. Bengtson, Phys. Fluids B1, 153~1989!.26R. Jha, B. K. Joseph, R. Kalraet al., Proceedings of 15th International

Conference on Plasma Physics and Controlled Nuclear Fusion, Seville, 26September–1 October 1994~IAEA, Vienna, 1995!, Vol. 1, p. 583.

FIG. 11. The time series of the particle flux. The time series is generated upto 1.23105 normalized time steps which is about few milliseconds at thelocation ~33,0! in the SOL.

FIG. 12. The PDF of the particle flux.

4024 Phys. Plasmas, Vol. 11, No. 8, August 2004 Bisai et al.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

128.114.34.22 On: Thu, 27 Nov 2014 07:20:24