Edge and scrape-off layer tokamak plasma turbulence simulation using two-field fluidmodelNirmal Bisai, Amita Das, Shishir Deshpande, Ratneshwar Jha, Predhiman Kaw, Abhijit Sen, and RaghvendraSingh Citation: Physics of Plasmas (1994-present) 12, 072520 (2005); doi: 10.1063/1.1942427 View online: http://dx.doi.org/10.1063/1.1942427 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/12/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Statistical analysis and modeling of intermittent transport events in the tokamak scrape-off layer Phys. Plasmas 21, 122306 (2014); 10.1063/1.4904202 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 Dynamics of turbulent transport in the scrape-off layer of the CASTOR tokamak Phys. Plasmas 13, 102505 (2006); 10.1063/1.2359721

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Edge and scrape-off layer tokamak plasma turbulence simulationusing two-field fluid model

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

�Received 19 January 2005; accepted 3 May 2005; published online 13 July 2005�

A novel two-dimensional �2D� fluid model is proposed for investigating flux-driven plasmaturbulence in the tokamak edge and scrape-off layer �SOL�. Unlike most previous turbulencesimulations of this region, the 2D model treats the two regions in a consolidated manner with asmooth transition region in between. The unified 2D model is simpler and less computer intensivethan 3D models, but captures most features of the 3D edge and 2D SOL turbulence. It also illustratesthe influence of tokamak edge turbulence on the SOL transport, something not captured by earlier2D SOL simulations. Existence of an equilibrium radial electric field in the edge and SOL regionshas been found. Two different plasma conductivity models have been used for the simulations.Turbulence in the edge is characterized by radially elongated streamers and zonal flows. Thestreamer structures occasionally break mainly in a region where the radial electric field changessign. A phenomenological condition for the breaking has been obtained. Effective diffusionco-efficient and density front propagation speed from the simulation have been calculated. Statisticalproperties of the particle transport obtained from this simulation are compared with earlierflux-driven 2D SOL turbulence simulations and also with Aditya tokamak results. © 2005 AmericanInstitute of Physics. �DOI: 10.1063/1.1942427�

I. INTRODUCTION

Edge and scrape-off layer �SOL� tokamak plasma turbu-lence has been studied using gradient-driven and flux-drivenmodels. In the gradient-driven model, profiles are not al-lowed to relax and are kept fixed �by unspecified sources�which drive instabilities to provide transport as a function ofgradients. The model assumes scale separation between fluc-tuations and equilibrium. This model has been successfullyused in theories and numerical simulations to account forexperimentally observed transport,1–3 but has been unable toexplain some important features such as nonlocality of thetransport phenomena observed in many tokamaks.4,5 In theflux driven model, on the other hand, the physical quantitiesare not separated into fluctuating and equilibrium parts. Anincoming flux of particles and/or heat drives the instability.In these models, the flux is maintained constant in time andprofiles fluctuate about their time average values. This modelhas been successfully applied in many tokamaks as it is ca-pable of explaining many experimentally observed facts suchas scale invariance in time of fluctuations, self-organizedcriticality type of flux spectrum behavior, and intermittentballistic transport events.

Flux-driven turbulence in the edge region and in theSOL tokamak plasma has been extensively investigated inrecent years. Edge turbulence studies have been mostly car-ried out with the help of flux-driven three-dimensional �3D�fluid simulation codes.6–11 The nonlinear patterns obtainedfrom the 3D simulations consist of radially extended convec-tion cells �streamers� which build up intermittently and giverise to large scale transport events. These simulations havealso shown a self-consistent generation of zonal flows whichtend to self-regulate the turbulence. In the case of SOL tur-

bulence studies, it has been customary to use flux-driven 2Dfluid models related to the interchange instability.12–16 Thesimulation results obtained from such models appear to ex-plain many experimentally observed facts like intermittentballistic transport together with scale invariance in time offluctuations and flux spectrum behavior similar to self-organized criticality. For a more detailed understanding ofthe physics of this transport a one-dimensional study has alsobeen carried out.8,17 Most recently, the influence of electrontemperature dynamics16 on 2D SOL turbulence has also beenstudied.

Most of the above simulation studies have been confinedsolely either to the edge or to the SOL region. A combinedsimulation of these two regions has been done by Garcia etal. using a 2D model.18 The dynamics in the parallel direc-tion �plasma resistivity� in the edge has been neglected bythem. Another major simplification, particularly in all 2DSOL turbulence12–16 studies, is the assumption that the par-ticle flux coming from the confined plasma into the SOLregion is constant in time and in the poloidal direction. Inreality the driving flux originates from the edge turbulenceand has spatiotemporal variations. Therefore, for a more ac-curate description of the SOL turbulence it is desirable tojoin the edge and SOL regions together. However, the phys-ics of the edge turbulence is quite different from the SOL. Inthe edge, magnetic field lines form nested flux surfaces andturbulence is related to resistive ballooning modes which areunstable if the pressure gradient scale length exceeds a cer-tain critical value. With finite resistivity, the parallel currentcarried by the electrons has a finite parallel divergence. Thisdivergence depends mainly on the electron collision fre-quency with the background. In the SOL region, the mag-

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netic field lines terminate on the limiter/divertor plate. As aresult, the electrons and ions follow the magnetic field linesand are lost to the limiter. This parallel current mainly de-pends on the sheath physics rather than on the collision fre-quency of the electrons with the background. Consequently,during cross field transport from the edge to the SOL, theplasma experiences two different plasma unstable regions.Edge plasma instabilities induce plasma injection into theSOL region, which modify the SOL equilibrium profile anddrive instability in the SOL. This phenomenon illustrates theinfluence of tokamak edge turbulence on the SOL transportwhich cannot be captured by the simple flux-driven 2D SOLmodel.

In this work, we have simulated the edge and SOL tur-bulence in a combined manner by using a single 2D simula-tion code. Our model consists of the two-field electron con-tinuity and current conservation equations which are valid inthe edge as well as in the SOL regions. The conductivity inthe edge and the sheath conductivity in the SOL are con-nected smoothly in the edge-to-SOL transition region. Twodifferent models of the conductivity have been used. Thesimulation results show the existence of a radial electric fieldand the concomitant existence of E�B zonal flows�0.5–3.0 km/s� mainly in the edge and edge-to-SOL transi-tion regions. An analysis of the spatial structure of the tur-bulent flux shows radially elongated �streamers� structures,which are similar to the earlier 3D simulation6,7,9 results. Thestructures obtained from our simulation extend from the edgeto the SOL and occasionally break mainly in the regionswhere the radial electric field changes sign. Using the simu-lation results, we have derived a condition for the breaking.The density front propagation speed in the SOL has beencalculated ��1/32th part of sound speed cs� from the simu-lation. The particle transport in the edge and the SOL hasbeen investigated. The SOL turbulence obtained from thiscode is different from the earlier flux-driven simple SOLturbulence.16 A comparison has been done between thesetwo. It is found that the SOL turbulence obtained from thepresent 2D simulation is closer to the tokamak experimentalresults compared to the simple flux-driven 2D SOL simula-tion. We have also compared simulated edge and SOL turbu-lence with Aditya tokamak experimental results.

We have organized the paper as follows. The modelequations have been derived in Sec. II. The input parametersand numerical simulation are also described in Sec. II. Thenumerical results are discussed and are compared with theexperimental results in Sec. III. Our results are summarizedand discussed in Sec. IV.

II. MODEL EQUATIONS

We have used two-field fluid model equations in theedge and the SOL regions. These are the flux-driven electroncontinuity and current conservation equations. The basic as-sumptions are quasineutral plasma �ni=ne=n�, zero electroninertia, uniform electron temperature, and negligibly smallion temperature in the edge and SOL. We have also ne-glected ionization in the edge and SOL regions. In our sim-plified approach, the edge and SOL geometry is a square box

as shown in Fig. 1, where x refers to the radial directionwhile y labels the poloidal direction at a given toroidal posi-tion of a tokamak. The 2D electron continuity and currentconservation equations which are valid in the edge and SOLare obtained from a parallel averaging of 3D equations12

dn

dt+ g� �n

�y− n

��

�y� − D���

2 n = � 1

L

�Je

+ Sn, �1�

d��2 �

dt+

g

n

�n

�y− ����

4 � = � 1

nL

�J

, �2�

where d /dt=� /�t+VE·��, ��= x�� /�x�+ y�� /�y� and VE=E�B /B2=−����B /B2, B is tokamak magnetic field andE is electric field �E=−���� for electrostatic approximationof the turbulence. Here n is density normalized by arbitrarydensity n0, � is potential normalized by T0 /e, Je and J areelectron current and total current normalized by en0cs �cs

=�T0 /mi, mi is ion mass�, �x ,y� is spatial coordinate normal-ized by ion Larmor radius �s, t is time normalized by ioncyclotron frequency �s

−1 �cs=�s�s�, L is parallel connectionlength normalized by �s. The term g is normalized gravitydenoted by g��s /R and D�, �� are normalized diffusioncoefficient and viscosity denoted by D��D /�scs and ��

�� /�scs. The driving source term Sn is denoted by Sn

�S0 exp(−�x /�s�2), where S0 is the maximum amplitude ofthe source and �s is the e-folding source width. The Sn isnormalized by n0�s. The region x�0 �x�0� with positive�negative� density gradient models the high�low�-field side ofa tokamak. It is to be noted that the Sn is independent of timeas is required by the flux-driven model. The form of Gauss-

FIG. 1. The modeling of a tokamak edge and SOL �HFS and LFS representhigh-field side and low-field side, respectively�. Hatched edge and SOLportions enclosed by four slanting lines in the HFS and LFS are joinedtogether for the simulation.

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ian source with an e-folding length �s is based on the esti-mates in typical tokamaks. Simulations have been carried outfor different values of S0 and �s. The simulation results aregiven in Sec. III A to show how results alter with �s and S0.

Here, our aim is to derive model equations using Eqs. �1�and �2� which will be valid both in the edge and in the SOLregions so that we can study turbulence in the entire edge toSOL regions using a single code. In the edge, first terms ofthe right-hand side of Eqs. �1� and �2� are determined byfinite parallel resistivity �=me�e /nee

2, where �e is collisionfrequency of electron with background, e is electron charge,and me is electron mass� of plasma and in the SOL; these aredetermined by both and mainly limiter sheath conductivity.In the edge-to-SOL transition region, a contribution fromboth and sheath conductivity has been considered bysmooth functions, which smoothly connects the edge andSOL regions. However, our aim is also to study turbulenceusing 2D model as it is less computer intensive than the 3Dmodel. The SOL turbulence has been studied successfullyearlier using 2D model by replacing the plasma dynamics inthe parallel direction by limiter sheath physics. But actuallythe edge turbulence is mainly 3D. The 2D approximation hasbeen done in a consolidated manner by replacing �

2→−k2

−�R0q�−2 in the �Je=−1�2��−ln nT0 /e�, where k, R0,

and q are parallel wave number, tokamak major radius, andsafety factor, respectively. It is to be noted that similar 2Dapproximation had been used earlier to investigate gradient-driven turbulence.19–21 The second term in the right-handsides of Eqs. �1� and �2� in the entire edge and SOL regionscan be written as

� 1

L

�Je

= − �x�ne�−� + ��x��� − n� , �3�

� 1

nL

�J

= �x��1 − e�−�� + ��x��� − n� , �4�

where n=n− �n�y, �=�− ���y, �.� refers to poloidal averageof the respective terms. The term � is floating potential de-noted by ��0.5 ln��4/ ��mi /me��. �x� is normalizedsheath conductivity denoted by �x�=0.50�2+tanh��x−x0� /�0�−tanh��x+x0� /�0��, where x0 and �0 are width ofthe edge and width of the edge-to-SOL transition regions,respectively. 0=�s /Lc is the maximum normalized sheathconductivity in the SOL region, where Lc is the connectionlength of the magnetic field lines in the SOL. The normalizedparallel plasma conductivity ��x�=�0f�x�, where �0

= ��s /qR0�2�mi /me���s /�e� and f�x� is a function which de-pends on the model. Two different models have been used. Inthe first model �model-I�, f�x�=1− (�x� /0) and in the sec-ond model �model-II�, f�x�= (1−�x� /0)exp�−�x� / l��, wherel� is e-folding length of f�x�. In model-I, f�x� 1 in theinside edge region and decreases in the edge-to-SOL transi-tion region and finally drops to zero in the deep SOL.Model-II is little more refined than the model-I and retainsthe near exponential dependence of �0��e

−1�Te3/2 /n in the

edge region. It is to be noted that for the derivation of Eqs.�3� and �4� cold ion approximation has been used so that

��1/nL��J� = ��Je�. Using Eqs. �3� and �4� the finalmodel equations are

dn

dt+ g� �n

�y− n

��

�y� − D���

2 n = − �x�ne�−�

+ ��x��� − n� + Sn, �5�

d��2 �

dt+

g

n

�n

�y− ����

4 � = �x��1 − e�−�� + ��x��� − n� ,

�6�

n = n − �n�y , �7�

� = � − ���y . �8�

Note that in the limit �0→�, Eqs. �5�–�8� reduce to the flux-driven SOL equations16 that have been studied earlier.

dn

dt+ g� �n

�y− n

��

�y� − D���

2 n = − 0ne�−� + Sn, �9�

d��2 �

dt+

g

n

�n

�y− ����

4 � = 0�1 − e�−�� . �10�

For a comprehensive study, we have also separately solvedEqs. �9� and �10� numerically and compared the results toour solutions of Eqs. �5�–�8�.

The input parameters for the simulation have been takencorresponding to typical tokamak edge and SOL parameters.The estimation of D�, ��, g, 0, �, S0, and �s has been donein Ref. 16. Here too we have used D�=��=0.01, g=8�10−4, �=3.9, S0=5�10−4, and 0=2�10−4 for the simu-lation. The width of the edge region x0 is approximated tox0=25. The smoothness parameter �0=3.0 has been used. Itis to be noted that a higher value of �0 yields a broaderedge-to-SOL transition region. We have also simulated dif-ferent values of �0. Parameter l�=15 �e-folding scale lengthof n /Te

3/2 of model-II� is used for simulation. The parallelnormalized conductivity �0 is approximated from the expres-sion ��s /R0q�2�mi /me���s /�0� and �0=6�10−4 is taken forthe simulation. We have also made the simulations usingdifferent values of �0. The radial position of source Sn, radialdependency of �x�, and ��x� is shown in Fig. 2. It is to benoted that the source is maximum at x=0 and has ane-folding length �s=5. The plot of ��x� for the model-I andII are shown by thick and thin dashed lines, respectively. Theedge-to-SOL transition region is shown by vertical dottedlines at x=25 and x=−25. It is to be noted that the turbulenceat the low-field side and high-field side of the edge and SOLwill be obtained from positive and negative values of x, re-spectively. The size of 2Ly is estimated from experimentallymeasured poloidal correlation length. The 2Ly is about two tothree times of the poloidal correlation length. The radial boxsize 2Lx is estimated from the experimentally observed typi-cal edge plus SOL width. The Lx is about three times of theexperimentally observed edge plus SOL width.

We solve Eqs. �5�–�8� using a pseudospectral code with128 Fourier modes each in the x and y directions. The timeintegration is performed using the Adams-Bashforth method.

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We have used random initial condition of n with a constantfloor density in space. The initial condition of � is random.The boundary conditions of n, and � is periodic in both xand y directions. The time step �t is equal to �s

−1. We havealso solved Eqs. �9� and �10� from the same code using �0 asa very large number.

It is to be noted that we have assumed uniform electrontemperature Te in the edge and SOL regions for the deriva-tion of Eqs. �5�–�10�. In reality, Te has a finite radial profile.If the Te profile is included into the simulation, it will intro-duce additional temperature gradient-driven instabilities22,23

into the turbulence. Furthermore, inclusion of temperatureprofiles in the SOL can induce internal spin/rotation of adensity blob �isolated high-density structure� which can re-duce the radial convection velocity due to charge mitigationof the blob.24,25 Inclusion of electron temperature is out ofscope of this paper and simulation results including tempera-ture profile effects in the edge and SOL regions will be pre-sented elsewhere.

III. NUMERICAL RESULTS

Numerical solution of Eqs. �5�–�8� describes turbulencein the high-field and low-field sides of a tokamak. The tur-bulence in the high-field side is stable and hence is excludedfrom our discussion. We shall be presenting only turbulencein the low-field side.

A. Electric fields and zonal flows

The simulation result shows a self-consistent generationof radial electric field. The plot of poloidal and time-averaged radial electric field �Ex�y,t is shown in Fig. 3�a�using model-I ��0=6�10−4, �0=3.0� and model-II ��0=6�10−4, l�=15.0�. Figure 3�a� shows that the shape of theradial electric field variation depends on the parallel resistiv-ity model. In both the cases, �Ex�y,t changes sign. The mini-mum of �Ex�y,t at x=20 for model-I is about 0.8 kV/m �Ex

�0.05 in normalized unit� for typical Aditya tokamak pa-

rameters. We have simulated the �Ex�y,t for different valuesof �0 and �0 for the model-I. It is found that �Ex�y,t increaseswith the increase in �0 and decreases with the increase in �0.Physically, higher values of �0 stabilize the resistive balloon-ing mode and hence a reduction of the turbulence occurs.Reduction of turbulence yields higher Ex. Simulation shows�Ex�y,t increases about 45% when �0 increases from 1�10−4 to 6�10−4. Keeping �0=6�10−4 fixed we havesimulated �Ex�y,t for different values of �0. For �0=3.0, 5.0,and 7.0 the minimum values of �Ex�y,t are about 0.045, 0.03,and 0.02, respectively. Dependence of �Ex�y,t on �0 and l� hasbeen simulated for the model-II. It is found that the variationis small compared to model-I. Simulation shows �Ex�y,t in-creases about 8% when �0 changes from 2�10−4 to 8�10−4. It is to be noted that the shape of �Ex�y,t is close tothe experimentally measured radial electric field of most to-kamaks. Figure 3�b� shows shear of radial electric fieldd�Ex�y,t /dx for the model-I �solid line� and model-II �dashedline�. Model-I indicates higher d�Ex�y,t /dx in the edge com-pared to model-II. Simulation shows that the shear increaseswith the increase in �0 and with the decrease in �0. In thecase of model-II, the shear varies more slowly for the varia-tion of �0 and l�. The self-consistent generation of the shearregulates the edge and SOL turbulence.

The typical Ex and Ey are shown in Figs. 4�a�–4�d� toshow how Ex and Ey varies in space. Figures 4�a� and 4�b�and 4�c� and 4�d� show Ex and Ey variations for model-I andmodel-II, respectively. In both the cases, Ey contours are ra-dially elongated in the SOL regions and are responsible for abursty transport.

The maximum value of Ex in the edge region can beestimated from zonal flow equation, which is obtained fromintegration and poloidal averaging of Eq. �6�,

FIG. 2. Particle source Sn, ��x�, and �x� as a function of radial x. For Sn,S0=5�10−4 and �s=5.0 are used. � is maximum at x=0 and is equal to �0

�6�10−4�. �x� is maximum in the SOL and is equal to 0 �2�10−4�. Itdrops to zero in the edge smoothly with parameter �0=3.0. The edge-to-SOLtransition region is shown by dotted vertical lines at x=25 and x=−25.

FIG. 3. Plots �a� and �b� show poloidal and time-averaged radial electricfields �Ex�y,t and its shear. Solid and dashed lines indicate model-I andmodel-II, respectively.

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�

�t�vy� = −

�

�x�vyvx� + ��

�2

�x2 �vy� , �11�

where �Ex�= �vy�=−� /�x���, Reynold stress term �vyvx� isobtained from z��� ��2� term in Eq. �6�, and �f� indi-cates poloidal average of f . Using Eq. �11� and mixing scalelength approximation, we obtain

�Ex� � 1

��kxln2� , �12�

where ln indicates radial density scale length and kx indicatesradial wave number. Using ln 25, kx 0.6 �estimated fromthe smallest structure obtained from simulation�, ��=0.01,we obtain �Ex��0.26. This is about five times higher thanthe simulation results �Fig. 3�a��. This difference is becauseof the fact that the turbulence generates much higher �� thanits input value.13 The theoretical estimation of radial electricfield in the SOL is given in Ref. 16.

In the numerical simulation, we have also observed E�B flows. Typical Ex�B values can be obtained from Figs.4�a� and 4�c� for the model-I and model-II, respectively. Inthe edge, the flows are mainly zonal flows �poloidal modenumber m=0� as is expected from Fig. 3. Near x=20 and x=40 the zonal flows are maximum and opposite in sign formodel-I. In model-II, the flows are maximum near x=8 andx=25 and change sign at about x=20. It is to be noted thatthe local maximum speeds for typical Aditya parameters nearx=20 �model-I� and x=10 are about 3.0 km/s along the iondiamagnetic drift direction. The zonal flows have also beenreported earlier in 3D simulations. The radial flows are alsoobserved because of Ey �B motion. The positive �negative�sign in Figs. 4�b� and 4�d� indicates that the velocity is alongradially outward �inward� direction. It is to be noted thattypical radially outward speed is about 0.1cs. The Ey contrib-utes to the radial particle transport and formation of streamerstructures.

The effect of viscosity and source on the radial electricfield are investigated. The viscosity coefficient �� generallyoriginates from the anisotropy of pressure tensor. In this fluid

code, the �� appears as an ad hoc parameter to ensure damp-ing of small-scale structure in the linear phase of the turbu-lence. But in the fully turbulent phase, the effect of �� issignificant. The zonal flow shear in the saturated turbulentstate can be obtained from Eq. �11� as ��vy� /�x��vyvx� /��.It indicates suppression of zonal flow shearing by the in-crease in ��. An efficient zonal flow shearing tends to reducethe radial correlation of convective cells and hence lowersthe transport.26 Therefore, viscosity appears as a key param-eter for the generation of zonal flow and hence a regulationof transport. Theoretical,27 numerical,7,28 and alsoexperimental29,30 studies show that the zonal flow regulatesthe turbulent transport.

However, it should be born in mind that the input �� hasusually a little or no impact on the code output as normallyturbulence generates larger viscosity and diffusion whichcontributes to an effective ��, D� which is much higher.13

We have carried out the simulations using D�+��=0.02with ���0.01 keeping all other input parameters same asgiven in Sec. II. The values of D�+�� has been kept con-stant to keep the linear growth rate same for all simulations,and only �� is varied. The simulations show that the Ex

remains almost the same for ��=1�10−2 to 4�10−3. For��=3�10−3 to 1�10−3 model-II provides a shape of Ex

similar to model-I. The magnitude of Ex increases with thedecrease of ��. At ��=1�10−3, the Ex is almost two timeshigher than the value of Ex obtained from model-I for ��

=1�10−2.We have carried out the simulation for different

e-folding length �s of the source Sn, keeping all other inputparameters same. The simulation shows that the density andthe Ex increase slightly by increasing �s from 5.0 to 10.0.However, increasing S0 from 2�10−4 to 1�10−3 �keeping�s=5.0� the Ex increases slightly but normalized value ofdensity increases almost two times.

B. Streamer structures and condition for streamerbreaking

Radially elongated streamer structure of the turbulentflux denoted by ��x ,y�=−n�� /�x has also been observed inour simulation. Similar streamer structures have also beenreported in earlier 3D edge simulations.6,7,9 As the edge andSOL regions are connected in our model, we are able toobserve the behavior of the streamers in the transition regionin our simulation. Formation of a streamer structure ismainly due to minimum poloidal flows and maximum radialflows. The simulation shows the streamers either break ordeform. A number of such structures obtained from both themodels have been analyzed to arrive at a condition of break-ing. It is observed that the shear �Ex /�x and the Ey within astreamer structure are responsible for the breaking. The shearmakes different parts of the streamer structure to move withdifferent poloidal velocities. Consequently, a net displace-ment among different parts of the structures can be obtained.If the resultant displacement between two major parts duringthe eddy turn over time is greater than the poloidal width ofthe streamer, the streamer will break into two. The streamer

FIG. 4. �Color online� Radial Ex�x ,y� and poloidal Ey�x ,y� fields at aninstant of time. Plots �a� and �b� are for model-I and plots �c� and �d� are formodel-II.

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will deform if the resultant displacement is lower than thepoloidal width. Mathematically, the condition of breaking is

��Ex/�x��Ey�

�x2 � �y , �13�

where �f� indicates radial average of f , and �x and �y indicateradial and poloidal widths of the streamer, respectively. Us-ing model-II, the temporal behavior of a streamer structurebefore and after the breakup is shown in Figs. 5�a�–5�d�. Thestreamer marked by arrow is seen to break up as is shown inFig. 5�b�. Figures 6�a� and 6�b� show the Ey and �Ex /�x in

the �0,−30� to �30, 5� box regions. Figure 6�a� indicates�Ey��0.05, and Fig. 6�b� indicates ��Ex /�x��0.01 and �x

�10. The �x is approximated to a radial length where theshear has its significant values. The �y is about 8–12. There-fore, above inequality is satisfied for the breaking. Afterbreaking, the SOL streamer structure moves mainly along theradial direction and edge streamer structure moves mainlyalong the poloidal direction as shown in Figs. 5�c� and 5�d�.It is to be noted that the streamer breaks preferentially at aposition where Ex changes sign, as in the neighborhood ofthis position the shear �Ex /�x is maximum. Therefore, mostof the streamer breaking has been observed near x=10 and36 �model-I�, and near x=18 and 38 �model-II�.

C. Density profile and density blob formation

Time evolution of poloidally averaged radial densityprofiles in the statistically stationary state is shown in Fig.7�a�. Figure 7�b� is the magnification of Fig. 7�a� to show thedensity profiles only in the edge region �x=0–25 for all y�.Both high- and low-density events traveling in the outwardand inward directions, respectively, are shown in Fig. 7�b�.This indicates a strong turbulent mixing of low density andhigh density due to the presence of Rayleigh-Taylor instabil-ity. It is to be noted that a similar turbulence had also beenreported in the earlier 3D simulation6–8 in the edge region.Figure 7�a� shows that the large bursts of density events ap-pear alternatively with a quiet period. The radial outwardburst or avalanche speed is measured from the straight line inFig. 7�a�. It is typically 1/32th part of cs, which is consistentwith the earlier simulation8,12,15,16 results and most tokamakexperimental results. It is also to be noted that using a simple

FIG. 5. �Color online� Radially elongated streamer of particle flux. �a� be-fore breaking. The streamers structures have been broken at x=15 in �b�.Both broken structures move in the radial as well as in the poloidal direc-tions, �c� and �d�. Each plot has been taken after 100 normalized timeintervals.

FIG. 6. Ey and �Ex /�x at position, �0,−30�– �30,5�, of streamers structurebreaking. �a� shows Ey and �b� shows �Ex /�x.

FIG. 7. �Color online� The time evolution of poloidally averaged radialdensity profiles in the edge �x=0–25� and SOL �x=25–88� regions. Thelines represent isodensity contours. A straight line corresponds to a velocityvaval�3�10−2. �b� is the magnification of �a� to show the turbulence mixingin the edge.

072520-6 Bisai et al. Phys. Plasmas 12, 072520 �2005�

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generic 1D model31 Tangri et al. had estimated the speedwhich is also of the same order of magnitude.

The time and poloidal averaged density profile’s usingmodel-I and model-II are shown in Fig. 8. The solid �dashed�line is obtained from model-I �model-II�. A vertical straightline at x=25 in Fig. 8 indicates the edge-to-SOL transitionregion. In the edge, model-I yields more peaked density pro-file than model-II as the model-I yields higher velocity shear�Ex /�x stabilization than the model-II. A valley �we call itvalley as the density profile is almost radially flat� near x=20 has been formed as ��Ex�y /�x=0 at this location. In themodel-II, valley near x=25 is negligibly small as this posi-tion is followed by a relatively low velocity shear. The SOLwidth is about 40�s which is about 1–2 cm and is typical formost tokamaks. Detachment of density structures from themain plasma has been observed. The detachment mainly oc-curs near x=20 region where the Ex changes sign �model-I�.The condition of detachment from the density streamer is thesame as given in Eq. �13�. The detached density forms like a“blob” and moves in the radial as well as in the poloidaldirections. The radial motion of the blob is mainly due tocurvature of the magnetic field lines. A detailed study of theblob dynamics will be given elsewhere.

D. Particle transport and statistical properties of flux

We have calculated the effective diffusion coefficient,Deff�x�=−����x , t��t / ��N�x , t� /�x�t, in the edge and SOL,and is shown in Fig. 9 using model-I �solid line� andmodel-II �dashed line�. The Deff is normalized by Bohm dif-fusion coefficient ��scs�. The term ���x , t� is denoted by���x , t��−�n�� /�y�y. It is to be noted that Deff has the peaksnear x=20 and x=40 because of zero velocity shear at thesepositions for model-I. The Deff for the model-II is alwayshigher than that for the model-I as the overall velocity shearof the model-II is always lower than that of the model-I.Figure 9 shows that the Deff saturates to �0.6 ��0.7� afterradial position x=45 �x=55� for the model-I �model-II� inthe SOL region.

The turbulent particle transport is simulated in the SOLat �45,0� and in the edge �15,0� regions using both the mod-

els. Simulation shows the transport properties obtained frommodel-I and model-II are similar. Here, we are presentingonly the model-I results. The particle flux obtained from thismodel is shown in Figs. 10�a� and 10�b�. It is to be noted thatthe flux time series �normalized by its standard deviation� inFigs. 10�a� and 10�b� shows both inward �negative� and out-ward �positive� flux but the net time-averaged flux is out-ward. The skewness and kurtosis obtained from the edge�Fig. 10�b�� and SOL �Fig. 10�a�� are about 1.6 and 8.0, and2.3 and 16.0, respectively. Occasional large positive flux inthe time series indicates the transport is bursty in nature.Therefore, SOL turbulence shows more burstiness as com-pared to the edge turbulence. The same has been observed ina tokamak experiment also. The probability distributionfunction �PDF� obtained from the flux time series of Figs.10�a� and 10�b� is shown in Fig. 11, the solid line indicates

FIG. 8. Time and poloidal averages of density �n�y,t. Solid and dashed linesare obtained from model-I and model-II, respectively.

FIG. 9. Effective diffusion coefficient Deff normalized to �scs for model-I�solid line� and model-II �dashed line�.

FIG. 10. Particle flux �model-I� obtained in the SOL region �45,0� and edge�15,0�. Plots show outward �positive� and inward �negative� flux. The skew-ness and kurtosis are about 2.3 and 16.0 in the SOL, and 1.6 and 8.0 in theedge.

072520-7 Edge and scrape-off layer tokamak plasma turbulence… Phys. Plasmas 12, 072520 �2005�

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PDF obtained from the SOL and the dotted line indicatesPDF obtained from the edge. It is to be noted that similarPDFs have been observed from the most tokamak experi-mental data.

The flux time series obtained from the Eqs. �9� and �10�is shown in Fig. 12. It shows that the transport is less burstyas occasional large amplitude bursts appear in comparision toFig. 10�a�. The skewness and the kurtosis of the flux timeseries are about 1.7 and 7.7, respectively, which deviatelargely from the experimental results. It is to be noted thatthe skewness and the kurtosis of the flux time series obtainedfrom Eqs. �5�–�8� are closer to the experimental results. Thesource term in the Eq. �9� is poloidally constant and indepen-dent of time. In actual tokamak, net flux may be fixed butprofiles fluctuate around their time average value and it mayhave poloidal variations. Therefore, the source in Eq. �9�which drives instability in the SOL is not a realistic one asthe source is spatiotemporally constant. The source whichdrives the instability in the SOL, obtained from Eqs. �5�–�8�,is more realistic as it has both spatial and temporal varia-

tions. The source term in Eq. �5� drives instability in the edgeand ejects plasma into the SOL. The SOL equilibrium ismodified and drives instability in the SOL. It illustrates aninfluence of tokamak edge turbulence on the SOL.

Aditya tokamak results and comparison with simulationare given here. Figure 13�a� shows the time series of particleflux measured 6 mm inside the SOL region of Aditya toka-mak. The experimental details are given in Ref. 32. The par-ticle flux measured 6 mm inside the edge region is shown inFig. 13�b�. It is observed that the net flux is outward in boththe cases, but the flux is more bursty in the SOL. The bursti-ness is also indicated by the estimated values of skewnessand kurtosis for the two regions. The skewness and kurtosisfor the flux in the SOL are 3.7 and 40.0 as compared to 2.6and 16.0, respectively, in the edge region. It is to be notedthat skewness and kurtosis obtained from the joint edge andSOL simulation deviate from Aditya experiments by a factorwithin 2. The PDFs for the two data sets are shown in Fig.14. It is observed that the PDF for the SOL data has muchlarger positive tail indicating larger bursts.

IV. SUMMARY AND DISCUSSION

We have simulated the edge as well as SOL turbulencefrom a single 2D code. Although it is a simple 2D, our simu-lation shows most of the basic features of the edge and SOLturbulence. Our simulation also shows an existence of radialelectric field and hence zonal flow. The radial electric fieldchanges sign between edge and SOL. The shear of the radialelectric field self-regulates the turbulence. Radially elongatedstreamers which build up intermittently and give rise to tur-bulent transport events are also observed. The streamer struc-

FIG. 11. Comparison of PDFs between PDF obtained from joint edge plusSOL simulation. The solid and dashed lines indicate PDFs obtained from theSOL and the edge regions.

FIG. 12. Time series of particle flux obtained from the numerical solutionsof the Eqs. �9� and �10�. This time series is measured at the same location aswas done in Fig. 10�a�. The skewness and kurtosis are about 1.7 and 7.7,respectively.

FIG. 13. The time series of particle flux estimated from the multipin mea-surements of ion saturation current and floating potential on alternate probesin Aditya tokamak. Probes are placed �a� 6 mm in the SOL region and �b�6 mm in the edge region. The mean and standard deviations of the particleflux are ���=0.94�1016 cm−2 s−1 and �=3.2�1016 cm−2 s−1, respectively.The Y axis shows particle flux normalized to the standard deviation.

072520-8 Bisai et al. Phys. Plasmas 12, 072520 �2005�

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tures occasionally break mainly at a position where the radialelectric field changes sign. A condition of the breaking ofstreamers has been derived. Formation of density blobs hasalso been simulated. The simulation shows the density frontpropagates in the SOL at a speed of about 1 /32th part of thesound speed. The effective diffusion coefficient in the SOL isclose to the Bohm diffusion. An influence of edge turbulenceon the SOL turbulence has been observed. The SOL turbu-lence obtained from our 2D simulation is more realistic andcloser to the experimental results than previously studied 2DSOL simulation. Lastly we compared our simulation resultswith Aditya tokamak results and found reasonable agree-ment.

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FIG. 14. A comparison of the probability distribution function �PDF� ofparticle flux in the SOL �solid line� and the edge region �dashed line� ofAditya tokamak

072520-9 Edge and scrape-off layer tokamak plasma turbulence… Phys. Plasmas 12, 072520 �2005�

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