One-dimensional model of detached plasmas in the scrape-off layer of a divertortokamakRajiv Goswami, Predhiman Kaw, Manoj Warrier, Raghvendra Singh, and Shishir Deshpande Citation: Physics of Plasmas (1994-present) 8, 857 (2001); doi: 10.1063/1.1342028 View online: http://dx.doi.org/10.1063/1.1342028 View Table of Contents: http://scitation.aip.org/content/aip/journal/pop/8/3?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 Predictive two-dimensional scrape-off layer plasma transport modeling of phase-I operations of tokamak SST-1using SOLPS5 Phys. Plasmas 21, 022504 (2014); 10.1063/1.4864628 Scrape-off layer tokamak plasma turbulence Phys. Plasmas 19, 052509 (2012); 10.1063/1.4718714 Influence of scrape-off layer on plasma confinement Phys. Plasmas 18, 032509 (2011); 10.1063/1.3566008 Generalized ballooning and sheath instabilities in the scrape-off layer of divertor tokamaks Phys. Plasmas 4, 1330 (1997); 10.1063/1.872309

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One-dimensional model of detached plasmas in the scrape-off layerof a divertor tokamak

Rajiv Goswami, Predhiman Kaw, Manoj Warrier, Raghvendra Singh,and Shishir DeshpandeInstitute for Plasma Research, Bhat, Gandhinagar 382428, India

~Received 18 October 2000; accepted 20 November 2000!

Numerical and analytical study of a detached divertor equilibrium is presented. The model usesone-dimensional equations for continuity, momentum and energy balance with radiation, ionization,charge-exchange, and recombination processes. A reasonably simple neutral model is alsoemployed. Analytical calculation, using a simple five-region model for a case with negligibleconvective heat flux and constant sources/sinks, captures the essence of detailed numericalcalculation for the same case. More general cases are handled numerically. The detachment isstudied as a function of the ratioQ' /S' @the ratio of power and particle volume source, comingfrom the core to the scrape-off layer~SOL! region#. For low values ofQ' /S' ~detached state!, atthe midplane and at the target, the ion temperature (Ti) is almost equal to the electron temperature(Te). As this ratio increases~attached state!, Ti is larger thanTe at the midplane. However at thetarget,Te is found to be slightly larger thanTi . It is also observed that asQ' /S' increases, theregion of most intense radiation shifts progressively from closer to the X-point towards the targetplate. © 2001 American Institute of Physics.@DOI: 10.1063/1.1342028#

I. INTRODUCTION

Charged particle power handling in a steady-state opera-tion remains a critical issue for the next step tokamak de-vices such as International Thermonuclear Experiment Reac-tor ~ITER!.1 Reduction of the heat flux to a technicallymanageable value;5 MW/m2 ~Ref. 2! before it reaches thedivertor plates appears imperative. There is a need to mini-mize the sputtering from divertor plates to decrease divertorplate erosion and also to keep the impurity backflow to thecore at a minimum. Thus, a means of reducing the parallelpower density and particle flux through volumetric losses~hydrogenic and impurity radiation, charge exchange, etc.!must be found.

There are two methods for high power dissipation in thedivertor region, namely, operating the device with a highrecycling divertor, or operating with a detached divertor.3 Inthe high recycling case, erosion associated with the largeincident ion fluxes may seriously limit the lifetime of thedivertor target,4 and the power deposited onto the divertorcan generally exceed the steady state power handling capac-ity of the target. In the detached divertor case, recombinationpower deposited on the target decreases with the reduction ofthe plasma flux.5 Since the largest particle flux now imping-ing on the surfaces is in the form of hydrogenic atoms andmolecules which are not accelerated by the sheath potential,and hence have lower energies than the corresponding ions,the amount of physical sputtering at the target is alsominimized.6 However, there are some disadvantages of thismode of operation. The chemical sputtering may increase~but being distributed over a larger area, its severity can bereduced!. Furthermore, the volumetric losses outside theseparatrix in the divertor region are decreased, the neutralcompression ratio is decreased,7 and the penetration effi-

ciency of impurities, both intrinsic and puffed, increases.8 Toameliorate these drawbacks the following measures havebeen implemented. The possible effect of neutrals on con-finement has lead to a ‘‘closed divertor geometry,’’9–11 lead-ing to reduced backflow and enhanced compression ratio.While, experiments on DIII-D have demonstrated the effi-cacy of augmenting the bulk plasma ion flow in the scrape-off layer ~SOL! sufficiently to overcome the thermal gradientforce, which otherwise drives impurities toward the coreplasma.12,13The detached divertor operation has thus becomean accepted technique for reducing heat loads to the divertorplates. It is therefore of interest to know the conditions underwhich divertor detachment occurs, and the physics of detach-ment.

Divertor detachment is primarily achieved by either in-creasing the core plasma density or by the addition ofimpurities.14 Experimentally, divertor detachment is definedas a loss of ion saturation current (I sat) and plasma heat fluxon the divertor plate, with additional effects such as, an in-crease in neutral pressure, large radiative losses in the di-vertor and X-point region, sharp drop in plasma density andpressure, and an electron temperature in the range of 1–3 eVnear the plate.15

Substantial research activity has taken place in theexperimental14,16–19 and modeling20–24 areas pertaining todetachment. It is now generally agreed that the strong drop inheat flux is obtained due to impurity radiation. Charge ex-change~CX! and elastic collisions between the ions flowingto the plate and the recycling atoms and molecules lead tothe observed drop in the pressure.25 Volume recombination,which can be significant in such low temperature plasmas,and the reduction in the ion source rate further upstream dueto the limitation or reduction in the power flow provide themechanism for the observed divertor plate current

PHYSICS OF PLASMAS VOLUME 8, NUMBER 3 MARCH 2001

8571070-664X/2001/8(3)/857/14/$18.00 © 2001 American Institute of Physics

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reduction.26 Since the physics of the edge plasma in a toka-mak is very intricate involving many atomic physics pro-cesses and geometrical effects, it is very difficult to constructreasonably simple mathematical models for the edge plasma.Large, multifluid, two-dimensional~2D! plasma fluid codessuch as B2~Ref. 27! and UEDGE ~Ref. 28! coupled toMonte Carlo/fluid codes for neutral species, e.g., EIRENE/NIMBUS ~Refs. 29,30! have been developed to investigatesuch problems. However, the complexity of the nonlinearequations and the various free parameters makes it necessaryto interpret these simulations in simpler analytical terms, soas to generate an understanding of the important processes.Such simple models can provide physical insight, qualita-tively useful information and also some simple limits thatcan be used to check large codes. Therefore, the incentive isstrong to extract as much information and insight as possiblefrom one-dimensional~1D! numerical and analytical model-ing.

In this paper we present some analytical and numericalsimulation results of the divertor detachment process basedon a time dependent 1D fluid transport model of the SOLplasma. In our model we have included the ion energy trans-port equation as against many other models which usuallyassume single temperature.31 The equal temperature assump-tion is questionable, because the ions and electrons havequite different source/sink terms, and their very differentmasses can thus produce a tendency to thermally decouple.Furthermore, impurity production and transport are strongfunction of the ion temperature, through the thermal force onimpurities. Since the SOL measurements of ion temperatureare quite sparse, this additional equation can help to assessvarious facets, such as the relative power sharing betweenelectrons and ions and ion temperature gradients along thefield. We also present a unified analytical description of de-tachment in terms of fronts associated with the variousatomic processes and compare it against a more complete 1Dnumerical model.

The plan of the paper is as follows: In Sec. II the physi-cal model used for 1D numerical simulation is presented. InSec. III we present the various results of the simulation. InSec. IV we briefly review the various existing analyticalmodels related to detachment. In Sec. V we analytically ex-amine the static problem and delineate the conditions for theformation of various fronts associated with radiation, ioniza-tion, and volume recombination. Finally, in Sec. VI our con-clusions are summarized.

II. PHYSICAL MODEL

A. Basic equations

Our physical model consists of ions, electrons, and hy-drogen atoms. A time dependent 1D code which solves theplasma fluid transport equations along the field lines, using aflux corrected transport algorithm32 has been used. The arclength,x, along the field lines from the midplane (x50) tothe target plates (x5L) is used as the spatial coordinate.Assuming charge neutrality and ambipolar conditions, i.e.,ne5ni5n andve5v i5v, the transport equations are givenas follows:

]n

]t1

]

]x~nv !5S'1nnnSi2n2~Srr 1S3b!, ~1!

]

]t~mnv !1

]

]x S mnv22h i i]v]xD

52]

]x@n~Te1Ti !#2mnnnSx~v2vn!

2mn2v~Srr 1S3b!, ~2!

]

]t S 3

2nTeD1

]

]x S 5

2nTev2k ie

]Te

]x D5Q'e2n2j IL~Te!2In~nnSi2nS3b!

2 32Ten

2~Srr 1S3b!2nnei~Te2Ti !, ~3!

]

]t S 3

2nTi1

mnv2

2 D1]

]x S 5

2nTiv1

mnv3

22k i i

]Ti

]x D5Q' i1

]

]x S h i i]v]x

v D2S 3

2Ti1

mv2

2 Dn2~Srr 1S3b!

2F3

2~Ti2Tn!1

mv2

2 GnnnSx2nnei~Ti2Te!, ~4!

wherem is the ion mass,n andv are the plasma density andvelocity, nn and vn are the neutral density and velocity,Te

andTi are the electron and ion temperatures,Tn is the tem-perature of the neutral atoms,Si , Sx , Srr , andS3b are theMaxwellian averaged electron impact ionization, ion-neutralcharge exchange, radiative, and three-body recombinationrate coefficients, respectively. In this calculation, optically-thin conditions are assumed for the transport of Lyman ra-diation, and thus this calculation overestimates the impor-tance of volume recombination, i.e., radiation trappingeffectively reduces the recombination rate.33 The k ie( i ) ,h ie( i ) , andnei are the electron~ion! parallel thermal conduc-tivities, viscosity, and thermal equipartition coefficients, re-spectively. The transport coefficients are classical and do notinclude the flux limits in order to account for kinetic effects,which are important when the scale length of the variation ofplasma parameters along the field is comparable to the rel-evant mean free paths. This would thus overestimate the heatflux for a given gradient and thus the radiation loss for de-tachment. The ionization potential of hydrogen isI, andS'

and Q'e( i ) are the uniformly distributed perpendicularsources of particles and heat into the SOL from the core~i.e.,from the midplane until the X-point!. A coronal equilibriummodel is used in calculating the radiation losses due to car-bon impurity, which is taken to be a fixed fraction (j I) of theion density. This simple impurity model provides no infor-mation on the impurity redistribution but it retains the featurethat is most essential for detachment, namely a strong reduc-tion of the heat flux through line radiation. The energy trans-port associated with the expenditure of 13.6 eV of electronenergy to remove the electron from the atom at ionizationand the subsequent redeposition of this 13.6 eV at recombi-nation is retained.

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B. Boundary conditions

At the midplane, assuming symmetry, we have em-ployed the following conditions:

]n/]x50,

]Te( i ) /]x50, ~5!

v50.

Hydrodynamic models of transport in a high-recycling SOLrequire that suitable boundary conditions be imposed at theentrance to the sheath region. The following outflow bound-ary conditions at the plasma–sheath interface have been em-ployed in this simulation:

v5cs5A~Te1Ti !/mi , ~6!

i.e., ions reach the sound velocity at the plates, which is theminimum velocity that satisfies the Bohm condition.

The energy fluxes are specified as

qie[5

2nvTe2k ie

]Te

]x5gencsTe ,

~7!

qi i[5

2nvTi2k i i

]Ti

]x5g incsTi ,

wherege53.5 andg i52.5, are the electron and ion sheathenergy transmission factors, respectively. Following standardpractice we fix these factors as constants in the computation.

C. Neutral model

A proper treatment of the neutral dynamics would re-quire a Monte Carlo routine accounting for realistic divertorand vacuum system geometries and incorporating detailedsurface and atomic physics like EIRENE.29 Such a treatment,though more accurate and preferable, is not useful for aqualitative study of the detachment process. Here we presenta simple neutral model that can be tailored to match theMonte Carlo code results or experimental observations of theneutral profiles along the field lines.

The major source of neutrals in the divertor region is thedivertor plate where the H1 ions get deposited, recombineand mostly get desorbed as H2 molecules. We assume thatexcitation and subsequent dissociation into Franck–Condonneutrals are the dominant processes when these moleculesenter the plasma fan. The energy loss resulting from thisdissociation is accounted for in the electron energy equation.The density of H neutrals at the divertor plate is howeverdependent on several other factors like how many neutralsbounce off other structures and reach the divertor plate~geo-metric effects! and how many neutrals are pumped off fromthe divertor region~pumping effects!. We therefore treat thedensity of H neutrals at the divertor plate as an input to thecode. The choice of this model is also motivated by the ex-perimental observation on Alcator C-Mod,3 that the neutralcompression decreases upon detachment. It was found thatthis reduction is primarily due to an increase in the neutralpressure at the midplane, as the divertor neutral pressure re-mains essentially constant. The neutrals are attenuated alongfield lines according to the local ionization and charge ex-

change mean free paths and also their transport across thefield lines away from the plasma region and into the plasmaregion. The first two attenuation factors~ionization andcharge exchange! are worked out from the local plasma den-sities and temperatures. The transport losses/gains acrossfield lines are difficult to account for and a free factorw isinput which helps us to tailor the neutral profile to matchMonte Carlo code or experimental observations. The neutralprofile is specified asnn( i )5nn( i 11)e2(dx/l(x)), wherenn(xd) is the neutral density at the divertor plate andl(x) isthe neutral penetration length which depends on the positionx through the local plasma densityn and temperaturesTe andTi and also the free factorw to account for transport acrossthe field lines. The neutral atom moving in the plasma with avelocity vneut corresponding to Franck–Condon~FC! energywill on an average penetrate a distance given by

l5vneut

n~Si1Sx!

1

w.

This relation holds forTe>10 eV. ForTe!10 eV, the neu-tral transport is diffusive, where many charge exchange col-lisions occur before the neutral is ionized, then

l5vneut

nA~Si3Sx!

1

w.

The above treatment of the neutrals qualitatively matches theneutral profiles obtained from more detailed treatments andthe factorw can be changed to match the results quantita-tively. To account for the significant ionization in the mainchamber due to neutral ‘‘leakage’’ out of the divertor or~undesired! plasma contact with the main vessel walls, weassume a uniform background density of neutrals (nn

bg), fromthe midplane to the location where the neutral profile falls upto this value. In this work we have chosennn

bg and the neutraldensity at the platenn(xd) to be equal to 131010 cm23 and531013 cm23, respectively. These values correspond to neu-tral pressure of;1023 Torr in the divertor and;1026 Torrin the main chamber, respectively. We also assumevn50 inEq. ~2!. This assumption corresponds to an upper limit on theamount of momentum loss due to CX collisions.

The kinetic temperature of neutralsTn in the simulationis taken to be equal to 2eFC/3, whereeFC53.5 eV is theFranck–Condon energy. This is important to account for thecharge exchange term in the ion energy equation~especiallyin the detached zone where theTi is comparable or evenlesser than 2eFC/3).

III. COMPUTATIONAL RESULTS

Results are presented for a problem that represents onehalf of a single null divertor with geometry scale lengthsclose to a typical tokamak divertor. The distance from themidplane to the divertor plate isL530 m. The distance fromthe midplane to the X-point is set at 21 m. All lengths andvelocities are normalized to the system lengthL and the ve-locity vT0

, respectively, wherevT05AT0 /mi , and T0510

eV. The total number of grids is 1000. The grid spacingD5131023. The normalized timestepdt is calculated from

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cD/vmax, where,c;0.001 is the Courant number andvmax isthe maximum normalized velocity. A schematic representa-tion of this model problem is presented in Fig. 1. We con-sider carbon as the only impurity species, and take its frac-tion to be j I.0.04. This fraction can be lowered ifdeviations from the coronal equilibrium due to CX reactionswith neutrals or rapid recycling of the impurities are takeninto account. The impurities are assumed to be localized inthe region between the X-point and the target. By carryingout the simulation with different values of particle sourceS'

and heat sourceQ'e( i ) , the change in plasma parameters areobserved from the attached states to the detached states. Theattached plasma is characterized by high temperature at theplate (Tt), and the detached one by very lowTt . High Tt

cases also have high particle currents to the plate, while lowtemperatures at the plate have low particle currents. For the

cases under discussion, the power flow at separatrix is as-sumed to be equally shared between the ions and electrons.

First, we show stationary solution for a detached casewith S'53.731022 m23 s21, andQ'52.6 MW m23 in Fig.2. An estimate of these values can be obtained from a typicalDIII-D discharge.34 The surface area of DIII-D is;50 m2

and a SOL width of;1 cm. The power inputQin is in therange of 1–15 MW. In steady state, approximately 80% ofthis input power flows across the separatrix into the SOL, therest being lost via radiation in the core. Using these valueswe obtain Qin for the detached case to be;1.625 MW,which is in the range of power injected. The total particleflux at the separatrix is in the range of 1000–3000 A forDIII-D. Our choice ofS' translates into;2950 A, which iswithin the observed range.

Figures 2~a! and 2~b! present theTe andTi profiles. Thetemperatures decrease slowly until the X-point. Furtherdownstream there is a sharp drop in temperatures at;24.3 mdue to radiation~radiation front! and CX energy losses. Theelectron and ion temperatures in this region fall from;10eV to ;2 eV in less than 0.2 m of parallel length. The strongT5/2 dependence of heat conduction results in such short spa-tial scales of the temperature gradients. The thermal front isseen to occur at some distance away from the plate towardsthe X-point, which is typical of detachment observations.35

The target electron and ion temperatures,Tet and Tit arefound to be almost equal at;1.6 eV and 1.7 eV, while theircorresponding midplane values,Tem andTim are;39.6 eVand 41.3 eV, respectively.

In Fig. 2~c! we plot the plasma density. Due to only aconstant perpendicular particle sourceS' until the X-point,the density remains almost constant. The sudden rise in den-sity in the vicinity of the thermal front is related to the ion-ization of the neutrals encountered by the plasma at the ion-ization front, and also from reduction in the flow velocitydue to ion-neutral collisions upstream. AsTe falls towards 2

FIG. 1. A schematic diagram of the one-dimensional simulation model. MPdenotes the midplane, X is the X-point, and DP is the divertor plate.

FIG. 2. Plots of plasma parameters along the field line.Here, a detached case is shown. Dotted vertical linesrepresent the X-point location. Here,Q'e5Q' i51.3MW m23, S'53.731022 m23 s21, nn

bg5131010

cm23, nn(xd)5531013 cm23, j I50.045.

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eV, the ionization sources decrease rapidly and because ofvolume recombination the plasma starts to recombine andhence there is observed a steady drop in density. At the tar-get, plasma densitynt is ;1.831013 cm23, lower than itsmidplane valuenm of ;5.831013 cm23.

Figure 2~d! shows the velocity profile. The velocity risesalmost linearly from the midplane until the X-point due tothe constant perpendicular particle source. This can be ex-plained from the continuity equation. In a steady state, wehavenv;S'x, and sincen; constant,v}x. Then, due topronounced CX collision of the plasma ions with the neu-trals, the velocity decreases. In Fig. 2~e!, the plot of neutraldensity is given and in Fig. 2~f!, we plot the ratio of the fluidvelocity to the local sound speed cs , where, cs

5A(Te1Ti)/mi . The plasma flow is subsonic over the entire

parallel length. For plasma flow to be subsonic up to thetarget in the presence of both friction and recombination, it isrequired that ‘‘friction be stronger than the volumerecombination.’’36 For vn50, this condition is given by

ns v̄ rec/nns v̄cx,1, and we find this ratio to be;0.001.We plot the experimentally observed signatures of a de-

tached plasma, which are a strong drop in the plasma flux(G), total pressure (P), and parallel heat flux (qi). In Figs.3~a!, 3~b!, and 3~c!, we plot the profiles ofG, P, andqi . It isobserved that there is a strong reduction of the ion flux,plasma pressure, and heat flux at the divertor plate. The totalpressure drops about 75 times of its midplane value. Suchlarge ratios;100 of the midplane to divertor pressure havebeen measured in Alcator C-Mod.3 This very large drop in

FIG. 3. The parallel particle flux, total pressure, andtotal heat flux profiles in a detached case. The param-eters are the same as in Fig. 2.

FIG. 4. Parallel variation of charge exchange and ion-ization rates, for same set of parameters as in Fig. 2.Dotted line represents charge exchange, while the solidline denotes ionization.

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pressure could be due to ignoring of the neutral–neutral col-lisions. When these collisions are infrequent, then the neutralwould have to ‘‘revisit’’ the solid surface between each col-lision to transfer momentum out of the system. On average,most of the ion’s momentum is removed by the neutrals whohave their first ion–neutral collision, and these are the oneswhich emanate from the wall. Here, after a CX event theneutralized ion is assumed to be lost from the plasma. Theincrease in the flux is fed by the strong ionization source.The flux then decreases due to strong recombination of theplasma ions. Moreover, since the flow near the target israther stagnant at detachment@see Fig. 2~d!#, the ions getmore time to recombine.

The pressure profile in Fig. 3~b! is constant until a sig-nificant concentration of neutrals is encountered, which re-

sults in a substantial loss of parallel momentum due to ion–neutral collisions. It is also to be noted that while thepresence of friction is a necessary condition to produce thepressure ratios observed in these discharges, a further condi-tion is also required, i.e., a strong reduction in the neutralparticle ionization rate (nnSi) in comparison to the rate ofion–neutral momentum loss (nnSx) is needed. This can beseen from Fig. 4.

In order to make a transition to an attached discharge,keeping the rest of the parameters fixed, we can do the fol-lowing: ~a! increase the perpendicular heat sourceQ' , ~b!decrease the perpendicular particle sourceS' , and ~c! de-crease the impurity fractionj I . We now define somewhatarbitrarily, the transition from detached to attached plasmasas being the power at which the plateTe rises to;4 eV. At

FIG. 5. Plots of plasma parameters along the field line.Here, an attached case is shown. In this case we simplyincrease the perpendicular heat source toQ'e5Q' i

52.44 MW m23, with the rest of the values as inFig. 2.

FIG. 6. The parallel particle flux, total pressure, andtotal heat flux profiles in an attached case for same pa-rameters as in Fig. 5.

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these temperatures, the neutral gas is efficiently ionized andthus cannot penetrate further towards the X-point. As a caserepresentative of this regime, keeping all other parametersfixed, we increase the heat input to;4.9 MW m23. Now thevalue of Tem and Tim are ;56.4 eV and 66.6 eV, whereasTet and Tit are ;4.1 eV and 3.2 eV, respectively. We plotthe various profiles of this case in Fig. 5.

As expected, we observe an increase in the particle flux@Fig. 6~a!#, due to a strong ionization sources and the ab-sence of recombination sink for plateTe;4.1 eV. However,a midplane to divertor pressure ratio of;4, which is morethan theclassicvalue of 2@Fig. 6~b!# is observed. This canbe attributed to the inclusion of CX momentum losses. Fig-ure 6~c! shows the parallel heat flux, and it is to be noted thatthe peak heat flux in this case drops only by a factor of 5 to;19 MW/m2, whereas in the detached case it was reducedalmost by 903 to ;0.5 MW/m2. This shows the efficiencyof detached operation over an attached one.

Figure 7 presents parameter scans to assess the impact ofvarying the ratio of perpendicular heat and particle sourcesQ' /S' , while keeping other parameters fixed. From Fig.7~a! it is observed that for lowerQ' /S' the midplane valuesof Te andTi are almost equal. Upon transiting to an attachedstate characterized by a higher ratio ofQ' /S' , the valuesshow a marked difference. At the midplaneTi is found to behigher thanTe , while Tet is slightly larger thanTit . Thiscould be predominantly due to the large charge exchangelosses in this region. It is known that there is a rapid decreasein the ratio of ionization rate to the charge exchange rate, asT drops below 10 eV. In Fig. 7~b! nt for a detached state isshown to be lower thannm , and vice versa for an attachedcondition. Moreover, the values ofnm , Tem, and Tim arelargely independent ofQ' /S' . This lack of any dramaticchanges in the upstream conditions during the transition sug-gests that it is conditionslocalized to the divertor that pre-cipitate the detached state. It is also to be noted that weobserve movement of the region of maximum radiation from

the target to locations closer to the X-point@Fig. 7~c!# as theratio is lowered. Further, from Fig. 8 it is to be noted that asQ' is increased, the total radiative losses~carbon and hydro-gen! Prad increases until it again decreases upon reattach-ment. This has been observed experimentally in the AlcatorC-Mod.37 Although conduction leads to the severe localiza-tion of the intense radiating region in Fig. 8,Prad is observedin regions with shallow temperature gradients. The reason isthe potential energy convected by the remaining plasmaflow, despite strong neutral collision rates upstream. Eachremaining ion can release 13.6 eV ionization energy in theregion where it recombines. In regions having temperatures;2 eV, Prad is possible from this collisional-radiative pro-cess.

As a check of these simulation results we calculate theSOL collisionality parameter (nSOL* ) defined in Ref. 38 as'10216nmL/Tem

2 . Consider a typical detached run havingQ' /S';439 eV,nm55.8531019 m23, L530 m, andTem

539.6 eV, we getnSOL* to be;111, which is greater than thelower limit 85 above which the flux tube will tend to detach.Similarly, for an attached run withQ' /S';824 eV, nm

55.1731019 m23, and Tem566.6 eV, we havenSOL* ;35,lower than the critical value and thus as expected a conduc-tion limited ~attached! regime results.

If the parallel plasma flow is significant over much of thelength of a flux tube, then parallel heat convection becomesimportant and the temperature ratioTm /Tt diminishes. Weobserve that the parallel plasma flow is enhanced for an at-tached casevis-a-visa detached one~Fig. 9! and we musttherefore get a smaller temperature ratio. Plugging in thenumbers we find this temperature ratio to be;13.6 and 27.6for attached and detached case, respectively.

As a final analysis, we calculate and compare the profilesof the two main forces that determine the impurity retention.These are~a! the friction force~directed toward the plate!given by mz(v2vz)/tz and ~b! the ion thermal force~di-

FIG. 7. Variation of ion and electron midplane tem-peratures, midplane, and target densities, and the frontlocation (xf) as a function of the ratio of perpendicularheat to particle sources (Q' /S').

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rected away from the plate! given bybz¹Ti . Both have beennormalized to the impurity densitynz . Heremz is the impu-rity mass,vz is the impurity velocity,tz is the Spitzer colli-sional time taken from Ref. 39 which is}Ti

3/2n21Z22, andbz is ;2.6Z2. We will plot the results for C13

(mz'123proton mass! and also assume that the carbon ionvelocity is uniform throughout the length. The comparison ofthe magnitudes between the two forces is shown in Fig. 10.The velocity chosen for the impurity parallel velocity areconsistent with Doppler-shift measurements in the divertor.35

From Fig. 10, we notice that the friction force is sufficient topush the impurities toward the plate, in both detached as wellas attached cases which we have examined.

IV. ANALYTICAL MODELING

There have been a number of analytical modeling effortson divertor plasma detachment. Flow reversal, i.e., flowaway from the target plate, has been predicted analyticallywith simple 1D ~Ref. 40! and more complete 2D~Ref. 41!considerations. Stangeby25 identified that apart from thepresence of ion–neutral friction to produce the large pressureratios observed under detached conditions, a condition fur-ther required is a strong reduction in the neutral particle ion-ization rate in comparison to the rate of ion–neutral momen-tum loss. The physics of pressure fronts have also beenapproximately dealt with by Ghendrih42 and Kesner.43 The

FIG. 8. Figure showing the radiation loss (Prad) as afunction of parallel distance (x), for various values ofQ' , keeping other values fixed as in Fig. 2. Here, a, b,c, and d representQ' equal to 2.6 MW/m3, 3.2MW/m3, 4.2 MW/m3, and 4.88 MW/m3, respectively.

FIG. 9. Comparison of parallel velocity in attached anddetached regimes. Solid line denotes detached regime,while broken line represents the attached regime.

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continuity and pressure balance equations were solved forthe length of the detached region in Ref. 42 assuming thatthe neutrals enter uniformly from the private region along theseparatrix, and a bifurcated solution was obtained consistingof attached and detached states, where the detached solutionhas the ionization front closer to the X-point. Kesner showedthat, for sufficiently low-temperature and high neutral den-sity, a region of greatly reduced power flux to the endplatecan also result from the simultaneous solution of the heatconduction and the pressure balance equations.

Hutchinson44 examined the stability of a thermal frontusing a simplified radiation function for the power loss andassuming that momentum conservation is satisfied and heatflow is via electron heat conduction. It was concluded that inthe detached state, the thermal front is stabilized near theX-point due to the cross-field heat transport experiencedabove the X-point, and that the thermal front is probablyonly stable at positions intermediate between the target andX-point for a narrow range of upstream densities.

Krasheninnikovet al.45 have shown analytically that anincrease in upstream radiation loss can account, simulta-neously, for a drop in divertor plate heat flux and particle

flux to the divertor plate both in the Knudsen and fluid neu-tral limits. Also there have been numerous forms of simple1D ‘‘two-point’’ models based on plasma pressure balanceand electron heat conduction.

In the majority of the above mentioned studies, the ef-fects of volume recombination on detached solutions wereignored. Moreover, each front has been individually exam-ined and there is no analytical treatment which takes intoconsideration the existence of all the fronts.

V. MODEL OF FRONTS

In this section we consider the coupled Eqs.~1!–~4! forstatic fronts and present their analytic solutions, connectingthe midplane to the divertor plate by a five-region analysisinvolving radiation, ionization, momentum loss, and volumerecombination effects. These sources/sinks are assumed to beconstants for analytical tractability. The solutions in the dif-ferent regions are obtained on matching the variables andtheir derivatives at the respective boundaries. We assume forsimplicity that the electron heat conduction dominates overconvection in the energy equation, along the entire parallel

FIG. 10. Plots showing the comparison between theparallel forces acting on the impurity, viz., frictionforce and ion temperature gradient force, in detachedand attached cases, with two different values of impu-rity ion velocity vz .

865Phys. Plasmas, Vol. 8, No. 3, March 2001 One-dimensional model of detached plasmas . . .

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length. Further we also assume thatTi50, and the plasmaflow is subsonic, i.e.,mnv2!p, which has been observed tobe the case in the simulation of detached regimes. Finally,simple ‘‘step-function’’ models are used for all sources andsinks with known ‘‘cut-off’’ temperatures.

Thus, in region I, assumed to exist from the midplane(x50) to the X-point (xx), the set of continuity, momentum,and energy equations are

d

dx~nv !5S' ,

dp

dx50,

d

dx S k0T5/2dT

dxD52Q' ,

~8a!

where p5nT, and k052400 W m21 eV27/2. The solutionscorresponding to Eq.~8!, subject to the boundary conditionsgiven by Eq.~5! andT(x50)5Tm , n(x50)5nm , are

nv5S'x, p5pm , T7/25Tm7/22S 7Q'

4k0D x2. ~8b!

The equations in region II, fromxx to the start of theenergy loss zone (x5xL), in the absence of sources/sinks,are

d

dx~nv !50,

dp

dx50,

d

dx S k0T5/2dT

dxD50. ~9a!

The solutions in this region are

nv5S'xx , p5pm , T7/25Tm7/22S 7Q'

2k0D Fx2

xx

2 Gxx .

~9b!

Next, in region III, which starts atxL and ends at thebeginning of the neutral zone (x5xc), the equations are

d

dx~nv !50,

dp

dx50,

d

dx S k0T5/2dT

dxD5L, ~10a!

where L is an unknown constant sink due to the radiativelosses. The solutions to these equations are

nv5S'xx , p5pm ,

T7/25Tm7/22S 7Q'

2k0D Fx2

xx

2 Gxx1S 7L

4k0D @x2xL#2. ~10b!

Region IV exists fromxc to the onset of the recombina-tion region (x5xr), and the equations take the followingform:

d

dx~nv !5G0d~x2xc!,

dp

dx52minxnv,

d

dx S k0T5/2dT

dxD50, ~11a!

wherenx[nnSx , G0 is an increment in the flux due to ion-ization. The solutions in this region are

nv5S'xx1G0Q~x2xc!,

p5pm2minx@S'xx~x2xc!1G0r ~x2xc!#,~11b!

T7/25Tm7/22S 7Q'

2k0D Fx2

xx

2 Gxx

1S 7L

4k0D ~xc2xL!@2x2xc2xL#,

whereQ is the Heaviside step function andr is the ‘‘ramp’’function. Finally, in the last region V, fromxr to the divertorplate at (x5xd), the equations are

d

dx~nv !52R,

dp

dx52minxnv,

d

dx S k0T5/2dT

dxD50,

~12a!

with R as the constant recombination loss term, assumed tobe known. In this region the solutions are as follows:

nv5S'xx1G02R~x2xr !,

p5pm2minx~S'xx1G0!~x2xc!1 12 minxR~x2xr !

2, ~12b!

T7/25Tm7/22S 7Q'

2k0D Fx2

xx

2 Gxx

1S 7L

4k0D ~xc2xL!@2x2xc2xL#.

The above solutions given by Eqs.~8b!, ~9b!, ~10b!,~11b!, and~12b! have the following unknowns;Tm , nm , L,and the locationsxL , xc , andxr . These locations are definedin terms of their respective temperatures, i.e.,T(x5xL)5TL , T(x5xc)5Tc , andT(x5xr)5Tr , with TL , Tc , andTr assumed to be known.

Now evaluating Eq.~9b! at xL and Eq.~10b! at xc , andsubtracting them, we get the following equation indL5xc

2xL :

dL52k0

7L F S 7Q'xx

2k0D2H S 7Q'xx

2k0D 2

27L

k0~TL

7/22Tc7/2!J 1/2G .

~13!

The discriminant of Eq.~13! when equated to zero gives anestimate of the maximum radiative lossLc , i.e.,

Lc57

4k0S Q'

2 xx2

TL7/22Tc

7/2D . ~14!

We then assume thatL is a fraction of this maximum radia-tive loss Lc . Substituting this value of L, we are able tocalculatedL . Now in order to obtain the temperature profilefrom the midplane to the divertor plate, we need to knowTm , and so we rewrite Eq.~12b! as,

Tm7/25Td

7/217Q'

2k0S xd2

xx

2 D xx

27L

4k0~xc2xL!@2~xd2xc!1xc2xL#. ~15a!

On subtracting Eq.~10b! and Eq.~12b! we get

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dc[xd2xc52k0

7 S Tc7/22Td

7/2

Q'xx2LdLD . ~15b!

The divertor temperatureTd is still an unknown, and weemploy the heat balance condition to evaluate it, i.e.,

Td51

gendvd@Q'xx2LdL#, ~Joules! ~16!

where from Eq.~12b!,

ndvd5S'xx1G02R~xd2xr !, ~17a!

and

d r[xd2xr52k0

7 S Tr7/22Td

7/2

Q'xx2LdLD . ~17b!

It is to be noted thatxd2xr is again a function of the divertorplate temperatureTd . But this recombination region exists ifand only if Td drops belowTr . We therefore define a tem-peratureTd1

, corresponding to the event of having no recom-bination, i.e.,

Td15

1

ge~S'xx1G0!@Q'xx2LdL#. ~18!

Now, if Td1.Tr , then recombination cannot happen and

henceTd is equal toTd1, whereas ifTd1

is less thanTr thenrecombination can take place and so to evaluateTd we sub-stitute Eq.~17! in Eq. ~16! to get

geTdFQ'xx~S'xx1G0!S 12L

LcD 1/2

22Rk0

7~Tr

7/22Td7/2!G

5Q'2 xx

2S 12L

LcD . ~19!

The roots of this equation give the value ofTd for thegiven values ofQ' , S' , G0 , R, TL , Tc , Tr , ge , xx , andxd . Then from Eq.~15a! we calculate the midplane tempera-ture Tm and using this in Eqs.~8b!, ~9b!, ~10b!, ~11b!, and~12b!, we thus generate the full temperature profile frommidplane to the plate.

On knowing the plate temperatureTd , we then evaluatethe plate densitynd from Eq.~17a!, and so the plate pressurepd (5ndTd) is known. Then from Eq.~12b! we get the mid-plane pressurepm . The whole density profile is then con-structed on dividing the pressure equation in various zonesby the respective temperature values. Finally, on getting the

FIG. 11. Analytical and numerical profiles of electrontemperature, density, and velocity. Solid line denotesnumerical solution, while broken line represents theanalytical solution. The parameters areQ'51.3MW m23, S'55.231022 m23 s21, k052380W m21 eV27/2, TL510 eV, Tc53 eV, Tr52 eV, Td

51.6 eV,nx52.53104 s21, andR5131023 m23 s21.

867Phys. Plasmas, Vol. 8, No. 3, March 2001 One-dimensional model of detached plasmas . . .

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density values in all the regions, we construct the velocityprofile from the flux solutions in those five regions.

As an example illustrating the analytical detached solu-tion, we choose the parameters used in the simulation de-scribed earlier, as,Q'51.3 MW m23, S'.5.231022

m23 s21 ~taking into the account the source due tonnbg),

k052380 W m21 eV27/2, TL510 eV,Tc53 eV, Tr52 eV,and Td51.6 eV. We fix the increment in the flux due toionizationG0 , as 40% of the flux built up until the X-point.The other variables are estimated from the numerical solu-tion, and we choosenx52.53104 s21 and R5131023

m23 s21. Using these parameters, we obtain the temperature,density, and velocity profiles and compare them with thenumerical results. As can be seen from Fig. 11, we find goodagreement between our simple model and the complete nu-merical simulation. The position of the front can be varied bychanging the fraction of the maximum radiative lossLc , andpredictably for lower fraction the front moves towards theplate. On increasing the collision frequencynx , we find thatthere is an overall increment in the density, while the veloc-ity drops throughout, but there is no change in the frontlocation. This is illustrated by Fig. 12, wherein we increasenx to 53104 s21, keeping all the other parameters fixed.

Next we simply increase the perpendicular heat sourceQ' to2.3 MW m23, and as expected we observe that the midplanetemperature goes up, while the midplane density dropsdown, the front location shifts towards the plate, and thevelocity increases. Figure 13 illustrates this behavior.

VI. CONCLUSION

Steady-state numerical solutions have been obtained forthe attached and detached regimes using a one-dimensionalmodel. The simulation has been able to reproduce the experi-mentally established general features of the attached and de-tached cases. A separate ion temperature equation has alsobeen included to probe the role of electron–ion equipartitionand parallel ion temperature gradient. It is found thatTi

;Te throughout the parallel length in detached situations,reflecting strong collisionality. In the attached caseTi is ob-served to be larger thanTe at the midplane, but slightlylesser at the plate. B2-EIRENE simulations46 of attachedphases showTi

SOL.TeSOL andTi

div;Tediv . The midplane den-

sity is found to be only a weak function of the input power.This could be because the radiation losses increase verysteeply (;n2) with the plasma density. It is observed that

FIG. 12. Plots showing the effect of higher ion–neutralcollision frequencynx in the analytical solution. Thisgraph hasnx553104 s21, with the rest of the param-eters same as in Fig. 11. The solid and broken linesrepresent numerical and analytical solutions, respec-tively.

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the region of maximum radiation progressively shifts fromthe vicinity of the plate towards the X-point as the ratio ofvolumetric heat to particle sources (Q' /S') is lowered. Thisis indicative of a typical transition from an attached case to aweakly/partially detached, strongly/fully detached and finallyan X-point multifaceted asymmetric radiation from the edge~MARFE!. This behavior has also been observed in experi-ments and UEDGE simulations.47,48The level of recombina-tion characterizes the type of detachment. In the case ofweak detachment, this level is small, and we find that^Srec/Si& to be ;0.3 for Q'54.2 MW/m3 and ;0.6 forQ'53.2 MW/m3. In case of strong detachment (Q'52.6MW/m3), this ratio increases to;1. We also find that radia-tion in the divertor is counteracting much of the effect ofincreasingQ' . Since the atomic physics of radiation occursat a constant temperature, higher power densities will requiregreater particle flux, through high density or flow speed. Weobserve increment in density for higher powers in our simu-lation of detached cases. Calculation of the profiles of theparallel forces on the impurities~the friction force and theion thermal force! has shown that impurities produced at thetarget plate will be pushed back toward the plate, in both theregimes. These results can be of use in choosing parameterssets for more efficient parameter-scans in more complex or2D models.

The limitations of the present analysis are the following:~a! A 1D analysis ignores the perpendicular transport or 2Deffects. Perpendicular transport can result in the SOL widen-ing from the X-point region to the divertor target. A widerSOL results in lower heat flux density, requiring a smallertemperature gradient for conductive transport.~b! Noncoro-nal effects due to CX or fast ion transport may allow morelow-Z radiation at higherTe . ~c! Neutral recycling physicscan affect the simulation. Recycling could enhance the ion-ization and with it the volumetric losses. If the ionization ofneutrals in a flux tube exceeds the particle losses through thesheath, flow reversal is predicted to occur, and this couldlead to impurity retention. Also for higher neutral densities,the neutral–neutral collisions become important, which leadto significant transport of energy with neutral convection,and lesser drop in plasma pressure and density.21 ~d! Due toexclusion of flux limits, the heat flux and radiation loss mayhave been overestimated.~e! The recombination rate has alsobeen overestimated due to neglect of radiation trapping. Thiscould lead to a less severe drop in the particle flux at theplate.

The analytical model seems to capture the essentialphysics and helps to explain qualitatively the code results.We observe that the front location depends on the amount ofradiative loss taking place, for higher losses the front moves

FIG. 13. Plots showing the effect of higher perpendicu-lar heat sourceQ'e on the analytical solution. In thiscaseQ'e52.3 MW m23, and keeping other parameterssame as in Fig. 11. Here also for comparison sake, thenumerical solution is plotted as a solid line, while theanalytical solution is displayed by the broken lines.

869Phys. Plasmas, Vol. 8, No. 3, March 2001 One-dimensional model of detached plasmas . . .

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inwards, away from the plate. The front location also shiftstowards the X-point as the perpendicular heat source is de-creased. The effect of more neutrals leads to higher densities,and lower velocities, throughout the parallel length.

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