Lipschultz, B.

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PFC/RR-83-25 DOE/ET-51013-96UC-20 basic,f,gA One-Dimensional Fluid Model for Transport inDivertor and Limiter Tokamak Scrape-off LayersB. LipschultzPlasma Fusion CenterMassachusetts Institute of TechnologyCambridge, MA 02139November 1983This work was supported by the U.S. Department of Energy ContractNo. DE-AC02-78ET51013. Reproduction, translation, publication, useand disposal, in whole or in part by or for the United States govern-ment is permitted.AbstractSingle-fluid transport in the plasma scrape-off layer is modeled forpoloidal divertor and mecahnically limited discharges. This numericalmodel is one-dimensional along a field line and time-independent. Con-ductive and convective transport, as well as impurity and neutral source(sink) terms are included. A simple "shooting" method technique is usedfor obtaining solutions. Results are shown for the case of the proposedAlcator DCT tokamak.IA ONE-DIMENSIONAL FLUID MODEL FOR TRANSPORT IN DIVERTOR OR LIMITERTOKAMAK SCRAPE-OFF LAYERSIntroductionRecent experimental measurements at the plasma edge in both diverted[1,2] and mechanically limited tokamaks [3-5] indicate that the electrondensity and temperature are not constant along a field line. In additionlarge local radiation losses are found in regions of the edge calledmarfes [5]. Understanding the mechanisms involved in these inhomogen-eities could be used to control the power flowing into the scrape-offlayer. If that power could be dissipated over a large area at the plasmaedge through charge exchange and radiation processes, then the remainingsputtering and heat loads to limiter/divertor surfaces would be greatlyreduced. Similar results occur when the high recycling condition, analo-gous to a marfe, is created in a divertor chamber.Various models have been developed elsewhere to understand edge tran-sport. These include 2-d [6], and 1-d [7-10] computer codes. Variationsin these models occur due to inclusion (exclusion) of impurity and neu-tral sources (sinks), viscosity terms and treatment of ions and electronsas two fluids or one.Model EquationsThe equations used in this model are time-independent and singlefluid (Te = Ti). They are solved in one dimension along a field line inthe scrape-off layer. This computer model is applicable to either mech-anically (fig. la) or magnetically (fig. 1b) limited plasma geometries by- 2 -simply changing the distribution of the source terms involving perpen-dicular transport out of the main plasma. Basically one end (x = 0) isdefined to be the symmetry point between limiter/divertor plates. Theother end, x = L, is then located at the limiter/plate surface. Theequations are:a- (nV) - SN (1)axa-(2nT + MinV2) - SP (2)ax- (5nTV + 1/2 MinV3 - KeT5/2VT) - SE (3)axip is the usual Braginskii electron thermal conduction coefficient [11].Sn, Sp and Se are respectively the particle, momentum and energy sources:SN - ne [no<v>ionization - ni<av>recombination] + SperpSp - -MiV[<clv>recombination + <av>cx] (5)SE - Qperp <0>recombination [13.6 eV + 3 kT + 1/2 MiV2]- <ov>cx[3/2 kT + 1/2 MiV 2]- nineLoxy(T) (6)These terms and others can be found with a more detailed description inMorgan [9]. Sperp and Qperp are the particle and energy sources origina-ting from the main plasma. NJ and Nb are the impurity (oxygen) and neutraldensities respectively. Loxy(T) is the radiative cooling rate for the ex-- 3 -amined impurity (oxygen) and V the bulk fluid velocity along a field line.Boundary conditionsThe end of the computational grid (figure 1), defined by x = 0, isa symmetry point. The boundary conditions there are determined accord-ingly:3T0 (7)ax x-0V Ix-0 0 (8)At the target plate or limiter surface (x - L) the boundary conditionsare determined by the Bohm sheath criterion. The form used here is thatderived by Emmert [12]. Equating the power flow along the field line atthe sheath edge to the power flow through the sheath using this criterionwe have:[5nTV + 1/2 Minv3 - ceT5/2] x-L r x (Wi + We )cT 1/2( ) I x [(ykT) + ( sheath - Iall + 2kT)] (9)2 Mir and wie are particle flux and energy respectively. 0 and V are definedin reference 12.Numerical method and assumptionsThe code uses a "shooting" method moving from x = 0 to x = L. Thefirst four points are determined using a Runge-Kutta extrapolation. Theremaining grid points are calculated with a Milne predictor-corrector [13].- 4 -Neutral density along the field line is set to:no(x) = nplate[(x-L)/XnI + nbackground (10)Typical values for nplate and nbackground are 1 x 1012 and 1 x 101 0 par-ticles/cm 3 respectively. The background impurity is chosen to be oxygen,but any other low-z impurity would produce similar results. The impuritydensity is set to be a fixed fraction of the electron density everywhere.Similar to Shimada [81, the cooling rate L(T) used is noncoronal. Qperpand Sperp are assumed uniform along the length of field line within themain plasma chamber. The electron density at x = 0 is chosen as aninput. T at the same point is guessed, followed by calculations at theremaining grid points. Successive guesses of T (x - 0) are made untilthe sheath criterion is satisfied at x = L. With 200 equally spaced gridpoints the cpu time required on a VAX 11/780 ranged from 8 to 60 secondsdepending on initial conditions.ResultsThis code has primarily been used to model the scrapeoff layer of adiverted discharge in the proposed Alcator DCT tokamak [14]. The basicparameters of Alcator DCT used in this model are provided in Table 1. Atypical profile of various quantities along a field line can be seen infigure 2. A divertor discharge with medium edge plasma density in themain chamber (x = 0) is used. Temperature at the plate drops rapidly asthe edge density (x = 0) is raised. The opposite dependence on edgedensity is exhibited by the density at the plate (figure 3). This strongdependence of the divertor chamber parameters on the main plasma edgedensity is suggestive of the marfe phenomenon. Basically, the inclusioni- 5 -of impurities in divertor simulations allows achievement of the highrecycling condition at lower edge densities. Illustration of the highrecycling condition and its effects can be found in figure 4. As theedge density is raised the flux multiplication factor, defined as theratio of particle flux at the plate to that at the divertor throat, alsorises. The average energy per incident ion drops quickly as well. Areduction of the sputtering coefficient follows from this. Power to theplate will be reduced due to the inclusion of charge exchange neutrallosses. Radiation from oxygen impurities also figure heavily in reducingthe heat flow to the plate. These effects are illustrated in figure 5.Future WorkOne possible next step is to apply this program to modeling of themarfe. This requires Qperp, and perhaps Sperp as well, to be nonuniformalong a field line [5]. Other improvements to the code that should bemade are: proper impurity and neutral transport algorithims; generaliza-tion of the single fluid description to two fluids; and relating theneutral pressure at the plate/limiter to the particle flux and recyclingcoefficient (R) there. In a steady state plasma the R of the plateitself must be 1. But inclusion of processes such as pumping or loss ofneutrals to the main plasma effectively reduces R to less than 1. Rela-ting plate neutral pressure to the incident charged particle flux willobviate the need to specify ne(x = 0). That edge plasma density will bedetermined by the transport of particles out of the main plasma (Sperp)and edge physics.An area of improvement that would not be a modification of the exist-ing program would be the creation of a completely new code. A problem ofthe shooting method used here is that there is strong control over the- 6 -boundary conditions only at one end of the computational grid. Further-more, impurity and neutral transport terms cannot be added easily to thepresent numerical algorithim. A matrix inversion solution of the equa-tions is the obvious choice. Time dependent equations will be usedbecause of the ability to model experimental data.SummaryA one dimensional, single fluid, edge transport code has been devel-oped. It includes conductive and convective transport as well as neutraland impurity sources. Previous divertor modelling results have been qual-itatively reproduced with a reduced expenditure in computer cpu time.Table 1Plasma major radius, ROPlasma minor radiusVacuum vessel boreToroidal magnetic field (R - RO)Toroidal field ripple on axisMaximum field in conductorTF conductorPoloidal field conductorTotal flux swing (R = RO)Plasma current (q = 3)HeatingMaximum discharge duration(minutes)2.0 m0.40 x 0.56 m1.1 x 1.56 m7.0 T.04%10.0 TNb3SnNbTi35 Wb1.0 MALH 4 MW(cw)ICRF 5 MW(cw)8 MW(30 sec)1.5 - 5, w/ohmicCo, w/LH currentdrive- 7 -AcknowledgementsThe author appreciates the time and stimulating discussions con-tributed by B. LaBombard, J. L. Terry, P. A. Politzer of M.I.T. and W.Schneider of Max-Planck-Institut fur Plasmaphysik.References[11 T. Koboyashi et al, "Langmuir Probe Measurements in Beam-Heated D-IIIDischarges", presented at Symposium on Energy Removal and ParticleControl, Princeton (1983).[2] ASDEX-team in Proc. of IAEA Technical Committee Meeting on Divertorsand Impurity Control, IPP Garching (1981), p. 23.[31 J. L. Terry et al., Bull. Am. Phys. Soc. (1981).[4] M. M. Pickrell, Ph.D. thesis - "The Role of Radiation on the PowerBalance of the Alcator C Tokamak." M.I.T. Plasma Fusion CenterReport PFC/RR-82-30, November 1982.[51 B. Lipschultz et al., Bull. Am. Phys. Soc. 27, 8 (1982).[61 M. Petravic et al., Phys. Rev. Lett. 48, 326 (1982).[71 W. Schneider et al., presented at Symposium on Energy Removal andParticle Control, Princeton (1983).[81 M. Shimada et al, "Modelling of Dense and Cold Plasma in D-III",presented at Symposium on Energy Removal and Particle Control,Princeton (1983).[9] J. G. Morgan and P. J. Harbour (1980) in: 'Fusion Technology',(Proc. 11th Symposium, Oxford) Vol. 2, Pergammon; p. 1187.[10] D. Post, "An Analytic One-Dimensional Divertor Model with NeutralSources", presented at Symposium on Energy Removal and ParticleControl, Princeton (1983).[111 S. I. Braginskii, Reviews of Plasma Physics 1, 205-311 (1965).[12] G. A. Emmert et al., Phys. Fluids 23 (1980) p. 803.[13] Handbook of Mathematical Functions. Edited by M. Abramovitz and I.Stegin. National Bureau of Standards Applied Math Series 55.[14] Alcator DCT design group, M.I.T. Plasma Fusion Center Report PFC/RR-83-18, August 1983.Figure CaptionsField line geometry for plasmas that are a) mechanically limitedand magnetically limited plasmas. x = 0 is the symmetry pointbetween limiter/divertor plate surfaces (x = L).Typical profiles of ne, Te, PTOT, Pcons, and Pcon v along afield line in divertor geometry.Ratio of plate (np) to plasma edge (ne) density vs. plasmaedge density ne.Flux multiplication (-x-x-) defined asr divertor plate/r divertor entrance and EAVE per incidention (0-0-0) vs. ne at the plasma edge.Total (-x-x-) and charge exchange power (0-0-0) losses versusne at the plasma edge.Figure 1:Figure 2:Figure 3:Figure 4:Figure 5:MAGNETICUNIFORM SOURCE OFENERGY+PLASMAPARTICLES SHEATHX:0 X = 1/2 L 1V11=0 VI =C 9UNIFORM SOURCE OFENERGY+ PLASMAPARTICLESX=1/2 L1V1 =CsENTRANCETO DIVERTORCHAMBERI(DIVERTORPLATEFigure 1LIMITERWIA'lFIELDX=ovilz 0Ten-Main Plasma Divertor50500 1000 1500 2000 2500 3000 3500RadiationconvectionConductionMain Plasma'7 K___500 1000 1500 2000Divertor2500 300012.08.0 'E0x4.03500DISTANCE ALONG FIELD LINE (cm). Figure 212010080 Va)4)I-604020 -01.0za:0CL0.5 f-0o-noxy/ne =.05X - noxy/ne= .005x/xI I I2.0 3.0n, (x101 3cni)Figure 3srs7.01-6.0 -5.01-4)4)0.C4.01-3.0 -2.0 -1.01-0 1.0noxy/no=.01Pedge =10 Mwatts- ,'NI xIx.0XN- X-X/2.0 4.0nf AT SYMMETRY POINT (X10 3 crmi 3 )Figure 4z0x-.U.10050-4600-44000zzzLU0.;0i-12000006.01-noxy/n.01Pedge U10 MwattsrIIIIIIIIIxIIIIRadiation4.0ne AT SYMMETRY POINT (X1013 ci )Figure 5S55.4.0 Total0%U)0.j2.0/I///x/IIII0 2.0 Ix'IxxCI I fXI

One-dimensional fluid model for transport in divertor and limiter tokamak scrape-off layers

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One-dimensional fluid model for transport in divertor and limiter tokamak scrape-off layers

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