Chen, K.I.

Luckhardt, S.C.

Mayberry, M.J.

Porkolab, M.

DSpace@MIT

PFC/JA-87-23Particle Transport Simulation of Density BehaviorDuring Lower-Hybrid Current Drive ExperimentsOn the Versator II TokamakK-I. Chen, S.C. Luckhardt, M.J. Mayberry,* M. PorkolabMay 1987Plasma Fusion CenterMassachusetts Institute of TechnologyCambridge, Massachusetts 02139 USAThis work was supported by DOE Contract No. DE-AC02-78ET-51013.Submitted for publication to Nuclear Fusion*Present address: GATechnologies Inc., San Diago, CA 92318- 1 -PARTICLE TRANSPORT SIMULATION OF DENSITY BEHAVIOR DURING LOWER-HYBRIDCURRENT DRIVE EXPERIMENTS ON THE VERSATOR II TOKAMAKK-I. Chen, S.C. Luckhardt, M.J. Mayberry,* M. PorkolabPlasma Fusion Center and Research Laboratory of ElectronicsMassachusetts Institute of TechnologyCambridge, Massachusetts 02139ABSTRACTPrevious analysis of the global particle balance has shown that signifi-cant density increases observed in Versator II with both 800MHz and 2.45GHzlower-hybrid current drive-experiments are the result of an improvement by afactor of two in the particle confinement time. In the present work the one-dimensional particle transport equation has been solved numerically to simu-late temporal and spatial evolution of the electron density. In order to fitthe 800MHz profiles, it is found that the main contribution to the profile.change is due to a severalfold increase of the inward pinch velocity. How-ever, for the 2.45GHz data case, a reduction of the diffusive term near theperiphery seems to be needed to expl~in the observed profiles.- 2 -1. INTRODUCTIONUnderstanding particle transport in tokamaks is an important physics is-sue as it relates to density control, fueling, etc. Previous studies of par-ticle transport have been performed predominantely in ohmically heated dis-charges [1-4]. The main result from those studies is that besides an out-ward diffusive particle flux, an inward particle flux is needed to explainthe observed density profiles. Furthermore, both transport coefficients(diffusion coefficient D and inward pinch velocity v) are 10-100 larger thanthe values predicted by the neoclassical theory.The energy transpot during discharges with the application of auxiliaryheating has been addressed before [5,6]; however, the particle transport .un-der the influence of auxiliary heating power has not been studied as exten-sively. In this paper, the particle transport during lower-hybrid currentdrive (LHCD) experiments on the Versator II tokamak will be analyzed. De-tails of the Versator II tokamak and RF experiments have been described else-where [7].In earlier experiments [8,9,10], it has been shown that the significantdensity increases observed in both Versator 800MHz and 2.45GHz LHCD experi-ments are due to a factor of two improvement in particle confinement time.Furthermore, the particle confinement improvement only occurs in the RF cur-rent drive regime. The purpose of this paper is to model the density pro-files obtained in the experiment with a particle transport code and obtainvalues for the spatial diffusion coefficient D and the inward pinch velocityv.2. MODEL SIMULATIONThe one-dimensional particle transport equation [2] isan e(r,t) 1amert) 1 r D + v n (rt) + S(r,t) (1)ar _F r ar e I- 3 -where ne(r,t) is the local electron density, D is the diffusion coefficient,v is the pinch velocity and S(r,t) is the electron source term which is main-ly due to ionization of hydrogen neutrals. The traditional way to simulateparticle transport problem is to convert the above parabolic differentialequation into a difference equation with the assumption that ne and S(r,t)are known. Model simulated profiles obtained through different trial valuesof D's and v's are compared with experimentally obtained profiles. Then Dand v are determined by the best fit of the profile. In the present case,ne (r,t) and S(r,t) are measured in the experiment. The density profiles havebeen obtained by Abel inversion of a twelve chord interferometry scan, andthe source term S(r,t) is inferred by spectroscopic measurement of the H.profile. Equation (1) is solved numerically by the Crank-Nicolson implicitmethod [10]. In modeling the anomalous D and v, the simplest radial depen-dences have been chosen as a starting point; namely, the diffusion coeffici-ent D is constant and the convective velocity is linearly proportional to theradius v(r)=vo-r/a. This simple model has been used by other authors [2,3]and appears to be adequate to explain some of the experimental results. How-ever, it will be necessary to introduce additional spatial variation of D toaccurately model the 2.45 GHz LHCD experiments. Details of this modifiedmodel will be discussed later.2.1. 800MHz LHCDFigure 1 shows the temporal evolution of the line-averaged density andthe density profile during 800MHz LHCD. In the absence of the RF currentdrive, the density profiles are relatively flat (ne (central)/ e(line average)'1.2). To model the temporal and spatial profiles in the initial ohmic phase,D=15000cm 2/sec and v(r)=600-(r/a)cm/sec are found to be optimum values. Theexperimental profile and model simulation profiles are shown in Fig. 2(a).The source term S(r,t) is represented by a Gaussian profile with an appropri-- 4 -ate width and time dependence to approximate the experimental profile. Theagreement between the modeling and the experimental values for the ohmicphase is good within 4% over the entire radial profile.During RF current drive, the density initially increases and the profilebecomes more peaked ((neo/fe)max'1.9) (see Fig. 1). Note the source term inFig. 2 is decreasing during the course of the RF pulse. The steady state so-lution ((an/at)=O) of a source free (S(r,t)=O) transport equation is a Gaus-sian density profile with a half width proportional to (v/D). Because thesource term is small or negligible near the central part of the plasma, theoverall density profile shape, especially near the center, is determined bythe ratio v/D. Putting this another way, the ratio v/D measures the rela-tive importance of the inward convective flow and the outward diffusive flow.Thus, the observed peaking of the density profile, indicates that the valueof v/D should be increased during RF current drive. An interesting questionis whether this increase is due to the increase of v and/or the decrease of D.Figure 3 shows the results of three numerical calculations with approximatelythe same v/D value. Simulation #1 yields the best fit (also see Figs. 2(b-e)),with a severalfold increase in v and unchanging D value. Note that this si-mulation correctly predicts both the initial increase and the subsequent de-crease of the electron density while the D and v are held constant. Hence,the density decrease in the later part of the RF pulse is due to the decreas-ing source term, not a change in transport.Simulations #2 and #3 predict continually increasing density behaviorwhich is not consistent with measured profiles. To test the effects of radi-al dependence of the transport coefficient, a more sophisticated transport mod-el was used, specifically, the diffusion coefficient D=DO-(1+Cl(r/a) C2) where3,C1,10, 1.5*C 244 and the pinch velocity v(r)=vo.(r/a) 2 [11,12]. These modelsdid not seem to make any improvement in the predictions relative to the results- 5 -obtained from the simpler model with D independent of r and v- r. Therefore,it appears that the density rise and confinement improvement is mainly a con-sequence of an increased inward particle convection during RF current drive.After the termination of the RF pulse, the particle transport deterio-rates. This is correlated with an anomalous Doppler relaxation mode burstwhich occurs after the RF pulse [8]. This instability manifests itself byshowing bursts in most diagnostics; e.g., loop voltage, hard x-ray, H. emis-sion. Using the pre-RF injection values of D and v results in profileswhich are in significant error (%75%) in the peripheral part of the discharge.However, if the transport model shown in Fig. 4(a) is used; i.e., the diffu-sive loss is larger near the outside, the predicted profiles are in consider-able better agreement (10%-.20%) with the experiment. Table I summarizes theparameters used in model simulations of the 800 MHz experiments.2.2. 2.45 GHz LHCDFigure 5 shows the temporal evolution of the line-averaged density andthe profile during 2.45GHz LHCD. In the initial ohmic phase, the density pro-file is more peaked (neo/6 e 2) compared to the profile in the lower density800MHz LHCD regime. D=5000cm 2/sec and v(r)=2100.(r/a)cm/sec produces a goodfit to the observed ohmic heating density profiles (see Fig. 6(a)).During the RF current drive pulse, the density increases and the profilebecomes broader (n e/i e1.5). Hence, near the center the ratio v/D must de-crease relative to the pre-RF values. Four representative models have beenused in simulating these experiments. Following the guidelines of decreasingv/D, simulation #4 uses a model shown in Fig. 4(b), which decreases the valueof v near the center by an order of magnitude and increases its value by afactor of two near the edge. This particle transport model does not success-fully model the time evolution of the density. In particular, the simulation densitydecreased more rapidly than the experimental results. When the value of D- 6 -(simulation #2) or the value of v (simulation #3) was increased by a factorof two, the experimental profiles could not be reproduced (see Fig. 7). Inboth simulations, the density increased more rapidly than experiment and wasmore peaked than experiment. Figures 6(b-d) show the much better results ob-tained in simulation #1. The model of D used in simulation #1 is shown inFig. 3(a). D was increased near the center, and decreased near the periphery.The code simulation is in much better agreement with the measured profile asshown in Fig. 6.To model the post RF-phase, the pre-RF D and v values have been used.This model is in good agreement with the experimental profile (see Fig. 6(e)).Table II lists values of D and v used in these simulations.3. DISCUSSION AND SUMMARYEquation (1) can be integrated to obtain the following equationr - S dr = D -n + vn (2)If for r<r', S=0, and (3n/at)=O, Eq. (2) yieldsD -2- + vn = 0Consequently, we obtainv - azn(n) (3)( arEquation (3) implies that for a source free, steady state particle transport,the ratio v/D is determined by the scale length of the density profile. TableIII shows the value of v/D from the profile measurement and from the modelused to yield the best fit. The values of v/D obtained from profile measure-ments and best fit models are in close agreement.Particle confinement improvement has been confirmed in both 800MHz LHCDand 2.45GHz LHCD cases which have quite different density profile shapes.- 7 -Some recent ASDEX results indicate the same trend [13,14]. It has been found inASDEX that improved particle confinement can be achieved in H-mode of neutralbeam heated plasmas which have the broader density profile, and in ohmic dis-charges with pellet injection which have the more peaked profile. Thus, theparticle confinement under the influence of auxiliary heating or fueling doesnot seem to be solely dictated by the degree of the peakedness of the densityprofile. One caveat here is that based on the data of the ALCATOR C and FThigh field, high density ohmic discharges, it has been suggested that themore peaked density profile seemed to be correlated with the better confine-memt [15].In summary the inward convective velocity was found to increase by a fac-tor of eight to model the 800MHz experimental result of improved confinement andpeaked profiles. In the 2.45GHz case, the diffusive loss had to be reduced nearthe plasma periphery also modeling improved confinement but with broadened pro-file. The difference in the particle transport in the two LHCD experimentsmight be related to the difference of wave penetration and current generationin the two different density and RF frequency regimes. In particular, the waveaccessibility condition is more stringent for the 2.45GHz experiment. Thus,the wave absorption profile might be quite different for these two LHCD experi-ments. However, the physical reason for particle confinement improvement dur-ing lower-hybrid current drive experiments in Versator II is not understood atpresent.ACKNOWLEDGEMENTSOne of the authors (K.C.) would like to acknowledge useful discussionswith James L. Terry. This work was supported by the US Department of EnergyCont. DE-AC01-78ET-51013.*Present address: GA Technologies Inc., San Diego, CA 92318- 8 -REFERENCES[1] COPPI, B. and SHARKY, N., Nucl. Fusion 21, (1983) 1363.[2] STRACHAN, J.D., et al, Cucl. Fusion 22, (1982) 1145.[3] RICHARDS, B., et al Bull. of Am. Phys. Soc. 30 (1985) 1567.[4] EFTHIMION, P.C., et al, Bull Am. Phys. Soc. 31 (1986) 1415.[5] KAYE, S.M. and GOLDSTON, R., Nucl. Fusion 25 (1985) 65.[6] TAKASE, Y., et al, Nucl. Fusion 27 (1987) 53.[7] LUCKHARDT, S.C., et al, Phys. Rev. Lett. 48 (1982) 152.[8] LUCKHARDT, S.C., et al, Phys. Fluids 29 (1986) 1985[9] MAYBERRY, M.J., CHEN, K.I., LUCKHARDT, S.C., PORKOLAB, M., (to be pub-lished in Physics of Fluids).[10] MITCHELL, A.R., GRIFFITHS, D.F., "The Finite Difference Method in Par-tial Differential Equations", New York, Wiley (1980).[11] FURTH, H.P., in "Fusion", edited by E. TELLER, New York, Academic Press,1981, p. 203.[12] BECKER, G., Nucl. Fusion 27 (1987) 11.[13] SINGER, C.E., et al, Nucl. Fusion 25 (1985) 1555.[14] VLASES, G., et al, Nucl. Fusion 27 (1987) 351.[15] High Field Tokamak Workshop, Boston, June 1981 (unpublished).- 9 -FIGURE AND TABLE CAPTIONSFig. 1. The temporal evolution of the line-averaged density and the densityprofile during 800MHz LHCD.Fig. 2. Model simulation #1 of 800MHz LHCD; (a) before the RF (b), (c),and (d) during the RF and (e) after the RF.Fig. 3. Three-model simulations of the density evolution on 800MHz LHCD atvarious radii.Fig. 4. Modified transport model of (a) diffusion coefficient and (b) pinchvelocity.Fig. 5. The temporal evolution of the line-averaged density and the densityprofile during 2.45GHz LHCD.Fig. 6. Model simulation #1 of 2.45GHz LHCD: (a) before the RF, (b), (c),and (d) during the RF and (e) after the RF.Fig. 7. Four-model simulations of the density evolution on 2.45GHz LHCD atvarious radii.Table I. Transport parameters used on model simulations of 800MHz LHCDTable II. Transport parameters used on model simulations of 2.45GHz LHCD.Table III. The ratio of pinch velocity (v) and diffusion coefficient (D)from experimental density profiles and from model simulations.- 10 -10 15 20TIME (MSEC)5r (cm)25 3010 LimiterFig. 1EQN'08 I I I (a)6-N Ni Nj c4-2- GASFEED GAS RFFEED10(b)23ms26 ms27.ms29ms7.5ES5.02.5005- 11 -Ne (r,t)(x IO 'cm' ) S(x IO/cmn-sec)Before RF -8 - EXp. N(r,t)ot 1 175ms.- Model Simulation* Source Term6-5 E16.5ms * 0.-17,0ns17.5 ms -0.53 02 0. '0 2 4 6 aRadius (cm)(a)tO 12 14Ne (r,865432t)(x IO 2 cm'-) S(xIO'cm3 sec)During RFE Exp, Ne(r,) at t120 ms0 20.oms19.5 ms* - e e19.0 ms18.5 ms,0 2 4 6 8Radius (cm)(b)10 10.514Ne(r,t)(x 1OA cm ~3)0 2 4S(xs '%m.sec)During RF '* , E Exp Ne(r.0) at f 23 ms1-=23 ms-0. .6 84. **,.* *,6 aRadius (cm)(C)10 12Ne (r,t)(x 10'cm'3)0.5014 2 4S(x1O'5/m=-sec)0.5Ouring RF8 t 23.5 ms * Exp. Ne(r,)at t=26 ms76 0=* *.5 -26 ms2 ,06-3-*6 a to 12 14Radius (cm)(d)Ne (r,t)(x lO'2cm~3)4320 -- 2S(xlo'/ci .sec)0.54 6 a 10 12 14Radius (cm)(e)After RFEDp Ne(tt) Ot to 29 msI * 28 ms00 2 *. . .Fig. 210 12II- 12 -Ne (r, t)(x1012Cm- 3 ) 800 MHz LHCC4-.r = I 1cm * Exp. - Model Simulation #12 --- " " .200r =8cm ...--4 - . -a---- --- .2-08 r=4cmr = 0 cm ..-----.--4 --RF ON01-20 25 30Time (ms)Fig. 3- 13 -N -0'IC,,- - - - qm~. - -.V0 LL.C0- - - - - - - - - - - NI h.I U,I - - - - - -I ~V -~I 0N0 0- 14 -25TIME (msec)5r(cm)-2.0o 1.50*20 3031 -19 ms-.-- --- 24 ms-, --- 26.5m(Ohmic)(R F)s (RF)Iv~8C.,Iv,-o6C105II00 10Fig. 5P 60kW-2Ne (r, )(x IO'3cm )An.S(xK/cm-sec)2.52.0k1.05-015-0.14.0353.02.52.01.51.00.58elore RF-. * Exp N(,t)01 1tl95msModel Simulation* Source Term..&5 m19.5 in- 8' 2 4 6Radius (cm)(a)Ne (r,1)(K 10 m n)1.53-02.52.01.5to05During RF- Eup Ne(r,t) ol tt24 ms1-. t.24 ms1 t22 ms* 0 002 4 . * 8* . .0 2 4 6 8Radius (cm)(c)a to 12 0 2 4 6 8 10 12Radius (cm)(b)S(x id''cm.sec)10 120150.0005N, (r,i)( 1O scm~3)3.53102.01.5I0D050S(xlODuring RF* Eap Net,toft 2a26.5 ms2 26.5 m6. "o- 24.5 ms *2 4 6 cmRadium (CM)10 12(d)N, (r,t)(x O 'cm -)403.5 E Exp Ne(rIQt t=290ms10 . 27ms2.52.0 1 29 \ms1.51.00.5 *0 * . ~0 2 4 6 8 10Radius (cm)(e)S(x0'/cnd-sec)Fig. 6-15 -0.05Ouring RF. Exp Ne,t) at t 215ms-* I20.5 ms- -. 21.5 me *CL150.1005During RF/cm .sec)0.150.I005D15010054n.Ne ( r, t)(X 10 3CM-3)- 16 -Ne (r,-t)(XlO 3cm -3) 2.45 GHz LHCD0 Exp.r=8cm * * p-Model Simulation*t0.5 -..----- * " " ..n .402- r=6cm4- --................... .........................04- r 3cm4-20 25 30Time (ms)Fig. 7- 17 -h..l0(0-010EuE0EU2USaoENuCY oEu0I Ilh..I0 ~. 101"'IN- -E0C0.5ENS0cE rn-h.II'-xL-U-Q'-4w-oa-- & -u00100CDLLU-oa,0a,E02- 18 -UUA2UMI2UEUA0;5h.IoNQto)- - ~-- I -USUI'2UA0UC~j0 h-jON:'CSE EE CONLO~ COL -CooUESc0-,.I4-SCaEV.2S02C\jEAUOC0i;-LLE5 N o0.22S0LLh..LLct004cnw-oI-C800 MHz LHCD- 19 -Time Radius v/D (cm- 1 )(ms) (cm) from profilemeasurement from model175 5 0.025 0.01623 5 0.14 0.1352.45 GHz LHCDTime Radius v/D (cm- 1)(Ms) (cm) from profilemeasurement from model19.5 4 0.154 0.1726.5 4 0. 04 0.10-Table III

Particle transport simulation of density behavior during lower-hybrid current drive experiments on the Versator II tokamak

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Particle transport simulation of density behavior during lower-hybrid current drive experiments on the Versator II tokamak

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