Kesner, J.

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PFC/JA-88-38Effect of Local Shear on Drift FluctuationDriven T'ransport in TokamaksJ. KesnerApril 1989Plasma Fusion CenterMassachusetts Institute of TechnologyCambridge, Massachusetts 02139 USASubmitted for publication in: Nuclear FusionEffect of Local Shear on Drift Fluctuation Driven Transport in TokamaksJ. KesnerMassachusetts Institute of TechnologyCambridge, MassachusettsABSTRACTA theory based electron thermal diffusivity has been proposed recently which is basedon electron drift wave turbulence. When account is taken for the effect of local shearchanges that are produced in high poloidal beta equilibrium this theory is seen to predictthe observed improvement of tokamak confinement with plasma current. Confinement isseen to be good at low poloidal beta, degrade as poloidal beta increases, and then enteran improved transport region at high poloidal beta.1I. IntroductionA theoretical understanding of tokamak thermal transport has not been well estab-lished and as a result the community has depended on empirical formulas. One empiricalform proposed by Goldston for strongly heated tokamaks [1] has the form= IR- 7 5Ki/20.037 PA.(1)a-117 D1/with I,= plasma current in MA, n the ellipticity, R and a the major and minor radii in m,and P. the auxiliary power in MW. It is evident from Eq. 1 that tokamak confinementimproves with current and degrades with auxiliary power, a troublesome result. Only afirst principle theory can give insight into the origin and implications of this result.The electron thermal diffusivity that results from both electromagnetic and electro-static drift wave turbulence in a collisionless plasma has been evaluated by Horton, Choi,Yushmanov and Parail [2]. They assume the presence of a fixed amplitude spectrum of elec-tron temperature gradient driven driven drift waves, the so-called 77, modes, and calculatethe resulting particle transport. For the short wavelength electrostatic and electromag-netic waves, they choose a wave spectrum with p, < 27r/ki < c/wp,. wp, is the electronplasma frequency and p, the electron gyroradius. Stochasticity results from the resonancebetween the trapped electron circulation frequency, 0k, with the trapped electron bouncemotion (for electromagnetic waves) or the wave difference frequency, AW [3]. The circula-tion frequency is the ExB drift frequency with which a trapped electron circulates aroundthe wave potential island structures (in r - 6 space). The circulation frequency is onlymeaningful for trapped electrons. The untrapped electrons will jump in the poloidal planean angle determined by the rotational transform with each transit around the torus andtherefore cannot be trapped in the wave potential structure. Ref. [2] finds an electronthermal diffusivity that has the form:X, = C1 / 2 (Dem cwb + De p . (2)~~~PC 'r~eB(2pe2where the Dem term comes from the short wavelength r7, modes and the D, term fromlonger wavelength drift waves. E is the inverse aspect ratio, c = a/R, wbe = 4/-ae /Roqthe electron bounce frequency, vj,, the thermal speed, and q is the safety factor, p,the ion gyroradius at the electron temperature, T. the electron temperature and r, thedensity gradient scale length. Dem and De, are adjustable constants of order unity forelectromagnetic and electrostatic waves respectively that we will determine from a best fitto a data base. Eq. 2 expresses a strong degradation of confinement as T rises. X, isshown to be independent of the wave amplitude over a wide range of amplitude.Dem is shown to be strongly reduced by shear and although global and local shearare not differentiated in [2], the local shear on the outer torus is the relevant shear sinceonly trapped particles can circulate on the wave potential islands. The effect of shear isto have the trapped electrons sample different potential islands as they drift in radius andthereby degrade the above mentioned resonance between fh and Wb or Lw.We will extract the shear dependence shown by [2] from Dem, i.e. Dem -+ fi(s)Dem,and consider that the relevant shear is the local shear (s -* s(Ec,)). The local shear, ai,is defined as the shear seen by the deeply trapped electrons; st = (0 = 0). q1 is thelocal safety factor in the region of the trapped particles, q1 = B - V4/B - VO and 4 and 0are the respective toroidal and poloidal angle variables. (si should more correctly by thebounce averaged local shear seen by trapped electrons but in order to observe trends mostsimply we have evaluates ai at the outermost point of the flux surface.) We furthermorepostulate that a similar factor can be extracted from D., for the longer wavelength driftmodes.We will show that equilibrium changes can have a marked effect on electron transportthrough the dependence on local shear. It is known that as poloidal beta increases the localshear on the outside of the torus decreases until at some critical beta , i3, a 1, the localshear reverses direction. Fig 1 shows contours of local shear for a modestly shaped plasmaat ef3, = 0.7. The local shear has reversed within the cross hatched region. Thereafterhigher poloidal beta leads to more strongly reversed shear. The detailed behavior oflocal shear with poloidal beta depends on the plasma current density profile as well as3on plasma shaping, but the reduction and reversal of local shear with increasing poloidalbeta is always observed in calculations with reasonable (monotonically decreasing) currentdensity profiles. For /3, > 1 this effect is responsible for the so-called "second stability"phenomena which has been predicted to restabilize MHD ballooning modes [4]. We shallsee presently that it is through the fo driven equilibrium changes of the shear appearingin the drift wave transport theory that an C,, and therefore a current dependence entersxe and rv,.Eq. 2 predicts an increase in thermal diffusivity with electron temperature and there-fore with auxiliary heating at constant current. This degradation of confinement is ac-centuated by the reduction of local shear that accompanies the increase in poloidal beta.However we see that at high enough poloidal beta the local shear reverses and then in-creases (in the reverse direction) leading to a recovery of confinement.Thus we expect two regions of good confinement at low and at high poloidal beta.Ohmic heated tokamaks always operate in the low poloidal beta region; more power re-quires more current and thereby reduces the poloidal beta. We would therefore not expecta degradation of confinement with ohmic power. On the other hand, experiments whichheat at constant current may be expected to exhibit a sharp deterioration of confinementto accompany increases in poloidal beta when , ,< 1.II. Theoretical model of TEe and its dependence on sl.A confinement time can be obtained from Eq. 2 by taking TE = a2 /4x.66 neaR2TEe =f(E#p)Dem VT(1/q± + A3e )with1 De, 1 m(3a)2 Dem /2 .The dependence on local shear, fi(si) is exhibited explicitly in Eq. 3 and 81 =81(c,).The confinement thus derived ignores profile factors and is approximate.4The dependence of the diffusion coefficient on magnetic shear is obtained numericallyin Ref. [2] for electromagnetic modes. We fit the ref [2] result with an exponential formand replace global by local shear to obtainft ;:: 0.65 exp(-3 I st 1) + 0.05. (4)To estimate the local shear dependence on poloidal beta we have performed equi-librium calculations using a version of the PEST code [5] that has been modified [6] toproduce current profiles having low edge current density for a fixed edge safety factor,q. In the first series of calculations we fix qo = 1, q_ = 4, A = 3 for both a circularand D shaped (elongation, K = 1.5, triangularity, 5 = 0.5) plasma. The ellipticity andtriangularity of the plasma boundary are defined by assuming the following form for theplasma boundary:Z(O) = K r sin(O - bsinO) , (5a)R(6) = Ro + (r) + a cos6. (5b)With (r) is the shift of the center of the elliptic flux surface, K parameterizes the ellipticityand 6 the triangularity.Fig. 2 shows the local shear vs. poloidal beta at the outer torus and at the flux surfacethat is halfway out in poloidal flux. A similar trend of reduced and reversed local shearis observed in both cases although the shear reduces sooner for the D-shaped equilibrium.The observed high beta shear reversal was observed to occur similarly for all reasonablecurrent profiles.In Fig. 3 we plot rEe vs T, at fixed current from Eq. 3 (solid line) for the D-shapedtokamak (q(a) = 4 , Ii, = 3x,1013 cm-, B = 1 T, e = 0.33, . = 1.5, 6 = 0.5). We chooseD., = 1.5 and Dem = 0.6 since these values minimize the mean square divergence fromexperimental D-III data in the data base [7] discussed below. As electron temperatureincreases the confinement time is observed to decrease until we reach a poloidal beta value5when the outer torus local shear disappears. Thereafter a further temperature rise leadsto an approximately constant rEe as the increasing reversed shear offsets the T drivendegradation. If we had assumed a larger q. value (lower plasma current) the confinementwould have been reduced in the low temperature (Te ,< 1.4 KeV) region but improved athigher T,.Fig. 4 plots rEe vs. plasma current for fixed plasma pressure (3 = 3 %). Above 300KA confinement increases approximately linearly with current due to the increasing localshear that accompanies decreasing ,. Below 300 KA however we are in the reversed shearregion and confinement increases with decreasing current. In fact at the confinement timeis equal at 200 and 600 KA. The region of minimum confinement corresponds to wherethe local shear goes through zero, and this generally corresponds to a region of MID(ballooning) instability. Thus both a stability and a transport barrier separate the thelow beta region, where confinement scales linearly with current from the high beta regionwhere we would predict confinement would scale inversely with current.In Fig. 5 we compare the scaling law gotten from Eq. 2 with the Kaye-Goldstondata base of auxiliary heated tokamaks [7]. Because of the complicated shear dependenceon plasma shaping we only consider data from the the circular cross section machines,Asdex, DITE, PDX, TFR and TFTR. The data base only includes data from getteredL-mode experiments with Ping ;> 2 POH. We assume that the electron loss channel isdominant (and therefore rE - 2rEe). The data base contains the measured rE values,plasma current, total input power and geometric parameters.To quantify the comparison we define v2  (rEC - rep)2 /Te. We find that v attainsa minimum value, v = 0.23 by setting De, = 8.5, D,/De,, = 2.7 in Eq. 3. If we leaveout the local shear dependence (i.e. set ft = constant) we would obtain v = 0.27. Forcomparison the Goldston L-mode fit gives v = .24. Thus we see that in this data base thelocal shear makes an observable improvement on the fit to the measured values.The agreement between theory and experiment is good, indicating that electron driftwave transport might indeed be the dominant loss channel. The lack of a stronger sheardependence is not surprising since the data base is confined to relatively low poloidal beta6shots. Other physical processes, including profile effects and losses in the ion channel,undoubtedly account for some of the scatter.We compared Eq. 3 with a D-III data base [7]. We did not carry out the appropriateequilibrium calculations for DIII-D but instead used the local shear from the K = 1.5 seriesof calculations discussed above. v 2 again was minimized for De,IDe = 2.7 with De,=1.5.III. Discussion and ConclusionsWe have shown that a self consistent treatment of the global equilibrium with thelocal transport problem may be a key to understanding tokamak transport. Taking x,[2] for electron drift waves and postulating that the local shear factor modelled by Eq.4 should be in front of xe leads to the conclusion that confinement can be improved ordegraded via the c,3p dependent local shear. We find that two regions of good confinementexist, at low and at high poloidal beta. Ohmic heated tokamaks always operate in the lowpoloidal beta region; more power requires more current and thereby reduces the poloidalbeta. Thus an increase of Te is offset by a decrease in poloidal beta.Since we have only discussed the electron channel the improvement or deteriorationof this channel is only meaningful if the ion channel does not dominate transport. Fur-thermore our conclusions are only valid if the plasma is collisionless.We have not attempted to match the radial dependence of Xe with experimentaldata. To do this properly one must not only know the radial temperature and densitydependences, but also the local shear. (The reduction of local shear will tend to make xeincrease locally with radius.)In strongly auxiliary heated discharges with co, < 1, we predict that adding powerwill decrease TEe and that adding current would normally increase rEe. The increase ofrE, with current results from a reduction of the outer torus poloidal beta driven sheardegradation. In the opposite limit of high poloidal beta (cp, > 1), reducing plasmacurrent would improve confinement. Additionally operating in the low current regime isdesirable for the following reasons: a. High ,3 can eliminate MHD ballooning modes (i.e.7second stability), b. Low current will reduce volt-second requirements and/or current driverequirements.Since a strong degradation of confinement accompanies auxiliary heating in high cur-rent discharges a transport as well as an MHD barrier must be overcome for entranceinto the second transport regime. Recent experiments on the Versator II tokamak[8] haveshown, that high beta equilibrium (c/3, ~ 1) can be achieved in low current dischargesin which the thermal plasma pressure remains low and current drive is employed. Boththe plasma current and pressure are carried by energetic passing electrons which wouldnot take part in the above mentioned transport. This leads to the possibility that onecan access the high confinement region by starting with a high poloidal beta dischargeand increasing the current (using RF current drive techniques) in proper proportion to thepressure during the heating phase so as to remain in the high confinement regime.In comparing with the data base for strongly auxiliary heated experiments (Fig 5 )one must keep in mind that we are only calculating one loss process, turbulent transportin the electron channel. The good agreement shown implies that electron channel driftwave turbulence may indeed be the dominant loss process.If we consider strongly auxiliary heated devices and therefore only keep the high betaterm in the denominator of Eq. 3 we can then apply the power balance (Eq. 6) to showTEe oc P-0 '. Thus we obtain a similar power degradation as observed by Kaye andGoldston [7]. The improvement of r-E with current in these empirical fits can be comparedwith Fig. 4 where we observe an approximately linear improvement of TEe with current inthe high current regime.8AcknowledgementsI would like to thank Dr. S. Kaye (PPPL) for supplying the data base for auxiliaryheated experiments. I would also like to thank P. Hakkarainen for use of his modifiedversion of PEST and J. Freidberg, B. Lane, D. Sigmar of MIT and L. Woods (Oxford) forhelpful and stimulating discussions. This work was performed under U.S. DOE contractDE-AC02-78ET-51013.9References:1 R.J. Goldston, Plasma Physics and Controlled Fusion 26, (1986), 87.2 W. Horton, D.-I. Choi, P.N. Yushmanov and V.V. Parail, Plasma Physics and ControlledFusion 29, (1987) 901.3 W. Horton, Plasma Physics and Controlled Fusion 27, (1985) 937.4 B. Coppi, G.B. Crew and J.J. Ramos, Comments Pl. Phys. and Cont. Fus., 6,(1981)109.5 R.C. Grimm, R.L. Dewar and J. Manickam, J. Comput. Phys. 4, (1983) 94.6 S.P. Hakkarainen and J.P. Freidberg, Bull. A.P.S., 32, (1987) 1772.7 S.M. Kaye and R.J Goldston, Nuclear Fusion 25, (1985) 65.8 S.C. Luckhardt, K.I. Chen, S. Coda, J. Kesner, R. Kirkwood, B. Lane, M. Porkolab andJ. Squire, Phys. Rev. Lett. 62,( 1989) 1508.10Figure Captions:FIG. 1. Contours of local shear for a modestly shaped plasma at O, = 0.7, 0 = 3%, qa = 3.Cross hatched areas contain negative shear.FIG. 2. Local shear at the outer torus at a flux surface halfway out in poloidal flux for a)A circular outer boundary and b) A D shaped boundary having . = 1.5 and 6=0.5 . Wefix e = 0.33, qo = 1 and q. = 4.FIG. 3. rEe vs T, at fixed current for drift wave scaling law. q(a) = 4 , i-e = 3 x1013 cm-3, Bt = 1 T, r = 0.33, K = 1.5, 6 = 0.5.FIG. 4. rEe vs. plasma current for fixed plasma pressuref = 3%, = 3x1013 cm-3, Bt =1 T, c = 0.33, K = 1.5, b = 0.5.FIG. 5. Experimentally measured rE vs theoretical value for circular boundary tokamaksin the Kaye [7] data base.11' +4.4.+44.4.4.4.=.1I.8 1.0 1.2124.20-.2'S..6 1.4I I-.4iI I i i0.2-o.100.0-S-A -0.10 2 3Pololdal Beta0.1 -0.0---0.2A 01 2 34Poloidal Beta13III II I I 1 1111 I I ~ IIII I I II III IC00 0 00(SW) w 1 :4uoWouljuoo1400C:)000 LCD L()E(A00-00 N(SW) O1 4UQWQU)JU0315L L*x 0xLx0>Wft 1-4xX+ ~N AN-000 7100 Lu0co00cc x 0.I--0 +0-W loUW3c03~6JU16

Effect of local shear on drift fluctuation driven transport in tokamaks

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Effect of local shear on drift fluctuation driven transport in tokamaks

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