Lidsky, L.

Politzer, P.A.

Potok, R.

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ION THERMAL CONDUCTIVITYIN TORSATRONSR. E. Potok, P. A. Politzer,and L. M. LidskyApril 1980PFC/JA-80-10ION THERMAL CONDUCTIVITY IN TORSATRONSR.E. Potok, P.A. Politzer, and L.M. LidskyPlasma Fusion CenterandDepartment of Nuclear EngineeringMassachusetts Institute of TechnologyCambridge, Massachusetts 02139ABSTRACTNumerical calculations of the ion thermal conductivityin helical toroidal (torsatron) magnetic configurations showthe presence of a plateau regime extending over two orders ofmagnitude in collision frequency. The value of the ionthermal conductivity is approximately equal to the neoclassi-cal plateau value for an equivalent torus without helicalmodulation. The predicted adverse 1/v behavior due toripple trapping is not seen.The family of plasma confinement devices that includesstellarators, torsatrons, and heliotrons is of considerableinterest for controlled fusion applications. These aretoroidal devices in which closed, nested magnetic surfaces aregenerated in the vacuum magnetic field by a helical configura-tion of the external windings. A major feature of this familyis the presence of a strong helical modulation, or ripple,of the field strength on the flux surfaces. Until now,neoclassical transport theory has associated large transportcoefficients with this modulation, due to particles which aretrapped in the helical magnetic wells.(1 ,2) These particlesare subject to a vertical drift resulting from the toroidalcurvature of the magnetic field and thus can, in some circum-stances, make large excursions from their initial flux sur-faces.An estimate of the ion thermal conductivity resultingfrom this mechanism has been given for stellarators by Connorand Hastie (2)2 2Xi=1. c3/2 Oi vti()62h 2where £h is the helical modulation of the field, pi is theion gyro-radius, vti = /2T /m is the ion thermal velocity,1 1R is the major radius, and vi is the ion-io n collision fre-quency. This expression is presumed to be valid when the- 2 -collision frequency is small enough for particles to completebounce orbits in the helical modulation but large enough thatthese trapped particles do not complete their poloidal driftorbits.This result is of particular concern for plasma conditionsappropriate to fusion reactors. In fact, the 1/v dependence ofthe transport coefficients has led recent torsatron reactordesign studies to concentrate on low temperature, high densityplasma regimes where collisionality is large and X. is accept-ably small.(3,4)Extensive orbit calculations for particles in model fieldsshow that the orbits are very much more complicated than assumed(5)in neoclassical theoretical models for rippled fields. Further-more, recent experimental results indicate that the expected1/v behavior may not be occurring. 6,7)In order to test thevalidity of the existing theory, we developed a computer programwhich follows test particle orbits, including all relevantcollisional effects, in complex magnetic fields. By examiningstatistically significant numbers of such orbits, we candetermine the thermal conductivity for various plasma conditions.The calculations reported here were performed for a torsa-tron magnetic field configuration; the torsatron is representa-tive of the heliotron/stellarator family and was chosen becauseof its particular suitability for scaling to a reactor size.- 3 -The magnetic field used was that generated by a set of specifiedexternal conductors. This method was used, instead of recourseto a model field, in order to ensure that no significantoversimplifying assumptions regarding field configurations weremade. The torsatron coil is specified by its major radius (R),minor radius (ac ), poloidal periodicity (f), toroidal period-icity (N), current (I) and winding law (#=$(O)). The fluxsurfaces obtained by following field lines are labeled withthe toroidal flux enclosed ($t).In order to follow particle orbits we use guiding centerequations accurate to second order in p. Collisional effects areincluded through energy scattering, pitch angle scattering,and drag terms dependent on the background density and tempera-ture (which can be functions of t). In order to evaluateXi, n and T were chosen to be independent of $t. To furthersimplify the calculation, we follow test ions and include onlyion-ion scattering effects. This simplification is possiblewhen thermal conductivity is being calculated; it would not becorrect in calculations of particle transport. For each run,360 test ions were launched on a given flux surface, $to*These test particles had a pitch angle and energy distributionappropriate to an isotropic Maxwellian with temperature equalto that of the background ions, and were uniformly spacedpoloidally. Each test particle was followed for 30 msec, and-4-after every 3 msec its energy and flux position were recorded.Thus, a test particle distribution function f . (x,v,t) was cre-ated, with x being defined by x = <r Sep> ($ t/ltsep) 1/2 where$tsep is the flux function at the separatrix and <r > is theaverage radius of the separatrix.The ion thermal conductivity can be determined from thetest particle distribution by1 fdxk (x-x) 2 u (x,t)X= 2 0 (2)fdx u (x,t)where u is the kinetic energy density of the test particledistribution3 1 2u (x,t) = Fd v f (x,v,t) 1 2 (3)Numerically, Xi was found by performing a least-squares fit toA + X.t. = y(t.)1 J Jwith El 2k (xkx o) uky(t.) =k(4)k kwhere the index j denotes the time and the index k denotes thespatial interval. Although all test particles in a particularcase are started on the same flux surface,there is a spreading- 5 -of the test particle distribution because the collisionlessdrift surfaces differ from the flux surfaces. This spread-ing results in a statistical fluctuation in y(t).The diffusive broadening manifests itself in a linear dependence ofy(t) on t. In fitting the data for y(t) to an expressionof the form A + BtP, we found <p> = 1.0 + .4 for the cases ofgreatest broadening (the six highest density cases for theR /a=6 torsatron). In the other runs, the statistical fluctua-tions are of the same order as the diffusive broadening, andthe best fits for p varied from 0 to 2. For these runs, themeasurements for Xi should be regarded as upper Limits, sincethe distribution broadenings were caused both by diffusive andnon-diffusive effects.In Figure 1 we plot the results of a scan of Xi versus vfor a large aspect ratio, reactor-sized torsatron with the follow-ing parameters:R = 48 m Bz (axis) = 5.5 Tac = 4 m $sep = 82.8 wZ = 3 <r > = 2.1 msepN = 32Both the test and background ions have a temperature of 8 keV, giv-MWWWMMWMW-6-20 -1ing v 125(n/102 ) sec for ions with mass 2.5. The testions for this case were started at $ = 0.25 Sp where E = 0.02,tosep th 0.015, and = l/q = 0.25; Et is the toroidal modulationgiven by the local value of the inverse aspect ratio and -C isthe local rotational transform. In the same figure we plot forcomparison the theoretical ion thermal conductivity for an axi-symmetric torus of otherwise identical parameters,(8 ) and theresult of the neoclassical calculation including helical trapp-ing (Eq. 1). In Figure 2 we plot the same quantitiesfor a torsatron with R = 24, N = 16. In Figure 3awe plot for.the conditions of Fig. 1, X vs p for v.. = 370i to 11sec~-. In Fig. 3b, the values of 6t and ch are shown.Apparently, the thermal transport for fixed collisionality isnot sensitive to the exact value of the modulation in theparameter range examined.The principal result of these computations is that thepredicted 1/v behavior does not occur; instead ion thermal con-ductivity is independent of collision frequency over a widerange of collisionality. This transport coefficient is approx-imately equal to that derived for the neoclassical axisymmetricplateau regime, and maintains this value over at least twoorders of magnitude in density (or collision frequency) variation.The "plateau" character of the thermal conductivity is confirmedby the 1/R dependence seen in Fig. 4. This result has favorableimplications for reactor design, as it indicates that the plasma- 7 -temperature, in moderate aspect ratio helical systems,may be raised to 15 keV (the value for minimum nT at ignition)without suffering from increased loss due to ripple transport.Although the reason for the absence of the predicted 1/vbehavior is not clear, and detailed calculations are beyondthe scope of this paper, a number of observations can be -aade.First, we suggest that ripple transport theory is not applicableto the torsatron geometry. The usual calculation of transportdue to ripple trapping relies on the assumption that a signifi-cant fraction (,r\.h ) of the particles in the system aretrapped in the helical ripples, and that the deviation of theircollisionless drift orbits from their initial flux surface posi-tion is determined primarily by the vertical drift caused bythe toroidal 1/R magnetic field gradient. However, detailedcalculations of collisionless particle orbits in a wide varietyof helical toroidal configurations indicate that this assumptiondoes not hold. As in axisymmetric systems, most particles arenever reflected, make small excursions from flux surfaces, anddo not contribute significantly to transport in low to moderatecollisionality regimes. In helical systems, the rest of theparticles undergo very complex motions. They can make frequenttransitions between quasi-circulating ("blocked") and helicallytrapped orbits (5). Even for large aspect ratios (ch >t) the-8-contribution from particles with simple, helically trappedorbits is negligible. The transition particles comprise a frac-tion of the total which is the larger of Y/ or .F Theirorbits do not conserve the adiabatic invariant J.We suspect that the observed transport results from anorbit resonance between the motion in the helical modulation andthe bounce motion in the toroidal modulation of the field.(9)Particles subject to these resonances do not conserve theadiabatic invariant, J, because their behavior depends criticallyon the phase of the bounce motion near the transition points.This model leads to a transport coefficient which is independentof collision frequency. Furthermore, we suggest that in largeaspect ratio systems (Eh > , it is not appropriate to con-sider the helical field structure as a perturbation on an axi-symmetric torus, which is the ordering common to existingtheories, but rather that one should consider orbits and trans-port in a two dimensional (straight) helix perturbed by a weaktoroidal curvature.FIGURE CAPTIONS1. Ion thermal conductivity as a function of plasmadensity for aspect ratio of 12. Curves correspondto the theoretical conductivity predicted by axi-symmetric neoclassical transport and ripple trans-port. The densities at which v*=l for collisionaldetrapping from the toroidal and helical modulationsare indicated. The estimated error on the observedconductivity measurements represents the 50% con-fidence level on a chi-square test of the fit inEquation 4. For 90% confidence error bars, multiplythe errors shown by 1.35.2. Ion thermal conductivity as a function of plasmadensity for aspect ratio 6.3. Ion thermal conductivity versus aspect ratio. Thesolid curve is the best fit to the equation: C0(a/R ) = X.0 1(a) Ion thermal conductivity versus fluxsurface position for aspect ratio 12 andn=3x1020 m 3(b) Peak-to-average magnetic ripple versusflux surface position. Open circlescorrespond to helical ripple, and solidcircles correspond to toroidal ripple.REFERENCES:1. T.E. Stringer, 3rd International Symposium on ToroidalPlasma Confinement. Garching, 1973.2. J.W. Conner and R.J. Hastie, Physics of Fluids, 17, 114(1974).3. A.V. Georgievskii, et al, CTO/1299 (Culham Translationof KhTFI 76-38), 1976.4. A. Iiyoshi and K. Uo, IAEA-CN-33/G-4, Conference onPlasma Physics and Controlled Nuclear Fusion,1974.5. A. Gibson and J.B. Taylor, Physics of Fluids, 10,2653 (1967).6. D.W. Atkinson, et al, CLM-P505, 1977.7. V.S. Voltsenya, et al, Third International Symposiumon Toroidal Plasma Confinement, Garching 1973.8. F.L. Hinton and R.D. Hazeltone, Reviews of ModernPhysics, 48, 239 (1976).9. R.H. Cohen, Comments Plasma Phys. Cont. Fusion 4,p. 157 (1979).0 CE~ofLI)OC- ge UC~0(1-5 ZlE~o000ItoE-0C3C(I0) za.8_ 60..25 -. 5 .75*k 4Sep.15-.1h.05~0.- .25 .5 *.75'k4SepCC\J. s w

Ion thermal conductivity in torsatrons

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Ion thermal conductivity in torsatrons

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