Mikhailov, M.

Nührenberg, J.

Shafranov, V.

Zille, R.

English

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Neoclassical Transport in Stellarators without Collisionless Ion Loss

Mikhailov, M. I.

Shafranov, V. D.

Max Planck Society eDoc Server

 ISSN 1063-780X, Plasma Physics Reports, 2008, Vol. 34, No. 6, pp. 525–527. © Pleiades Publishing, Ltd., 2008.Original Russian Text © J. Nührenberg, R. Zille, M.I. Mikhailov, V.D. Shafranov, 2008, published in Fizika Plazmy, 2008, Vol. 34, No. 6, pp. 573–575. 525 * Without radial electric field, ions in conventionalstellarators would be lost through a loss cone [1].Quasi-neutrality leads to the ion root [2] with inward-pointing electric field. Under special conditions [3](e.g.,  T i      T e ) an electron loss cone can lead to an elec-tron root.Quasi-isodynamic [4] configurations can collision-lessly confine particles, i.e., are without a loss cone.The contours of the second adiabatic invariant  J  arepoloidally closed and, at finite  β , exhibit maximum  J behavior [5]; i.e., reflected ions have poloidal driftswith a sign corresponding to a radial electric field point-ing inward.The alignment of  J  surfaces with magnetic surfacesdoes not have to be perfect (and in practice is not [6])for collisionless confinement; on the scale of collision-less electron motion, this may lead to substantial elec-tron deviation from magnetic surfaces and, thus, trans-port ( 1/ ν ) in the lmfp regime. This led to the conjecturethat electrons may dominate over ion transport in thelmfp regime even without electric field. This expecta-tion was verified with monoenergetic global MonteCarlo [7] neoclassical transport calculations [8] for theconfiguration described in [6]. In Fig. 1, a result isreproduced here, slightly augmented, for illustration ofthis situation. In the case considered and for vanishingradial electric field, ions are confined for about twice aslong as electrons, so that quasineutrality requires anoutwardly directed radial electric field, i.e., an electronroot, which is a stable root at a relatively small potentialheight of about 2 keV. This radial electric fieldincreases the ion transport. The mechanism is thereduction of poloidal  B   ×   ∇ B  drift by  E   ×   B  drift. Com-plementarily, it is also obvious from the change of the * The text was submitted by the authors in English. contours of the second adiabatic invariant due to theelectric potential; see Fig. 2.The purpose of this simulation was qualitativeinsight; here, a more quantitative verification is BRIEFCOMMUNICATIONS Neoclassical Transport in Stellarators without Collisionless Ion Loss*  J. Nührenberg a, R. Zille a , M. I. Mikhailov b , and V. D. Shafranov b a  Max-Planck-Institut für Plasmaphysik, EURATOM Association, D-17491 Greifswald, Germany b  Russian Research Centre Kurchatov Institute, Moscow, Russia Received December 19, 2007 Abstract —For quasi-isodynamic stellarators with poloidally closed contours of the magnetic field strength andhigh plasma beta neoclassical transport coefficients are computationally obtained over a large range of meanfree paths. In particular, a long-mean-free-path regime is found in which quasi-neutrality obtains without radialelectric field.PACS numbers:  28.52.-s, 52.55.Hc, 52.65.Pp, 52.25.Fi DOI:  10.1134/S1063780X0806010X 7–5 5 Loss rate, s –1 Φ (0), kV06543210 10 15 20 25 IonsElectrons Fig. 1.  Losses of monoenergetic 50-keV ions and electronssubjected to pitch-angle scattering with a normalized meanfree path  L * = 3  ×  10 3  ( L * =  λ / L c ,  λ  mean free path,  L c  halfthe connection length) following the procedure of [9] versusthe height of the electric potential applied in the quasi-iso-dynamic configuration of [6] with  〈β〉  = 0.1, magnetic fieldstrength  B  = 5 T, and a plasma volume of 10 3  m 3 . 526 PLASMA PHYSICS REPORTS        Vol. 34        No. 6        2008 N Ü HRENBERG et   al. attempted by considering a thermal plasma locally anddemonstrating that, without electric field, the diffusioncoefficient  D 11  of neoclassical theory (characterizingdensity-gradient-driven particle flux; see, e.g., [1]) isindeed larger for the electrons than for the ions in thelmfp regime; also, the ion and electron  ν  regimes areobtained.First, monoenergetic diffusion coefficients areobtained for orientation. Figure 3 shows such coeffi-cients for ions and electrons over a very wide range ofmean free paths to demonstrate the various regimes.While the ions clearly exhibit their PS and  ν  regimes,their  1/ ν  regime barely exists, in keeping with the near-coincidence of the contours of the second adiabaticinvariant with the magnetic surfaces [6]. On the con-trary, in addition to the PS and  ν  regimes, the electronsexhibit a  1/ ν  regime extending across about two ordersof magnitude in the mean free path and, accordingly,diffusion coefficient, because the deviation of the con-tours of the second adiabatic invariant from the mag-netic surfaces is large on the electron gyroradius scale.Within these simplified simulations the quasineutralitycondition is satisfied at about  L * = 2  ×  10 3 .Second, Maxwellian ion and electron distributionswith the same average energy, i.e., with a temperaturetwo-thirds as high, are simulated; Fig. 3 shows theresults. In keeping with results in [1], the ion diffusioncoefficients deviate very little from the monoenergeticresults. For the electrons, the  1/ ν  regime is shifted byabout a factor of 2 to a shorter mean free path so as tobe qualitatively expected from the contribution of theMaxwellian-tail particles. Thus, at  L * > 10 3  the electron D 11  dominates over the ion  D 11 . Again in keeping withthe results in [1], one will expect  D 12  (characterizingtemperature-gradient-driven particle flux; see, e.g., [1])to be larger than  D 11  so that a further small shift of thequasi-neutrality point to a shorter mean free path is tobe expected.Thus, for the parameters used here, quasineutralityrequires an electron root that should be weak in view of Fig. 2.  Contours of the second adiabatic invariant in mag-netic coordinates  s  (normalized toroidal flux) and  θ  (poloi-dal coordinate) of a medium-deeply reflected particle with-out electric potential (top) and with potential differenceacross the plasma radius of one-tenth of the particle’senergy and the sign chosen so that the  E   ×   B  drift counter-acts the diamagnetic effect of the plasma  β  in the strength of B  (bottom), i.e., the electric field pointing radially outward. 10 2 Diffusion coefficient  D 11 ,  m 2 /s 1234 10 8 10 1 10 0 10 –1 10 –2 10 –3 10 6 10 4 10 2 10 0 10 –2 L * Fig. 3.  Diffusion coefficients  D 11  for ions and electrons ver-sus a normalized mean free path (the ion mean free pathwith respect to  90°  pitch-angle scattering is used; electronresults corresponding to the respective ion results areobtained at a lower mean free path (because electron–elec-tron and electron–ion collisions need to be considered) butplotted at the ion mean free path). Symbols:   , + for elec-trons;  × ,    for ions;   ,  ×  monoenergetic results; and +,   results with energy scattering. The configurational parame-ters are the same as in Fig. 1. The diffusion coefficients areobtained at half the plasma radius. For monoenergetic par-ticles, 50 keV are again used; this value is used as the aver-age energy of the Maxwell distributions for the ions andelectrons so that  T e  =  T i  ~ 33  keV. Thus,  L * ~ 2.4  ×  10 3  cor-responds to the density  n  ~ 1.5  ×  10 20  m –3 , which is consis-tent with the  β  value. –1.0 19.220.83020292827262524232221 1.0–1.00.80.60.40.20–0.2–0.4–0.6–0.8 s sin θ s cos θ 1.0–1.00.80.60.40.20–0.2–0.4–0.6–0.8–0.5 0 0.5 1.0 20.620.420.220.019.819.619.4PLASMA PHYSICS REPORTS      Vol. 34      No. 6      2008NEOCLASSICAL TRANSPORT IN STELLARATORS 527the above-cited result [8], but, nevertheless, helpful inavoiding impurity accumulation.In summary, a new type of electron root for neoclas-sical transport in quasi-isodynamic high-β stellaratorshas been found.REFERENCES1. W. Lotz and J. Nuehrenberg, Phys. Fluids 31, 2984(1988).2. H. E. Mynick and W. N. G. Hitchon, Nucl. Fus. 23, 1053(1983).3. H. Maassberg, C. D. Beidler, U. Gasparino, et al., Phys.Plasmas 7, 295 (2000).4. J. Nuehrenberg, Europhys. News 6, 216 (1998).5. M. I. Mikhailov, V. D. Shafranov, A. A. Subbotin, et al.,Nucl. Fusion 42, L23 (2002).6. A. A. Subbotin, M. I. Mikhailov, V. D. Shafranov, et al.,Nucl. Fusion 46, 921 (2006).7. R. H. Fowler, J. A. Rome, and J. F. Lyon, Phys. Fluids 28,338 (1985).8. J. Nuehrenberg and R. Zille, in Proceedings of the IAEATechnical Meeting on Innovative Concepts and Theoryof Stellarators, Madrid, 2005.9. W. Lotz, J. Nuehrenberg, and A. Schlueter, J. Comp.Phys. 73, 73 (1987).

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Neoclassical Transport in Stellarators without Collisionless Ion Loss

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