Baldwin, D.E.

Kesner, J.

Lee, Y.C.

Post, R.S.

Smith, D.K.

DSpace@MIT

PFC/JA-81-17TANDEM MIRRORHOT ELECTRON ANCHORJ. Kesner, R.S. Post, and D.K. SmithMassachusetts Institute of TechnologyPlasma Fusion CenterCambridge, MA 02139D.E. BaldwinLawrence Livermore National LaboratoryLivermore, CAY.C. LeeUniversity of MarylandCollege Park, MDBy acceptance of this article, the publisher and/orrecipient acknowledges the U.S. Government's rightto retain a nonexclusive, royalty-free license inand to any copyright covering this paper.AbstractAn ECRH generated hot-electron plasma is shown to provide an attractive alternative to a neutral-beamformed plasma for the MHD "anchor" of a tandem mirror. This mirror cell would provide a Finite-Larmor -Radius-like stabilization for m = 1 modes. For a significant MHD fluid-like response, a small ion core betais seen to be required and stabilization results from the response of the core beta in the hot-electron depressedmagnetic field. At a higher core-beta the hot electrons behave in a fluid-like manner. Utilization is mostpromising in an outboard anchor tandem mirror arrangement.11. IntroductionAs originally conceived, the tandem mirror confines a long solenoid of plasma in a uniform magnetic field[1,2]. The solenoid called the central cell is bounded at each end by mirror "plugs" that run at a relatively highdensity, whose purpose is to provide electrostatic plugging of the central cell as well as MHD stability. Thelatter function results from the plugs being "minimum-B" cells which create an average minimum-B overallcon finement system.More recently proposed tandem mirror configurations separated the plug into two separate mirror cells[3-5], one providing electrostatic central cell ion confinement as well as an electron "thermal barrier" [61; theother serving as the MHD anchor. The MHD anchor was usually envisioned to be a neutral beam driven cellproducing a high ion beta plasma which is localized in the good field curvature of the minimum-B "anchor."In this paper, we propose a new method for producing the tandem mirror MHD anchor; creation of anECRH heated high-beta hot electron plasma in the minimum-B anchor cells. This scenario will be seen toprovide a considerable reduction in anchor cost, technology and power requirements. Since the gas source andpower are much reduced from the neutral beam driven anchor case, ECRH anchors in present day experimentsmay be run steady state, aiding in tandem mirror startup. The ability to produce high-Beta ECRIT mirror plas-mas has been demonstrated experimentally in the INTEREM series experiments [7] which used 10.6 GC (X-band) ECRH heating to produce - 30to50% plasmas at densities in the 10" cm-3 range and hot electronenergies above 100 KeV.The TARA tandem mirror magnetic geometry [4,51 is particularly appropriate for use of an ECRH anchorsince in this configuration the anchors are exterior to the plugs and can be nn at the low densities required forECRH accessibility. Additionally, in this configuration the quadrupole fields of the "outboard" anchors do notupset the axisymmetry of the central cell and plug provided the anchor is properly designed so that equilibriumdoes not require axial currents to flow through the central cell.Stability of the hot electron plasma as an MHD anchor is akin to the EBT ring problem [8,9]. At low corebeta (the core beta is the beta that results from ions and unheated electrons) the hot electrons are not expectedto respond to central cell drift frequency MHD fluctions but enhance the core beta generated stabilization dueto the magnetic field depression that they generate. When this stabilization is insufficient an increase in corebeta resulting from, for example, application of ion cyclotron resonance frequency (ICRF) heating will enhancestability by causing the hot electrons t. behave in an MHD fluid-like manner.In section II we will describe the physics of the hot electron anchor using an electrostatic "local" analysis.2Section 111 will detail the integration of the anchor cell within the tandem mirror. Section IV summarizes theconclusions of this study.1I. Physics ConsiderationsA number of the physics issues can be illuminated through the use of an electrostatic (low-beta) localplasma description.In cylindrical (r, z) coordinates assuming derivatives along field lines vanish (4 -+ 0), we can write thegyrophase averaged, fourier analyzed distribution function fluctuation as- efo W - .Wj # B 2 a'fj(r, z) = T 1 - J+ ) (1)For T the temperature, the fluctuating potential, Jo the Bessel function of order zero, k_ the azimuthalwave number (k1 = m/r), w. the drift frequency, w. = , wD the magnetic drift frequency, WO =T , , p is the magnetic moment and p the ion gyroradius. For a low beta plasma = , with R, thefield line radius of curvature.For the central cell and plug, we can take w >> WD to perform the velocity space integration whichyields the density fluctuation, whereas for the hot electron component in the anchor we will make the oppositeapproximation, namely w < wi). In this way we obtain the following dispersion relationJ W d9{(- )p2_I+ _ -+ 2W Ii(2with wpa the plasma frequency and T the temperature for species a. We assume w./T is constant through thedevice. Wph and woh are the respective plasma and curvature drift frequencies for the hot electrons. The firstterm represents the integral over the central cell, plug and anchor "core" plasma and contains the well knownFLR response terms. The last term is the anchor term which contains the core plasma term (1) which resultfrom the imbalance between the anchor ion density, nia and the core electron density, nla(lca a= ni - nchfor na, the hot electron density) and the anchor hot electron response term (1/wO) which may be ignored. Theanchor core plasna thus enters in two ways, first in the MHD response term wx)j ; second, through the chargeimbalance term.3The local approximation ! = 0 will now give a resulting "local" quadratic dispersion relationaw2+ bw+ c = 0 (3)witha= [nfk p2]IT,.+ 2[nik Ip2 ]T. (3a)b = 2(w./T) nch - nrk p2 - [n k 2p2] - [nik 2P2] (3b)I L a - cc = (w./T) [D + 2Dp + 2(Dia + Dea)] (3c)for1 2 kP2] = t f (k~ [di 0nik-1 p2 = kp)2 ni,[nil,=f lpha ffniandD,,= f = mc df (3d)JB Wre JarR,D:, Dp, Di. and De. represent the appropriate drive terms from the central cell, plug and thermal ions andelectrons of the anchor respectively. The stability criterion is then b2 > 4ac . The inertial term, 3a, and theMHD drive term, 3c, come principly from the plug and central cell. The term 3b contains the anchor chargeimbalance term plus the finite Larmor radius stabilization terms (to be referred to as FLR terns). These termsenter with opposite signs and therefore could cancel for some mode number, in. This reflects the fact that inthe hot electron anchor the ions drift with the wave uncovering electrons whereas in the rest of the device theelectrons uncover ions which cannot repond fully to the wave since they see a finite gyroradius averaged wavefield.Avoidance of this cancellation requiresni(kp)> nc (4)When this condition is met, the net response is Fl-R-like and an FLR stabilization is possi le. Sincek rn/r stability of the r = 2 mode implies stability of all higher modes. This effoct would bc enhanced4by ICRF heating in the anchor which will be seen to be desirable for other reasons. For m = 1, a proper radialtreatment would eliminate the FLR terms leaving an "FLR-like" stabilization term that results from the chargeimbalance term.Stability is then determined by the balance of the FLR terms with the drive terms and a necessary condi-tion for instability is c > 0. If inequality 4 is not met, a cancellation in the FLR term can occur and forsome mode number bnL -+ 0. (This cancellation is not seen for a "hot-ion" anchor because of a reversal insigns). In this situation the real frequency of the mode would drop to near zero (recall w ~ b/2a) and eventhe core plasma would have to be treated as "rigid." However, this event would probably only result in a smallincrease in plasma radius which would serve to decrease k±. Thus the cancellation would be upset and the wavefrequency would rise to the range of the central cell drift frequency.An electromagnetic treatment in slab geometry has recently been carried out by Baldwin and Pearlstein[10] and yields similar results. Additionally they find that for sufficiently low core beta, and provided that thecore plasma gradient-B drift frequency is much less that the wave frequency (w ~ w.), stabilization resultsfrom the core plasma drifting in the well created by the hot "rigid" electrons, that is 3e should be replaced by-+ f with A _(B/AB). For sufficiently high core beta, the hot electrons can be treated as anMHD fluid and Da -+ f PCdt/rR,. The core beta at which this change of behavior occurs is the so-called VanDamm-Lee condition [918# = 2(1 + T 1 /TI)(rp/Rc)a/(1 +#1h).The fluid response of the core beta can be elucidated by reference to recent work on kinetic energy prin-ciples. Antonsen and Lee have derived a fully electromagnetic energy principle which, if we ignore potentialvariations and assume an eikonal approximation (high mode number approximation), reduces to [111:AW = - dad# f -{VS1 (b. V(XB))2 + _ L2 f B -41 4dr-IXB2(g- K) e- P + . P + AW, (6)with a, 3 the Clebsch co-ordinates B = Bb = Va X VO, t the gradient on a and #(V VaA + VP J-)and K the field line curvature (K b Vb). S is the eikonal and we require the field line displacement E tobe = (x)ezpi(S(x)). We decompose as = Xb x VS + YVS. a I  + 4ir(P_ - P)/B2 andS= I + J4'' . A WK is the kinetic term which contains the perturbed distribution function appropriatelyaveraged over bounce and drift motion [11]. For P(B) hot electron equilibrium (PIH = PLH(B)) P±LH0 and for rapidly drifting hot electrons (w <' wr)) the kinetic term can be shown to be small by the ratio Kra,with r,, (denPc/dr)'. For a, r > 0 a necessary condition for instability then becomes5[dIXBI2(g -K)[(f - P ) +*- VP 11] > 0 (7)with e = B-'b X VS.For interchange modes (IXBI = constant), Eq. 7 takes the approximate formfdt 0 (0/7-) < 0 (7a)with R, = (t -K/e)-'. Eq. 7a exhibits the form of the MHD drive term of Eq. 3e, except for the additionalfactor (a/r). Using the MHD magnetostatic force balance and assuming a P(B) equilibrium from hot electrons,we findRe 2r,with_ B P_A V and r = P_LB V I PeThe drive then becomesD= ff 1+fHIR ) (8)r,(1 + xkc/2r,) rp(14 + 1ARewhere we have set a - 1 and estimated /3, ~ 2rp/A. (A similar form can also be shown to result froma non-P(B) hot electron distribution [12].) Eq. 8 reproduces the already mentioned response in the limit ofhigh and low core beta as well as indicating the response in the intermediate core beta regime. For smallcore beta (3c < r,/R,) the core plasma responds in an MHD fluid manner except that the relevant scalelength becomes that of the field depression brought about by the hot electrons rather than the vacuum radiusof curvature (D -+ f d1e#/r,,A); for high core beta, the hot electron response is similar to that of an MITDfluid (D -+ f def1i/Rcrp). In the intermediate case when P* - A/ReIOj1 = 2rp/R, (the Van Damm-Leecondition for loss of hot electron ring rigidity in the EB1' case), we find D - f dtfn /2Rerp. Note that ifthe hot electrons were in a maximum rather than a minimum-B cell, the sign in the denominator of 8 wouldbe reversed. In the minimum-B cell situation presently being considered, nothing dramatic occurs at the VanDamm-Lee beta condition. 0* represents the beta value at which optimum stabilization will occur. Clearly, astrong stabilizing effect implies small A (large hot electron field depression), small vacuum radius of curvatureand core beta in the range of 2r,,/Rc. At this core beta, the hot electrons weight the vacuum curvature as if theywere an MHID fluid.6III. Tandem Mirror with Hot Electron AnchorOne tandem mirror arrangement that could utilize the hot electron anchor is embodied in the TARAexperiment [4,5]. Here the central cell is terminated by axisymmetric plugs and the flux tube then maps througha transition region into a quadrupole anchor. The outboard location of the anchor permits the anchor densityto be low enough for easy ECRH access. The plugs are formed by neutral beam injection utilizing sloshing-ions for microstability and a thermal barrier can be formed at the midplane of the plugs. The schematic axialmagnetic field strength profile is shown in fig. la and the potential profile in fig. lb.Following the results of the INTEREM [71 experiments, the anchor electrons could be heated at X-bandfrequency (10.6 GC for an anchor resonance at 3.8 kG) with a density in the low to mid 10" c1M- rangeto attain an electron beta of 30 to 50% . The requirement for a significant core fluid response requires a corebeta - 2% for r = 0.2, requiring ions in the 5-10 keV range (fi* = 2% for r,/R, = .01). These ionenergies would be readily attainable through the use of ICRF heating by evanesent fields, as demonstrated inthe Phaedrus experiment [13].With no ion heating, the neutralizing ions in the anchor are cold (Ti ~ 2 - 6 eV due to Frank-Condonionization). Baldwin [14] has shown that in this situation the potential shown schematically in fig. 2b wouldbe expected to peak at the mirror throats and have a small midplane depression so as to provide electrostaticconfinement for the cold neutralizing ions. The potential well that forms in the transition region would fill upto close to the peak ion density. When ICRF is applied, ions would be heated along velocity space trajectoriesdefined by the resonance field ORF = Sin-' v/B/B, with B the minimum field and B, the resonance field.For diffusive ion heating, the potential depression would then deepen so that the loss cone boundary permitsthe core ions to be fuelled and heated with sufficient confinement for neutralizing the hot electrons. As indi-cated in Fig. 2d by the solid curve, the loss cone boundary will remain close to the velocity space RF heatingtrajectory. For low ion energies, the single pass kick an ion receives in passing through resonance with the RFfield can be several hundred cV and in this case the cell potenital could rise positively without cutting off thefuel source (dashed curve, fig. 2d). The exact potential level depends on the microstability of the trapped ionpopulation but we can expect the potential to rise to a level such that the ion density will just neutralize theelectrons.At the projected densities (mid 10 1 cm- 3 range) little ICRF power will be required and good penetrationof antenna near fields is assured [13]. An ion density of 5 X 10" 1cm-1 , energy of 10 KeV and estimatedconfinement time of 5 ms requires < 2 kW of power. Even a significant degradation of confinement would stillonly result in modest ICRF power levels.7IV. ConclusionsWe have presented here a new method for creating a tandem mirror MHD anchor, utilizing electron andion RF heating. In this scenario, the ECRH heated hot electron population digs a diamagnetic well in theanchor which enhances the fluid-like stabilization of the core ion (and electron) beta. Additionally, a FLR-like charge imbalance stabilizing term appears to be operative for m = I flute perturbations. For the nextgeneration of tandem mirror experiments, such as TARA, this innovation offers the possibility of replacingseveral Megawatts of neutral beam power with - 1 KW of ECRH in X-band and < 10 kW of ICRF power.Thus a substantial reduction in power requirements, as well as technology, would result.In a reactor scenario use of the hot electron anchor is particularly promising. In the reactor parameterrange, the disparity in drift frequencies between hot electrons and central cell plasma drifts is not large and thehot electrons would be expected to behave in a MID fluid-like manner [15]. The anchor could still consist of arelatively low-field, large volume cell and therefore not require the development of very high field quadrupolecoils.An interesting arrangement suggested by this work would be an axisymmetric maximum-B outboard hot-electron anchor creating a totally symmetric geometry. With the hot electrons behaving in a rigid fashion(wDh > w., low core beta) the system could be FLR stabilized for modes with m > 2 if Eq. 4 is satisfied andthe hot electron anchor could stabilize the m = I mode via the charge imbalance term. Unlike the EBT modeof stabilization, this method would not require the presence of core beta in the ring-produced field depression.However, an additional EBT-like stabilization would occur in the presence of some core beta (provided the VanDamm-Lee limitation on core beta is not violated).8AcknowledgementsThe authors wish to thank B. D. McVey and M. J. Gerver (MIT) for helpful discussions and commentsand B. G. Logan (LILNL) for pointing out expected onset of fluid-like hot-electron response as core ions areheated. We furthermore acknowledge the independent observation by A. J. Lichtenberg (U. C. Berkeley) of thestabilizing effect of the charge imbalance term. This work was carried out under D.O.E. contracts DE-AC02-78ET-51013T.O. 15 (J.K., R.S.P. and D.K.S.), W-7405-ENG-48(D.F.B.), and DE-AC05-79-ET-53044 (Y.C.L.).99References1. Dimov, G.I., Zakaidakov, V.V., Kishinevsky, M.E., Fiz. Plasmy 2(1976) 597.2. Fowler, T.K., and Logan, B.G., Comm. Plas. Phys. and Cont. Fus. 2 167 (1977).3. Baldwin, D.E., Logan, B.G., Simonen, T.C., editors, "Physics Basis for MFTF-B," Lawrence NationalLaboratory, report UCID-18496 (1980).4. Kesner, J., Nucl. Fus. 20 (1980).5. Post, R.S., Kesner, J., "Construction of the TARA Tandem Mirror Facility," MIT Proposal submitted toU.S. DOE (Nov. 1980). See also Kesner, J., Post, R.S., McVey, B.D., Smith, D.K., "A Tandem Mirrorwith Axisymmetric Central Cell Ion Confinement," MIT Report PFC/JA81-11 (1981), to be published inNucl. Fus. (1982).6. Baldwin, D.E. and Logan B.G., Phys. Rev. Lett. 43 1318 (1979).7. Ard., W. B., et al. Plasma Phys. and Cont. Nucl. Fus. (Proc. 4th International Conference, Madison,Wisconsin) 2, IAEA Vienna 619 (1971), also, Haste, G.R. and Lazar, N.H., Phys. Fluids 16 683 (1973).8. Nelson, D.B., Phys. Fluids 23 1950 (1980).9. Van Damm, J.W. and Lee, Y.C., In Proc. of the Workshop on EBT Physics (ORNL, Oak Ridge, 1979),Conf. 791228,471.10. Baldwin, D.E. and Pearlstein, L.D., paper in preparation (1981).11. Antonsen, T.M. and Lee, Y.C., "Electrostatic Modification of Variational Principles for AnisotropicPlasmas," submitted to Phys. Fluids (1981).12. Antonsen, T.M. and Lee, Y.C., private communication, manuscript in preparation (1981).13. Breun, R., Conrad, J., Golovato, S., Kesner, J., Post, R.S., et al., Plasma Phys. and Cont. Nucl. Fus. Res.1980, IAEA-CN-38/F-2-1, 105 (1981); Smith, D.K., "Fundamental Resonance Ion Cyclotron Heating inthe Phaedrus Plug Mirrors," U. Wisc. Ph.D. Thesis (1980) (article in preparation).14. Baldwin, D.E., "Model for Ion Confinement in a Hot-Electron Tandem Mirror Anchor," LLNL ReportUCID-18802,(1980).15. Logan, B.G., private communication (1981).10Figure Captions1. Schematic of axial variation of a) magnetic field and b) electrostatic potential for an outboard anchortandem mirror arrangement. Only right half of machine is shown.2. a) Schematic magnetic field in a mirror cell. b) Potential variation in the presence of a hot electroncomponent with a cold ion background (Tj < 10eV). c) Expected potential when ion heating is used.d) Velocity space boundaries at cell midplane indicating the loss cone and RF heating trajectory. Dashedline is expected loss cone boundary if the cell develops a midplane potential maximum.11B PLUGCENTRAL ANCHORCELLzNO ANCHOR-IICRFFIG 1.- -mB R FI I ITi*> TeeRF = Sin- B./ BRFVeOLC = Sin - B V.V/1FIG. 2B/2a2b2c2d

Tandem mirror hot electron anchor

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Tandem mirror hot electron anchor

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