PPPL 4150

Summary

Princeton Plasma Physics Laboratory PPPL-4150 PPPL-4150 Advances in the Numerical Modeling of Field-reversed Configurations Elena V. Belova, Ronald C. Davidson, Hantao Ji, and Masaaki Yamada February 2006 Prepared for the U.S. Department of Energy under Contract DE-AC02-76CH03073. Princeton Plasma Physics Laboratory Report Disclaimers Full Legal Disclaimer This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government n…

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Princeton Plasma Physics Laboratory Report Disclaimers Full Legal Disclaimer This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, nor any of their contractors, subcontractors or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or any third party’s use or the results of such use of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof or its contractors or subcontractors. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. Trademark Disclaimer Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof or its contractors or subcontractors. PPPL Report Availability Princeton Plasma Physics Laboratory This report is posted on the U.S. Department of Energy’s Princeton Plasma Physics Laboratory Publications and Reports web site in Fiscal Year 2006. The home page for PPPL Reports and Publications is: http://www.pppl.gov/pub_report/ Office of Scientific and Technical Information (OSTI): Available electronically at: http://www.osti.gov/bridge. Available for a processing fee to U.S. Department of Energy and its contractors, in paper from: U.S. Department of Energy Office of Scientific and Technical Information P.O. Box 62 Oak Ridge, TN 37831-0062 Telephone: (865) 576-8401 Fax: (865) 576-5728 E-mail: [email protected]

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Advances in the numerical modeling of field-reversed configurations Elena V. Belova, RonaldC. Davidson, HantaoJi andMasaaki Yamada Plasma Physics Laboratory, Princeton University, Princeton, NJ 08543 Abstract The field-reversed configuration (FRC) is a compact torus with little or no toroidal magneticfield. A theoreticalunderstandingofthe observedFRCequilibriumandstabilityproper- tiespresentssignificantchallengesduetothehighplasmabeta,plasmaflows,largeiongyroradius, andthestochasticityoftheparticleorbits. Advancednumericalsimulationsaregenerallyrequired to describeand understandthe detailed behaviorofFRC plasmas. Results ofsuch simulations are presented in this paper. It is shown that3D nonlinear hybridsimulations using the HYM code [E. V. Belova et al., Phys. Plasmas 7, 4996 (2000)]reproduce all major experimentallyobserved sta- bility properties of elongated (theta-pinch-formed)FRCs. Namely, the scaling of the growth rate ofthen = 1tiltmodewiththeS(cid:3)=E parameter(S(cid:3)istheFRCkineticparameter,E iselongation, and n is toroidal mode number), the nonlinear saturation of the tilt mode, ion toroidal spin-up, and the growth of the n = 2 rotational mode have been demonstrated and studied in detail. The HYMcodehasalsobeenusedtostudystabilitypropertiesofFRCs formedbythecounter-helicity spheromak merging method. A new stability regime has been found for FRCs with elongation E (cid:24) 1, whichrequiresaclose-fittingconductingshelland energeticbeamionstabilization. 1

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I. INTRODUCTION The field-reversed configuration (FRC) is a compact torus with zero toroidal magnetic field and high plasma beta. The FRC is an example of a self-organized plasma state, in which no external coils are used to confine the plasma, and confinement is achieved by means of the self- inducedpoloidal magneticfields. Althoughmagnetohydrodynamic(MHD)theorypredictsstrong instabilities, experimentally it is found that FRC configurations are surprisingly robust, and have been shown to survive, forexample, duringtranslationfromone chamber to another and multiple reflectionsfrommagneticmirrors[1]. The traditionaltheta-pinchformationmethodusually produceshighlykineticFRCs with rel- atively low magnetic flux, small S(cid:3) (the FRC kinetic parameter S(cid:3) is the ratio of the separatrix radius to the ion skin depth) and large elongation E. Experimentally, these FRCs are observed to suffer the n = 2 rotational instability, which can be suppressed by the application of weak multipolemagneticfieldsafterFRCformation[1,2]. Then = 2rotationalmodeistheonlyexper- imentallyobservedglobalmode,whichoftenprematurelyterminatesthetheseFRCconfigurations [1,3]. The n = 1 tilt mode, which has been theoreticallyshown to be strongly unstable within an MHD model,and whichis consideredtobe themostdangerous globalmodeinprolateFRCs, has notbeen observed at all, orit has beenseen to“decay away” withoutdestroyingtheconfiguration [3–5]. Theintriguingstabilitypropertiesofprolate(theta-pinch-formed)FRCshavebeenthesubject ofnumeroustheoreticalstudies[6–12]. AtheoreticalunderstandingoftheobservedFRCbehavior presents significant challenges due to the high plasma beta, plasma flows, large ion gyroradius, andthestochasticityoftheparticleorbits. Advancednumericalsimulationsaregenerallyrequired to describe the self-consistent stability properties of kinetic FRCs [10–13]. The results of such 2

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simulationsarepresentedinthispaper. Itisshownthatreasonablygoodagreementbetweenkinetic simulations using the HYM code and experimentally observed FRC stability properties are been obtained. In particular, thermal ion finite-Larmor-radius (FLR) effects have been shown to play a major role in reducing the growth rates of unstable modes in prolate kinetic FRCs, and the nonlinearsaturationofthe n = 1tiltinstabilityhas beendemonstratednumerically[12]. Forlarge-scale,reactor-relevantconfigurations,newFRCformationmethodsarebeinginves- tigated,includingthecounter-helicityspheromakmergingmethod[14–17],and rotatingmagnetic field(RMF) currentdrive methods[18]. Since FLR stabilizationdoes not scale to largeFRC con- figurations,additional methodswillbe needed in ordertostabilize the low-nMHD modes. These methods may include, for example, stabilization by a close-fitting conducting shell, and by RMF effects [19]. Injection of energetic ion beams may also provide an additional stabilizing mech- anism, as well as plasma heating and current drive in future planned experiments [20,21]. This paper investigates the stability properties of oblate FRCs formed by counter-helicity spheromak merging,includingtheeffectsofaconductingshellandneutralbeaminjection(NBI)stabilization. This paper is organized as follows. The numerical model is described in Section II. The results of nonlinear hybrid simulations of prolate FRCs are summarized in Section III. Nonlinear simulations of oblate FRCs including the effects of a conducting shell and NBI stabilization are describedin SectionIV,and theconclusions aresummarizedinSectionV. II. NUMERICAL MODEL The stability properties of FRC configurations have been investigated numerically using the hybrid version of the 3D nonlinear simulation code HYM [10]. A full-orbit kinetic description is used for both the thermal ions and the beam ions, and the electrons are treated as a cold fluid. The numerical scheme implemented in HYM code has been described elsewhere [10]. The basic 3

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equations for the magnetic field B, electric field E, total plasma current J, and electron flow velocityv aregivenby e @B = −cr(cid:2)E; (1) @t E = −v (cid:2)B=c+(cid:17)J; (2) e J = (c=4(cid:25))r(cid:2)B; (3) v = −(J−J −J )=en : (4) e i b e HereJ +J isthetotalioncurrentdensity(thermalionplusbeam),electroninertiaeffectsandthe i b displacementcurrentareneglected,andquasineutralityn = n +n isassumedforsingly-charged e i b ions. The ion density and current density are calculated from the distribution of the simulation particles,whichfollowtheiontrajectoriesusingtheLorentz-forceequations. Thenonlineardelta- fparticlesimulationmethod[22,12]isemployedinordertoreducethenumericalnoiselevel. The nonlinear HYM simulations have been performed with a cylindrical grid size of 50 (cid:2) 32 (cid:2)60 in (r;(cid:30);z) using 2(cid:2)106 simulation particles. The region outside the separatrix (open field lines) is modeledas a low-density,high-resistivityplasma. In the simulations, the thermal ion equilibrium distribution function is taken to be a local Maxwellian with f (cid:24) exp(−“=T ), where ” = m v2=2+e(cid:30) is the ion energy, (cid:30) is the equilib- i 0 i 0 0 riumelectrostaticpotential,andT =constisthethermaliontemperature. Theequilibriumthermal 0 iontoroidalflowis neglected. In Sec. IV, an exponential rigid-rotor distribution function is assumed for the beam ions, corresponding to f (“;p ) = Aexp[−(” − Ω p )=T )], where A is a normalization constant, b (cid:30) 0 (cid:30) b p = m Rv − e is the canonical toroidal angular momentum, ” is the beam ion energy, T (cid:30) i (cid:30) b is the beam ion temperature(assumed constant), and Ω =const. The beam ion distribution func- 0 4

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tioncorrespondstoalocal, shiftedMaxwelliandistributionwithconstanttoroidalangularrotation frequencyΩ . 0 Self-consistentequilibriahavebeencalculatedincludingthecontributionofthebeamionsfor simulationsdescribed inSectionIV.The generalizedGrad-Shafranovequationforpoloidalflux can beexpressed as [23] (cid:1)(cid:3) = −R2p0 +RJ ; (5) i b;(cid:30) where(cid:1)(cid:3)istheGrad-Shafranovoperator,p = n T isthethermalionpressure,J isthetoroidal i i 0 b;(cid:30) component of the beam current density, and the prime (0) denotes differentiation with respect to thepoloidalflux. The equilibriumcanbe calculated providedthe magneticsurfacefunctionp ( ) i is specified, and the beam ion currentdensity is calculated using the fast-iondistributionfunction f . The solution to Eq. (5) has been obtained iteratively with the total toroidal current used as an b integralconstraintduringthe iterations. III. NONLINEAR STABILITY PROPERTIES OF PROLATE FRCs NumericalstudiesofthenonlinearevolutionofMHDmodeswithtoroidalmodenumbersn (cid:21) 1 in kinetic, prolate FRCs (with E (cid:21) 4) have been performed using the 3D nonlinear hybrid and MHDsimulationcodeHYM[10]. IthasbeendemonstratedthatduetothestrongFLRstabilization of the higher-n modes, the n = 1 tilt mode is the most unstable mode for S(cid:3) (cid:24) < 50 −80, i.e., for nearly all experimentally-relevantnon-rotatingFRC equilibria [12]. An empiricalFLR scaling of the tilt-mode linear growth rate in elongated elliptical FRCs has been obtained from a fit to the numerical results (Fig. 1) according to γ = γ exp(−3E(cid:26) =R ), where γ = CV =R E is mhd i s mhd A s the MHD growth rate, and C (cid:25) 2 is a constant. Other FRC parameters are defined as follows, R s istheseparatrixradius, (cid:26) isionthermalLarmorradius, V isacharacteristicAlfve´nvelocity,and i A 5

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theelongationE is definedas theratiooftheseparatrixhalf-lengthto itsradius, E = Z =R . s s A significant reduction of the growth rate occurs in the kinetic regime with small S(cid:3) and largeE, andthisreductiondependsontheparameterS(cid:3)=E, becauseE(cid:26) =R (cid:24) (S(cid:3)=E)−1. More- i s over,nonlinearhybridsimulationshaveshownthatthetiltinstabilitysaturatesnonlinearlywithout destroying the configuration, when its linear growth rate is sufficiently small. In particular, the saturation of the n = 1 tilt mode is found for FRC parameters S(cid:3) < 20 and E (cid:24) 6 (Fig. 4 of Ref. [12]). These numerical results explain the experimental data for prolate FRCs, which show that stabilitywith respect toglobal MHD modes depends on the S(cid:3)=E parameter[5],and that the stable FRCs areobservedforS(cid:3)=E (cid:24) < 3−4. Figure1shows thatthedeviationfromS(cid:3)=E scalingoccurs forS(cid:3)=E < 2forconfigurations with elongations E (cid:20) 6 (i.e., for smaller values of S(cid:3) < 12). Linearized hybrid simulations have shown that this deviation is due to resonant-particle destabilization of FLR-stabilized MHD modes [11]. The growthrateof theresonant instabilitydepends, in particular,on the stochasticity of the ion orbits. Namely, the wave-particle resonances are shown to occur only in the regular regionsofthephase-space. Therefore,thestochasticityoftheequilibriumparticleorbitsinprolate FRCs, which occurs due to the large curvature of the magnetic field lines near the FRC ends, has an important effect on the FRC stability properties. Analysis of the particle orbits in long configurations, for differentvalues of S(cid:3), have produced ratherunexpected results [11]. Contrary totheusual assumptionofalargedegreeofstochasticity ofthe ionorbitsinFRCs, wehave found thatasignificantfractionoftheorbits(upto60%ofallconfinedorbits)isregularinconfigurations with S(cid:3) (cid:24) 10. Furthermore, the number of regular orbits has been shown to scale approximately linearly with 1=S(cid:3) , independent of E (see Fig. 9 of Ref. [11]). Although the magnetic moment is not conserved for most of the ion orbits, another adiabatic invariant, based on the smallness of 6

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the1=E parameter,is conserved inlow-S(cid:3) FRCs. A largernumberof regularion orbitsinlow-S(cid:3) configurationsresultsinastrongerresonantdriveforthen = 1mode,whichexplainsthedeviation from S(cid:3)=E scaling for the E (cid:24) 6 case in Fig. 1. For a given value of S(cid:3)=E, a larger value of E implies a larger S(cid:3) value, and therefore a larger number of ergodic orbits. For these reasons, the FLR-fluid-likebehavioris evidentforthe E = 12 case (Fig.1). Nonlinear kinetic simulations performed for a set of FRC equilibria with E = 4 − 6 and S(cid:3) = 10 − 80 show that the n = 1 tilt mode saturates nonlinearly without destroying the con- figuration, provided the FRC kinetic parameter is sufficiently small, S(cid:3) (cid:24) < 20. In addition to the saturation of the tilt mode, the simulations show that the ions spin-up toroidally in the ion dia- magnetic direction, and the n = 2 rotationalmode growsin the nonlinearphase ofthe simulation (Fig. 4 of Ref. [12]). Initial conditions for the simulations are set at t = 0 so that the ions have a non-rotatingMaxwelliandistribution,whichisconsistentwithexperimentalobservationsjustafter FRC formation. However, as the simulationproceeds, the ions gradually begin to rotate, and near the end of the simulation run the ion toroidal flow velocity is comparable to the ion diamagnetic velocity. The saturation of the tilt instability occurs in the presence of a significant ion toroidal rotation,andinthenonlinearphase, theionrotationrateiscomparabletothelineargrowthrateof thetiltmode. Therefore,theiontoroidalspin-up,inadditiontodrivingtherotationalinstability,is likelyto contributesignificantlytothesaturationofthe tiltinstability[24]. A separate set of 2D (axisymmetric) nonlinear hybrid simulations has been performed in order to investigate the mechanism of the ion spin-up [25]. The simulations show that there is a significant particle loss associated with the resistive decay of the poloidal flux. Namely, the magnetic field decay results in a slow change in the particle trajectories, and the initiallyweakly- confined ion trajectories eventually change into open-field-line trajectories. It is also found that 7

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most of the lost particles have negative toroidal velocity, so that there is a net flux of negative momentum away from the separatrix region, and therefore a net positive ion rotation inside the separatrix (where the positive direction corresponds to the direction of the toroidal current). The simulations show that the ion toroidal spin-up is related to the resistive decay of the internal flux, and the resulting loss of weakly-confined particles, and the details of the ion toroidal spin-up determinethenonlinearevolutionofrotationalinstabilities. Both 2D and 3D simulations with zero initial ion rotation demonstrate the formation of an approximaterigid-rotorvelocityprofileinsidetheseparatrixinabout40-60Alfve´ntimes,depend- ing on the plasma resistivity. Radial profiles of the ion toroidalflow velocity are shown in Fig. 2. The linear velocity profile (V (cid:24) R) suggests that the distribution function f of the ions inside (cid:30) i the separatrix evolves towarda shifted local Maxwellian distribution,which may indicate that the stochasticity ofthe ionorbitsplaysa significantroleinFRC relaxation. IV. STABILIZATION OF OBLATE FRCs BY A COMBINATION OF CONDUCTING SHELL ANDBEAM IONEFFECTS Thecounter-helicityspheromakmergingmethodtypicallyproducesFRCconfigurationswith relatively small elongation with E (cid:24) 1. It is known that stability properties of oblate FRCs with E (cid:24) < 1 are different from prolate FRCs (E (cid:29) 1). In particular, earlier studies have shown that the n = 1 tilt mode is an external mode in oblate FRCs, and that this mode can be completely stabilizedbyaclose-fittingconductingshell evenintheMHDregime[26]. Withconducting-shell stabilization, the n > 1 internal co-interchange (kink) modes become the most unstable MHD modes [26]. In contrast to prolate FRCs, the thermal ion FLR stabilization of the low-n modes in oblate FRCs is weak even for low values of the S(cid:3) parameter. However, the localization of the low-nkinkmodesnearthemagneticnullsuggeststhatneutralbeaminjection(NBI)maybeavery 8

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effectivestabilizingmechanism foroblate FRCs. The equilibrium solutions of the generalized Grad-Shafranov equation have been calculated forthermalplasma andbeam ion parameters consistent with the proposedMRX-FRC experiment [20], i.e., for E = 1:1, S(cid:3) = 18, and for the following beam ion parameters: beam toroidal velocity (at the magnetic null V = R Ω ) V (cid:25) 6V , normalized peak density n =n =3%, and 0 0 0 0 A b e normalized temperatureT^ = T =(m V2=2) = 10. It is found that the beam ions tend to coalesce b b i A between the magnetic null and the separatrix near the FRC midplane, in agreement with earlier calculations of FRC-beam equilibria [27,28]. Due to localization, the peak beam current density can be comparable to the local plasma currentdensity, even when the fractionof the total current carriedbythebeam ionsis small. The calculated FRC-beam equilibria have been used as initial conditions for the linearized and nonlinear simulations using the hybrid version of the HYM code. Stability properties of the MHDmodeswithtoroidalmodenumbersn = 1−4havebeenstudiedwithandwithouttheeffects of a close-fitting conducting shell and the NBI ions. The growth rates of the most unstable mode for each toroidal mode numberfor three differentsets of simulations are shown in Fig. 3a (where thegrowthratesarenormalizedtoγ = V =Z ). 0 A s Foran oblateFRC configurationwiththermalion parameter S(cid:3) = 18 and elongationE (cid:24) 1, the stability parameter S(cid:3)=E (cid:24) 18 is well above the empirical stability boundary [1] at S(cid:3)=E = 3−4. Then = 1tiltmodeandothern > 1MHDmodesareexpectedtobestronglyunstableinthis regime. Indeed, Fig. 3a shows that in the absence of a conducting shell and beam ion effects, the n = 1modeisthemostunstablemode, withgrowthrateγ = γ = 0:83γ ,whereγ = 1:2γ 0 mhd mhd 0 is the growth rate of the n = 1 mode in the MHD regime. The growth rate of the n = 2 mode is comparable to that of the tilt mode, whereas the growth rates of the n = 3 and n = 4 modes are 9

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reduced, probablydueto strongerthermalionFLR stabilizationathighervalues ofn. Analysis of the linear mode structure shows that the n = 1 tilt mode is an external mode, whichhasalargeperturbationamplitudeattheseparatrix,whereasthehigher-nmodes,n > 1,are more localized and have smaller radial extent. The growth rate of the n = 1 tilt mode is reduced almost by an order-of-magnitude when a close-fitting conducting shell is used for stabilization (Fig. 3a, green curve). The growth rates of the n > 1 modes are also reduced due to conducting- shell effects. Conducting-shell stabilization for the above FRC parameters is stronger than had beenpreviouslyfoundinMHDplasmas[26]duetothechangeinthelinearmodestructurecaused by thermal ion kinetic effects. With conducting-shell stabilization, the n = 2 axially polarized kink mode becomes the most unstable mode, and all unstable low-n modes are localized near the magneticnull. Figure3aalso showsthe simulationresults withthe combinedeffectsoftheconductingshell and energetic beam ion stabilization (blue curve). It is evident that the beam ions have a strong stabilizing effect on the n = 1 and n = 2 modes, which are stabilized completely. For the same set ofbeamionparameters,the growthratesofthe n = 3 andn = 4 modesremainapproximately the same, with then = 3 modebeing moreunstable thanthe n = 4 mode. Numericalsimulations with larger beam ion density, n =n = 0:05, show a reduction of the growthrates of these modes b e bya factor1.5, butnotcompletestabilization. A set of3D nonlinearhybridsimulations has been performedin orderto study the nonlinear evolutionof the FRC configurationin the presence of a conducting shell and energetic beam ions withn =n =3%(Fig.3b). These simulationsshowthattheresidualn = 3(andn = 4) instability b e saturates nonlinearly at low amplitude (Fig. 3b), and thereforeit is not a dangerous mode. Simu- lation runs which model a “sustained FRC” (i.e., without decay of the equilibrium current) show 10

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that after the n = 3 mode saturates, the resulting configuration remains stable with respect to all global MHD modes. Note that this is a new stability regime (i.e., a combination of small elon- gation, conducting shell, and beam ion effects), which has not yet been studied experimentally. Thestabilizingeffectsoftheneutral-beam-inducedbulkionrotationhavenotbeenincludedinthe present numerical study, but these effects will likely contribute to the stabilization of the low-n MHD modes. V. CONCLUSIONS The hybridsimulations presented here demonstratethat ion FLR effects determinethe linear stabilitypropertiesofnon-rotatingprolateFRCs. Ithasbeenshownthattheinclusionofnonlinear and ion-toroidal-floweffectsis necessary fora satisfactorydescriptionofplasmabehaviorin low- S(cid:3) FRC experiments. In particular, nonlinearhybrid simulations have shown that the ion toroidal spin-upplaysan importantroleinFRC nonlinearevolution,includingthatofthen = 1 tiltmode. The 3D hybrid simulations have been able to reproduce all major experimentally observed stabilitypropertiesofkinetic(theta-pinch-formed)FRCs. Namely,thescalingofthelineargrowth rateofthen = 1tiltinstabilitywiththeS(cid:3)=E parameterhasbeenobtainedforaclassofelongated elliptical FRCs [12]; and ion toroidal spin-up, the nonlinear saturation of the tilt mode, and the growth of the n = 2 rotational mode have been demonstrated. It has been shown that the loss of ions with a preferential sign of toroidal velocity due to the resistive decay of the poloidal flux results in the ion toroidal spin-up, which reproduces very well the experimentally-observed ion rotation. StabilitypropertiesofFRCswithsmallerelongationE (cid:24) 1,whicharetypicallyformedusing counter-helicity spheromak merging methods, have also been studied. A new stability regime has been discovered which requires a close-fitting conducting shell and energetic beam ions for 11

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stabilization. It has been shown that in this regime, the n = 1 tilt mode and the n = 2 mode are linearly stable, whereas the residual weak instabilities of the n = 3 and n = 4 modes saturate nonlinearlyatsmall amplitude. TheresultingFRCconfigurationremainsstable withrespecttoall globalMHD modes, providedthat the FRC currentis sustained. Futuretheoreticalstudies will be performedin orderto optimizethe beam and bulkplasma parameterswith respect to stabilization ofglobalMHD modesinoblate FRCs. 12

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ACKNOWLEDGMENTS This research was supported by the U.S. Department of Energy under Contract No. DE- AC02-76CH03073. Calculations were performed at the U.S. National Energy Research Super- computingCenter. 13

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References [1] M.Tuszewski,Nucl.Fusion28,2033(1988). [2] S. Ohi,T.Minato,Y.Kawakamietal., Phys.Rev. Lett.51,1042(1983);A.L.Hoffman, J.T.Slough, D. G. Harding, Phys. Fluids 26, 1626 (1983); R. E. Siemon, W. T. Armstrong, D. C. Barnes et al., FusionTechnol.9,13(1986). [3] J.T.SloughandA.L.Hoffman,Phys.FluidsB5,4366(1993). [4] A.L.Hoffman,L.N.Carey, E.A.Crawfordetal.,FusionTechnology23,185(1993). [5] M. Tuszewski,D. C. Barnes, R. E. Chrien, J. W. Cobb, D. J. Rej, R. E. Siemon, D. P. Taggart, B. L. Wright,Phys.Rev.Lett.66,711(1991);M. Tuszewskietal.,Phys.FluidsB3,2856(1991). [6] R.D.Milroy,D.C.Barnes,R.C.Bishop,R.B.Webster,Phys.FluidsB1,1225(1989). [7] A.Ishida,H.MomotaandL.C.Steinhauer,Phys.Fluids31,3024(1988). [8] L.C.SteinhauerandA.Ishida,Phys.FluidsB2,2422(1990). [9] D.C.Barnes,Phys.Plasmas9,560(2002);8,4856(2001). [10] E.V.Belova,S.C.Jardin,H.Ji,M.Yamada, R.M.Kulsrud,Phys.Plasmas7,4996(2000). [11] E.V.Belova,R.C.Davidson,H.Ji,M.Yamada, Phys.Plasmas10,2361(2003). [12] E.V.Belova,R.C.Davidson,H.Ji,andM.Yamada, Phys.Plasmas11,2523(2004). [13] H.Ohtani,R.Horiuchi,andT.Sato,Phys.Plasmas10,145(2003). 14

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[14] Y.Ono,M.Inomoto,T.Okazaki,andY.Ueda,Phys.Plasmas4,1953(1997). [15] Y.Ono,T.Matsuyama,K.Umeda,andE.Kawamori,Nucl.Fusion43,649(2003). [16] C.D.Cothran,A.Falk,A.Fefferman,M.Landreman,M.R.Brown,andM.J.Schaffer,Phys.Plasmas 10,1748(2003). [17] S.P.Gerhardt,M.Inomoto,M. Yamada,etal.,Bull.Am.Phys.Soc.50,164(2005). [18] H.Y.Guo,A.L.Hoffman,R.D.Brooks,A.M.Peter,Z.A.Pietrzyk,S.J.Tobin,andG.R.Votroubek, Phys.Plasmas9,185(2002). [19] H.Y.Guo,A.L.Hoffman,R.D.Milroy,K.E.Miller,andG.R.Votroubek,Phys.Rev.Lett.94,185001 (2005). [20] M.Yamada, andH.Ji,J.PlasmaFusionRes.Series2,62(1999);ibid195(1999). [21] T.Asai,M.Inomoto,N.Iwasawa,S.Okada,andS.Goto,Phys.Plasmas10,3608(2003). [22] S.E.ParkerandW.W.Lee,Phys.Fluids5,77(1993). [23] E.V.Belova,N.N.Gorelenkov,andC.Z.Cheng,Phys.Plasmas103240(2003). [24] H.Ji,M.Yamada, R.Kulsrud,N.Pomphrey,andH.Himura,Phys.Plasmas5,3685(1998). [25] E. V. Belova, R. C. Davidson, H. Ji, M. Yamada, C. D. Cothran, M. R. Brown, and M. J. Schaffer, “Numerical study of the formation, ion spin-up and nonlinear stability properties of field-reversed configurations”,tobepublishedinNucl.Fusion46(2006). [26] E.V.Belova,S.C.Jardin,H.Ji,M.Yamada, R.M.Kulsrud,Phys.Plasmas8,1257(2001). [27] D.C.Barnes,andR.D.Milroy,Phys.FluidsB3,2609(1991). 15

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[28] A.F.Lifschitz,R.Farengo,andN.R.Arista,Nucl.Fusion42,863(2002). 16

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FIGURE CAPTIONS FIG.1. Normalizedgrowthrates obtainedfromlinearizedhybridsimulationsofthen = 1tilt instability for three different elliptical equilibria with E = 4, 6.25, and 11.6. The dashed curve correspondstothe scalingγ = γ exp(−3E(cid:26) =R ). mhd i s Fig.2. Radial profiles of the ion toroidal flow velocity at the FRC midplane at t = 20, 40, and 80t obtained from2D hybridsimulations with S(cid:3) = 20 and E = 4. The separatrix radius is A R =R (cid:25) 0:6. s c Fig.3. (a) Normalized growth rates of the n = 1−4 modes obtained from linearized hybrid simulationsincludingthermalionkineticeffects(red),theeffectsaofconductingshell(green),and the combinedeffectsof a conductingshell andNBI stabilization(blue)foran FRC withE = 1:1. (b) Plots of the time evolution of the n = 0 − 4 Fourier harmonics of the ion kinetic energy obtained from 3D nonlinear hybrid simulations including the effects of the energetic beam ions andthe close-fittingconductingshell. 17

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1 E=4 E=6.25 0.8 E=11.6 S*/E scaling g g 0.6 / mhd 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 (S*/E) - 1= Er /R i s Figure1. Belova, Physics ofPlasmas 18

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0.3 t= 80t A 0.2 t= 40t A V / V 0.1 j A t= 20t A 0.0

0.1 0.0 1.0 R / Rc Figure2. Belova, Physics ofPlasmas 19

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1.2 With conducting shell With shell and NBI 1 0.8 g /g 0.6 0 0.4 0.2 0 1 2 3 4 n 0.01 n=0 0.0001 n=1 n=2 n=3 n=4 1e-06 |V |2 i 1e-08 1e-10 0 200 400 600 800 1000 tw ci

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External Distribution Plasma Research Laboratory, Australian National University, Australia Professor I.R. Jones, Flinders University, Australia Professor João Canalle, Instituto de Fisica DEQ/IF - UERJ, Brazil Mr. Gerson O. Ludwig, Instituto Nacional de Pesquisas, Brazil Dr. P.H. Sakanaka, Instituto Fisica, Brazil The Librarian, Culham Science Center, England Mrs. S.A. Hutchinson, JET Library, England Professor M.N. Bussac, Ecole Polytechnique, France Librarian, Max-Planck-Institut für Plasmaphysik, Germany Jolan Moldvai, Reports Library, Hungarian Academy of Sciences, Central Research Institute for Physics, Hungary Dr. P. Kaw, Institute for Plasma Research, India Ms. P.J. Pathak, Librarian, Institute for Plasma Research, India Dr. Pandji Triadyaksa, Fakultas MIPA Universitas Diponegoro, Indonesia Professor Sami Cuperman, Plasma Physics Group, Tel Aviv University, Israel Ms. Clelia De Palo, Associazione EURATOM-ENEA, Italy Dr. G. Grosso, Instituto di Fisica del Plasma, Italy Librarian, Naka Fusion Research Establishment, JAERI, Japan Library, Laboratory for Complex Energy Processes, Institute for Advanced Study, Kyoto University, Japan Research Information Center, National Institute for Fusion Science, Japan Professor Toshitaka Idehara, Director, Research Center for Development of Far-Infrared Region, Fukui University, Japan Dr. O. Mitarai, Kyushu Tokai University, Japan Mr. Adefila Olumide, Ilorin, Kwara State, Nigeria Dr. Jiangang Li, Institute of Plasma Physics, Chinese Academy of Sciences, People’s Republic of China Professor Yuping Huo, School of Physical Science and Technology, People’s Republic of China Library, Academia Sinica, Institute of Plasma Physics, People’s Republic of China Librarian, Institute of Physics, Chinese Academy of Sciences, People’s Republic of China Dr. S. Mirnov, TRINITI, Troitsk, Russian Federation, Russia Dr. V.S. Strelkov, Kurchatov Institute, Russian Federation, Russia Kazi Firoz, UPJS, Kosice, Slovakia Professor Peter Lukac, Katedra Fyziky Plazmy MFF UK, Mlynska dolina F-2, Komenskeho Univerzita, SK-842 15 Bratislava, Slovakia Dr. G.S. Lee, Korea Basic Science Institute, South Korea Dr. Rasulkhozha S. Sharafiddinov, Theoretical Physics Division, Insitute of Nuclear Physics, Uzbekistan Institute for Plasma Research, University of Maryland, USA Librarian, Fusion Energy Division, Oak Ridge National Laboratory, USA Librarian, Institute of Fusion Studies, University of Texas, USA Librarian, Magnetic Fusion Program, Lawrence Livermore National Laboratory, USA Library, General Atomics, USA Plasma Physics Group, Fusion Energy Research Program, University of California at San Diego, USA Plasma Physics Library, Columbia University, USA Alkesh Punjabi, Center for Fusion Research and Training, Hampton University, USA Dr. W.M. Stacey, Fusion Research Center, Georgia Institute of Technology, USA Director, Research Division, OFES, Washington, D.C. 20585-1290 05/16/05

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