— - !979 S he r wo od (Deet ing — A§pect$ ef Centyotted C h € y m e a H c ! e a . y Q)ovat #ecem#$ #ean$g!¥ania A p r -i! 1 8 ^ § 0 , 1 9 7 9 ',-''-j'...':,r !''r BY ' . ' j, L.ABOr^':*OH Y PROCEEDINGS OF THE SHERWOOD MEETING THEORETICAL ASPECTS OF CONTROLLED THERMONUCLEAR FUSION April 18 - 20, 1979 Pocono Manor, Mt. Pocono, Pennsylvania Sponsored by Plasma Physics Laboratory, Princeton University Princeton, New Jersey 08544 EXECUTIVE PROGRAM LOCAL ARRANGEMENTS COMMITTEE COMMITTEE COMMITTEE H. Weitzner, Ch. A. H. Boozer, Ch. J. L. Johnson, Ch 1. Bernstein D. Barnes A. H. Boozer C. K. Chu H. L. Berk R. Donald J. M. Dawson W. Grossmann A. H. Glasser G. Guest J. Hogan P. H. Rutherford A. Kaufman N. Krall K. E. Weimer H. R. Lewis R. Lovelace M. Weissenburger D. Nelson R. E. Price L. D. Pearlstein A. Ware D. Ross P. H. Rutherford W. L. Sadowski A. Simon (,P General Information All sessions will be held at the Pocono Manor. The morning oral sessions will be in the Terrace Ballroom; the afternoon (or evening) poster sessions will be in the Plymouth Meeting Center. Coffee and other refreshments will be available during both the oral and poster sessions. There will be two consecutive poster sessions on Wednesday afternoon and Thursday evening and one on Friday afternoon. Thursday afternoon is free. A Cocktail Hour will be held in the Horizon Lounge Wednesday at 5:30. Two drinks are included in the registration fee. The Registration and Travel desks are in the Fountain Room. If you need assistance in planning transportation out, check with Travel early. There were 255 papers submitted of which 21 were chosen for oral presentation. Provisions have been made for these authors to present the details of their work in a subsequent poster presentation. SCHEDULE Tuesday, p.m. 5:00 - 7:00 Registration & Complementary Punch 7:00 - 8:00 Dinner 8:00 - 11:00 Registration 9:00 - 11:30 Snack for late arrivals Wednesday 7:30 - 9:00 Breakfast 8:30 Welcome 8:45 Oral Session 1A Wednesday 10:25 - 10:45 Coffee 10:45 Oral Session 1A 10:00 Coffee Hour 12:00 - 1:00 Lunch for Guests of 1:30 - 3:15 Poster Session IB Conference 3:00 - 4:00 Refreshments (Horizon Lounge) 3:30 - 5:15 Poster Session 1C 5:30 - Cocktails 6:30 - 8:00 Dinner Thursday 7:30 - 9:00 Breakfast 8:45 Oral Session 2A 10:25 - 10:45 Coffee 10:45 Oral Session 2A 12:00 - 1:00 Lunch FREE AFTERNOON 6:30 - 8:00 Dinner 7:30 - 9:00 Poster Session 2B 8:45 - 9:30 Refreshments 9:15 - 10:45 Poster Session 2C, Friday 7:30 - 9:00 Breakfast 8:45 Oral Session 3A 10:25 - 10:45 Coffee 10:45 Oral Session 3A 12:00 - 1:00 Lunch 1:30 - 3:00 Poster Session 3B 1979 SHERWOOD MEETING THEORETICAL ASPECTS OF CONTROLLED THERMONUCLEAR RESEARCH April 18-20, 1979 — Pocono Manor — Mt. Pocono, Pa. TUESDAY APRIL 17 5:00-7:00 REGISTRATION AND COMPLIMENTARY RUM PUNCH Fountain Room 7:00-8:00 DINNER Main Dining Room 8:00-11:00 REGISTRATION Fountain Room 9:00-11:30 SANDWICHES FOR LATE ARRIVALS Sam's Place WEDNESDAY APRIL 18 7:30-9:00 BREAKFAST Main Dining Room 8:15-12:15 REGISTRATION (Registration and Travel Desk open during aLL sessions) 8:30 Welcome M. B. GottLieb Announcements J. L. Johnson 8:45 ORAL SESSION 1A Terrace BaLLroom D. Ross and H.R. Lewis Chairmen 1A1. CaLcuLation of the Kolmogorov Entropy for Motion ALong a Stochastic Magnetic FieLd A. B. Rechester, M. N. RosenbLuth, and R. B. White. 1A2. Finite Beta sub e Universal. Mode Turbulence and ALcator ScaLing. K. MoLvig. 1A3. Energy Cascade in Drift-Tearing Modes. J. F. Drake and C. S. Liu. 1A4. RenormaLized Induced Scattering and Nonlinear Damping of CoLLisionLess Drift Waves J. A. Krommes. 10:25-10:45 COFFEE BREAK 1A5. Theoretical. Studies for the ELmo Bumpy Torus (EBT) Device. D. A. Spong, D. B. BatcheLor, C. L. Hedrick, and E. F. Jaeger. 1A6. Enhancement of Tandem Mirror PLug Potentials by Thermal. ParticLe Pumpout. D. E. BaLdwin and B. G. Logan. 1A7. Effects of Toroidicity on the NonLinear Interaction of Tearing Modes. H. R. Hie s, B. Carreras, and S. J. Lynch. 12:00-1:00 LUNCH (Dining Room doors cLose promptLy at 1:00) 1:30-3:00 POSTER SESSION 1B (ALL Poster Sessions are in the Plymouth Meeting Center) ORAL Papers 1A1-1A4 wiLL be given in Patrick Henry C ManorHaLL Auditorium 181. Charge Exchange as an Impurity Recombination Mechanism. R. A. HuLse, D. E. Post, and D. R. MikkeLsen. 182. RadiaL Scaling in the QuasiLinear ModeL of Drift Cyclotron Loss Cone (DCLC). L. D. Pearlstein, J. J Stewart, T. D. RognLien, andH. L. Berk. 183. ToroidaL Effects on the Accessibility of Lower Hybrid Slaves. P. 1. BonoLi, E. Ott, and J. M. Wersinger. 184. A FuLLy Two-DimensionaL Transport ModeL. M. H. Emery, N. Winsor, and J. Boris. 185. Lower Hybrid ELectron Landau Damping and Current Drive in the Presence of an Applied DC ELectric Field and Transport Losses. K. D. Marx, R. W. Harvey, V. S. Chan, and J. M. RawLs. 186. Current Profile Stabilization of D-Shaped Tokamaks to Ideal MHD Modes. L. C. Bernard, D. Dobrott, F J. HeLton, and R. W. Moore. 187. A Compact Form of the Integral Equation for Waves in an Inhomogeneous PLasma. S. P. Auerbach. 188. NonLocaL Hybrid-kinetic Stability Analysis of the Mirror Drift-Cone Instability. H. S. Uhm, R. C. Davidson, and R. E. Aamodt. 1B9. Stability Properties of a FieLd-Reversed Ion Layer in a Background PLasma. R. C. Davidson and H. S. Uhm. 1B10. Thermal Equilibrium Properties of an Intense Ion Beam With Rotational and AxiaL Motion. J. Chen and R C. Davidson. 1B11. Geometric Optics in Inhomogeneous Isotropic and Anisotropic PLasmas and on Their Boundaries. L. FriedLand and I. B. Bernstein. 1B12. Higher Order Chapman-Enskog Theory for Electrons: Application to Temperature Gradient-Driven Modes. A. B. Hassam. 1B13. Dissipative Drift Modes Driven By The ELectron Temperature Gradient In A Sheared Magnetic FieLd. C. L Chang, J. F. Drake,N. T. GLadd, andC. S. Liu. 1B14. Microtearing Modes and Anomalous Transport in Tokamaks. N. T. GLadd, J. F. Drake, C. S. Liu, and C. L. Chang. 1815. Observation of Transport in Tokamaks of Arbitrary Shape and Approximate NumericaL Description. M. SoLer. 1S16. Equilibrium NumericaL Study of the Formation of the PLasma in Tormac. A. Aydemir and C. K. Chu. 1817. Two-Way Diffusion Equations and Diffuse Reflection of Lower-Hybrid Waves. N. J. Fisch. 1B18. EBT NeocLassicaL Ion Transport with Non-MaxweLLian f and Higher Order PoLoidaL Expansions. R. B. Campbell, R. J. Kashuba, and T. Kammash. 1819. Argonne Beam Propagation and Target Experimental Program for Proposed Heavy Ion Facility. G. R. MageLssen. Jefferson Room 1B20. A Finite Element Solution of a Reduced Fokker-PLanck Equation. 1. Bernstein, A. Weiser, S. Eisenstat, and M. Schultz. 1B21. ALpha-ParticLe Heating in Tokamaks. D. R. MikkeLsen and D. E. Post. 1822. Tearing Modes in a Braided Magnetic FieLd. P. K. Kaw, E. J. VaLeo, and P. H. Rutherford. 1823. CoupLing of Lower Hybrid to Acoustic Modes. E. J. VaLeo and L. Chen. 1B24. A PossibLe Strange Attractor in MHD Convective InstabiLities. Y. M. Treve and 0. P. ManLey. 1B25. Cubic TurbuLence. D. R. NichoLson and D. F. DuBois. 1B26. The SLpw Ion CycLotron Wave in Tokamaks. C. Chu. 1B27. Guiding Center PLasmas in the Presence of GravitationaL (DeL B) Drifts. G. Joyce, C. S. Liu, and D. Montgomery. 1B28. FieLd Reversed PLasma Rotation and Transport. L. C. Steinhauer. 1B29. NonLinear Saturation of BaLLooning Modes for Tokamaks. F. Bauer, 0. Betancourt, and P. Garabedian. 1B30. Free Boundary EquiLibria with MuLtipoLe Expansion of ExternaL FieLd in NoncircuLar Tokamaks. 0. Okada S. DaLhed, J. DeLucia, and M. Okabayashi. 1B31. Spectrum and Eigenfunctions for a Fietd Equation With Stochastic Ray Trajectories. S. W. McDonald and A. N. Kaufman, 1B32. MagnetohydrodynamicaL Interchange InstabiLity in Low-Beta PLasmas in Sheared Systems. S. Ydshikawa and R. B. White. Monroe Room -1B33. Current-Driven Drift-Wave InstabiLity of a Finite-Beta PLasma in a Sheared-Magnetic FieLd. T. Tange, C. Ueno, H. Irie, T. Watanabe, S. Inoue, K. Itoh, K. Nishikawa, and S. Yoshikawa. 1B34. Two-DimensionaL Eigenmode Analysis of the Trapped-Ion InstabiLity. R. Marchand, G. RewoLdt, and W. M. Tang. 1B35. AnaLysis of PLT Discharges with High NeutraL Injection. A. L. Sutton, M. Cotsaftis, and H. H. KLein. 1B36- Conducting ShetL StabiLization of FCT Equitibria. L. A. CharLton, R. A. Dory, Y-K. M. Peng, D. J. StrickLer, S. J. Lynch, and D. K. Lee. Patrick Henry A 1B37. Optimization of Transition CoiL Design in Tandem Mirror Systems from the Point of View of Interchange Stability. T. B. Kaiser. 1B38. Cross-FieLd Electron Transport Due to Thermal Electromagnetic Fluctuations. A. T. Lin, J. M. Dawson, and H. Okuda. 1B39. Magnetohydrodynamic InstabiLities in a High Shear HeLicaL System. M. Wakatani, T. Yoshioka, K. Hanatani, 0. Motojima, A. Iiyoshi, and K. Uo. 1840. NonLinear Kink InstabiLities in Force-Free FieLds. H. C. Lui. 1841. AnomaLous Diffusion and PLasma Leakage Through Open FieLd Lines in FieLd ReversaL Configurations. S. Hamasaki. 1B42. EquiLibrium and StabiLity of Tokamaks with Tensor Pressure. A. Cooper, D. B. NeLson, G. Bateman, and T. Kammash. 1B43. Impurity Control by NeutraL Beam Injection. W. M. Stacey and D.J. Sigmar. . 1B44. StabiLity of NeutraL Beam Heated EquiLibria to BaLLooning Modes. R. W. Moore, R. L. MiLLer, and R. E. WaLtz. Patrick Henry B 1845. Resonant Second Harmonic Generation of Upper Hybrid Radiation in a PLasma. D. P. Tewari and V. K. Tripathi. 1B46. ELectron CycLotron Resonance Heating Rate in EBT PLasma. T. Uckan. 1847. Finite Temperature Effects on Microwave Propagation in EBT. D. B. BatcheLor and R. C. GoLdfinger. 1848. A SimpLe AnnuLus Power BaLance in EBT-1. S. K. Borowski, N. A. Uckan, E. F. Jaeger, and T. Kammash. 3:00-4:00 REFRESHMENTS Manor GriLL 3:30-5:15 POSTER SESSION 1C OraL Papers 1A5-1A7 wiLL be given in Patrick Henry C Manor HaLL Auditorium Resonance Wave-Wave Coupling and Ponderomotive Effects in Lower-Hybrid Heating. K. Matsuda, Y. 1C1. Matsuda, G. E. Guests and T. Ohkawa. 1C2. Effects of Ion Dynamics on Tearing Modes. X. S. Lee, S. M. Mahajan, and R. D. HazeLtine. 1C3. Nontinear Interactions of Drift-ALfven Waves. E. A. Frieman and L. Chen. 1C4. Bum Control Via Regulated RippLe Applied to Reactor-Grade Plasmas. J. M. RawLs, T. W. Petrie, and W. Chen. 1C5. Electron Landau Damping of Instabilities in Short, Fat, FieLd-Reversed Ion Rings. M. J. Gerver. 1C6. Kink Instabilities of a Field Reversed Ion Ring with a Toroidal Magnetic Field. J. M. Finn. 1C7. Stability of Low Beta Axisymmetric Mirror Machines. H. Weitzner. 1C8. Spectrum Cascade in Drift Wave Turbulence. A. Hasegawa, C. G. MacLennan, and Y. Kodama. 1C9. Thermal Fluctuation Levels and Convective Amplification. R. R. Dominguez, R. E. Waltz, and W. Pfeiffer. 1C10. Simulations of DCLC Modes Near Linear Marginal Stability. 8. 1. Cohen and N. Maron. 1C11. Stability Analysis of Runaway Distribution Function. 0-1. Choi, J. C. Wiley, and W. Horton, Jr.. 1C12. The Nonlinear Evolution of the Ion Mirror Instability. A. G. Sgro, D. W. Hewett, and T. C. Cayton. 1C13. Ion-Temperature-Gradient Instability in Toroidal Plasmas. P. N. Guzdar, L. Chen, W. M. Tang, and p. H. Rutherford. 1C14. High Beta Stellarator Stability Theory. M. J. Schmidt. 1C15. PLasma Diffusion in the Presence of Strong Turbulence. H. Okuda and C. Z. Cheng. 1C16. On the Cylindrical Limit of Various MHD Phenomena. E. Canobbio. 1C17. Magnetohydrodynamic Stability Analysis Using Approximate Codes. D. Dobrott, J. A. Tataronis, and R. W. Moore. 1C18. Drift Wave Turbulence in a Sheared Magnetic Field. S. P. Hirshman, J. C. Whitson, and K. MoLvig. 1C19. Particle Simulation of Drift-CycLotron Instability. J. K. Lee and C. K. Birdsall. Jefferson Room 1G20. Medium-Beta, Medium Aspect-Ratio SteLLarators. J. Nuhrenberg. 1C21. Anomalous Reconnection in Disruptive Processes in Tokamak Like Plasmas. H. WeLter and D. Biskamp. 1C22. Similarity Solutions of Partial Differential Equations Using MACSYMA. P. Rosenau and J. L. Schwarzmeier. 1C23. AxiaL CoLLisionaL Heating of Linear Magnetic Fusion Systems. P. McKenty, R. Morse, and G. Sowers. 1C24. Electron Stability Analysis of the Inhomogeneous Beam PLasma System— Application to the Electrostatic Double Layer. P. J. Morrison. 1C25. "Pinch-Tormac" - A New Fusion Device. T. Hatori and A. K. Sen. 1C26. Two Dimensional Structure and Variational Principles for Toroidal Ballooning Modes. S. MigLiuoLo and B. Coppi. 1C27. The Trapped-Untrapped Electron Boundary Layer in Tokamak Geometry. J. F. Santarius, F. L. Hinton, and D. W. Ross. 1C28. Stability of Field Reversed Theta Pinches. D. C. Barnes, C. E. SeyLer, and D. V. Anderson. 1C29. Solid Material End Plugging of Linear Magnetic Fusion Systems. F. L. Cochran, P. McKenty, R. Morse, and G. Sowers. 1C30. Characteristics of Ignited, High-WaLL-Loading Catalyzed Deuterium Tokamak Plasmas. M. Katsurai and D. L. Jassby. 1C31. Alpha Particle "Pumping" in a Toroidal Fusion Reactor by Magnetic RippLe Effects. J. D. Callen, R. H. FowLer, and J. A. Rome. 1C32. FCT Heating of Free Boundary Equilibria. M. Azumi and D. 8. NeLson. 1C33. TEH - A NumericaL SimuLation of the Time EvoLution of Drift Waves. C. 0. Beastey, W. 1. van Rij, and J. Denavit. 1C34. Curvature Drift Resonance Effects on Trapped-Etectron Modes. T. L. CrystaL and J. Denavit. 1C35. Mathematicat Probtems Arising in Adiabatic Compression of Ptasma. G. Vigfusson. 1C36. The Hami Ltonian for a Charged ParticLe in an ELectromagnetic FieLd. H. K. Meier and J. A. Rome. Patrick Henry A 1C37. Reduced Set of Resistive MHD Equations in ToroidaL Geometry. B. Carreras, H. R. Hicks, and J. A. HoLmes. 1C38. Free and Forced m = 0 OscitLations of a Sharp-Boundary VLasov-FLuid Screw Pinch. T. E. Cayton and H R. Lewis. 1C39, ParticLe Orbits in FieLd-Reversing Ion Rings: Erogdic or Not?. D. A. Larrabee and R. V. LoveLace. *tC40. Numericat Approaches to a Time-dependent Non-Linear Fokker-PLanck Equation in Two Vetocity Coordinates D. Fyfe, S. Eisenstat, M. SchuLtz, and 1. Bernstein. 1C41, RenormaLized Dispersion Tensor for ELectromagnetic VLasov Turbulence. R. V. Jensen and J. A. Krommes. 1C42. Wave ParticLe Transport From ELectrostatic InstabiLities: An Overview. S. P. Gary. 1C43. Studies of Current Due to RF Induced Runaway in the DIIA Lower Hybrid Experiment. R. W. Harvey, J. C. Riordan, J. L. Luxon, and K. D. Marx. 1C44. Convective Drift Wave InstabiLity in a Sheared Magnetic FieLd. W. M. Nevins, L. Chen, and C. Z. Cheng. Patrick Henry B 1C45. Transient AmpLification of Shear ALfven Waves. Y. Y. Lau. 1G46. NumericaL Simulation of PLasma Confinement and Heating by FieLd-Reversed Ion Rings. A. Mankofsky, R. N. Sudan, and J. Denavit. 1C47. PLasma TurbuLence Near A Magnetic FieLd ReversaL Point. D. Winske. 1C48. NumericaL SimuLation of Impurity Transport and PLasma Decontamination by Impurity Driven Modes. N. Sharky, B. Coppi, and T. Antonsen. 5:30-6:30 COCKTAILS Horizon Lounge 6:30-8:00 DINNER Main Dining Room THURSDAY, April 19 7:30-9:00 BREAKFAST 8:45 ORAL SESSION 2A Terrace BaLLroom A.H. GLasser and K.Tsang Chairmen 2A1. SeLf-HeaLing of BaLLooning Modes. A. Ferreira, B. Coppi, J. W-K. Mark, J. J. Ramos, and L. Sugiyama. 2A2. The Coupling of the Resistive-g and Ion Temperature Gradient InstabiLities in a Sheared Magnetic FieLd J. 6. Cordey, E.. M. Jones, and D. F. H. Start. 2A3. Resistive Instabilities in the Reverse FieLd Pinch. J. P. Freidberg and D. Hewett. 2A4. NumericaL CaLcuLations of Necessary and Sufficient Conditions for MHD StabiLity of a Stationary FieLd Reversed Mirror Plasma. D. V. Anderson, W. A. Newcomb, D. c. Barnes, and C. E. SeyLer. 10:25-10:45 COFFEE BREAK 2A5. Theoretical Intepretation of PLT Density Fluctuation Measurements. G. RewoLdt, R. Marchand, and W. M. Tang. 2A6. The Trapped Ion Mode in the Presence of Drift Have Fluctuations. W. Horton, D-1. Choi, D. Biskamp, andP. Terry. 2A7. ion Temperature Drift Instabilities in a Sheared Magnetic Field. W. W. Lee, W. M. Tang, W. M. Nevins, and H. Okuda. 12:00-1:00 LUNCH (Dining Room doors close promptly at 1:00) FREE AFTERNOON 6:30-8:00 DINNER Main Dining Room 7:30-9:00 POSTER SESSION 2B Oral Papers 2A1-2A4 wiLL be given in Patrick Henry C Manor HaLL Auditorium 2B1. Suppression of Current-Driven Ion Cyclotron Waves by a Lower Hybrid Pump in a <3 Machine. C. S. Liu and V. K. Tripathi. 2B2. ECRF Absorption ReLated to EBT. J. F. Pipkins and R. L. Hickok. 2B3. Magnetic Field Diffusion through a Magnetic Conducting WaLL. K. Evans, Jr. and E. M. GeLbard. 2B4. Variational Principle for Magnetohydrodynamic EquiLibrium States. A. Bhattacharjee and R. L. Dewar. 2B5. Ponderomotive Effects of an Electromagnetic, Wave in a Nonuniform Magnetic Field. C. Grebogi, A. N. Kaufman, and R. G. Littlejohn. 2B6. A Guiding Center Hamiltonian Using Physical Variables. R. G. Littlejohn. 2B7. MuLtipoLe Equilibria With Beta Equal to One. R. L. Spencer. 2B8. LH-Quasimode Parametric Excitation at the Edge of a Tokamak PLasma. E. ViLLaLon. 2B9. SimuLatiOn of Adiabatic Compression in Reversed Field Plasmas. W. Grossmann and E. Hameiri. 2B10. Electron Heating by Lower Hybrid Waves in the Presence of Anomalous Transport. V. s. Chan, S. C. Chiu, and T. Ohkawa. 2B11. Finite Beta Trapped Electron Instabilities. J. c. Whitson, K. T. Tsang, P. J. Catto, and M. N. RosenbLuth. 2B12. Stability of High Beta Tokamaks to Ballooning Modes. D. A. MonticeLLo, H. R. Strauss, W. Park, R. B. White, S. C. Jardin, M. S. Chance, A^ M. M. Todd, and A. H. GLasser. 2B13. A Nonlinear Mode BeLow the Electron PLasma Frequency. V. Krapchev and A.- Ram. 2B14.^EquiLibrium and ThermaL StabiLity Properties of Ignited Plasmas with Advanced Fuel Cycles. J. H. SchuLtz, L. Bromberg, and D. R. Cohn. 2815. Features of Ignited Operation. L. Bromberg, D. R. Cohn, and J. Fisher. 2B16. Electron Transport in Random Magnetic Fields. M. S. Chu and C. Chu. 2B17. Coupling of Drift Modes in a Torus. R. E. WaLtz, W. Pfeiffer, and R. R. Dominguez. 2B18. BaLLooning Stable Profiles in Circular Tokamaks. D. Lortz and J. Nuhrenberg. 2619. A Numerical Study of the Effect of Impurities on PLasma and Magnetic FieLd ProfiLes in the Reversed FieLd Pinch. E. J. Caramana and F. W. Perkins. 2B20. Ion Cyclotron Resonance Heating in a Tandem Mirror. J. E. Howard and J. Kesner. 2B21. The Continuous Spectrum and Ballooning Modes. E. Hameiri. 2B22. Drift-Wave Eigenmodes in Toroidal Plasmas. L. Chen and C. Z. Cheng. 2B23. Cherenkov Resonance as an FLR Effect on the ALfven-Ion-CycLotron Mode. J. Goedert and J. P. Mondt. 2B24. The Microwave Spheromak. J. L. Shohet. 2825. Initial Results of Tandem Mirror Transport Calculations. J. M. GiLmore and R. H. Cohen. 2B26. Preliminary Results of a Tandem Mirror Transport Code. A. A. Mirin, R. H. Cohen, M. E. Rensink, and J. KiLLeen. 2B27. Transport Equations for Tandem Mirror Machines. R. H. Cohen, M. E. Rensink, and J. H. Foote. 2B28. Particle Motion in a Cyclotron Resonant Field. Y. Matsuda and H. L. Berk. 2B29. Interaction of Lower Hybrid Fields with the Drift-Cyclotron Loss-Cone Mirror Instability. K-C. Shaing R. W. Conn, and j. Kesner. 2B30. Three Dimensional Fluid Simulations of Drift Waves. D. Biskamp, R. Estes, and W. Horton. 2B31. Magnetohydrodynamic Particle Code With The Lax-Wendroff Method. F. Brunei, J. N. Leboeuf, T. Tajima, and J. M. Dawson. 2B32. Lower Hybrid Heating in Tandem Mirror Geometry. J. T. Woo and K. A. Connor. Monroe Room 2B33. Simulation of Multi Impurity Species Transport in Tokamaks. E. C. Crume Jr. and D. E. Arnurius. 2B34. Real-Time MHD Computations for Noncircular Tokamaks on a High-Speed Array Processor. T. S. Wang. 2B35. Effects of Shear on Drift-Cyclotron Instability. P. Satyanarayana and P. Bakshi. 2B36. A Monte Carlo Model of Particle Motion in Field-Reversed Mirrors - MCFRM. D. E. Driemeyer, G. H. MiLey, andW. C. Condit. Patrick Henry A 2B37. A Numerical Investigation of the Evolution of the Electron Distribution Function in Tokamaks. W. H. Miner, N. K. Winsor, and I. B. Bernstein. 2B38. Alpha Particle Orbits in SteLIarators and Torsatrons. J. A. Derr and J. L. Shohet. 2839. Computa ;;0PR: - 7600 EST DN FOR 30 MIN. CJS tional and Analytic Study of Ballooning Modes in Highly Elongated Tokamaks. C. H. An and G. Bateman. 2B40, Nonlinear Magnetohydrodynamics in Three Dimensions. J. U. BrackbilL. 2B41;, Transport of Electron Thermal Energy in Confined Plasmas. B. Coppi and E. Mazzucato. 2B42. Towards a Complete Theory of Field Reversed Equilibria. B. McNamara, J. K. Boyd, and H. L. Berk. 2B43. Generalized WKB Method in One Dimension. H. L. Berk and R. R. Dominguez. 2B44. Stability and Force-Free Fields in an Elliptical Cylinder. G. VahaLa. Patrick Henry B 2B45. Stability of Drift and Drift-AIfven Waves in Sheared Magnetic Field. Y. C. Lee, L. Chen, and W. Nevins. 2B46. Vortices In 2-D Guiding Enter Plasma With Gravity. H. H. Chen, Y. C. Lee, C. S. Liu, and D. Montgomery. 2847. Shape Optimization of Tokamak Plasmas to Localized MHD Modes. R. L. Miller, R. W. Moore, and L. Bernard. 8:45-9:30 REFRESHMENTS Manor GriLL 9:15-10:45 POSTER SESSION 2C Oral Papers 2A5-2A7 wiLL be given in Patrick Henry C Manor HaLL Auditorium 2C1. The Effects of Lou Frequency Electromagnetic Turbulence on Toroidal Plasmas. D. A. Hitchcock. 2C2. On Mode Conversion of Lower Hybrid Haves. S. C. Chiu, V. S. Chan, and G. E. Guest. 2C3. Stabilization of Trapped-ELectron Shear-ALfven Instabilities by Temperature Gradient. D. H. Ross, S. M. Mahajan, R. D. HazeLtine, and H. R. Strauss. 2C4. StabLe Spheromak Current Profiles. H. SeLberg and A. H. Gtasser. 2C5. Lower Hybrid Heating and Current Generation in Versator II. R. EngLade, T. Antonsen, and M. PorkoLab. 2C6. Refinements and Applications of the RINGHYBRID Code. A. Friedman, R. N. Sudan, and J. Denavit. 2C7. The Distribution of and Classical Transport by Alpha Particles in a Thermonuclear Plasma. J. D. Gaffey and R. S. Schneider. 2C8. Parametric Decay Heating with an Electron Cyclotron Have. 6. B. Elder and F. H. Perkins. 2C9. Particle Simulation of X-Point Dynamics. J. N. Leboeuf, J. M. Dawson, T. Tajima, and A. T. Lin. 2C10. Simulation Study of Thermal Versus Particle Diffusion. R. W. Huff, J. M. Dawson, and T. Kamimura. 2C11. Nonlinear Behavior of BaLLooning Modes in Tokamaks. C. C. Hu, P. L. Pritchett, and J. M. Dawson. 2C12. Coalescence of Magnetic Islands. P. L. Pritchett and C. C. Hu. 2C13. Stability of Drift Haves in a Field Reversed Configuration. A. S. Sharma and R. N. Sudan. 2C14. NeocLassicaL Transport in EBT. H. H. Klein, R. D. HazeLtine, N. A. KraLL, and P. J. Catto. 2C15. Linearized Simulation of an Axis Encircling Ion Gyro Instability. J. A. Byers. 2C16. Electron Cyclotron Resonance Heating of Tokamaks at Omega = 2 Omega sub ce. B. H. Hui, K. R. Chu, E. Ott, and T. M. Antonsen. 2C17. A New Trapped-Ion Instability with Large Frequency and Large RadiaL Wavenumber. M. Tagger and R. PeLLat. 2C18. Axisymmetric Sharp-Boundary Toroidal Equilibria and Stability with High Pressure and SmaLL Aspect Ratio. T. Mizoguchi and T. Kammash. 2C19. Analytic Theory of the Trapped Electron Mode. S. K. Hong, S. Inoue, and K. Itoh. Jefferson Room 2C20. GATO. F. J. HeLton, L. C. Bernard, and R. H. Moore. 2C21. Two Transport Models for NoncircuLar Axisymmetric Devices. D. E. Shumaker,-M. G. McCoy, J. KiLLeen, andA. A. Mirin. 2C22. Finite-Length Theory of Collective Free-ELectron Lasers. S. Johnston. 2C23. High-Beta Tokamak Transport Modelling Studies. J. T. Hogan. 2C24. Shear Damping of Drift Haves in Toroidal Geometry. J. H. Connor, R. J. Hastie, K. H. Hesketh, and J. B. TayLor. 2C25. Ionic Cross Section Relevant to PLasmas. A. L. Merts. 2C26. A Transport Estimate for EBT in the Banana Regime. P. J. Catto, M. N. Rosenbluth, and K. T. Tsang. 2C27. Anomalous Loading of RF Antennae Due to Near FieLd-ParticLe Interactions. G. J. Morales. 2C28. Turbulent Model of Magnetic Braiding Part 1: Resonance Broadening Effects on Stochastic Magnetic Fields. 0. TetreauLt, P. Diamond, and T. Dupree. 2C29. Turbulent Model of Magnetic Braiding II: Pressure Correlation Function and SeLf-Consistency. P. Diamond, D. TetreauLt, andT. Dupree. 2C30. The ELectric Sheath and Pre-Sheath in a CoLLisionLess Finite Ion Temperature Ptasma. G. A. Emmert, R. M. WieLand, A. T. Mense, and J. N. Davidson. 2C31. Orbits and Transport in Three-DimensionaL Geometries. A. H. Boozer and L. G. Kuo-Petravic. 2C32. EvaLuation of Orbits and Transport in Three-DimensionaL Geometries. L. G. Kuo-Petravic and A. Boozer. Monroe Room 2C33. A Second StabiLity Region for a Sequence of Finite-Beta FLux-Conserving Tokamak Equitibria. L. Sugiyama, B. Coppi, A. Ferreira, and J. W-K. Mark. 2C3A. AnaLytic Treatment of BaLLooning Mode ModeL Equations in the Vicinity of the Magnetic Axis. J. J Ramos, T. Antonsen, B. Coppi, and A. Ferreira. 2C35. BaLListic Damping - Some Physics Considerations. R. F. Post and H. L. Berk. 2C36. Linear Theory of High-M Tearing Modes. M. Rosenberg, R. R. Dominguez, W. Pfeiffer, and R. E. WaLtz. Patrick Henry A 2C37. Kinetic Equations for Low Frequency InstabiLities in Axisymmetric PLasmas. B. Lane and T. M. Antonsen Jr.. 2C38. Finite Beta Trapped ParticLe Modes. T. M. Antonsen. 2C39. Current Drive With Energetic Etectrons. D. K. Bhadra and R. W. Harvey. 2C40. WKB Theory of the BaLLooning Mode Spectrum. R. L. Dewar, M. S. Chance, and A. H. GLasser. 2C41. NumericaL Studies of Resistive BaLLooning Modes. M. S. Chance and A. H. GLasser. 2C42. The RoLe of the Continuous Spectrum in Ideat MHD BaLLooning Mode Theory. A. H. GLasser. FRIDAY, APRIL 20 7:30-9:00 BREAKFAST Main Dining Room 8:45 ORAL SESSION 3A Terrace BaLL Room B. Cohen and O.ManLey Chairmen 3A1. PoLoidaL Rotation InstabiLity in Tokamaks. A. A. Ware, R. D. HazeLtine, and J. C. wiLey. 3A2. PeLLet AbLation Rate Modifications for Large PeLLets in Tokamak PLasmas. W. A. HouLberg. 3A3. Effect of the QuadrupoLe FieLd on Ion Motion in the Presence of an Electrostatic Wave in a Mirror Machine. G. R. Smith, H. L. Berk, J. A. Byers, and Y. Matsuda. 3A4. Diffusion of Ions in VeLocity Space by a Coherent Lower Hybrid Wave. C. F. F. Kamey. 10:25-10:45 COFFEE BREAK 3A5. A Kinetic Theory of EvoLution of Anisotropic PLasma. Y-P. Pao. 3A6, Adiabatic Compression of a Rotating PLasma. H. 6rad and E. Hameiri. 3A7. StabiLity of FieLd Reversed, Force Free PLasma EquiLibria with Mass FLow. R. N. Sudan. 12:00-1:00 LUNCH (Dining Room doors cLose promptly at 1:00) 1:30-3:00 POSTER SESSION 3B OraL Papers 3A1-3A4 wiLL be given in Patrick Henry C OraL Papers 3A5-3A7 wiLL be given in Patrick Henry B Manor HaLL Auditorium Equilibrium of Low-Aspect-Ratio PLasma Configurations and ImpLications Concerning StabiLity. 6. K. Mori kawa. 3B OPR:- 7600 DOWN APPROX 30 MNS - TINA Inst; biLity Driven by the Electron Return Current in a Field Reversed Ion Ring. A. Reiman and R. N. Sudan. 383. Transition from CoLLisionaL to Pastukhov Ion Confinement for TMX. T. D. RognLien, R. H. Cohen, and T. A. Cutler. 334. Chaotic, Strange Attractor-Type Behavior in Instability Saturation by Mode Coupling. J. M. Wersinger J. M. Pinn, andE. Ott. 385. NonLocaL Investigation of the Lower-Hybrid-Drift Instability in Reversed Field Configurations. J. D. Huba, J. Drake,and N. T. GLadd. 386. Resistive Diffusion of FCT Equilibria. D. B. Nelson. 387. Ion Streaming Instabilities. R. W. Landau. 388. Nonlinear Stabilization of the Ion Beam-CycLotron Instability. J. R. Myra and C. S. Liu. 3B9. Stochastic Heating in a Large-AmpLitude Standing Wave. J. Y. Hsu, K. Matsuda, M. Chu, and T. Jensen. 3B10. Computer Simulation of Current Generation by Lower Hybrid Waves. V. K. Decyk and G. J. Morales. 3811. Ion Beam Fusion: Beam Transport, The Penultimate Problem. S. Jorna and W. B. Thompson. 3B12. Magnetic Fluctuations Excited by ALpha-ParticLes. F. Pegoraro and B. Coppi. 3813. Shear Modifications of Ion Cyclotron Modes. G. GanguLi and P. Bakshi. 3B14. Current Penetration Stage in a Tokamak. P. L. Mascheroni, L. Matteson, and A. L. Sutton. 3B15. Simulation of Axisymmetric Alfven Resonance Heating of Tokamaks. J. DeLucia, S. C. Jardin, and F. W. Perkins. 3816. Beam-TurbuLence Electron Heating. M. C. VeLLa. 3B17. 1-D Reverse FieLd Pinch Bum Simulations. R. A. Nebet, G. H. Mitey, and R. W. Moses. 3B18. Neoclassical Diffusion in Plasmas of HeLicaL or Toroidal Symmetry. A. Pytte and A. H. Boozer. 3819. Low Frequency Wave Propagation in a Hot Toroidal PLasma. M. Cotsaftis. Jefferson Room 3B20. Low Density Ignition Scenarios Using Injection Heating. J. A. HoLmes, J. A. Rome, Y-K. M. Peng, W. A. HouLberg, andS. J. Lynch. 3B21. Tokamak PLasma Variations Under Adiabatic Compression to SmaLL Aspect Ratios. Y-K. M. Peng, J. A. HoLmes, D. J. StrickLer, andS. J. Lynch. 3B22. Interchange Stability of Axisymmetric FieLd Reversed EquiLibria. L. Sparks, J. M. Finn, and R. N. Sudan. 3B23. EquiLibrium and StabiLity of Finite-Beta MuLtipoLes. D. A. D'IppoLito, E. A. AdLer, and Y. C. Lee. 3B24. TurbuLent EvoLution of the CoLLisionLess Tearing Mode due to Stochastic Magnetic FieLds. R. G. KLeva J. A. Krommes, and C. Oberman. 3B25. Diffuse VLasov-FLuid Screw Pinch. C. E. SeyLer and H. R. Lewis. 3B26. Crescent Shape Orbit Diffusion in EBT. K. T. Tsang, J. D. CaLLen, C. L. Hedrick, S. P. Hirshman,andD. A. Spong. 3B27. One-DimensipnaL Transport SoLutions for EBT-11. E. F. Jaeger and C. L. Hedrick. 3B28. Enhanced TaiL for Ions in EBT. C. L. Hedrick, R. A. Dory, E. F. Jaeger, and D. A. Spong. 3S29. A One-FLuid ModeL of Magnetic FieLd FLuctuations In A Magnetized PLasma With A Temperature Gradient. I M. Tkachenko. 3B30. Rotation of a ToroidaL PLasma. S-L. Wen and Y-P. Pao. 3831. The NonLinear EvoLution of Resistive InstabiLities in Finite Beta Reversed FieLd Pinches. D. Schnack andJ. KiLLeen. 3B32. MHD EquiLibrium and StabiLity of the Levitated OctupoLe. M. W. PhiLLips. 3B33. Interaction Between Anomalous Loss and Neoclassical Impurity Transport in Tokamaks. T. E. Stringer. 3B34. Power Requirements of EBT Electron Rings. G. W. Stuart. 3B35. Quasi Linear RadiaL Transport Simulation of TMX Plugs. J. J. Stewart, Y. Matsuda, and H. L. Berk. 3B36. Most ProbabLe MHD Equilibria And Their Stability. J. Ambrosiano and G. VahaLa. Patrick Henry A 3B37. Anomalous Current Penetration. S. M. Mahajan, D. A. Hitchcock, and R. D. HazeLtine. 3B38. PEST II. R. C. 6rimm and R. L. Dewar. 3B39. Plasma Transport by Stochastic Magnetic Fields in Axisymmetric Geometries. H. E. Mynick and J. A. Krommes. 3B40. Neutral Beam Heating Calculations for Torsatrons. D. T. Anderson, J. L. Shohet, J. A. Tataronis and S. Rehker. 3B41. Computer Model of a Slow RFP. R. N. Byrne and C. K. Chu. 3B42. Effect of Toroidal Curvature on Stability Windows for MHD Kink Modes. J. Manickam, J. M. Greene, J L. Johnson, and A. E. MiLter. 3B43. The Goodness of Ergodic Adiabatic Invariants. E. Ott. 3B44. Fueling of a Long-Putse Divertor Tokamak. H. C. Howe. Patrick Henry B 3B45. Modulations). Theory of the Cubic NonLinear Schrodinger Equation. A. E. WaLstead and W. A. Newcomb 3B46. Toroidal Pinch Equilibria With Flow. R. Y. Dagazian. 3B47. Coupling and Penetration of Whistler Waves in Inhomogeneous Ptasma. K. S. TheiLhaber. 3B48. Orbit-Averaged Particle Codes for Long-Time Simulations. T. A. BrengLe, B. 1. Cohen, D. B. Conley, and R. P. Freis. 3:00 WE HOPE YOU ENJOYED THE MEETING Aamodt, R. E. - 138 Canobbio, E. - 1C16 2B31, 2C9, 2C10, 2C11 Grimm, R. C. - 3B38 AdLen, E. A. -3B23 Caramana, E. J. - 2B19 DeLucia, J. - 1830, 3B15 Grossmann, kt. - 2B9 Ambrosiano, J. - 3B36 Carreras, 8. - 1A7, 1C37 Decyk, V. K. -3B10 Guest, G. E. - 1C1, 2C2 An, C. H.-2S39 Catto,P. J. -2B11, Denavit, J. - 1C33, 1C34. Guzdar,P. N. -1C13 Anderson, D. T. - 3840 2C14, 2C26 1C46, 2C6 Hamasaki, S. - 1B41 Anderson, D. V. - 1C28, Cayton, T. C. - 1C12 Derr, J. A. - 2B38 Hameiri, E. - 2B9, 2B21, 2A4 Cayton, 1. E. - 1C38 Dewar, R. L. - 2B4, 2C40 3A6 Antonsen Jr., T. . - Chan, V. S. - 1B5, 2B10, 3B38 Hanatani, K. - 1839 2C37 2C2 Diamond, P. - 2C28, 2C29 Harvey, R. W. - 1B5, Antonsen, T. - 1C48, 2C5, Chance, M. S. - 2B12, Dobrott, D. - 1B6, 1C17 1C43, 2C39 2C34 2C40, 2C41 Dominguez, R. R.- 1C9, Hasegawa, A. - 1C8 Antonsen, T. M. - 2C16, Chang, C. L. - 1B13, 2B17, 2843, 2C36 Hassam, A. B. - 1812 2C33 1B14 Dory, R. A. - 1B36, 3B28 Hastie, R. J. - 2C24 Amurius, D. E. - 2B33 CharLton, L. A. -1B36 Drake, J. - 385 Hatori, T. - 1C25 Auerbach, S. P.-1B7 Chen, H. H. - 2B46 Drake, J. F. - 1A3, 1B13 HazeLtine, R. D. - 1C2, Aydemir, A. - 1B16 Chen, J. - 1B10 1814 2C3, 2C14, 3A1, 3B37 Azumi, M. -1C32 Chen, L. - 1B23, 1C3, Driemeyer, D. E. - 2B36 Hedrick, C. L. -1A5, Bakshl, P. -2B35, 3S13 1C13, 1C44, 2B22, DuBois, D. F. - 1B25 3B26, 3B27, 3B28 BaLdwin, D. E.-1A6 2845 Dupree, T. - 2C28, 2C29 Hetton, F. J. -186, Barnes, D. C.-1C28, Chen, U. -1C4 Eisenstat, S. - 1B20, 2C20 2A4 Cheng, C. Z. - 1C15, 1C40 Hesketh, K. M. - 2C24 BatcheLor, D. B. - 1A5, 1C44, 2B22 ELder, G. B. - 2C8 Hewett, D. - 2A3 1B47 Chiu, S. C. -2B10, 2C2 Emery, M. H. -1B4 Hewett, D. U. -1C12 Bateman, G. - 1B42, 2B39 Choi, D-1. - 1C11, 2A6 Emmert, G. A. - 2C30 Hickok, R. L. - 232 Bauer, F. - 1329 Chu, C. - 1B26, 2B16 EngLade,R.- 2C5 Hicks, H. R. - 1A7, 1C37 SeasLey, C. 0. -1C33 Chu, C. K. - 1B16, 3B41 Estes, R. - 2B30 Hinton, F.L.-1C27 Berk, H. L. -1B2, 2B28, Chu, K. R. -2C16 Evans, Jr., K. - 2B3 Hirshman, S. P. - 1C18, 2B42, 2B43, 2C35, 3A3, Chu, M. - 3B9 Ferreira, A. - 2A1, 2C33, 3B26 3B35 Chu, M. S, - 2B16 2C34 Hitchcock, D. A. - 2C1, Bernard, L. C. - 1B6, Cochran, F. L. - 1C29 Finn, J. M. - 1C6, 3B4, 3B37 2847, 2C20 Cohen, B. 1. - 1C10, 3B22 Hogan, J. T. - 2C23 Bernstein, 1. - 1820, 3348 Fisch, N. j. - 1B17 Hotmes,J.A.-1C37, 1C40 Cohen, R. H. - 2B25, Fisher, J. - 2B15 3B20, 3B21 Bernstein, 1. B. - 1B11 2B26, 2B27, 3B3 Foote, J. H. - 2B27 Horton, Jr., M. - 1C11 2B37 Cohn, D. R. - 2B14, 2B15 Fowter, R. H. - 1C31 Horton, W. - 2A6, 2630 Betancourt, 0.-1329 Condit, W. C. - 2B36 Freidberg, J. P. - 2A3 Hout.berg, M. A. - 3A2, Bhadra, D. K. - 2C39 ConLey, 0. B. - 3B48 Freis, R. P. - 3B48 3B20 Bhattacharjee, A. - 234 Conn, R. M. - 2B29 FriedLand, L. - 1B11 Howard, J. E. - 2B20 BirdsaLL, C. K. - 1C19 Connor, J. W. - 2C24 Friedman, A. - 2C6 Howe, H. c. - 3844 Biskamp, D. - 1C21, 2A6, Connor, K. A. - 2B32 Frieman, E. A. - 1C3 Hsu,J.Y.-389 2B30 Cooper, A. - 1B42 Fyfe, D. - 1C40 Huba, J. D. - 3B5 Bonoti, P. T. - 183 Coppi, 8. - 1C26, 1C48, Gaffey, J. D. - 2C7 Huff, R. U. - 2C10 Boozer, A. H.-2C31, 2A1, 2341, 2C33, 2C34, Ganguti, G. - 3B13 Hui, B. H. -2C16 2C32, 3B18 3812 Garabedian, P. - 1829 HuLse, R. A. -1B1 Boris, J. - 1B4 Cordey, J. G. - 2A2 Gary, S. P.-1C42 Iiyoshi, A. - 1B39 Borowski, S. K. * 1348 Cotsaftis, M. - 1B35, GeLbard, E. M. - 2B3 Inoue, S. - 1833, 2C19 Boyd, J. K. - 2B42 3B19 Gerver, M. J. - 1C5 Irie, H.-1B33 BrackbiLL, J. U. - 2B40 CrumeJr., E. C. -2B33 GiLmore, J. W. - 2B25 Itoh, K. -1B33, 2C19 Brengte, T. A. - 3B48 CrystaL, T. L. - 1C34 GLadd, N. 1*. - 1B13, Jaeger, E. F. - 1A5, Bromberg, L. - 2814, Cutter, T. A. - 3B3 1B14, 3B5 1B48, 3B27, 3328 2B15 D'IppoLito, D. A^ - 3B23 GLasser, A. H. - 2B12, Jaroin, S. C.-2B12, Brunet., F.. - 2B31 Dagazian, R. Y. - 3B46 2C4, 2C40, 2C41, 2C42 3B15 Byers, J. A. - 2C15, 3A3 Dathed, S. - 1830 Goedert, J. - 2B23 Jassby, D. L. - 1C30 Byrne, R. N. - 3B41 Davidson, J. N. - 2C30 GoLdfinger, R. C. - 1B47 Jensen, R.V.-1C41 Catten, J.. D. -1C31, Davidson, R. C. - 188, Grad, H. - 3A6 Jensen, T. - 3B9 3B26 189, 1310 Grebogi, C. - 2B5 Johnson, J. L. - 3B42 CampbeL L, R. 8. - 1B18 Dawson, J. M. - 1B38, Greene, J. M. - 3B42 Johnston, S. - 2C22 Jones, E. M. * 2A2 Manickam, J. - 3842 Nishikawa, K. - 1B33 Schneider, R. S. - 2C7 Joma, S. -3B11 Mankofsky, A. - 1C46 Nuhrenberg, J. - 1C20, SchuLtz, J. H. - 2B14 Joyce, 6. - 1B27 ManLey, 0. P. - 1B24 2B18 SchuLtz, M, - 1B20, TC^O Kaiser,T.B. -1B37 Marchand, R. - 1B34, 2A5 Oberman, C Schwarzmeier, J. L. ' Kamimura, T. 2C10 Mark, J. M-K. - 2A1, Ohkawa, T. 1C22 ' ' Kammash, T. - 1B18, 1B42, 2C33 Okabayashi SeLberg, H. - 2C4 1B48, 2C18 Maron, M. - 1C10 Okada, 0. Sen, A. K. - 1C25 Karney, C. F. F. - 3A4 Marx, K. b. - 1B5, 1C43 Okuda, H. SeyLer, C. E. - 1C28, Kashuba, R. J. -1B18 Mascheroni, P. L- - 3B14 2A7 2A4, 3B25 Katsurai, M. - 1C30 Matsuda, K. * 1C1, 3B9 Ott, E. - 1B3, 2C16, 364, Sgro, A. 6. - 1C12 Kaufman, A. N. - 1831, Matsuda, Y. - 1C1, 2B28, 3B43 Shaing, K-C. - 2B29 2B5 3A3, 3B35 - 3A5, 3B30 Sharky, N. - 1C48 Kaw, P. K. -1B22 Matteson, L& - 3814 . - 2B12 Sharma, A. S. - 2C13 Kesner, J. - 2820, 2B29 Mazzucato, E. - 2B41 ;in, L. b.-1B2 Shohet, J. L. - 2B24, KiLLeen, J. - 2B26, 2C21, McCoy, M. 6. - 2C21 i, F. - 3B12 2B38, 3B40 3B31 McbonaLd, S. M. - 1B31 R. -2C17 Shumaker, b. E. - 2C21 KLein, H. H. - 1B35, McKenty, P. - 1C23, 1C29 -K. M. -1B36, Sigmar, b.J. - 1B43 2C14 McNamara, B. - 2B42 3B21 Smith, 6. R. - 3A3 KLeva, R. 6. -3B24 Meier, H. K. - 1C36 Perkins, F. M. - 2B19, SoLer, M. - 1B15 Kodama, Y.-1C8 Mense, A. T. - 2C30 2C8, 3B15 Sowers, G. - 1C23, 1C29 KraLL, N. A. -2C14 Merts, A. L. -2C25 Petrie, T.M.-1C4 Sparks, L. - 3B22 Krapchev, V. - 2B13 MigLiuoLo, S. -1C26 Pfeiffer, W. - 1C9, 2B17, Spencer,R.L.-2B7 Krommes, J.A. -1A4, MikkeLsen, b. R. - 1B1, 2C36 Spong, 0. A. - 1A5, 3B26, 1C41, 3B24, 3B39 1B21 PhiLLips, M. W. -3B32 3B28 Kuo-Petravic, L. 6. - Mi Ley, 6. H. - 2B36, Pipkins, J. F. - 2B2 Stacey, M. M. - 1B43 2C31, 2C32 3B17 PorkoLab, M. - 2C5 Start, b. F. H - 2A2 Landau, R. - 3B7 MiLLer, A. E. -3B42 Post, b.E.- 181, 1B21 Steinhauer, L. - 2B48 Lane, 6. -2C37 MiLLer, R. L. -1B44, Post, R. F. - 2C35 Steinhauer, L. C. - 1B28 Larrabee, 0. A. - 1C39 2647 Pritchett, P. L. - 2C11, Stewart, J. j. - 1B2, Lau, Y. Y. -1C45 Miner, N. H. - 2B37 2C12 3B35 Leboeuf, J. N. - 2B31, Mirin,A. A. - 2B26, Pytte, A. - 3B18 Strauss, H. R. - 2812, 2C9 2C21 Quimby, b. - 2B48 2C3 Lee, b. K. - 1B36 Mizoguchi, 1. -2C18 Ram, A. - 2813 Strickter, b. J. - 1836, Lee, J. K. - 1C19 MoLvig, K. - 1A2, 1C18 Ramos, J. J. - 2A1, 2C34 3821 Lee, W. w. - 2A7 Mondt, J. P. - 2B23 RawLs, J. M. - 185, 1C4 Stringer, T. E. - 3833 Lee, X. S.-1C2 Montgomery, b. - 1B27, Rechester, A. B. - 1A1 Stuart, 6. W. - 3B34 Lee, Y. C. - 2B45, 2B46, 2B46 Rehker, S. - 3B40 Sudan, R. N. - 1C46, 2C6 3B23 MonticeLLo, b. A. - 2B12 Reiman, A. - 382 2C13, 3A7, 3B2, 3B22 Lewis, H.R.-1C38, Moore, R. W. - 1B6, 1B44, Rensink, M. E. - 2B26, Sugiyama, L. - 2A1, 2C33 3B25 1C17, 2B47, 2C20 2B27 SuLton, A. L. - 1B35, Lin, A. T. -1B38, 2C9 MoraLes, G. J. - 2C27, RewoLdt, 6. - 1B34, 2A5 3B14 LittLejohn, R. 6. - 2B5, 3B10 Rlordan, J. C. - 1C43 Tagger, M. - 2C17 2B6 Morikawa, 6. K. - 3B1 Rogniien, 1. b. - 1B2, Tajima, T. - 2B31, 2C9 Liu, C. S. - 1A3, 1B13, Morrison, P. J . -1C24 3B3 Tang, M. M. - 1B34, 1C13 1B14, 1B27, 2B1, 2B46, Morse, R. - 1C23, 1C29 Rome, J. A. -1C31, 1C36, 2A5, 2A7 3B8 Moses, R. W. -3B17 3B20 Tange, T. - 1B33 Logan, B. G. - 1A6 Motojima, 0. - 1B39 Rosenau, P. - 1C22 Tataronis, J. A. - 1C17, Lortz, D. - 2318 Mynick,H.E. - 3839 Rosenberg, M. - 2C36 3B40 LoveLsce, R. V. - 1C39 Myra, J. R. - 3B8 Rosenbtuth, M. N. -1A1, TayLor, J. B. - 2C24 Lui, H. C. - 1840 Nebet, R. A. -3817 2B11, 2C26 Terry, P. - 2A6 Luxon, J. L. - 1C43 Netson, b. B. -1842, Ross, b. w. - 1C27, 2C3 TetreauLt, b. - 2C28, Lynch, S. J. - 1A7, 1B36, 1C32, 386 Rutherford, P. H. - 1B22, 2C29 3B20, 3B21 Nevins,W. M. -1C44, 1C13 Tewari,b.P.-1B45 Mactennan, C. 6. - 1C8 2A7, 2B45 Santarius, J. F. - 1C27 TheiLhaber, K. S. - 3B47 Magetssen, G. R. - 1B19 Newcomb, W. A. - 2A4, Satyanarayans, P. - 2B35 Thompson, B. -3B11 Mahajan, S. M. - 1C2, 3B45 Schmidt, M. J. -1C14 Tkachenko, 1. M. - 3B29 2C3, 3B37 NichoLson, b. R.-1B25 Schnack, b. - 3B31 Todd, A. M. M. -2B12 Treve, Y. M. - 1B24 VaLeo, E. J. - 1B22, 1B23 Watanabe, T. - 1B33 WieLand, R. M. - 2C30 Tripathi, V. K. -1845, vanRij, W . 1.-1C33 Weiser, A. - 1B20 WiLey, J. C. -1C11,3A1 2B1 VeLLa, M. C. -3816 Weitzner, H. - 1C7 Winske, D. - 1C47 Tseng, K. T. - 2B11, Vigfusson, G. - 1C35 WeLter, H. - 1C21 Winsor, N. - 184 2C26, 3B26 Vi LLaLon, E. - 2B8 Wen, S-L. - 3830 Winsor, N. K. - 2B37 Uckan, N. A. - 1848 Wakatam', W. - 1839 Wersinger, J. M. - 183, Wong, S. K. - 2C19 Uckan, T. - 1346 Watstead, A. E. -3B45 384 Woo, J. T. - 2B32 Ueno, C. - 1833 WaLtz, R. E. -1B44,1C9, White, R. B. - 1A1, 1B32, Wu, C. C. -2C11, 2C12 Uhm, H. S. -1S8, 189 2B17, 2C36 2B12 Yoshikawa, S. - 1832, 1833 Uo, K. - 1839 Wang, T. S. - 2834 Whitson, J. C. - 1C18, Yoshioka, T. -1839 VahaLa, G. - 2B44, 3B36 Ware, A. A. - 3A1 2B11 CALCULATION OF THE KOLMOGOROV ENTROPY FOR MOTION ALONG A STOCHASTIC MAGNETIC FIELD* A. B. Rochester Bell Laboratories, Murray Hill, New Jersey, 07974 M. N. Rosenbluth Institute for Advanced Study, Princeton, New Jersey, 08540 R. B. White Plasma Physics Laboratory, Princeton University, Princeton New Jersey 08540 We have developed a statistical theory for stochastic magnetic fields. A formula for the Kolmogorov entropy has been derived. Excellent agreement between a probability descrip­ tion and direct dynamical computations has been found. & This work was supported in part by DoE contracts No. EY-76-C-02-3073, and No. EY-(76-S)-3237. FINITE 5 UNIVERSAL MODE TURBULENCE AND ALCATOR SCALING* e Kim Molvig (MIT); S.P. Hirshman and J.C. Whitson (ORNL) A self-consistent resonance broadening theory for finite g universal 1 ^ ' mode turbulence is presented. Saturation results from resonance broadening of the electron response due to radial diffusion in combination with streaming along the lines in the presence of magnetic shear, as described in the preceding paper. Electron diffusion, for g > m /m., e e l is due to the magnetic part of the fluctuations. The island width exceeds the rational surface spacing at fluctuation levels of order B /B - 10^, giving ergodic behavior of the lines on a fine scale. Accordingly, the theory constitutes an example of a self-consistent theory of stochastic magnetic fluctuations. The anomalous electron thermal conductivity at saturation, X, - O.inyd, + I,)J has many similarities with experimental observations, including absolute magnitude, and scaling with density, electron and ion temperatures, magnetic field, aspect ratio, and ion mass. lx. Molvig, S.P. Hirshman, J.C. Whitson, MIT Research Report PFC/RR-79-4 (1979). *This research was sponsored in part by the Office of Fusion Energy (ETM), U.S. Department of Energy, under contract No. W-7405-eng-26 with the Union Carbide Corporation and the U.S. Energy Research and Development Administration Grant No., EG-77-G-01-4108. ENERGY CASCADE IN DRIFT-TEARING MODES* J. F. Drake and C. S. Liu Department of Physics & Astronomy University of Maryland College Park, Maryland 20742 The importance of magnetic field fluctuations in producing anomalous cross-field energy transport has been recognized recently and, in particular, the temperature-gradient driven drift-tearing mode is likely to be the source of these magnetic fluctuations.^ An investigation of the non­ linear interaction of these drift-tearing modes has been carried out which demonstrates that wave energy cascades from long to short wavelength. The temperature gradient is of crucial importance in producing this energy flow. The quantity ^]A^)^ is conserved in the nonlinear interaction. Damping of the long wavelength are unstable) by the short wavelength modes^(m*>Vg^) leads to a saturation of the instability when )B)/B=.3pg/l^,. The results are in good agreement with recent measurements of magnetic field fluctuations on the macrotor tokamak^ in which the spectrum of IB^I extended to m*v2-3 and the amplitude IBI/B was independent of the density. ^Research supported by the Department of Energy. 1) J. F. Drake and Y. C. Lee, Phys. Fluids 20, 1341 (1977); D. D'Ippolito, J. F. Drake and Y. C. Lee, BAPS _2_3, 867 (1978); N. T. Gladd, et. al. (this meeting). 2) S. J. Zweben, C. R. Menyuk and R. J. Taylor, UCLA Report #PPG-383. Renormalized Induced Scattering and Nonlinear Damping of Collisionless Drift Waves John A. Krommes Plasma Physics Laboratory, Princeton University Princeton, NJ 08544 A kinetic theory of the turbulent damping of collisionless drift waves is presented. The Direct Interaction Approximation for the nonlinear dielectric 1 1 function is reduced to a renormalized version of induced scattering. In contrast to classical resonance broadening theory, the theory reduces correctly to the weak turbulence limit and is energetically consistent; it includes both propagator broadening by turbulent collisions as well as turbulent corrections to the mean distribution function. Explicit calculations are given for shear- free geometry in the approximation which reduces to Compton scattering on the ions; these systematize, correct, and extend to finite ion gyroradius the 2 earlier calculations of Dupree and Tetreault. For long wavelengths where the Markovian approximation ]k] ->-0 is valid, the nonlinear ion "growth" rate is 2 large and positive, proportional to k,D, . Nevertheless, energy conservation between the waves and particles is demonstrated explicitly by eschewing the Markovian approximation and summing <5j*6E> over all modes. The net power flow into the ions is small, proportional to the square of a typical parallel wavenumber. Extensions of the theory which describe sheared geometry and electron nonlinearities are discussed. *Work jointly supported by U.S. DoE Contract No. EY-76-C-02-3073 and U.S- AFOSR Contract No. F 44620-75-C-0037. ^*D.F. Dubois and M. Espedal, Plasma Phys. ^0, 1209 (1978) I J.A. Krommes and R.G. Kleva, Princeton Plasma Phys. Lab. Rept. PPPL-1522 (1979) ^T.H. Dupree and D.J. Tetreault, Phys. Fluids _21, 425 (1978). THEORETICAL STUDIES FOR THE ELMO BUMPY TORUS (EBT) DEVICE* D. A. Spong, D. B. Batchelor, C. L. Hedrick, and E. F. Jaeger Oak Ridge National Laboratory, Oak.Ridge, Tennessee 37830 ABSTRACT The ELMO Bumpy Torus (EBT) is a closed line device consisting of a core plasma confined within 24 toroidally linked mirror sectors and heated by microwaves. In the toroidal or T-mode of operation, rings of hot electrons are formed in the midplane at each mirror and produce an average minimum in the magnetic field, thus stabilizing the toroidal plasma against flute and interchange modes. Energy and particle loss rates in the toroidal core plasma are predominately due to random scat­ tering of particles onto drift orbits with greater displacements from the plasma center (neoclassical diffusion). The toroidal plasma is heated by the strong damping of extraordinary mode microwaves at the fundamental cyclotron resonant surfaces. Theoretical understanding of this device has rapidly evolved during recent years in a number of areas. These include: equilibria, particle orbits, MHD and drift wave stability, heating (micro- wave and neutral injection), transport, and ring physics. Work in a number of these areas will be described. * Research sponsored by the Office of Fusion Energy (ETM), U.S. Department of Energy under contract W-7405-eng-26 with the Union Carbide Corporation. ENHANCEMENT OF TANDEM MIRROR PLUG POTENTIALS BY THERMAL PARTICLE PUMPOUT David E. Baldwin and B. Grant Logan Lawrence Livermore Laboratory, University of California Livermore, California 94550 ABSTRACT The solenoid ions of a tandem mirror are confined by the potential between the solenoid and the more dense plugs which is generated by Maxwellian electrons. This potential barrier increases only logarithmically with plug density, although (classically) the power to maintain the plugs increases as their density squared. The high peak magnetic fields and neutral injection energies formally seen as required for a tandem mirror reactor are direct consequences of these confinement characteristics. In principle, plugs could alternatively be formed by magnetically confined electrons. However, in the face of electron scattering rates, to do so in the presence of Maxwellian ions filling the resultant negative potential leads to densities having prohibitive power requirements. Provided the neutralizing, oppositely charged species can be maintained non-Maxwellian, dramatic reductions can be achieved in the density (and thus power) required to maintain either ion (1) or electron (E) plugs. Versions of bounce resonance heating or transit-time magnetic-pumping appear promising in the role of a pump-out mechanism. The requirements for frequency, penetration, coupling, and side effects differ for I and E plugs <and ultimately will determine the superiority of one type. This paper will cover a number of topics related to the pump-out, energetics, and stability of such plugs with enhanced potentials. The physics issues have important antecedents in both the Mirror and ELMO-EBT Programs, so that credible cases can be described which require little extrapolation from past experiments. In general, this technique appears to offer promising means for significant reduction in the technology requirements on fields and beams here-to-for thought necessary for tandem mirrors and to improve Q at higher power density. *Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore Laboratory under contract number W-7405-Eng-48. Effects of Toroidicity on the Nonlinear Interaction of Tearing Modes* H. R. Hicks, 8. Carreras", and S. J. Lynch Oak Ridge National Laboratory Oak Ridge, Tennessee 37830 The nonlinear evolution of tearing modes proceeds much the same in toroidal geometry as in cylindrical.^ The most significant difference is that, in our toroidal calculations, about 90% of the plasma volume is encompassed by a stochastic magnetic field line region when pre-disruption profiles^* (q > 1.) are studied. The time for the development of this region corresponds to the growth time of the linear 2/1 tearing mode. The negative voltage spike has a similar character in both cylindrical and toroidal cases. A survey of profiles shows that the extent of the stochastic region depends on the profile assumed. In the toroidal case, the set of profiles which evolves to a large stochastic region may be somewhat larger than in the cylindrical case. The results are obtained with a new computer program, Lobeto, which advances the low 8 toroidal reduced resistive MHD equations.^ When a flux coordinate system is employed, the equations are formally very similar to the cylindrical ones. Consequently, we have used numerical techniques similar to our cylindrical code RSF.^ The three-dimensional functions are expanded in a trigonometric series in poloidal and toroidal angles, thus converting the problem to a large number of coupled one­ dimensional partial differential equations. Research sponsored by the Office of Fusion Energy (ETM), U.S. Department of Energy under contract W-7405-eng-26 with the Union Carbide Corporation. 'Visitor from Junta de Energia Nuclear, Madrid, Spain. ^*B. V. Waddell .et al., Phys. Rev. Lett. 41, 1386 (1978). 2 . . B. Carreras, H. R. Hicks, abstract submitted to this conference. 3 . - . H. R. Hicks et al., Computational Plasma Physics Meeting, Monterey, CA, June 1978, C0NF-780614. Charge Exchange as an Impurity Recombination Mechanism R. A. Hulse, D. E. Post, and D. R. Mikkelsen Plasma Physics Laboratory, 'Princeton University Princeton, New Jersey 08544 -14 2 Relatively large (' 10 cm ) cross sections for charge exchange between neutral hydrogen atoms and highly stripped impurity ions can yield an important recombination mechanism for such ions present in tokamak plasmas. The result can be a marked alteration of the ionization balance, along with enhanced radiative losses in some circumstances. The linear dependence of the conventional electron-ion recombination processes on electron density (together with the typically decreasing penetra­ tion of neutrals into the plasma interior with increasing density) makes the charge exchange recombination process particularly important for low density, intensely neutral beam heated plasma environments. Results are presented from zero-dimensional time independent (coronal equilibrium) and time dependent atomic physics codes in which charge exchange recombination has been included. Particular attention is given to modeling the recent PLT high temperature experiments, where the behavior of the iron impurity at the center is calculated to be dramatically altered by this process. The implications for other tokamak experiments are also discussed. * * RADIAL SCALING IN THE QUASILINEAR MODEL OF DRIFT CYCLOTRON LOSS CONE (DCLC) L. D. Pearlstein, J. J. Stewart, T. D. Rognlien, H. L. Berk Lawrence Livermore Laboratory, University of California Livermore, California 94550* ABSTRACT Linear theory of DCLC predicts that, as the plasma radius (relative to the Larmor radius) increases, stability requires less warm plasma 'dwelling in the uncontained hole in velocity space. To realize this reduction the temperature of this plasma must also drop. However, prior attempts to simulate this effect with the quasilinear code failed in that in all cases marginal stability occurred where the distribution function filled the entire velocity space hole as was the case for the small plasma radius configuration. This behavior was due to the fact that the stable point typically requires more phase space density in the hole as its temperature climbs. In the first quasilinear models the self generated turbulence affected the stream only by heating, consequently reducing its ability to stabilize; as a result, the simulation ran away to the regime where the turbulence was high enough to spill particles out of the trapped part of phase space (2XII-B scaling). The latest calculations improve upon this model by incorporating the effect that a heated plasma stream (due to the turbulence) increases the penetration over the ambipolar barrier thus raising the supply of warm plasma. For the tandem mirror configuration we adopt the model stream J stream ^ "c where n^(np) is the density in the central cell (plug) and the exponent is the standard Boltzman factor. The quasilinear code has been run to compare steady state parameters relevant to TMX and the proposed MFTF B configuration. For the latter, typical plug parameters at 80 kev neutral injection are T^ ^ 1 kev, T^ ^ 50 kev, 6 ^ .5 and e^ ^ 30-50 ev. *Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore Laboratory under contract number W-74G5-ENG-48." Toroidal Effects on the Accessibility of Lower Hybrid Waves * P. T. Bonoli, E. Ott, and J.-M. Wersinger Department of Electrical Engineering, Cornell University Ithaca, New York 14853 We use a lower hybrid wave, toroidal ray tracing code which includes electromagnetic effects, thermal effects, and damping to study the accessibility and energy deposition of lower hybrid waves. Comparisons are made with the limiting case of a cylindrical plasma. In toroidal geometry the pploidal wave number is no longer a conserved quantity. As a result the wave number parallel to the magnetic field kj j changes. This modifies considerably the straight cylinder picture of accessibility and electron Landau resonance. In general the picture of accessibility is quite different than for a straight cylinder. For example, the lower hybrid wave can mode convert to a fast wave and the fast wave can mode convert back to a lower hybrid wave several times before (due to toroidal changes in k^), becoming accessible. Results showing energy deposition for typical tokamak situations of interest (e.g., Alcator C) will be presented, and the implications of different ion damping mechanisms will be discussed (e.g., ion cyclotron and unmagnetized ion Landau damping). *Work supported under U.S. Department of Energy Contract EY-76-S-02-3170, (Task II). A Fully Two-Dimensional Transport Model* by Mark H. Emeryf Science Applications, Inc. McLean, VA 22101 Niels Winsor and Jay Boris Naval Research Laboratory Washington, DC 20375 A fully two-dimensional Eulerian-Lagrangian computer simulation model of tokamak discharges has been developed. The model is based on the quasi­ static evolution of force balance in the plasma and steady-state flow along the flux surfaces. The coordinate system incorporates a general connecti­ vity triangular grid which can simulate non-circular flux surfaces, multiple magnetic axes (including a separatrix) and limiters. The Lagrangian dynamics^ are based on the assumption that the force- balance equation is always satisfied which permits the spatial motion of the flux surfaces to be tracked directly. The transport and diffusion equations are solved in an Eulerian fashion. Since the magnetic fluxes diffuse through the surfaces, the separatrix remains rigorously defined. The perpendicular velocity is found from Ohm's law and the parallel velocity is found from the momentum equation assuming steady-state flow. The triangular grid structure allows the current densities to be deter­ mined in closed form. Results will be presented illustrating the dynamical evolution and trans­ port of. a circular tokamak discharge. Both high and low poloidal beta discharges will be considered and the resulting differences in the induced currents will be discussed. *Work supported by U. S. Department of Energy. iPresent address: NRL, Code 6020, Washington, DC 2210L ^M. Emery, et al., NRL Memorandum Report 3744. LOWER HYBRID ELECTRON LANDAU DAMPING AND CURRENT DRIVE IN THE PRESENCE OF AN APPLIED DC ELECTRIC FIELD AND TRANSPORT LOSSES* K. D. Marx National Magnetic Fusion Energy Computer Center Lawrence Livermore Laboratory Livermore, California R. W. Harvey, V. S. Chan and J. M. Rawls General Atomic Company San Diego, California An applied dc electric field T5 can give rise to a distortion of the electron velocity distribution f^(v) sufficient to significantly modify the (quasilinear) electron Landau damping of lower hybrid waves. In particular, the electron tail will be enhanced in the direction antiparallel to E and depleted in the direction of E_. This asymmetric distortion of f^(v) will result in different absorption rates of the two LH ray channels arising from a symmetric standing wave antenna. In addition, if the electron energy loss channels are velocity dependent, further deviations from a Maxwellian electron velocity distribution will result. In some instances, a portion of the LH energy which is deposited on the electrons will not thermalize but instead will be lost directly by transport processes. The effects on LH heating and current drive of both an applied dc electric field and electron transport due to braided magnetic fields^ are examined by means of a Fokker-Planck code containing a quasilinear electron Landau diffusion term. The key issues addressed are: 1. Electron Landau damping of LH waves propagating in the parallel and antiparallel directions with respect to E. 2. Transport losses of the absorbed LH energy relative to the energy thermalized on the bulk of f^(v). 3. Effects of transport on LH current drive. *Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore Laboratory under contract number W-7^05-ENG-48. -Molvig, J. Rice, and M. Tekula, Phys. Rev. Lett _^1, 12^0 (1973). CURRENT PROFIEE STABILIZATION OF D-SHAPED TOKAMAKS TO IDEAL MHD MODES L. C. Bernard, D. Dobrott, F. J. Helton, and R. W. Moore General Atomic Company San Diego, California 92138 ABSTRACT A complete numerical study of ideal MHD modes, including both internal and external modes, has been carried out. A fixed D shape has been chosen (aspect ratio 2.4, elongation 1.7) in order to concentrate on current pro­ file optimization. The stability analysis of internal modes is done with the interchange and ballooning mode criteria. External modes are studied with the global code ERATO including the effect of a conducting wall arbit­ rarily located. The axisymmetric mode is shown to be easily stabilized by an external wall. By contrast, it is shown that it is difficult to sta­ bilize external kink modes by an external wall. No wall stabilization is used for the external kink mode. Instead the current profile is varied to obtain stability. Under these conditions, plasma equilibria with beta above 8% are found which are stable to all modes. The n = 1 external mode appears to be the most restrictive, where n is the toroidal wave number. The optimal current profile is rather flat and has a poloidal beta less than unity. Work supported by Department of Energy, Contract No. EY-76-C-03-0167, Project Agreement No. 38. A COMPACT FORM OF THE INTEGRAL EQUATION FOR WAVES IN AN INHOMOGENEOUS PLASMA Steven P. Auerbach Lawrence Livermore Laboratory, University of California Livermore, California 94550 ABSTRACT Consider an electrostatic wave propagating in a one-dimensional plasma confined by an external field. The electric field of the wave, E(x) expiMt, obeys an integral equation E(x) =fdx' K(x,x';t*))E(x'). The kernel K(x,x';a)) 1 ^ is determined by the unperturbed distribution function f^(x,v). By making use of Liouville's theorem, which states that phase space is conserved, a simple compact form of K can be derived: CO 0 where v(x,x,*r) is the velocity required for a particle which starts at x, at time t = o to arrive at x at t = T. (There may be multiple values of"v; a sum over such values is implied.) One noteworthy feature of this form is that 3fp/3v does qot appear. In words, this states that the influence of point x on point x is an integral over T of the number of particles which can travel from x to x in time T, weighted by the appropriate wave phase expim. The proof of this result, applications, and generalizations to three-dimensional magnetized plasma will be presented. *Mork performed under the auspices of the U.5. Department of Energy by the Lawrence Livermore Laboratory under contract number W-7405-Eng-48. 3-H. L. Berk and D. L. Book, Phys. Fluids 649 (1968) NONLOCAL HYBRID-KINEIIC STABILITY ANALYSIS OF THE MIRROR DRIFT-CONE INSTABILITY* Han S. Ehm" and Ronald C. Davidson Plasma Fusion Center, Massachusetts Institute of Technology Cambridge, Massachusetts 02139 Richard E. Aamodt Science Applications, Inc. 934 Pearl St., Boulder, Colorado 80302 This paper develops a fully self-consistent nonlocal theory of the mirror- drift-cone instability with emphasis on the influence of large ion Larmor radius and axis-encircling orbits on stability behavior. The analysis is carried out within the framework of a hybrid Vlasov-fluid model. The electrons are described as a macroscopic, cold (T^O) fluid immersed in a uniform axial magnetic field On the other hand, we adopt a fully kinetic model for the ions in which the ions are described by the Vlasov equation. This allows for the possibility of large ion orbits with characteristic thermal Larmor radius (r^) comparable to the radius of the plasma column (R^) . The stability analysis assumes electrostatic flute perturbations about a cylindrical ion equilibrium f?(H,-M.Pa, v ) where to.=const.=angular velocity of mean rotation. 1 - l o z a The radial eigenvalue equation for the potential amplitude $(r) is solved exactly for the particular choice of loss-cone equilibrium, f?=(nQnu/2m) x 5(H,-u)^Pg-lL)G(Vg), which corresponds to a sharp-boundary (rectangular) density profile and a parabolic temperature profile. The resulting dispersion relation for the complex eigenfrequency LO is an algebraic equation of order Z+2, where i is the azimuthal mode number. The dispersion relation is solved numerically for a broad range of system parameters including the important influence of large ion orbits and ion thermal effects. It is found that the nonlocal growth rate exhibits a sensitive dependence on m. ./R , R /R , etc. Moreover, the Li p p c stability growth rate is typically more severe for fast rotational,equilibria (m.=<jjt) with axis encircling orbits than for slow rotational equilibria (M.=oj.), l i i i Stability results are presented for the entire range of r^/R^ allowed by the equilibrium model (0<2r^./R < 1). Li p * Research was supported by the Department of Energy. The research by one of the authors (H.S.U.) was supported in part by the Office of Naval Research under the auspices of a Joint Program with the Naval Research Laboratory. i Naval Surface Weapons Center, Silver Spring, Maryland, 20910. STABILITY PROPERTIES OF A FIELU-REVERSED ION LAYER IN A BACKGROUND PLASMA* Ronald C. Davidson and Han S. Uhm+ Plasma Fusion Center Massachusetts Institute of Technology Cambridge, Massachusetts 02139 Stability properties of an intense proton layer (P-layer) immersed in a background plasma are investigated within the framework of a hybrid model in which the layer ions are described by the Vlasov equation, and the background plasma electrons and ions are described as macroscopic, cold fluids. Moreover, the stability analysis is carried out for frequencies near multiples of the mean rotational frequency Og of the layer. It is assumed that the layer is thin, with radial thickness (2a) much smaller than the mean radius (R^). Elec­ tromagnetic stability properties are calculated for flute perturbations (3/3z=0) about a thin P-layer described by the rigid-rotor equilibrium distri­ bution function f^=(m^n^/2'rr)6(H,-MgPg- T)G(v^), where n^,o)gand T are constants. Moreover, it is assumed that the background plasma has a step-function density profile. Stability properties are investigated including the important effects of (a) the equilibrium magnetic field depression produced by the P-layer, (b) transverse magnetic perturbations (6^0), (c) small (but finite) transverse temperature of the layer ions, and (d) the dielectric properties of the back­ ground plasma. All of these effects are shown to have an important influence on stability behavior. A detailed analysis of the radial eigenvalue equation is carried out for eigenfrequencies near multiples of the mean P-layer rotational frequency, i.e.,](jO-%.d)g)<<(i)^, where M is the complex eigenfrequency, f is the aximuthal harmonic number, is the mean rotational frequency of the P-layer, and is the radial betatron frequency of the layer ions. It is found that the instability growth rate exhibits a sensitive dependence on layer density n, , background plasma density n , the degree of magnetic field depression D Q P p=B (r=0)/B , and the transverse temperature of the beam ions. For example, z ext for a dense background plasma, the system can be easily stabilized by a suffi­ ciently large transverse temperature of the layer ions. Moreover, for -1<P<1, the instability growth rate is significantly reduced whenever the back­ ground plasma density is sufficiently large that tu^Rg/c>>l. * Research was supported by Department of Energy. The research by one of the authors (H.S.U.) was supported in part by the Office of Naval Research under the auspices of a Joint Program with the Naval Research Laboratory. T Naval Surface Weapons Center, Silver Spring, Maryland, 20910. [THERMAL EQUILIBRIUM PROPERTIES OF AN INTENSE ION BEAM WITH ROTATIONAL AND AXIAL MOTION* J. Chen and R. C. Davidson Plasma Fusion Center Massachusetts Institute of Technology Cambridge, Massachusetts 02139 This paper investigates the thermal equilibrium properties of an intense ion beam that has rotational as well as axial motion in an externally applied guide field B^j^. The ion beam propagates through a background plasma that provides partial charge neutralization, and the ion beam equilibrium is described by 1 0 ^ 3/2 ^ the thermal equilibrium distribution function f^=[h^/(2tm^T) ]exp(-m_,V^/2T) x 2 ^ h l 1 "z,Q exp[- (H+t^Pg-V^Pg)/T]. Here, H y /2m^+e^ is the energy, Pg = r[p^ + eAg(r)/c] is the canonical angular momentum, P^=p^+eA^(r)/c is the axial canonical momentum, y and T are constants, -y=const. is the angular velocity of mean rotation, and Vyconst. is the mean axial velocity. Introducing the effective potentials, ig(r)= -nutr^r^/2T + et^Ag(r)/cT and Q^(r) = (e/l)[c})Q(r)-^A^(r)], yields the coupled nonlinear equations.for ^ and I JL /i -Lst () exp [-(Q^+i^)] 1 r 9r \ r 3r r ' 1 z E-yexp [-(ig+Q^)], (2) r or 3r b 2 2 -2 ^2 where n^(r)=yexp[-(ig+i^)] is the density, 6 ^=(u^^^/c^vP, andb ^=tJ^[P^-(l-f))/4v^ 2 2 Here vt=2T/m., 6 =V /c, (D , =4fn^e /nn , f=const. is the fractional charge neutrali­ 1 ' i' z z ' pb zation, and E=sgn[g^ - (1-f)]. The solutions to Eqs. (1) and (2) are investigated z 2 2 analytically and numerically for 0 _< o /b <. °°, and the necessary and sufficient conditions for the existence of radially confined equilibrium solutions [nP(r*^°)=01, and for the onset of field reserval B^(r=0)/B^(r^°°) < 0 are derived. D 2 2 z z As a general remark, for o^«b , the inequality ]is satisfied, and the ^ ^ . . 0 azimuthal rotation and associated influence on the axial field profile B (r) .2 2 dominates the equilibrium behavior. On the other hand, for o >>b , we find the axial motion and equilibrium space charge fields play the dominant role. arch.was supported by the Dec artment or .by one of authors (J.C.) was supported in part by .esearch r the ausnices of a Joint Prosram with oratory. GEOMETRIC OPTICS IN INHOMOGENEOUS ISOTROPIC AND ANISOTROPIC PLASMAS AND ON THEIR BOUNDARIES* L. Friedland and I. B. Bernstein Yale University, New Haven, CT 06520 This paper exploits a general approach to geometric optics in inhomogeneous plasmas based on the properties of the local dielectric tensor E. We express E in terms of its eigenvalues Ej and eigenvectors e^. Then to zero order in the geometric optics approximation the determinant D-E^E^e^ vanishes giving in gen­ eral three branches of the dispersion relation. The possibility of branching makes the formulation of the geometric optics equations different in an aniso­ tropic plasma, where only one eigenvalue vanishes, from that in an isotropic (degenerate) plasma with more than one zero eignevalues. In the nondegenerate case, one can trace the rays by solving equations r = -D. /D ; k = D /D (1) — k w — jr w These equations, however, are singular in the degenerate plasma. Here one can use the fact that the sum F=E^2+^Eg+E2E2 of the second order minors of E also van­ ishes and, therefore, can be used in defining nonsingular ray equations r = -F, /D k = F /F (2) — k w — r w The transition from (2) to (1) on the boundary between degenerate and nondegenerate regions presents numerical difficulties, since D=D =D =0 on the boundary. It will be shown that 1'Hospital's rule applied to (1) as one approaches the boundary from the nondegenerate side, gives in general two values for k, corresponding to dif­ ferent branches of the dispersion relation. On using these derivatives, one can split the rays on the boundary, make a small step into the nondegenerate region, and then follow each of the modes by solving (1). We will demonstrate this method in a case where radiation from a vacuum region enters an inhomogeneous magnetized plasma. The details of our general geometric optics code, which traces the rays and finds the amplitudes, polarization and absorption of the waves along the rays, will also be reported. A Work supported by the U.S. Department of Energy, contract EG-77-S-02-4349. HIGHER ORDER CHAPMAN-ENSKOG THEORY FOR ELECTRONS: APPLICATION TO TEMPERATURE GRADIENT-DRIVEN MODES* A. B. Hassam Department of Physics & Astronomy University of Maryland College Park, Maryland 20742 The Chapman-Enskog expansion is carried to second order for the electron fluid. The resulting equations, an order more accurate than those of Braginskii, are employed to obtain a class of temperature gradient-driven modes. These modes were heretofore not derivable from fluid theory. In particular, the Collisional^'^ and semi—Collisional^ ver­ sions of the tearing mode are recovered. Likewise, the higher-m temperature gradient-driven drift and drift-tearing modes^ are also considered. ^Research supported by a fellowship from the Center for Theoretical Physics, University of Maryland. 1) R. D. Hazeltine, D. Dobrott, and T. S. Wang, Phys. Fluids 18, 1778 (1975). 2) J. F. Drake and Y. C. Lee, Phys. Fluids ^(3, 1341 (1977). 3) N. T. Gladd, J. F. Drake, C. L. Chang, and C. S. Liu, this meeting. DISSIPATIVE DRIFT MODES DRIVEN BY THE ELECTRON TEMPERATURE GRADIENT IN A SHEARED MAGNETIC FIELD* C. L. Chang, J. F. Drake, N. T. Gladd and C. S. Liu Department of Physics & Astronomy University of Maryland College Park, Maryland 20742 We have investigated the Collisional electrostatic drift wave in.slab geometry and have demonstrated both numerically and analytically that this mode can be driven unstable by a positive electron temperature gradient = d In T /d In n. Collisions have been included with a velocity-dependent Lorentz collision operatorThe temperature gradient produces "wells" on either side of the rational surface which localize the radial eigenmode. For a reasonable shear (1^/1 ^ 20), the dissipative drift waves has a threshold n % 3. As the collisionality is reduced, these n driven modes become stable, making a smooth transition to previous collisionless results.^ Analytic expressions for the growth rate have been obtained and are in good agreement with the numerical results. ^Research supported by the Department of Energy. 1) J. F. Drake and Y. C. Lee, Phys. Fluids _20, 1341 (1977). 2) N. T. Gladd and C. S. Liu, to be published in Phys. Fluids. IB 14 MICROTEARING MODES AND ANOMALOUS TRANSPORT IN TOKAMAKS* N. T. Gladd, J. F. Drake, C. S. Liu, and C. L. Chang Department of Physics & Astronomy University of Maryland College Park, Maryland 20742 A numerical and analytic study of the temperature- gradient-driven tearing mode^ has been carried out which demonstrates that this mode is unstable in a slab model for realistic tokamak parameters. Collisions have been included with a velocity-dependent Lorentz collision operator. Modes with are found to be unstable with the growth rate peaking around Details of growth rates, eigenfunctions, and the stability boundaries w^.ll be presented. Radial magnetic field fluctuations ]B)^/B^ associated with this instability lead to enhanced electron thermal transport by effectively converting Xejj toXei* In a nonlinear calculation, we have shown that' this instability saturates by transferring energy from unstable long wavelengths to stable short wavelengths, a form of eddy viscous damping. We find that the resulting perpendicular transport Xgj_ scales as Tgl/2/n, which is consistent with Alcator scaling, and is of the same order of magnitude as the experimentally observed transport coefficient. ^Research supported by the Department of Energy. 1) J. F. Drake and Y. C. Lee, Phys. Fluids 1341 (1977); D. D'Ippolito, J. F. Drake and Y. C. Lee, BAPS 23, 867 (1978). 2) J. F. Drake and C. S. Liu, see separate abstract at this meeting. Observation of Transport in Tokamaks of Arbitrary Shape and Approximate Numerical Description* Mario Soler' Oak Ridge National Laboratory Oak Ridge, Tennessee 37830 Techniques for accurately measuring local values of heat transport coefficients have been reported previously.They involve the analysis of electron temperature perturbations produced by internal disruptions, made apparent in fluctuations of the soft x-ray Bremstrahlung emission. The advent of elaborate multi-channel x-ray diagnostics looking at the plasma cross section in two orthogonal directions will allow to obtain, by means of the quoted methods, local measurements of heat transport flowing in these two directions. For elliptical cross section plasmas such as those to be produced in ISX-B, values of local transport differing by a large (2 to 4) factor should be observed depending on the direction of observation, because of the influence of geometry. The descriptions of transport in noncylindrical geometry usually imply a surface averaging of transport, effectively effacing local features. The comparison of these descriptions with accurate local measurements does not seem very meaningful. In particular, the influence of transport on the evolution of geometry is not clear. Some preliminary work is presented on approximate computational methods being developed with the purpose of taking into account the effects of local values of transport. ^Research sponsored by the Office of Fusion Energy, U. S. Department of Energy under contract W-7405-eng-26 with the Union Carbide ^Corporation. 'Visitor from Junta de Energia Nuclear, Madrid, Spain. 1. Callen, J. D. and Jahns, G. L., Phys. Rev. Lett., 33, 491 (1977). 2. Jahns, G. L., Soler, M., Waddell, B. V., Callen, J. D., and Hicks, H. R., Nucl. Fus. 18, 609 (1978). 3. Soler, M. and Callen, J. D., to be published in Nucl. Fus. 4. Soler, M., Callen, J. D., Navarro, A. P., Granetz, R., Seguin, F., Petrasso, R., Bull. Am. Phys. Son., 23, 759 (1978). EQUILIBRIUM NUMERICAL STUDY OF THE FORMATION OF THE PLASMA IN TORMAC* A. Aydemir, C.K. Chu Columbia University Results from our two-dimensional, single fluid resistive magnetohydrodynamic simulation of TORMAC are presented. We find that with the existing design, TORMAC suffers from a serious "start-up" problem. Starting with some reasonable initial condi­ tions, the plasma fails to implode towards the desired min-B cusp equilibrium. The reasons for this failure are presented. We also present some modifications to the external current distribution and timing, which greatly improve the behavior of the plasma during the implosion phase. With these modifications, the plasma is_ able to implode to a state which resembles the desired cusp equilibrium, and the overall performance, in terms of confinement and temperature, is better than the original TORMAC's. * Work supported by U.S. Department of Energy under contract EY-76-S-02-2456. Two-Way Diffusion Equations and Diffuse Reflection A of Lower-Hybrid Waves Nathaniel J. Fisch Plasma Physics Laboratory, Princeton University Princeton, NJ 08544 We consider the general two-way diffusion equation, h(6)3f/3x = (3/3e)(D3f/36), for 0<x<L, which is well-posed when initial (in x) con­ ditions are given where h is positive and final conditions where h is negative. Here separation of variables does not yield a complete set of eigenfunctions; however, we prove that supplementing that set with a linear (in x) eigenfunction obtains completeness. This eigenfunction expansion has been used in the special case of diffusion through a slab.*** Another special case of interest is the propagation of lower-hybrid waves through density fluctuations, which can be described by diffusion in perpendicular (to B) velocity space. ' The transmitted power falls off only algebraically as the inverse fluctuation thickness, i.e., -1/L . We use the eigenfunction expansion to numerically find the transmitted and reflected spectra. A Work supported by U.S. DoE Contract No. EY-76-C-02-3073. "**H.A. Bethe, M.E. Rose, and L.P. Smith, Proc. Am. Philos. Soc. 78, 573 (1938). ^A. Sen and N.J. Fisch, M.I.T. PRR 78/16 (May, 1978). ^E. Ott, Cornell U. LPS 253 (August, 1978). EBT Neoclassical Ion Transport with Non-Maxwellian f^ and Higher Order Poloidal Expansions* *- R. B. Campbell, T. Kammash, The University of Michigan, Ann Arbor, Michigan 48109, and R. J. Kashuba, McDonnell Douglas Corporation, St. Louis, Missouri 63166 There are two basic assumptions inherent in all neoclassical transport calculations for bumpy tori reported on to date (1,2). The first is that the distribution function can be Fourier decomposed in poloidal angle 0 , and only terms up to the first harmonic (1, cos & , sin 9 ) need be retained. The second states that the Q independent distribution is Maxwellian, f =f . For a simple BGK collision model, relatively high collisionality ) , and for a 9 independent poloidal drift frequency , these assumptions are justified. However, in the lossy regions of velocity space, where J2. — a , the poloidal structure of becomes very impor­ tant. Moreover, for a collisionless plasma < J ; the zero- order distribution exhibits non-maxwellian features specifically a "loss cone" distribution. Our calculation is based on the bounce-averaged drift kinetic equation for ions, in which we use a simple BGK collision model. The collision frequency is velocity dependent and modeled to crudely take into account the large velocity space derivatives near the /la: o loss cone. With this model, and with the restriction (2), we have calculated four varia­ tions of the neoclassical particle diffusion coefficient, D^. Two of the coefficients are computed using poloidal terms up to and including the first harmonic (cos 9 , sin 6 ) , one with f^=f^, and the other with f^=f^.. The other two coefficients are similar to these, but we have retained up to and including the second poloidal harmonic. Sample results are sketched below for typical EBT-S operating conditions. The loss cone results can be interpreted physically by noting that the regions where o are de­ populated, thus reducing the driving term in the neoclassical fluxes in this region. The plateau result in ref. 2 relys on this lossy area con­ tributing greatly to D', as it should in that case, since the assumption ^o"^M made. This, however, implies that the bulk distribution neither sees nor responds to the loss cone. The introduction of non-maxwellian f^ driving terms seems rather important at low collisionalities since it appears to allow to scale like , decreasing with decreasing col­ lisionality . o *7" — f =f.. 2nd harmonic o M --f =f„ 1st harmonic f =f-^ lstt 2nd harmonic o LC (1) C. L. Hedrick, D. A. Spong, L. W. Owen, Bull. Am. Phvs. Soc. 23_ Sept. 1973 p. 376 papers 8P1 through 8P3. (2) . R. D. Hazeltine, X. A. Krall, H. H. Klein, "Neoclassical Transport in EBT" March 1979 (to be published). *wcrk suncarted bv DCE IB 19 Argonne' Beam Propagation and Target Experimental Program for Proposed Heavy Ion Facility G. R. Magelssen Argonne National Laboratory Abstract Argonne National Laboratory is proposing to test the feasibility of heavy ion beam drivers for inertial confinement fusion. The first proposed facility would irradiate small targets, and study beam propagation in a background gas with Xe beams. The proposed program structure contains two phases. The Phase I facility could be used to create Xe beams with particle energies of 1.6 to 6.4 GeV and pulse lengths of 2 to 10 nsec. The total beam energy on target would be 1 to 8 kJ. For Phase II the total energy would be increased to 40 to 50 kJ. Me will discuss the direction and goals for the associated programs. Simple unclassified ion beam driver compression targets as well as foil targets will be described, and implosion calculations presented The important heavy ion background plasma beam instabilities will be reviewed. A discussion of diagnostics for the experimental programs will also be given. A Finite Element Solution of a Reduced Fokker-Planck Equation 1. Bernstein, A. Weiser, S. Eisenstat, and M. Schultz Yale University- New Haven, Ct. 06520 In this paper we compare several methods for the numerical solution of a reduced Fokker-Planck equation, and conclude that one method, the Rayleigh-Ritz finite element method, has innate advantages over the others. In particular, a minimum principle yields approximations to the flux across the boundary, the curved portion of the domain has a natural boundary condition which needs no special treatment, and a singularity in the solution can be handled easily using nonuniform meshes. We describe an efficient implementation of the Rayleigh-Ritz method using tensor products of one-dimensional piecewise-polynomial basis functions and numerical quadrature rules. We present numerical results comparing the flux estimates obtained using the Rayleigh-Ritz method to previous analytic and numerical flux estimates. Work supported by Department of Energy contract EG-77-S-02-4349 * Alpha-Particle Heating in Tokamaks D. R. Mikkelsen and D. E. Post Plasma Physics Laboratory, Princeton University Princeton, New Jersey 08544 A Monte Carlo alpha-particle heating routine for use in 1-D tokamak transport codes has recently been developed. The algorithm models first-orbit losses, spatial spreading of the plasma heating profile which is due to wide banana orbits, and the time delay in the heating due to the finite slowing-down time. The alpha-particles follow orbits prescribed by the first- order guiding-center drift equations for axisymmetric tokamaks. The average drag caused by Coulomb scattering is used to compute the bulk heating rates for the electrons and ions and self- consistently slow down the sample alpha particles; changes in the orbits caused by the drag are included. We compare our results to previous calculations of the influence of the plasma current and of changes in the alpha-particle orbits on the heating profiles. We then present results from an extensive parameter survey in which we varied the aspect ratio, the current profile, the density profile, the Tg profile, the alpha birth profile, and the position of the wall which acts as the alpha-particle limiter. * Work supported by U. S. DoE Contract No. EY-76-C-02-3073. Tearing Modes in a Braided Magnetic Field P.H. Kaw, E.J. Valeo, and P.H. Rutherford Plasma Physics Laboratory, Princeton University Princeton, NJ 08544 Magnetic braiding, together with large electron mobility parallel to B , has been suggested as an explanation for the anomalously large electron ther­ mal conductivity observed in toroidal confinement experiments. Simple esti­ mates demonstrate that the anomalous electron viscosity p that should be an additional consequence of the magnetic braiding can yield tearing mode growth rates much larger than those determined by the (observed) classical resistivity. We calculate the growth rate of the m = 1 tearing mode as well as that of the (constant-^) m>_2 tearing mode."** These rates scale as p ^ and p ^ , respectively. The nonlinear behavior of the constant-^ mode is calculated in analogy 2 with a previous calculation for the resistive problem. When viscosity dom- V? inates, the island width w increases in time as t at a rate much faster than the constant rate determined by resistivity alone. Our estimates indi­ cate that this should typically occur during the initial growth of the island, when its size is a small fraction of the discharge radius; at later times, the usual resistive growth dominates. We also examine the possibility that elec­ tron viscosity of this type could play a role in the disruptive instability. Recent calculations suggest that field line stochasticity may onset abruptly as the island width exceeds a critical value. We have modeled this effect by rapidly increasing p(t) during a short interval. Immediately upon this increase, w(t) accelerates, followed by a return of w(t) to the resistive rate. The impulsive increment in w can be a moderate fraction of the dis­ charge radius for plausible values of p and could be responsible for triggering disruptions. & Work jointly supported by U.S. AFOSR Contract No. F 44620-75-C-0037 and U.S. DoE Contract No. EY-76-C-02-3073. "**H.P. Furth, P.H. Rutherford, and H. Selberg, Phys. Fluids 1(5, 1054 (1973). ^P.H. Rutherford, Phys. Fluids 16, 1903 (1973). A Coupling of Lower Hybrid to Acoustic Modes E.J. Valeo and Liu Chen Plasma Physics Laboratory, Princeton University Princeton, NJ 08544 We calculate the convective amplification factor and absolute instability thresholds and growth rates for decay of lower hybrid lower hybrid + acoustic quasimode. Density and temperature gradients establish the thresholds by limiting the spatial extent over which appreciable ion lA, response occurs. Relative pump bandwidths greater than (m/M)can sub­ stantially reduce the convective growth factor. Work supported by U.S. DoE Contract No. EY-76-C-02-3073 and by U.S. AFOSR Contract No. F 44620-75-C-0037. A POSSIBLE STRANGE ATTRACTOR IN MHD CONVECTIVE INSTABILITIES Y. M. Treve, La Jolla Institute and 0. P. Manley, Office of Fusion Energy ABSTRACT We discuss the possibility that in a tokamak for sufficiently large temperature gradients, the convective motion driven by those gradients (2 3) evolves into a strange attractor ' . Such dissipative (non-Hamiltonian) motion is intrinsically stochastic, similar to that previously encountered (4) by E. N. Lorenz . This model of MHD turbulence is expected to lead to enhanced heat transport. We begin by introducing the concept of a strange attractor as it arises in fluid mechanics. Next, motivated by physical considerations, we present a theorem from which the level of turbulent fluctuations may be estimated^"^. Finally, in the context of a simple model, we discuss the possible impli­ cations for tokamaks. References 1. E. K. Maschke, R. B. Paris, and B. Saramito, Calculs Non Lineaires de Stabilite MHD, Eur-Cea-FC-938, Fontenay-Aux-Roses (France). 2. D. Ruelle and F. Takens, Comm. Math. Phys., 20^ (1971) 120. 3. Y. M. Treve, in "Topics in Nonlinear Dynamics," AlP-Conf. Proc., No. 46, American Institute of Physics, NY, 1978, Ed. S. Jorna. 4. E. N. Lorenz, J. Atm. Sci., _20 (1963) 130. 5. Y. M. Treve, J. Math. Phys. (submitted for publication). Cubic Turbulence* by D. R. Nicholson and D. F. DuBois** Department of Physics and Astronomy The University of Iowa Iowa City, Iowa 52242 Hie nonlinear interaction of a wave with itself is often described by a cubiciy nonlinear partial differential equation. Important examples include the nonlinear Schrodinger equation model of Langmuir turbulence, 2 and a model for the nonlinear interactions of drift cyclotron modes which has been applied to mirror plasmas. We develop an approximate statistical theory for such equations. While our development is new, 3 the results are implicit in the elegant formalism of Martin et al. Our theory is analogous to Kraichnan's direct interaction approximation for quadratically nonlinear equations. We present a progress report of an investigation of the properties of this approximate theory. For example, when applied to the nonlinear Schrodinger equation, the theory yields the oscillating two stream in­ stability as an almost trivial consequence. * Work supported by U.S. D.O.E. and KSF Atmospheric Research Section. ** Address: Los Alamos Scientific Laboratorv 1. V. E. Zakharov, Sov. Phys. - JETP 35, 90S* (1972). 2. A. K. Nekrasov, Nucl. Fusion 14, 865 (1975); R. E. Aamodt, Y. C. Lee, C. S. Liu, M. N. Rosenbluth, B. 1. Cohen, and D. R. Nicholson, to be published. 3. P. C. Martin, H. A. Rose, and E. D. Siggia, Phys. Rev. AS, 423 (1973). THE SLOW ION CYCLOTRON WAVE IN TOKAMAKS Cheng Chu General Atomic Company San Diego, California 92138 ABSTRACT The propagation properties of electromagnetic waves in a bounded plasma imbedded in a tokamak type nonuniform magnetic field is studied numerically in the ion cyclotron frequency range. By solving the coupled wave equations, it is found that, in addition to the global fast cavity mode with (D > the slow ion cyclotron wave with M can also be a global mode provided that the parallel wavelength is sufficiently / 2 2 2 2 2 2 2\ short ^k^C == - Lo) > (Vp^/(ML - LO )^. This slow mode has a dominant left-hand polarized electric field and can propagate within the plasma. A unique feature of this mode is that it always has a small poloidal mode number with respect to its center. Since the ion heating is primarily from the left-hand polarized electric field, this slow mode gives a more efficient heating and better energy deposition pattern than that from the fast mode, which has a small and usually ill-placed left- hand polarized component. A recent experimental observation of the increase of loading at M =0.8 $2. seems to indicate the excitation of this *** 1 slow ion cyclotron mode in a tokamak type device. The ramifications of this mode will be discussed in the context of RST and PLT. Supported by the Electric Power Research Institute, EPRI Contract No. RP 323-3. ^Y. Yasaka, S. Komori and R. Itatani, Paper C3-1, Third Topical Conference on Radio Frequency Plasma Heating, Pasadena, California, Jan. 11-13, 1978. GUIDING Di: Glenn uoyce (Universiuy of ..cva, , Lu Jniversiu*.* Maryland, and 3. Mcncrcmerv The nest common feature of the two-dimensional guiding cenuer plasma model has been its propensity for the nonlinear formation of large- scale vortices and for exhibiting associated enhanced transport. The large vortices can result from a variety of stimuli: high initial energies, external electric fields, or microscopic instabilities driven by gradients. Here we point out another, essentially universal, mechanism: magnetic field gradients, or more simply, a uniform gravitational field. The essential feature is a guiding-center drift which depends upon the sign of the charge. A slab geometry is considered, with an electric field E = -Vd>, <j)=<j)(x,y,t), B = B e , g = -ge^, with 3/3z = 0. Simulations — — o z — y involving 10,000 particles, with periodic boundary conditions in x and <{) = 0 on the y boundaries, are started from a condition of zero electro­ static energy by randomly loading pairs. Electrostatic energies develop which reach maxima above the random loading value (the threshold value for the onset of the negative temperature regime in the g = 0 case), and then execute large fluctuations. A maximum in the electrostatic energy' as a function of g is observed, and is unexplained. A most-probable-states analysis leads to the Poisson equation = -47fe{n^exp[-(etp + nugy)/e] -n ^expt-(-e^) + m gy)/Gj) ' oe e which has no_ spatially uniform solution for either- sign of 0. Probably the most interesting feature of the analysis is that it appears possible to pass from negative to positive temperatures by changing only the boundary conditions. Clamping the potential between the two faces normal to g at the same value can enhance the vortex formation in contrast to the situation where a finite potential jump is permitted. This suggests the possibility sunnressinr ;enuiaa i; Lan: rermauion m multipcies by applying appro­ priate potential differences between the current-carrying ods and the walls. FIELD REVERSED PLASMA ROTATION AND TRANSPORT Loren C. Steinhauer Mathematical Sciences [Northwest, Inc. Bellevue, Washington 98009 Field Reversed Plasmas (FRP) have been observed to spin up and sub­ sequently develop a destructive m=2 stability. While the stability limit for spin velocity has been characterized approximately, the cause of rota­ tion remains unclear although several mechanisms have been proposed. We present theoretical evidence for one of these, i.e., the shorted, rotating, open field line plasma causes plasma on unshorted closed lines to spin up by viscous friction. This model seems to explain several observations; the time delay in onset of spin, the rapid spin up rate, and the achieve­ ment of rotation' rates faster than the diamagnetic drift frequency. The decay of stable FRPs is necessarily a two-step process; particle diffusion from closed to open field lines followed by convective stream­ ing along field lines. Since the latter process is typically much faster, it will rapidly deplete the open line plasma, leading to steep density gradients near the separatrix; this will accelerate the cross-field dif­ fusion and possibly excite anomalous processes. We describe a model based on a quasi static open field line plasma. The results indicate that the global confinement is. a geometric combination of the diffusive and streaming timescales (the geometric mean for the case of classical transport). In addition there is an extremely sensitive dependence on aspect ratio; low. aspect ratio plasmas have much faster instantaneous decay rates. This work was supported by USDOE contract no EY-76-C-06-2319 1B29 NONLINEAR SATURATION OF BALLOONING MODES FOR TOKAMAKS F. Bauer, 0. Betancourt and P. Garabedian New York University, New York, N.Y. 10012 A CRAY version of our equilibrium and stability code for plasmas in three-dimensional toroidal geometry has been written that runs 30 times faster than the published version for the CDC6600 (cf. F. Bauer, 0. Betancourt and P. Garabedian, "A Computational Method in Plasma Physics," Springer Series in Computational Physics, Springer-Verlag, New York, 1978). Fourth order accurate estimates of the energy landscape can be calculated. The improved code has enabled us to study nonlinear saturation of instabilities for screw pinches with realistic distributions of pressure and rotational transform. After difficulties stemming from truncation error in the variational method are overcome, results are obtained that go beyond what has been learned from linear stability theory. An investigation of the saturation cf ballooning modes for Tokamaks is in progress. Free Boundary Equilibria with Multipole Expansion of External Field in Noncircular Tokamaks 0. OKADA**, S. DALHED, J. DELUCIAand M. OKABAYASHI Plasma Physics Laboratory,Princeton University Princeton, New Jersey 08544 The external magnetic field for noncircular equilibrium is studied in terms of multipole moment coefficients, and these co­ efficients are expressed in terms of geometrical parameters and the plasma properties. The noncircularity for a given external field configuration is compared with the exact numerical calcul­ ation. The Grad-Shafranov equation is solved for arbitrary noncir­ cular tokamaks. Noncircularity, e.g., b-a/b+a for ellipticity, is assumed to be of the order of the inverse-aspect-ratio; therefore, the elongation ratio of up to about 2 is covered with­ in the framework of the present study. The external poloidalflux is expressed in terms of multipole moment coefficients. By ex­ pressing the external field requirement with the moment coefficients, we can directly invert the problem, namely, the geometrical para­ meters such as a major radius, ellipticity, triangularity, etc., are obtained from the given external fields. The analytic result is compared with the result of numerical calculation by the Princeton Equilibrium Code, which gives a satisfactory agreement. * This work is supported in part by United States Department of Energy Contract No. EY-76-C-02-3073. On leave from Central Research Laboratory, Hitachi Ltd. , Tokyo, Japan. SPECTRUM AND EIGENFUNCTIONS FOR A FIELD EQUATION WITH STOCHASTIC RAY TRAJECTORIES* Steven W. McDonald and Allan N. Kaufman Physics Department and Lawrence Berkeley Laboratory University of California, Berkeley, California 94720 As a model for linear wave equations arising in plasma physics, with non- separable geometry, we investigate the two-dimensional Helmholtz equation^ (V + k )^(x) = 0. For a racetrack boundary, whose only parameter is its aspect ratio, the ray trajectories (of geometrical optics) are stochastic, except for the limiting case of a circular boundary. We examine the eigenvalue spectrum (for a given parity), and find the eigenvalues well-spaced, with no near degeneracies; in contrast, for the circle, the eigenvalues are highly clustered, with frequent accidental near-degeneracies. The eigenfunctions are qualitatively different for the racetrack and the circle: the racetrack's eigenfunctions have random-looking nodal curves, which almost never cross. Also, they fill the whole area uniformly, in contrast to the Bessel eigenfunctions of the circle. The sensitivity of these features to the aspect ratio will be presented. Work supported by the Office of Fusion Energy of the U.S. Department of Energy under contract No. W-7405-ENG-48. 1. S. McDonald and A. Kaufman, LBL-8587, submitted to Phys. Rev. Letters. MAGNETOHYDRODYNAMICAL INTERCHANGE INSTABILITY IN LOW-S PLASMAS IN SHEARED SYSTEMS Shoichi Yoshikawa and Roscoe White Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08544 ABSTRACT The stability criterion of a magnetized plasma with respect to interchange in sheared configuration was reexamined. By retain­ ing the finite growth rate, the singularity at the magnetic surface, where the perturbation is constant along the magnetic line, is removed. The resultant wave equation yields the result that the Suydam condition is both a necessary and a sufficient condition for stability at least for the plasma pressure less than (1 - A) 8^.. Here 8^ is the maximum 8 for the original Suydam criterion and A is a small positive number (less than 0.03), which presumably can be made arbitrarily small by improving the numerical approximation. Work supported by U.S. DoE Contract No. EY-76-C-02-3073. Current-Driven Drift-Wave Instability of a Finite-6 Plasma in a Sheared-Magnetic Field T.Tange, C.Ueno, H.Irid, T.Watanabe, S.Inoue*, K.Itoh*, Kyoji Nishikawa and S.Yoshikawa** Institute for Fusion Theory and Faculty of Science, Hiroshima University, Hiroshima, Japan * Department of Physics, University of Tokyo, Tokyo,Japan ** Princeton University, Princeton, N.J., U.S.A. We have made an analytical and numerical analysis of the drift-wave instability in a sheared magnetic field by taking both the current and the finite-p effects into consideration. Recent analysis has shown the stability of the collisionless drift wave in a sheared magnetic field. The plasma current which produces the magnetic shear can, however, drive an instability. In low-p plasmas where the drift wave can be treated as electro­ static, the stability criterion is obtained numerically as shown in Fig.l. Here u is the electron drift velocity along the mag­ netic field and the other notation is standard. For 6>m/M, the electron-to-ion 0-2 t !— )— !— r— r* — !— !— - mass ratio, the drift wave couples to the Alfven wave. We found that al­ though this coupling tends to stabilize UNSTABLE the short wavelength modes, as shown previously,1) it destabilizes a long & wavelength mode. Our method is to > o-i first derive a coupled set of equa­ tions for the electromagnetic drift wave in the presence of both the STABLE - current and the finite-6 value, and then apply the technique similar to that of Antonsen for the electro­ 00 __[__i__' 1 '__!_ ! ) ! 00 0-5 static mode to the coupled equa­ 1-0 k pi tions. The result indicates the Fig.l existence of a new-type of insta­ Stability criterion for elec- bility driven by the combined effect trostatic drift wave of low-; of the current and the coupling to plasma([TT„ ==TT^.,, M/m=1836, the Alfven mode. We then obtained L /L =32)^ ^ numerical solutions for the local­ s n ized mode of the coupled equations and derived the stability criterion 0.1 which is shown in Fig.2. In the numerical analysis, we developed a new technique, i.e. the Newton method, in order to obtain the UNSTABLE eigenvalue by the shooting method. 0.05 This technique has considerably improved the convergence of the ...."I iterative solution. The result shows that the drift wave, when - I--. coupled to the Alfven mode, can STABLE be driven more unstable by ,a cur­ rent of substantially smaller 0.6 1.2 electron drift velocity than the electrostatic case. - B C*/. 3 Fig.2 1) S.Inoue, K.Itoh, T.Tange, Stability criterion for elec­ Kyoji Nishikawa and S.Yoshikawa, tromagnetic drift wave IAEA-CN-37/W-3 (1978). (T =T., M/m=1836, L /L =32). e i s n Two-Dimensional Eigenmcde Analysis of the Trapped-Ion Instability^ R. Marchand, G. Rewoldt, and W. M, Tang Plasma Physics Laboratory, Princeton University, Princeton, N. J, 0.8544 An analysis of the two-dimensional eigenmode structure of the trapped- ion instability in axisymmetric toroidal geometry is presented- The approach is based on the drift-kinetic equation in which collisions are modeled by an energy and pitch-angle-dependent Krook operator. The perturbed electro­ static potential i is expanded in a Fourier series in 9 to account for the poloidal structure and each harmonic <j)^ is expressed as a truncated Taylor series in the minor radius to account for the radial structure. The governing equations take into account the spatial variations in the equi­ librium profiles (e.g., density, temperature, etc.). They also allow for the analysis of eigenfrequencies both less than and of the order of the average ion transit frequency. In addition to the two-dimensional problem, Our basic analysis is applied to the familiar radially local problem, in which radial derivatives of the perturbed potential are ignored, and to the one-dimensional radial analysis of Gladd and Ross,^ in which the mode is assumed to be nearly flutelike. Results corresponding to these two special cases are found to be in reasonable agreement with previous calculations. The main original contribution of this work, however, is that it is capable of treating the full two-dimensional structure of the instability over the whole plasma cross section. A comparison of a full two-dimensional calcula­ tion with its corresponding one-dimensional counterpart shows significant quantitative and qualitative differences in the mode structure and eigen- frequency. Our approach, assumes the large aspect ratio limit with circular, concentric magnetic surfaces. It is limited to electrostatic, long wave­ length perturbations for which ^r^bi^ ^ ^ well-satisfied, where k^ and p . are the typical mode radial wavelength, and the ion hanana width., bi * * Work supported by U.S. Department of Energy Contract #EY-76-C-02-3073, "**N. T. Gladd, D. W. Ross, Phys. Fluids 16, 1706 CL973). ANALYSIS OF PLT DISCHARGES WITH HIGH NEUTRAL INJECTION* A. L. Sulton, M. Cotsaftis, ^ and H. H. Klein Science Applications, Inc., La Jolla, California 92037 ABSTRACT We have analyzed a PLT discharge with high neutral injection (P = 2.4 MW) with the 2-D tokamak transport code, G2M, including the previously developed global transport model. ^ The model accounts for the anomalous electron and ion heat and particle transport arising from drift-tearing islands in the outer region, from trapped electron modes in the central region, and from sawtooth modes inside the region, q = 1. Also, the programmed input rate of neutral gas was taken in order to study the density evolution. We find that the build up of the electron and ion temperatures, T^ and T., and the density as well as the displacement of the flux surfaces is in close agreement in space and time with experimental results. During the heating process where the ion temperature jumps from 1 keV up to 5. 7 keV, the transport improves in the center due to the particle loading increasing the density, though the safety factor remains below 1. As a consequence, the energy lifetime, after a deterioration at the beginning of the turn on of the neutral beam, shows a slow recovery, varying from 37 ms down to 18. 5 ms, and then back to 23. 2 ms. Various properties of the system during its time evolution are likewise discussed. * Work supported by the U. S. Dept, of Energy. ^Permanent address Fontenay-aux-Roses. *M. Cotsaftis and H. H. Klein, APS Meeting, Colorado Springs, November 1978. CONDUCTING SHELL STABILIZATION OF FCT EQUILIBRIA* L. A. Charlton, R. A. Dory, Y-K. M. Peng D. J. Strickler, S. J. Lynch and D. K. Lee Oak Ridge National Laboratory Oak Ridge, Tennessee 37830 The computer code ERATO^* has been used in an ongoing study of the stability properties of D-shaped flux conserving tokamak (FCT) equilibria with aspect ratio of 4 and elongation of 1.65. As reported previously , equilibria were found with g in excess of 10% and stability for values n=l,2,3 and 4 of the toroidal mode number. These equilibria had 3^=2.5 (about half the aspect ratio) and a q ratio (safety factor at the edge/ safety factor at the axis) of 2.0. A conducting shell was assumed, having a radius 20% larger than the plasma radius. Additional studies have now been completed with the conducting shell removed to infinity. The resulting change in stability is about as one might expect on the basis of conducting wall stabilization: the critical i3 value falls from 16% to 10% for n=l, but varies only slightly for the higher modes whose radial wavelengths are small enough to provide isolation of plasma effects from the shell. To minimize the higher modes and maximize [3, these equilibria have relatively broad current profiles; they rely on the D-shapedness of the cross section to keep q above the Mercier criterion limit at the axis and above the empirical limit q surface H 2-3. Because plasma stability is determined by the most unstable model, we conclude that wall stabilization is not required to obtain stability at finite n for 3 values as high as 10%. ^Research sponsored by the Office of Fusion Energy (ETM), U. S. Department of Energy under contract W-eng-26 with the Union Carbide Corporation. ^D. Berger, R. Gruber and F. Troyon, Second European Conference on Computa­ tional Physics, Garching, Germany (1976). ^L. A. Charlton, R. A. Dory, Y-K. M. Peng, D. J. Strickler, S. J. Lynch and D. K. Lee, Bull. Am. Phys. Soc. 23 897 (1978). OPTIMIZATION OF TRANSITION COIL DESIGN IN TANDEM MIRROR SYSTEMS FROM THE POINT OF VIEW OF INTERCHANGE STABILITY Thomas B. Kaiser Lawrence Livermore Laboratory, University of California Livermore, California 94550 ABSTRACT Tandem mirror systems are potentially subject to curvature-driven ballooning-interchange modes because of unfavorable field-line curvature in the transition regions between the minimum-B end cells and the axisymmetric central cell. One way to alleviate this problem is to shape the transition coils in a way that optimizes the stability of the system with respect to these modes. A technique is presented for optimizing the coil design with respect to a particular subclass of curvature-driven instabilities, viz., those with a purely interchange character. The calculation uses a variational procedure to find the transition field configuration that maximizes the value of the quantity ^da(K^/B) ) 3 ^ + p„)/3^[, whose positivity is a sufficient condition for stability. is the normal component of curvature, p^,„ the plasma pressures perpendicular and parallel to the magnetic field, and ^ the magnetic flux. This approach to improving interchange stability generalizes to finite S and anisotropic pressure one based on the low-g, isotropic pressure stability criterion Vp - VjfdR/B >_ 0 used by Riordan et in the design of the Multiple Mirror Experiment. *Work performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore Laboratory under Contract number W-7405-Eng-48. ^J. C. Riordan, A. J. Lichtenberg and M. A. Lieberman, Nuc. Fusion 1J3, 21 (1979). CROSS-FIELD ELECTRON TRANSPORT DUE TO THERMAL ELECTROMAGNETIC FLUCTUATIONS* A. T. Lin and J. M. Dawson Center for Plasma Physics and Fusion Engineering University of California, Los Angeles, California 90024 and H. Okuda Princeton Plasma Physics Laboratory Princeton University, Princeton, New Jersey 08540 The cross-field electron particle and energy transport for a high 3 thermal plasma is investigated using a magnetostatic plasma model. The diffusion process due to magnetic fluctuations alone is observed to have a Bohm-like scaling, whereas in the presence of both electrostatic convec­ tive cells and magnetic fluctuations, the diffusion, because of the strong interaction between these static modes, becomes smaller than the diffusion- assuming these modes are independent. *h!ork supported by USDOE. Magnetohydrodynamic Instabilities in a High Shear helical System M. Wakatani, T. Yoshioka, K. Hanatani, 0. Motojima, A. Iiyoshi, K. Uo Plasma Physics Laboratory Kyoto University Gokasho, Uji, Kyoto A rotational transform exceeding unity at -+*he plasma surface, i-(a) > 1, and a high shear can be produced in a helical heliotron ^ (l) configuration with 3 = 2 short pitch helical coils . Current driven kink and tearing modes and pressure driven low m modes are (2 3) investigated by applying the Stellarator expansion ' to the heliotron configuration. For the low 3 current carrying heliotron configuration, the kink and tearing modes become unstable along the line of ^ ( ) + 1 0 ip(0) = n/m in the (i^(a), i^(a)) stability diagram. The m = 1 modes with n > 2 give wide unstable regions and large growth rates. Eigenfunctions of kink modes are fairly localized inside the plasma column when the resonant surface exists outside the plasma column. (4) Comparison is made with the results of Heliotron-D experiments For finite g plasmas, the pressure terms destabilize the low m modes and give larger growth rates and wider unstable regions than the low g case. The validity of the Stellarator expansion is examined by comparing our results with the initial and boundary value problem of linearized MHD equations for the heliotron con­ figuration^^. For current-less finite g plasmas, g limit due to low m pressure driven modes is estimated and g>5 % may be expected by tailoring the pressure profile according to the shear parameter. 1. K. Uo, Nucl. Fusion 1_3 (1973) 661. 2. J.L. Johnson, C.R. Oberman, R.M. Kulsrud and E.A. Frieman, Phys. Fluids 1 (1958) 281. 3. K. Matsuoka, K. Miyamoto, K. Ohasa and M. Wakatani, Nucl. Fusion 17 (1977) 1123. 4. K. Uo et.al., Phys. Rev. Lett. 3_1_ (1973) 986. 5. K. Uo et.al., in 7th Int. Coni, on Plasma Phys. and Contr. Nucl. Fusion Res. (Innsbruck, 1978) CN-37/L-1. NONLINEAR KINK INSTABILITIES IN FORCE-FREE FIELDS* H.C. Lui Plasma Physics Laboratory Columbia University New York, N.Y. 10027 The theory of Pao^ for the nonlinear behavior of linearly unstable kink modes in a sharp boundary plasma surrounded by a vacuum is extended to include force-free and distributed plasma currents. Nonlinear stabilization of the external kink (m = 1) instability by force-free fields is enhanced by increasing the current strength in the force-free region up to an optimum value. For fixed force-free currents, nonlinear kink stability is always decreased by increasing the strength of the distributed plasma current. In general, the optimum value of the force-free current increases as the plasma current increases. A condition to deter­ mine the nonlinear kink stability is derived for long wave length perturbations. 1) Y.P. Pao, Phys. Fluids 2j^, 765 (1978). * Work supported by U.S. Department of Energy under contract EY-76-S-02-24S6. ANOMALOUS DIFFUSION AND PLASMA LEAKAGE THROUGH OPEN FIELD LINES IN FIELD REVERSAL CONFIGURATIONS* S. Hamasaki Science Applications, Inc., La Jolla, California 92037 ABSTRACT Using GIM/HYBRED code, which features self-consistent anomalous resistivities due to various microinstabilities, we have studied anomalous diffusion in reversed field configurations like field reversal experiments (FRX) of LASL. In the FRX, the open field lines at outer regions support the inner closed reversed field configuration. The plasma contained in the closed field region gets diffused out into the open field region due to the lower hybrid drift or drift ion cyclotron instability. The diffused-out plasma leaks out through open field lines, which enables the plasma edge near the open field lines to sustain, in spite of large diffusion, a finite gradient which feeds the microinstabilities. In this manner, the microinstabilities pump out the plasma from the closed field region to the open field region. Our calculation of these leakage rates due to the lower hybrid drift instability and the drift ion cyclotron instability showed a rough agreement, in order of mag­ nitude, with the experimental data of the FRX even though the FRX usually develops the rotational m = 2 MHD instability which terminates the containments. *Work by U. S. Dept, of Energy. supported the EQUILIBRIUM AND STABILITY OF TOKAMAKS WITH TENSOR PRESSURE'' A. Cooper, D. B. Nelson, Glenn Bateman Oak Ridge National Laboratory Oak Ridge, Tennessee 37830 T. Kammash University of Michigan Ann Arbor, Michigan 48105 Tensor pressure equilibria are computed for small aspect ratio tokamaks of arbitrary cross section. The perpendicular and parallel pressures are evaluated from a distribution function that models neutral beam injection. The stability of these equilibria to ballooning modes of large toroidal mode number are examined by numerically solving an Euler equation derived from a guiding center fluid energy principle. This second order ordinary differential equation is similar in form to the corresponding ideal MHD equation. The criteria obtained are either necessary or sufficient for the stability of a guiding center plasma depending on whether the double adiabatic contribution to the Euler equation is kept or ignored. Research sponsored by the Office of Fusion Energy (ETM) , U.S. Department of Energy under contract W*7405**eng**26 with the Union Carbide Corporation. IMPURITY CONTROL BY NEUTRAL BEAM INJECTION W. M. Stacey, Jr. School of Nuclear Engineering Georgia Institute of Technology Atlanta, Georgia 30332 and D. J. Sigmar Fusion Energy Division Oak Ridge National Laboratory Oak Ridge, Tennessee 37830 The use of momentum transfer due to neutral beam infection to drive impurities out of a plasma was apparently first suggested by Ohkawa and has subsequently been 2—5 6 examined by several workers. Stacey showed that a general external momentum source could drive radial particle fluxes, in the Collisional regime. While the direct effect of external momentum transfer was treated in these earlier works, two important indirect effects were omitted: the modification of the first-order flows in the flux surface and the adjustment of the radial, ambipolar potential gradient in response to beam injection and external drags. We have used a recently developed generalization of neoclassical theory,^ which is valid when external momentum sources and drags are present, to make a self-consistent calculation of particle transport in a tokamak plasma in the presence of neutral beam injection and external drags. Our formalism is valid in all collisionality regimes and for arbitrary geometry and beta, although we specialize to the low- ^limit for clarity. We establish the conditions for which coinjection drives impurities out of a plasma. We estimate that order-unity effects could be observed in PLT and ISX-B, for example, and that beam injection might be a feasible means of impurity control in a reactor-type plasma. Three important subsidiary results are contained in our work. Since the formalism is developed for a general momentum input, it is generally applicable for the analysis of impurity control by other forms of toroidal momentum input (e.g. 8 9 rf). As an auxiliary result, we have extended the theory ' for impurity control by asymmetric particle sources to all collisionality regimes. Finally, we have worked out the general theory for particle transport (subject to the Lorentz form of the friction) in a two-species plasma. Acknowledgement: This work was sponsored by USDOE. References 1. Ohkawa, T., General Atomic Report GA-A12926 (1974). 2. Connor, J. W., Cardey, J. C., Nucl. Fusion 14, 185 (1974). 3. Callen, J. D., private communication. 4. El-Derini, Z., Earnert, G. A., Nucl. Fusion 16, 342 (1976). 5. Fomenko, V. V., Sov. J. Plas. Phys. 3, 775 (1977). 6. Stacey, W. M., Jr., Phys. Fluids 21, 1404 (1978). 7. Stacey, W. N., Jr., Sigmar, D. J., ORNL/TM-6575 (1978); submitted to Phys. Fluids. S. Burrell, K. H., Phys. Fluids 19, 401 (1976). 9. Wong, J. K., Phys. Fluids 21, 299 (1978). STABILITY OF NEUTRAL BEAM HEATED EQUILIBRIA TO BALLOONING MODES R. W. Moore, R. L. Miller, and R. E. Waltz General Atomic Company San Diego, California 92138 ABSTRACT Neutral beam heated PLT equilibria are modeled using the General Atomic 1-1/2-D transport code.* The stability of these equilibria to localized ballooning modes is studied as a function of time during the neutral beam heating. The ballooning stability dependence on the transport generated current profile is illustrated by changing transport coefficients, plasma density, and neutral beam injection parameters. Special attention is paid to current broadening caused by poor beam penetration and its effect on ballooning stability. Work supported by the Department of Energy, Contract No. EY-76-C-03- 0167, Project Agreement No. 38. *R. L. Miller, "Shape Control of Doublets," General Atomic Company Report, GA-A15186 (November 1978), submitted to Nucl. Fusion. RADIATION IN A PLASMA D. P. Tewari* Department of Electrical Engineering Drexel University Philadelphia, Pennsylvania 19104 and V. K. Tripathi** Department of Physics and Astronomy University of Maryland College Park, Maryland 20742 In the vicinity of upper hybrid resonance (in the higher density regime) an upper hybrid EM wave resonantly excites a second harmonic upper hybrid radiation. The efficiency of har­ monic generation is limited by the wave number mismatch created by the density gradient. However, the yield of harmonic conversion for typical laser produced plasmas could be as high as 5% which is in order of magnitude agreement with the experimental observations. This mechanism of harmonic generation is operative in tokamak type plasmas also. * On leave from Physics Department, Indian Institute of Technology, New Delhi, and supported by Indo-American Exchange Program. ** Supported by the Department of Energy. ELECTRON CYCLOTRON RESONANCE HEATING RATE IN EET PLASMA* T. Uckan Oak Ridge National Laboratory Oak Ridge, Tennessee 37830 - The perpendicular energy gain, AtV^, of electrons from the applied extraordinary microwave field in EBT is calculated by means of the stochastical model^ for the field-plasma cyclotron resonance interac­ tions. In these calculations, the inhomogeneous external bumpy field is chosen to be ^ ( r ') B(z) = B„ which simulates the field strength reasonably well for the EBT. Here M is the mirror ratio and L is the distance between the mirror sectors. The effects of the initial energy of the electrons as well as the mirror ratio on the trapped and untrapped electrons in the bumpy field are discussed. Then the heating rate AW^/At, At being one reflection time from the mirror, is estimated for the trapped electrons. Research sponsored by the Office of Fusion Energy, U. S. Department of Energy under contract W-7405-eng-26 with the Union Carbide Corporation. ^H. Grawe, Plasma Phys. 151 (1969). ^D. A. Spong et al., Oak Ridge National Laboratory Report ORNL-TM 6215 (1978). FINITE TEMPERATURE EFFECTS ON MICROWAVE PROPAGATION IN EBT D. B. Batchelor and R. C. Goldfinger Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830 ABSTRACT A three dimensional ray tracing code, RAYS, has been developed as a part of the ongoing theoretical study of microwave heating in the Elmo Bumpy Torus device. Recently this code has been improved to include the effects of finite temperature on the ray paths as well as cyclotron damping. Our dispersion relation allows us to study effects at the second and higher harmonic resonances. In particular, we observe conversion of the extraordinary mode to electrostatic Bernstein waves at the upper hybrid resonance and second harmonic resonance. Total absorption rates are given for ordinary and extraordinary mode waves at first and second harmonic resonances for plasma parameters appropriate for EBT-I and pro­ jected parameters for EBT-11. We have investigated the influence which the choice of direction for the imaginary part of the refractive index s 1 ^ has upon total absorption (/ ds k. * V ). It was found that when the o *"**- S absorption is weak jk^l << jk^) the total absorption is virtually indepen- dent of the direction of k.. *"i Research sponsored by the Office of Fusion Energy (ETM), U.S. Department of Energy under contract W-7405-eng-26 with the Union Carbide Corporation, g A Simple Annulus Power Balance in EBT-1 -L S. K. Borowski,' M. A. Uckan, E. F. Jaeger 1 T. Kammash Oak Ridge National Laboratory Oak Ridge, Tennessee 37830 An essential feature of the ELMO Bumpy Torus (EBT) concept is the presence of a relativistic electron annulus in each of the toroidal mirror sections. These high-beta annuli are formed and sustained by microwave heating and are of sufficient density and temperature that diamagnetic currents produce the necessary minimum in the magnetic field required for MHD stability of the toroidal core plasma. Since the electron rings play an important role in the confinement characteristics and performance of EBT, the trade off between the formation of the rings and the power required to sustain them represents an important problem in a fusion reactor. A power balance for the rings which includes drag cooling in addition to the radiation (synchrotron and bremsstrahlung) and annulus electron-electron scattering losses, indicates that drag dominates the annulus energy balance in EBT-1. Drag cooling of the relativistic annulus electrons on the toroidal core plasma appears to provide a reasonable explanation for the decrease in the annulus electron temperature in going from ELMO to EBT-1, Theoretical estimates of the microwave power required to sustain the annulus are found to be within a factor of 2 of the experimentally determined value. Scaling projections are shown for EBT-S and parametric study results for ELMO, EBT-1 and EBT-S are presented for various microwave power levels. The Maxwellian distribution function is assumed in these calculations. The results are found to be sensitive to the details of the hot electron distribution function as well as geometric and scaling parameters. Improvements to the model are underway in order to increase its capability and accuracy in assessing the overall power balance. % Research sponsored by the U. S. Department of Energy under contract W-7405-eng-26 with the Union Carbide Corporation. ^University of Michigan, Ann Arbor, MI. 1C 1 RESONANCE WAVE-WAVE COUPLING AND PONDEROMOTIVE EFFECTS IN LOWER-HYBRID HEATING ' Kyoko Matsuda, Y. Matsuda,f G. E. Guest, and T. Ohkawa General Atomic Company San Diego, California 92138 ABSTRACT The electrostatic particle simulation code "EZOHAR"^ has been applied to study plasma response to high-power, lower-hybrid heating. In general, strong edge heating of electrons and ions, and tail heating of electrons in the interior have been observed. Spatial density modulation by pondero- motive forces does not occur at the edge due to large velocities of par­ ticles;^ but such ponderomotive effects may appear inside the plasma, depending on the density profile. New phenomena, such as resonance excita­ tion of Langmuir waves, have been found where the Langmuir frequency is a harmonic of the external frequency. The electric field associated by these higher harmonics may become comparable with the fundamental modes at the interior. Work supported by Department of Energy in part under Contract No. EY- 76-C-03-0167, Project Agreement No. 38, and in part under Contract W-7405- ENG-48. ^Y. Matsuda, W. M. Nevins, and M. Gerver, in Proc. 8th Conf. Numerical Simulation of Plasma, Monterey, CA, June 1978. ^A. Bers, Bull. Am. Phys. Soc. 15 (1978) 765. tPermanent address: Lawrence Livermore Laboratory, Livermore, CA 94550. Effects of Ion Dynamics on Tearing Modes X.S. Lee, Swadesh M. Mahajan, and R.D. Hazeltine Fusion Research Center The University of Texas at Austin Austin, Texas 78712 Abstract Using a simple self-consistent derivation of ion dynamics, we find two terms contributed by ions to the parallel electrical conductivity. One corresponds to the familiar ion acoustic term, and the other, which we call the 'frictional term', is due to the ion-electron collisions. The tearing mode equations are modified by adding these terms to the parallel conductivity, and are then solved to analyze their effects. The stabilizing tendency of the ion acoustic termt , earlier found on the semi-collisional drift-tearing mode, is found to be true for several other tearing modes. The effect of the ion frictional term is also investigated, and is found to be destabilizing in one case. Quantitative expressions for the change in mode frequency are given. We further show that most known unstable tearing modes are, in fact, the manifestations of the same mode in different regimes of plasma parameters. ^Bussac, et. al., Phys. Rev. Lett. 40, 1500(1978). This work is supported by the U.S. Department of Energy Contract DE-AC05-79ET53036. Nonlinear Interactions of Drift-Alfven Waves* E. A, Frieman and Lin Chen Plasma Physics Laboratory, Princeton "University, Princeton, N. j. Q8544 We present a general gyro-kinetic formalism for the nonlinear interactions of kinetic drift-Alfven waves. The nonlinear equations thus derived include full finite ion-Larmor radius (FILR) effects and are valid in the strong-turbulence regime. Applying this formalism to the parametric decays- of kinetic drift-Alfven waves, it is found that the nonlinear decay processes are modified, both qualitatively and quantita­ tively, by the FILR and diamagnetic-drift effects. Work supported by U.S. Department of Energy Contract #EY-76-C-02-3073. BURN CONTROL VIA REGULATED RIPPLE APPLIED TO REACTOR-GRADE PLASMAS? J. M. Rawls, T. W. Petrie and W. Chen General Atomic Company San Diego, California Control of a reactor-grade plasma may be lost if a sizable thermal excursion occurs after ignition is achieved. Because of its strong tem­ perature dependence and its finite value in the critical center region of the plasma, the magnetic ripple produced by the toroidal field (TF) coils is a promising means of inhibiting such a thermal runaway.^ However, achieving this ripple via a conventional TF-coil configuration provides little flexibility for subsequent modification of startup operation. Fur­ thermore, the degradation of confinement associated with ripple transport renders heating to ignition more difficult. Although there is a range over which fixed ripple values lead to a satisfactory burn, i.e., a stable, ignited plasma within prescribed beta limits, the size of this "window" suggests a somewhat restrictive operating mode. These difficulties can be relieved by a TF-coil network characterized by a small number of large superconducting TF-coils supplemented by copper 2 pull-back coils, a TF-array of the type employed in the recent NUMAK q design. The dynamical scenario proposed is to activate the "correction" coils during startup to minimize the impact of ripple on both plasma con­ finement and auxiliary heating demands, and to reduce the current in these coils when ignition is achieved, thus enhancing ripple losses to a level sufficient for burn control. In this way, the ignition requirement is con­ siderably relaxed, resulting in a much larger effective operating "window". Furthermore, the small number of superconducting coils provides a spatial distribution of ripple more suitable for burn control. This approach may provide the flexibility needed to track the plasma and to control the reactor power level. — — — — — Work supported by Department of Energy, Contract EY-76-C-03-0167, Project Agreement No. 38. ^T. W. Petrie and J. M. Rawls, "Burn Control Resulting From Toroidal Field Ripple," General Atomic Report GA-A15218, March 1979. ^C. Baker and T. Ohkawa, Kakuyugo-Kenkyu, Vol. 35, No. 3, 224 (March 1976). 3 R. W. Conn, gt al., Trans, of Third Topical Meeting on the Technology of Controlled Nuclear Fusion, Santa Fe, New Mexico, 351 (May 1978). Electron Landau Damping of Instabilities in Short, Pat, Field-Reversed Ion Rings* M. J. Gerver Laboratory of Plasma Studies Cornell University Ithaca, New York 14850 The integral equation for the most dangerous (oj - v^/L) normal modes of a high energy, low density field-reversed ion ring immersed in a cool, high density background plasma is simplified to a form allowing practical numerical solution, for an arbitrary (axisym- metric) geometry. Che qualitatively different feature of the short, fat ring, as opposed to the long layer or bicycle tire, is that electron Landau damping is much less effective, because there are far fewer resonant electrons. In a long layer or bicycle tire, the magnetic field is uniform along a given field line, and resonance requires M = kj jV i.e. Vi v^. In a short, fat ring, the magnetic field varies along a field line, and electrons with V] j < Vj_ are trapped; for trapped electrons the resonance condition is a) = (where - v/L), i.e., v - v^. For moderately large p, v » v., there are far fewer electrons with v - v, than with Vi . - v,. e a A I I A This result does not apply to ion damping since v^ < v^. & This work supported under U.S. Department of Energy Contract EY-76-S-02-3170. Kink Instabilities of a Field Reversed Ion Ring with a Toroidal Magnetic Field J. M. Finn Laboratory of Plasma Studies, Cornell University Ithaca, New York 14853 The low frequency MHD stability of an axisymmetric field reversed ion ring in a current-carrying background plasma with a toroidal magne­ tic field is studied. A generalization of the energy principle^ which treats' the background plasma by fluid equations and the beam by kinetic theory is employed. The major effects upon stability are the MHD response of the plasma and beam, a collective reaction of the beam, and betatron resonances. It is found that, if the beam and plasma currents are roughly equal, and if the exterior region contains a cold but highly conducting plasma, there is a window of stability for kinks with safety factor q as low as 1/2. The results may also explain the anomalous trapped current losses in the RECE-Christa 3 electron ring experiment. * *Work supported under U.S. Department of Energy Contract EY-76-S-02-3170. lR. N. Sudan and M. N. Rosenbluth, Phys. Fluids 22_, 282 (1979). 2j. M. Finn and R. N. Sudan, Phys. Rev. Lett. 41, 695 (1978). 3R. A. Meger, Ph.D. Thesis, Cornell University, 1977. STABILITY OF LOW BETA AXISYMMETRIC MIRROR MACHINES Harold Weitzner Courant Institute of Mathematical Sciences New York University With the use of an energy principle of W. Newcomb (LLL Report UCID 17182 (1976)) the stability of an axisymmetric mirror machine with anistropic pressure is studied.* The analysis is particularly simple in the low beta limit. For a mirror machine the low beta limit may be taken in several ways. Unpublished results of W. Newcomb on a wide class of unstable systems are recovered. Other unstable configurations are also described. In the appro­ priate limit a stable configuration is given for a plasma with a very weak singularity in the magnetic field, B is smooth but VB tends to infinity at a point. If such a singularity is smoothed out, then the plasma will be unstable only on the outer edge, where line tying may stabilize the system. Some numerical examples will be given and extensions to high beta systems will be considered.