Copyright © 1979, by the Author(S). All Rights Reserved

Home , Sceptre (fusion reactor)

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission. INFORMAL CONFERENCE ON

PARTICLE AND HYBRID CODES FOR FUSION

C. K. Birdsall and A. Friedman (Coordinators)

Memorandum No. UCB/ERL M79/79

December 10 and 11, 1979

Napa, California

ELECTRONICS RESEARCH LABORATORY

College of Engineering University of California, Berkeley 94720

/ •^%

Research sponsored by the Department of Energy Contract DE-AS03-76F00034- DE-AT03-76ET53064. PURPOSE INFORMAL CONFERENCE OK

PARTICLE AND HYHRID CODES FOR FUSION This conference was held in order to facilitate an exchange of ideas December 10, 11, 1<<7? and results on plasma particle and particle-fluid hybrid codes in an Informal, Napa, California unpublished manner, with frank and open discussion among about fifty people. The only printed record is this collection of copies of transparencies used CONTENTS by the thirty-five speakers. In keeping with this informality, the results presented here are being distributed to the participants only and are not I Purpose to be quoted without explicit permission from the author.

II Attendees In December 197I1 a similar unpublished conference was held in Berkeley III Schedule at the request of Robert Price of ERDA. The present conference is a follow-up IV Panel Discussion and is due in large part to the interest of David Nelson (Chief, Fusion V Conclusions; Recommendations Theory and Computer Services Branch, Div. Applied Plasma Physics,Office of VI Papers Fusion Energy, DOE). Dr. Nelson presented his view from DOE of contributions by computation and simulation toward furthering plasma research leading to ACKNOWLEDGMENTS fusion reactors; he received considerable feedback from participants. We This conference was conceived during a telephone call with David are most grateful to him for his active and stimulating participation. Nelson at DOE roughly two months ago. His suggestions as to purpose The conference is also due to the pressure of accumulated interest and focus, as well as his active participation in the meetings are among those who work with particle and hybrid codes, who wished to exchar.je acknowledged most gratefully. new ideas, methods and results, in an informal setting. By no means was My colleague, Alex Friedman, has been invaluable in all stages of this meeting intended to conflict with the Ninth Conference on Numerical conference preparation, from the suggestion of topics to be covered, and Simulation of Plasmas scheduled for June 30 - July 2, I98O at Northwestern their grouping, consultation with many outside Berkeley, LBL, and I.LL, University, sponsored by Professors J. Denavit and C. Knorr, which covers the formulation cf Invitations, to many details of the meeting itself. I a much wider area and is to be published. am most grateful for his help. Lastly, the particle-hybrid code community had concern that the very Giiiger Pletcher, my part time assistant, devoted nearly all of her real contributions of particle and particle-fluid simulations to the under time for two months to finding a suitable meeting place near Berkeley standing of fusion plasmas, in support of both theory and.experiment, might (but far enough away to keep participants from straying to LBL, LLL, just have been underestimated in Washington. This conference is, in part, Stanford etc. during the meetings) for the meetings and a pleasant a reaction to recommendations of the DOE committee for computer time place for the conference dinner. She also did the typing, letter writing, allocations for FY80 which included: major labs receiving 63^ of their and mailing to the potential and actual participants (with help from requests, but universities receiving only Uo£; the decision that large Steve Au Yeung on the initial computer mailing), made many phono calls to particle pushing or kinetic codes can be afforded only sparingly; basic ensure that the details of the conference were handled properly, and plasma theory was given approximately a 20^ absolute cut; ability to follow arranged the get-together Sunday evening. We owe Cinger an immense out unexpected results was cut to a bare minimum; studies of alternate vote of thanks for her help. concepts were cut more than were mainline studies. The challenges to the My thanks go to the speakers and panelists for their excellent community are clear: make our contributions known; make our codes more presentations, which are already stimulating us to better work. The efficient in terms useful physics per unit of computer time (optimize, interchange of ideas that was sought was achieved; the interaction with use minimum number of dimensions and particles etc.); check code physics David Nelson that was desired was also achieved.

C. K. Blrdsall Berkeley, Dec. 17, '979 -r-r r T~

PURPOSE (continued) by analysis (time and spatial grid effects, fluctuations) and by com Final Schedule for parison with known linear and noalinear results so as to increase confidence Informal Conference on Particle and Hybrid Codes for Fusion December 10-11, 1979, Napa. California in the simulations.

Editorially we note that the field has matured considerably from the Conference coordinated by Charles K. (Ned) Blrdsall, Alex Friedman, assisted by Ginger PI etcher, at the Electrical Engineering and early desk calculator work of Duneman and'Itartree on magnetrons in the early Computer Science Department, University of California, Berkeley 911720 19'tO's, the one demensional electrostatic plasma work on modern computers by Buncman and Dawson in the late 1950'a, to the current 3d fully electro There will be four main sessions. Talks will be 20. 15 and 10 minutes with adequate time for discussion. If certain subjects draw magnetic codes of Buneoan and others. The early art has become more of a great interest, additional time will be available. science. Simulation for magnetic fusion has now become strongly applications Of general interest is the efficiency of particle and particle- oriented, with DOE devoting the bulk of FY 60 computing resources to the fluid codes, in terms of physics output per unit of computer resource direct support of existing experiments and the design of next generation (time, memory, volume of output, etc.). Of comparable interest Is the progress toward working at lower and lower frequencies, with larger mass devices, Including calculations of transport, impurities, heating, stability, ratios, eto. equilibrium, coil design etc.,a milestone-Indeed. Coupled with this new emphasis Is the current plateau in large computer time for magnetic fusion, with no relief planned until at least summer I98I. These factors present HONDAT. DECEMBER 10 Session 1A - 8:30 am to 12:00 noon Chairperson: C. Nlelson a challenge to all of us to be aore efficient and more effective in our

siitulations. * Long-time-averaging (LTA). Particle simulation or slow transport phenomena. " Particle KHD vis a vis fluid MUD and other fluid codes. * Large time step problems (ml/me>>1, omega dt»1, digital filtering, stiff equation Integrators). Co-ordlnators: C. K. (tied) Blrdsall Alex Friedman U. C. Berkeley, December 17, 1979 Paper 1 J. Denavlt, T. L. Crystal, C. E. Rathmsnn, J. L. Vomvorldls. "Appllca- E|9£ *~ tlons of Long-Time-Scale Particle Simulations."

2 B. Cohen, R. P. Frels, T. A. Brengle, "An Orbit-Averaged Particle Code." 22

3 W. Fawley. "Low Pass Filtering In Time." 44

^ D. Anderson, "Toward a Full 3d Hybrid Transport Code." 51

5 T. Tajlma, J. H. LeBoef, F. Brunei, J. H. Dawson, "Recent Efforts in 56 ~ Particle HUD Code Development."

7 H. Gkuda. "Steady State Drift Turbulence and Anomalous Diffusion, 3d." 101

6 C. K. Chu, "Vortex Ring Formation by Vortex-ln-Cell Method." 93

9 C. Toll, "Code Exchange Mechanics." 121 ft B. t'cFanara, "Remarks on the Use of Integrals of Motion" 120 CONFERENCE ATTENDEES AND ADDRESSES

Dr. W.W. Lee Dr. J. W. Poukey Princeton Plasma Physics Laboratory Division 521*1 Princeton, New Jersey 085>»0 Sandla Laboratories Dr. Jones H. Albritton L-477 Albuquerque, New Mexico 87115 Lawrence Llvermore Laboratory Dr. Kent Estabrook I.U77 Dr. Paulett Liewer P.O. Box 808 Dr. Jeff Ojiintenz Liveroore, Ca. 9U550 Lawrence Llvermore Laboratory Physics Department P.O. Box 80S University,.of California, Los Angelea Division 52>«1 Los Angeles, California 90021* Sandla Laboratories Dr. David Anderson Liveroore, Ca. 91*550 Albuquerque , New Mexico 87115 L-«»39 Dr. William Fawley Dr. Irv Lindeouth Laurence Llvermore Laboratory Dr. Abraham Sternlleb Liveroore, California 9>)558 L-»»39 Los Alamos Scientific Laboratory Lawrence Liveroore Laboratory P.O. Box 1663 Physics and Astronomy Department Los Alamos, New Mexico 875^5 University of Maryland Dr. Robert Berman P.O. Box 808 College Park, Md. 207*40 36-227 Liveroore, Ca. 91*550 Dr. Rodney Mason Massachusetts Institute of Technology Dr. T. Tajima Cambridge, Ma. 02139 Dr. Charles Finan Los Alamos Scientific Laboratory L-22 P.O. Box I663 Physics Department Los Alamos, New Mexico 975**5 University of California, Los Angeles Dr. Oscar fttn^mnn Lawrence Llvermore Laboratory Los Angeles, Ca. 9002U ERL P.O. Box 808 Liveroore, Ca. 91*550 Dr. Yoshi Matsuda Stanford University Dr. Carol Tull Stanford, California 91*305 L-l*39 Dr. Bob Fries Lawrence Liveroore Laboratory L-l*02, Bldg. 219, Room 108 Lawrence Liveroore Laboratory Dr. Jack Byers L-J»39 P.O. Box 808 Liveroore, Ca. 9*550 P.O. Box 806 t-^39, Bldg. 315, Rm. 285 Lavrence Liveroore Laboratory Liveroore, Ca. 9^550 Laurence Llvermore Laboratory P.O. Box 80S Liveroore, Ca. 94550 Dr. Homer Meier P.O. Box 808 Dr. Thomas Tuoolillo Llvermore, Ca. 91*550 P.O. Box y, Bldg. 9201-2 Dr. Brendan Godfrey Oak Ridge National Laboratory JOYCOR Oak Ridge, Tennessee 37830 P.O. Box 370 Dr. Frank Chambers Mission Research Corporation Del Mar, California 921Xw L-321 Santa Barbara, Ca. 94102 P.O. Box 808 Dr. Barry Moore Dr. Dennis Hewett Austin Research Associates Dr. Dan Winske Lawrence Liveroore Laboratory I9OI Rutland Drive University of Maryland Llvermore, Ca. 91*550 Mail Stop 61*2 College Park, Md. 20708 Los Alamos Scientific Laboratory •Austin, Texas 78758 Dr. C.Z. Cheng P.O. Box 1663 * Dr. David Nelson Dr. Wee Woo Princeton Plasma Physics Laboratory Los Alamos, New Mexico 8751*5 Applied Science Department Princeton, New Jersey 08590 Magnetic Fusion Energy Division Department of Energy University of California, Davis Dr. John Kulp Davis, Co. 93616 Dr. Chia K. Chu 36-227 Geroantown, Md. 20767 Mechanical Engineering Dept. Massachusetts Institute of Technoloior Columbia University Cambridge, Mass. 02139 Dr. William Nevins Bew York, N. Y. ll*850 L-l*39 Lawrence Liveroore Laboratory Dr. Bruce Langdon Dr. Bruce Cohen P.O. Box 808 H77, Bldg. 381, Room 1228 Liveroore, Ca. 9U550 I~»»39, Bldg. 315, Rn.277 Lawrence Liveroore Laboratory YOUR HOSTS Yu Jiuan Chen Lawrence Liveroore Laboratory -—————"• Vincent Thomas P.O. Box 808 ^ Dr. Clair Nielson P.O. Box 808 Liveroore, Ca. 91*550 Douglas Horned Liveamore, Ca. 91*550 Group CTR-6 Stephen Au Yeung Los Alamos Scientific Laboratory Niels Otani Dr. Barbara Laslnski P.O. Box 1663 Dr. Jacques Denavit 1-477, Bldg. 381, Room 1232 Los Alaaos, New Mexico 97^5 Dr. C. K. Blrdsall Mechanical Engineering Dept. Lawrence Liveroore Laboratory EECS Dept, Cory Hall Northwestern University P.O. Box 808 University of California, Berkeley Evanston, 111. 60201 Dr. Hideo Okuda Liveroore, Ca. 91*550 Princeton Plasma Physics Laboratory Berkeley, California 91*720 Princeton, New Jersey 085to Dr» Adam Drobot Dr. Jae Koo Lee Code 7750 P.O. Box 81608. Dr. Douglas Potter Dr. Alex Friedman U.S. Kaval Research Laboratory General Atomic Company EECS Department Washington D.C. 20375 Space Sciences Laboratory Cory Hall, University of California, Berkeley San Diego, Ca. 92130 University of California, Berkeley Berkeley, Ca. 9«*720 Berkeley, Ca. 9^720 Y

Dr. Tony Lin Department of Physics University or California, Los Angeles Los Angelea, California 9002U Session IB - 1:30 pm to 5:00 pm Chairperson: C. K. Blrdsall

Dr. Viktor Decyk • David Nelson; view from DOE. Department of Physics * Improvements in simulation 1971 to 1979, and future. University of California,. Los Angeles * Code production vs development running times; allocations and priori Los Angeles, California 9002U ties. • Hardware, e.g. array processors, graphics, class VII computers. Dr. Robert Huff * Efficient 3d simulation; 3d grid vs 2d«Fourler representation. Department of Physics University of California, Los Angeles Los Angeles, California 90021* C. K. Blrdsall, Comments on 1971 Berkeley Meeting; hopes for this meet ing. Page Dr. Thomas Ollphant D. Nelson, "Role of Particle and Hybrid Codes Present and Future; Com Los Alamos Scientific Laboratory ~T4T P.O. Box I663 puter Availability; Possibility tff Adding Special Purpose Computers Los Alamos, New Mexico 875'*5 for Particle Codes."

11 A. B. Langdon, "Tradeoffs Among Code Development vs Hardware Costs vs 145 Dr. Brandon HcNamara __Elapsed Calendar Time; Future Hardware Needs; ZED Postprocessor." L-«»39 Lawrence Llvermore Laboratory P.O. Box 808 Llvermore, Ca. 9^550 12 J. Kulp, "High Performance Array Processor for LIST Machine} Architecture, Impact on Particle Simulation." 152 Dr. Tony Sgro Los Alamos Scientific Laboratory 13 . B. Moore, W. Drummond, "Particle Simulation o n the VAP." 159 P.O. Box I663 Ik T. Brengle, N. Maron, G. Sutherland, nVae of an Array Processor 165 Los Alamos, New Mexico 8751»5 With PDP-10." Thomas Brengle 15 R. nerman, "Macrocell Algorithm for Efficient Particle Pushing, L-1*U0 for CRAY and AP in 2d, 2}d, 3d." 169 Lawrence Llvermore Laboratory P.O. Box 808 Llvermore, Ca. 9U550 "l6 "c"."T.""Cheng, H. Okuda, "3d Simulation of Trapped Electron Instabilities 101 ~~" In Toroidal Systems."

17, 0. Buneman, "Data Management for a Million-mode 3-d e-m Code; Use of 196 Tetrahedral Mesh."

18 T. Tunollllo, "MEEC-3D: Description of an Existing Self Consistent Par 210 ticle Pusher." Page 1*1*0 TUESDAY, DECEMBER 11 32 B. Godfrey, "Electromagnetic Numerical Instabilities in Two-Dloensional Session IIA - 6:30 am to 12:00 noon Chairperson: J. Denavit Relativlstic Beam Sioulations." 33 a. Codes 1>5U • Linearized codes. Sternlleb, "Coupling of Particle to Electric Circuits." "Modified particle codes (use of linear susceptibility or Boltzmann response. Id stretched to 2d or 3d). 3l» J. Poukey, J. P. Quintenz, "Preliminary Simulations of Ion BeamNeutral- 1*90 » Quaslneutral, hybrid, and Darwin codes (applications to confinement, izatlon." compression, equilibrium, stability, and transport). 503 • Buildup and plasma trapping, neutral beam Injection, and wave heating 25 y. Chen, "Multi-Beam Instability Interference with Lower Hybrid Drift (laser-pellet plasmas, pinches, mirrors, tokamaks). Instability." Paper Pa9e W. W. Lee, H. Okuda, "Particle Simulation Models for Low Frequency 233 12 Hlcroinstabllities. «iu»." 4:00 pa to 5:00 pa Moderator: B. Cohen 20 A. G. Sgro, "Hybrid Simulation of Non-MHDPhenomena. 251 D. Hewett, "A Global Method of Solving the Electron Field Equations in a 263 • Panel Discussion: When to use a particle, fluid, or hybrid code (or Zero-Inertia Electron Hybrid Simulation." none at all). All participants invited to contribute.

22 D. Wlnske, "Particle Simulation of Reversed Field Configurations." 281 Panelists: J. Denavit, A. B. Langdon, B. HeNamara, C. Nlelson, H. Okuda

23 R. Mason. "Monte Carlo (Hybrid) Model for Electron Transport in User 288 Plasmas."

?.U J. Byers, "Id Linearized Particle Model for Tandem Mirrors and Field 301 Reversed Mirrors." 25 A. Friedman, J. Denavit, R. N Sudan. "Linearized 3d Hybrid Simulations; 320 Ergodlc Orbits in Simulation." 26 V.Decyk, "Diagnostics for Bounded Plasma, with Applications." 350

27 B. Cohen, N. Haron, G. R. Smith, W. M. Nevlns, "DCLCSimulations with a 365 Stretched Id Code."

28 R. Huff. "Particle Hybrid Codes on the CHI Computer." 389

Session IIB - 1:30 pa to 4:00 po Chairperson: 0. Buneman • Inhoaogeneous plasmas (fluctuations, initialization techniques in 2d and 3d). * Undesirable instabilities in warm plasma sioulations (e.g. multi-beam and multi-ring). • Grid effects (e.g. curvilinear coordinators, dlv B nonzero).

20 W. Nevlns, "Fluctuations in Inhoaogeneous Systems." 407

30 V. Thomas, C. K. Blrdsall, "Alias Growth in Hybrid Oscillations due to 419 Initiation at kax * «." 31 A. Drobot, A. Palevsky, "E-*i Simulation of Strongly Radiating Systems.1 427 DRAFT • Panel Discussion (Continued) IV. Panel Discussion

Plasma simulation is one tool in the attack on understanding plasmas to then:. The question In not whether, but how to continue, effectively. We and eventually designing fusion reactors. Simulation using many particles snist poy more attention to iracroscopic work; as we do, we are likely to find vies for usefulness with simulation using fluids. Particle codes con that distinctions between particle ond fluid work will disappear. The new deliver the full dynamics which is sometimes needed; fluid codes use hardware is revolutionary, but there still Is on open question as to using parameters from theory, particle simulation, and/or experiment, and deliver one big computer or many email machines; his view ia thot special computers average or long-term information. Particle-fluid hybrids attempt to use ore still too small, too complicated to use. the best features of both, mixing fast and slow time scales. Particle McNomoro presented a list of unresolved problems as follows: codes, in some quarters, hove gained a reputation for using much more Tokomaks computer time per unit of useful physics delivered than used by fluid codes. 1. Anxnalous electron transport This reputation is considered undeserved by many particle simulators in - ia it really classical? 2. Nonlinear effects of ballooning and tearing most national lobs and universities; the opinion may ring true occasionally. Fortunately, porticle simulators, especially those with limited budgets, fork Field Reversal very hard to optimize their codes, work in the least number of dimensions 1. Non-axisymmetrlc states In RFP for'the problem at hand, with the least number of particles necessary, aid ?. Equilibria and stability or lorge orbit FRM use as much fluid simulation and guide from theory as can be fit in and Tandem Mirrors still do acceptable physics. Those with larger computing budgets, please follow. Generally speaking, the commitment Is to the physics sought, and 1. Nonlinear saturation of cyclotron modes

not to the local method or local code available. 2. MID stability and ballooning Okuda addressed the progress made In understanding anomalous transport 3. Energy transport In tandems, thermal barriers, etc. indigenous to tokomaks, moving from o eorly fully dynamic (hence, expensive) model to guiding center electron models in 3d, both electrostatic and magneto- plus Computing - an Upbeat View, as follows: static, with long time steps (less expensive), toward o 3d toroidal model 1. Simulation very effective and increasingly realistic. Therefore, push for steady atate, Including turbulence and transport, poloidol dlvertors, major design and physics efforts for fusion systems. and hybrid heating. ?. Provide more software and graphics support for designated activities. Denavit observed that hybrid codes, 3d particle codes, with time -othera benefit by fallout. filtering, long time scales, all appear to be problem dependent. In conflict 3. Hncourage more collaboration - from three day advisory groups for with this, there is the need for optimization via assembly language, which new problems to joint efforts and publications. Implies loss of flexibility. We ore fortunote, in porticle simulation, to be 1|. Improve the numerical analysis input - without a $200K tax to pay for it. oble to work on Interesting problems and to obtain useful results. Simulation 5. Write more reviews of the state of the art and advertise good results identifies physical mechanisms, suggests theoretical work ond rejects ideas that more vigorously. do not fit. (>. WKi TERM GOALS - DESIGN A FUSION REACTOR ! Are we getting there?? Nielsen stated that simulation must continue In order to keep theorists plus the Computational Attack on Plosno Parameters, on the attached chart. honest; simulation is the only econo-.ical means of checking Uicory directly open Uotc the marriages of all kinds, across Otzzy boundaries: particle plus fluid; wlcro plus macro; dynamics plus equilibria, etc. McNomoro ended with stressing the need for more advertising of accomplishments. * Tills Is a draft from my notes, to be checked over by the panelists before submitting formally to DOE for their internal use. The errors, omissions etc. ore mine olone. C.K. Blrdsall Panel Discussion (continued)

Lon^don noted that the MFE computational effect has grown greatly in the HEM/TiC last five years. He repeated his advice of the last meeting that, if we wont to work with long time scales, then most time integrations will require im IPTKhWoAT plicit time averaging, which may not be attractive, but is needed, because v-t explicit methods will end up unstable at large «•>&«. He suggests turning ft; o to continuum methods for modeling kinetic physics - painful, but needed (Rod Mason has done Borne, with multigroup transport). He is interested in 4P partially kinetic models. ;/ Cohen concluded the panel discussion with the observations that scaled I variables probably always be with us , that having computervariables match f/teTiO-E. _._ POINT.! experiment is probably impossible in the foreseeable future. He presented a KHP - _ Wo PELS summary of his philosophy on hybrid simulations, attached. (Everyone, please %RJtti), mfc/w.-.. sl ... _ note that Cohen has given good example in this move to more efficient com fXAc5L0 ATAxtwr^ \* putation. ) o I Bil,,P—, _ Blrdsall sketched on the blackboard the DOE hierachy of models, with increasing expense reading left to right, IOLE. $ j/ fluid multifluid drift-kinetic Maxwell-Boltzmann single particle JiXCfQ hybrid L0M6- Tirtt I and his inteactive model . AvfcMG€._ CARLO

coper : 20rlAfl, t making the point that these models interacting are mutually supportive and are es±. seldom wholly independent: over eoiphasizing one tool or removing another might j^f^jLmiLlA.0:) well reduce the effectiveness of the remaining system. cooes 1 -

rVrl L X J. X t"h "",=T "i I p — L/C; V"*73 i CVCLOTAON BouWt S>urt. DRAFT » •*»«/ discussion V. CONCLUSIONS: RECOMMEIiUATIONS Hybrid ^t'ûl^-fjo^s - philosofky I Many of these are already in the I"and Plscussion. The l dsJrojQ natures p«rh'dcs. \or Pfv) eftSenpfroMjJ. linear and quadratic weighting widely used and understood (multlpole use Isoloted); hybrid codes were J«st storting then and now exist In a variety of forms; 3d 4. /£ =O-Qty /«»ê.- orWf Aaui/ftrVuM£-^*nyz>r£ codes, then done by one or two groups ore now done by several; one special purpose computer was mentioned then, with many now. However, several authors from I97I* gave up-dates in 1979 on problems still not fully resolved, such as linearized particle codes, long-time-scale particle ^'\w*>~ C'CI-) rH'bro»Vfabili'fy 4- drift >**Wes simulations, the plea for use of implicit schemes for large time steps, and 3d economical codes. The 197'i conference called for more use of hybrids (happening), more emphasis on simulations including lnhomogeneous density, magnetic field, temperature etc. (happening) and for honest use of two species, large mi/tnc 1.3. v*we (*) di'ssifKh'on^mppiy (happening somewhat). Perhaps tfce long term needed to resolve these problem s**rUf> of r&omioi. areas indicates their toughness and raises the question of pursuing them , rurther - some of them obviously must be. When Kinewc Jehitl i& c^jwecesfcry.-eAwtiiKiTSoVt ofcwlic-ks <*"<& ^•••e- David Nelson's view from DOE, paper #10, was provocative. The loud and clear 4f**a. 4j spues, <&*h*. and «dcfhe*. of fluid cfescr'iphon eon anjuJ-ly i-essages were l!«at magnetic fusion computing capability is fixed, with no relief expected before l°8l, with Increasing allocations to device specifics and less j^fjha^d ceyriûT^r'tayxAK poww. n\'"\.«l offen szp*r«1vs iam if ekc/i-ons. to large particle pushing or kenetlc codes. His list of problems areas needing more attention was: uv*oe5 >a/i?p«| Diverlors.-axi + son axisymmetric sheaths, potentials, boundry layers, amblpolarlty, Plasma edge including shadow of llmlter neutrals, atomic physics, reflux, metal surface, « «o/k kydrt}V**£y&S~i'c, Transport in non axisymmetric systems RF heating including nonlinear effects >7 oi/k <».+rc. Optimization of gyrotron geometry linear" phenomena li«€Ar- tfaiechrtc. "X (u>~iJ lc\, Stability or E fl T ring-plasma 'î°TT: ^Matyfio rcdtuoiSow of flutl dss Formation + tearing problems in RFM, RF6P, spheromak All the old unsolved problems Including bump on toll (otparticles) Specific pqnuriefar' realties omJL problems (i) result i* Utye. His cost-hierarchy of models was repeated at the end of the panel write-up; &\viviJ5 of ccv*tx4,hr *rK**ory £t^W tuxA (2.) qr&xTly e./bznc{ fh this ranking, while possibly taken as self-evident in some quarters, was not readily accepted by the participants; or, even IT accepted in a fuzzy way, It was considered highly undesirable to use In some form of cost-effectiveness arguement for ond against fluid versus particle simulations. That Is, both kinds need the birr»<* simu'ATioKsirriulAttcm aidi '>v\i'oro'\rt$fj ti.e panelists before suiKiltthif. formally to DOE for their Internal use. The errors, o'>:sslons etc. arc mine olone. CKli eovne-frt'es. © CONCLUSIONS: RECOMMENDATIONS (continued) other and the best is in the middle; indeed, the progress made in the past five years and noted in this conference was strong evidence that both kinds of /. LONG-7WE-SCALE Am* LOu- FREQUENCE simulations are moving toward particle-fluid hybrids. Simulations A. Bruce Langdon, in Paper If 11, i.ade several strong suggestions aboit doing MFE computing more effectively and more efficiently. Optimization pays off very quickly (he gave experience with ZOHAR showing that optimization paid •77Z..£ry*"rVt/ , XDCnai/itj C.E. XAfhi*a.»rty J. I. tfimvorljis off almost as it was done!) Optimization must be done selectively, so as not to hurt in making clianges (e.g. ZOHAR particle boundary conditions are handled 'ntvtreitY in Fortran). He advocated use of monitor, intervening, postprocessing, and linking to other codes; these rely heavily on interactive codes, sharing the (large) data set of the simulation code, doing considerable competing and OUTLINE driving rapid graphic displays. Portprocessing is very helpful in evoking good physical understanding, such as spectra, filtering, finding spatial and temporal correlations etc. Fast graphic display, like LLL's TMDS, is extremely invaluable. /. L0N6~TIME-SCALB PAHTICLS SIMULATES The pitch to MFE was" not only to update software, but also to update hardware. Recommendations are both explicit and implicit in the Panel Discussion and Alj}*r%W*vr% in the above. To all: in computing be more effective, oore efficient; in physics, make results better known. To DOE: be more aware of the interactions within the particle siinulation-theory-experiment net; be more aware that most particle simulation workers have moved toward particle-fluid models to stay; plug, on your end, for more efficient equipment; include particle simulation people on your computer time allocations committee; reward inventions, like long-time averaging, orbit-averaging,hybrid codes, MHD-particle codes, use of global Z. LOW-FREQUENCY SIMULATIONS integrals of motion, use of mocrocells, etc., when these result in real savings; keep alive support of the smaller university efforts, which in turn aid much in keeping honest the larger notional lab efforts; penalize users with larger budgets who are slow to use new invention to save time (do away with "if you C leci)r** titxpftna Astof Tr-+>jof>ei/-&iccjr#f% modes don't use it, you lose it" policy). ©

HYBRID REPRESENTATION ^rs" Algorithm T\N0-COH*ON*Nr PLASMA For oArnew { MlirrueWnfu>4hw*ve ft! f (JertfMjColJl) l/J(O"1^(t)- |fI £ J*-/*,fcli).(t'-tty*)) VArVi»i« fvyicfieni of Tt***C. (L « ^ £.)

FLUID MODGL Sotmct? TERHS

$e/H

lt-j PAfrn F*?F<3 \JBNCf s m f*T" : C*)r4<«/ (trKn*e.f>tt>l«LS*r>a.) 0.-«V-«p.—£T «r«C^f « «* £r/v***y:5C4=^S,At i«.t JCP 2_f,-94?^,177*. £ «•-&♦')to* •/Sir ©

L r\=n0(i+A) a« I. co _ \Jrr for AssO.

10 -

Sa.fiJr4.fi0n -rhitrig.*: A 0Vtf// (PFjfff}/Z*^ /*7i) CJL^/ **&& £.. fimtfrl - RoSCrtLloTh •/t J ,'. 10 t • •—• » i 11 n 1 1—* * , i ,J 0.61 0-1 1-0

^/mi»/a//w r*c**A«l}

.LZzJkt

Ft** velocity ttsolvti** T* j *-r~

~~~0«/0v/o*«-f»*/«At-2,^j."1 6?. a.13 f> a 1^=0./~~

~~AMPLITUDE Vs. TIME TRAPPING IN NON-UNtFORH fig-MUtf~~

~~/ 2* ft?~~

P* i>«2jti/0"*

~~2*11**1 36oo 4«oo~~

~~Y2oo~~

~~t-\2oo{t,-* OAW "5~~

~~\HI~~

~~EUcLTRjoU DtSTIb.FNCT. o.?W~~

~~I20D 2400 0.1 0.15 UU *Nc ® 1 d>~~

~~&XAMPL.E~~

~~M0UUN6AR WAVE &ROWTH WH5V R<\~~

~~Linear disp- refafiVH R>0 tv U) l\jv JL-Cd Pesoncwr' velocity~~

~~S/tnv laiiuu joorh'cks 2£ * ttAf* At* U.L X? Wo tm« • 72*7 -K* decfnpu A/We.- fT3^ iT-ft^o pac-h'dti /eawiV* reio«W 77, * **•" CJf~~

~~% * 0.OZ See: PF 22 , 367 r'^7; to.o I) (10**) PF J** /^o ©~~

~~LlNEA\fi\*FO OHE FOR ELECTRONS .FOUfilER-TGAUSFoRMBD O/tC ***** : e''"*"*>~~

~~«f *5?=^tru.t? + i &(*-*)± to * \ at * 'm ^ $u Urn*/ viw-i/ \~~

~~%<^t 6s£ /»»•. a&f>*tT •B e js £ $*ftbj feu/or~~

~~i IN hArt: J *1 -U> P* (n~~

~~grV/Tcoty)co>e^ /, e/.oj * 2r*SL >e {t nr ' See : P.P. 2/ , IST33 (/f78) also T.L.tctftUl a*d 7.bi*\ovit 'CoeraTuH. a*>M. 6tr>diu%f f>cif+ Effeds o>* T~~

~~It J~~

~~a. i~~

~~IS o a It~~

~~3 in fr it 5^ 3 I?~~

~~i i & ^4 C Hi -P 2~~

~~~~Si Q o sJ II .*. J 3 o2 K- Uj "fi I' 1 1 *_U ii y» -P o -a- r - » ft "a Q-T-I 5: 05 a. V7< w3 • /-fl%~~~~

~~~~to 3 tl i i» c~~~~

~~~~-... , •••• i~~~~

~~~~-«l '^;=^^_ 1 mm o « \ ^^> -CM ' J» O SRI C~^^** x ° / q~~~~

~~~~• i ..~~~~

~~~~i= S1 r 3> I~~~~

~~~~3 II s- ^ CO 5 Si. 3« o 3 0 s ? ii a £ I 7% I ii A. 3 fl I A .* * 6- si. W* II 5 i I 2 r i > 3T Tf 1 4 i- r s *> «€> tl If r« C~ J* lM «P~~~~

~~~~© i © r4 Q~~~~

~~~~II o I a II~~~~

~~~~i o 1 , N I o ll + \ ™l II ii k \ I \ a"\\\\ Uj i 53 Ji d 2: > o o_* II •«VJ~~~~

~~~~♦ E + N~~~~

~~~~I H M Q I* ii~~~~

~~~~a o -J 6 CO i 8 V~~~~

~~~~,^_ ^~- 8r *> K CM a a t .» * CO •»•• •* It 3 o ii c 3~~~~

~~~~LO -J fr r-~~~~

~~~~$£ O —» Uj 6 • -< -* <3 ***%.~~~~

~~~~ii tl 14~~~~

~~~~I ~ ra * b 5 * ^ i~~~~

~~~~1 c II ll II O~~~~

~~~~ll 1~~~~

~~~~3 i IS 5r *-* f* F ltt» ft- r ^3~~~~

^ ll *

~~~~i 0 irtu c DlS I? Ux? ^ It V* & 1 i pi*-* ft -fr o at • 3* ST* » J2 ^ it~~~~

~~~~1 -^l VNi~~~~

~~~~J "^5~~~~

~~~~5 13 O •♦•i il M 3 ll~~~~

~~~~"4- s Co 4 ^•^ 32 5* a~~~~

~~~~Ii < O . I Jo It ii~~~~

~~~~k> i i Ut -9-« -». II n lis -r o I o IS o CM II -8~~~~

~~~~si n L~~~~

~~~~JIJ~~~~

~~~~rJ CD i II CO •J Uj -lis s~~~~

~~~~Ul ^4 0 <-. , ~-o CV l~~~~

~~~~W V) V •J- ""^ j Hit SKi —«_* *•* ii eg 3 Lb * .» -^ I X / *? pr "3 <3~~~~

~~~~r^°~~~~

~~~~oj|y2> Q I .at I 3 i II ll* i' lit r Tl A3~~~~

~~~~£ ORBlT-AVERAGHlPfflTinFrnnFR e.g. Orbit-Averaged Magneto-Inductive Codes~~~~

~~~~Bruce Cohen, Tom Brengle. and Bob Freis 2D SUPERLAYER and ID MAGIC Lawrence Liverhore Laboratory~~~~

~~~~MFE • Magneto-inductive physics model v*8 = 1jt1t+ yty~~~~

~~~~Qqau Perform numerically stable and accurate~~~~

~~~~particle sihilations of very slowly varying phenomena -* - 0~~~~

without loss of kinetic details. i.e. ot -> —> —* _^ T « J", electrons are cold ions : f-s.rn a «_' »* • Filter high frequency self-consistent field Open magnetic field lines terminate on CHANGES; conductors which short-out vjrf X B%

~~~~• Follow PARTiaE orbits on their natural time~~~~

~~~~scale: If magnetic field lines aosE, as in field-reversal,~~~~

~~~~electrostatic and electron dynamical effects must be • average partiae currents and charge densities included. j. hay no longer be negligible. OVER MANY ORBIT TIMES: AND~~~~

~~~~• SOLVE FOR THE SELF-CONSISTENT FIELDS MUCH LESS~~~~

~~~~OFTEN THAN ADVANCING THE PARTiaES SO AS TO~~~~

~~~~• REDUCE THE PARTiaE STATISTICAL REQUIREMENT AND~~~~

~~~~IMPROVE THE ECONOMICS OF THE SIMULATION. SUPERLAYER - J.A. Byers, Phys. Rev. Lett. 13. (1977). 1176~~~~

~~~~MUSIC - T.A. Brengle 8 B.I. Cohen. LCID-17795 Rev.1 (1978) ~7~m. *\ a? • y4v«ioaê J^ m Yê. ovev- m«cro fime 5êp AT»dt~~~~

~~~~• Advance, policies o* »*wcro -fi^e s^p a£ e~~~~

~~~~a/2 4T/4t n's0 J 2c J IV 5 fata window dvu( aîf^/ s»*,oo+h,Vij £cuc4t>v ^WfrV^V^rVj* a-. *" +vÂt • 5*olv«. /Howell'* eûaKom for £*«d § us'ma <\7 >~~~~

~~~~wUrC £s g^ fa/erpoW,bn ^cfo^ forZ^J. K?ri* If M+'/-i pr*Al'crbv* hfe, use Boru' scheme - 2*U orvfeh /«apfvxg (&'U) * r/Hf/ • /law^aWe T fro>*'ioY& J* exxck At *J +- 0-«)V~~~~

~~~~0J i ; ( J * J */ '~~~~

~~~~**d. C^^)AT/2-) —> (tohl'AT) corrector where of H^^«W *wguW mo^enfavn ^2 r*g « -fAft i^aflttchc. pax d^ *~~~~

* /U-**"<* pu^f/c/t ve'ociAis wM (e-j?)" "J (EE) Mtopoht**/&iv*pohjh#/ r j?} r 47

~~~~Flow Chart lor Magneto-Inductive Coda with Orbit Averaging and Time Splitting~~~~

~~~~Initialiiation and Data input~~~~

~~~~B» VXAfl 4 B0 Predict (x,*2,t-t+AT v|~~~~

~~~~9 * at at t + AT I ~ Inject Ions .- -..nAt (x.v) Calculate Ag * from~~~~

VXvXAj4«,)'+4T/2 Boris Push All lorn Boris Push nAt-»(n+ 1) At, Injectsd lom •* -* nAt -+nAt ^.In-ll At (E,B) -const. x • v *

too tui*

~~~~+ Ww tm ,v/(«) tuj M~~~~

0 ATA X \J 0 Vwjjr CaJAT/o- Boris Push ,WCc) Injected Ions Boris Push -jnAt -*(n-|)At All Ions nAt-»(n+1) At Predict Aj*AT/2 from 0 V 2ir 3ic uitYnb. with(E.B)nAI VXVXA«,-£u(,>t +AT/2 interpolated V/(»)* Inject lorn (x, v) B-VXAj+Bj

~~~~0 I (aiAT/%. Correct (k,iT|?M-*t + AT 9 c dt at t+AT/2~~~~

~~~~Diagnostics «nd Data Output MAT («Hl)6T T t -a~~~~

v»

*Jl-~

~~~~? ]~~~~

~~~~.v» d x •<~~~~

~~~~1 3 i s 1! 3 ->&•~~~~

~~~~r F % §l~~~~

~~~~7* r C3 1~~~~

~~~~rf 1- 4 "75 > -«C V* £ 0 ~r -r O - I «l«* Or* •a- •£ y -a 1 Hi 'l I - I I—I V V Hi" L O i • / T - A. ^ 4) S» 3 s *' ST~~~~

~~~~. s: •5 b r 31. 1"J) fyUgnelo -Xwiuc^iv^- Si'wuWfiovis fV vr vtt) MAGIC and AiA^'C.L (ovVil*-averaged.)~~~~

~~~~MAGIC u£.t=WQ~~~~

~~~~04 - 'flfcj - I~~~~

~~~~- :|§ - Compress ioy\<\1 Alive* Wove ibispcrsion RalaTion~~~~

~~~~1 —•—i—»_... • «~~~~

~~~~t£;At< I~~~~

~~~~2 0- ? •0.4 0 0.4 -0< 0 04 «(ttOacmhl ».IIO*cmft|~~~~

~~~~Cl"~~~~

~~~~JL 2 4 6 8 10 Effective Alfven Frequency kvA,HAi~~~~

V/M orbif-rtVtna^iViJ, modes witJv ia)>tt/AT oajl Suppressed. /vl0d«5 w.ft u)« At""' are. UMctiriVW ok«A unetompe/A Jim to oc - (Re goat) j. Orh\l-<\W4.<\\»i4 dutucks +he e^en-odd* ivijfabilify 4twvje"t'io*i work equally well. t^jAts.-Z AT/Afc-fc1* ,0°° ,W 43

~~~~MAGIC MAGICL u£t--fr20~~~~

~~~~Square Data Window Rounded Data Window~~~~

~~~~-10s-~~~~

~~~~800 0 800~~~~

~~~~r = 4cm r = 4cm~~~~

~~~~800 0 «S«. 0.01 0 v r «* 4 cm •\N 800 0 800~~~~

~~~~(cl~~~~

~~~~)4 - ^\-~~~~

~~~~800 0 800 "St 800~~~~

~~~~(d) 0.01~~~~

~~~~800 0 800 ««t~~~~

~~~~°a/Qa °A/»A I 0 t i—i—i—i—i—i—i—r i i i i i i i i i i r i i i i i i i i 0 -a|oq je|od|qun~~~~

~~~~I '?<•»8U1U. OZI ••!'' . •'••vlV--i',";«r~~~~

~~~~OZt~~~~

(U13) J U3AV1U3dDS (UJ3) J 01 01 Trm-p—r—i—r—r i I I r t—i—i—r—i—n—i—rT 0 V0 f0 kjwg-| S>| vhJ^-| 5"2 UJ3Q9t 80 UJ3QBI 80 |toS^Aiypftij ^Varyof dnppng ajj/fc u»sr8l- = * ui3 s£«bi- »i 21 21 tt.1»:*!3*i39VU3AVU3dns U3AVTU3dnS

~~~~uouMttu ^)OS£b =a Oit?A *rw«|ov A»o5sl 5|»mjti9vivoot-j 4q Aiti2*- a>~~~~

~~~~ST ^ 3( "J 37~~~~

~~~~Code PerformanceCharacteristics E.G. SIMULATION OF 2X1IB ON REALISTIC TlME SCALES~~~~

~~~~Optimizeaccuracy,ensure stability,and reduce numberof simulation particles by adjusting •• Successfuldemonstrationof SUPERAVERAGE~~~~

~~~~SIMULATION OF BUILDUP AND TRANSPORT OVER LONG •ATAt - ASLARGEASPOSSIBLE,LIMITEDBY TIMES. E.G. 60 TO 1000~~~~

~~~~• oCAwdp - ACCURACY:fl^J-"!- STABILITY:0.6£<*^ • SUPERAVERAGEaccommodateslargeLarmorradius~~~~

~~~~EFFECTS AND DISPARATE TIME SCALES: • NUMBEROF CORRECTORITERATIONS - FEWERTHE BETTER FOR~~~~

~~~~ECONOMICS. STABILITY: ACCURACYIMPROVES:HO. n0.t\ > | ORBITTIMES SEC. PARTiaE ~«JT~10~8Cl~~~~

~~~~• INTERPOLATION/EXTRAPOLATIONOF(6,g) FORPARTiaE PUSH- PLASMABUILDUPTIMES 10"^SEC. PREDICTORPASS:t,B= CONST.ISADEQUATE~~~~

~~~~• ORBIT-AVERAGING CORRECTORPASS:b,B= LINEARINTER POLATION & EXTRAPOLATION NECESSARY TO (l) FILTERSOUT HIGH FREQUENCYFLUCTUATIONS.~~~~

REMOVEJERKS. BIASING INTERPOLATION (2) REDUCESTHE NUMBER OF SIMULATIONPARTICLES, BACKWARDSINFAVOROFOLDfcTjB*IMPROVES AND ACCURACY& STABILITY. WlTH BIAS ONE CORRECTOR PASS IS ENOUGH. (3) ELIMINATESARTIFICIALACCELERATIONOF SLOW

PROCESSES,E.G.,DRAG,RF DIFFUSION.AND • NUMBEROF PARTiaES - REDUCTIONPOSSIBLE DEPENDSON INJECTION. AT/At>*»l ANDPHASE-SPACEDISORDER,IS PROBLEM DEPENDENT BUT CAN BE QUITE DRAMATIC. • NON-ADIABATICIONLOSSESIN2X1IB AREREPRODUCED

~~~~IN OUR SIMULATIONS. A NEW RESULT. • DATAWINDOWW- ROUNDED"CORNERS"REDUCESEftNOISE BUT NOT DRAMATICALLY.~~~~

~~~~TlHE-FILTERING IS ACCOMPLISHED;THERE IS SUBSTANTIAL SMOOTHINGOFAt DISCRETENESSNOISE. E.G. INJECTIONNOISE; AND WAVES ARE DAMPED. c n r 39 aXJEB BailcJluLp SUPERAVERAGE Physics Model~~~~

~~~~• 2-D (r.^) AXISYMMETRIC~~~~

~~~~• SELF-CONSISTENT Br ANDB4. AND INDUCTION E 1.00 (Darwin model) . I 6 Inm t Charge neutral and Jg 0 .30 -~~~~

~~~~• Electrostatics negligible on open field lines for rr\ ' '.10~~~~

~~~~- .4o~~~~

~~~~• 3-D DESCRIPTION OF NEUTRAL BEAM DEPOSITION IN- 60~~~~

~~~~aUDING CHARGE EXCHANGE AND IONIZATION .40~~~~

~~~~• RF TURBULENT DIFFUSION BASED ON OUASILINEAR I 4o~~~~

~~~~THEORY ft •2D J9~~~~

*/ • Ions drag on electrons (*£ is an input parameter) •lo */ tune

~~~~i i, _. >„ 2 4 0 I 10 \t M & It 10 AT t .. -* |1JD00«^; s,Ci»/o $ec FlELP REVERSAL VS *tlME~~~~

~~~~COMPARED FOE. TklO ACCELLEftlvnoN f Acvro^s (FT*) ifO 2Xrj UilAiOf~~~~

~~~~^i\\\i ys-^r/HE-^—r- .1. ,:l;. .i~:„ „l T~~:. ^MJE_McZ5^ f ii~irfiR-B7T/2A£y~UTOTS~~~~

~~~~T7MQ~~~~

~~~~4 6 c? /o /2. J4 /fc /& ZOat iioco<;-.UP~~~~

~~~~s > 3. CI" 70 a > < 1 m TO it ! s o m i o u o O $ "5 O X o o (A) o o i o o i U a:~~~~

~~~~o o ii 2 §- i> w * •+- o o 2' o C O 70~~~~

~~~~i < u 2 .& Uj r s c? «-•"'•'"Ct«» 1*1 0*1 6*0 8*0 i'O 9'0 S'O VO CO Z'O TO O'O ro- • ' ' i i i~~~~

~~~~• Qq Qq jjP^ °a. B 3 i~~~~

~~~~1 eg~~~~

~~~~5: C* w. L¥ ^] + v„ A -- A~~~~

~~~~wy 3 t +.3 ^ i-v.3 ' u t >*E)+>.^ --«A IV • .A + ,-Mx - v.x~~~~

~~~~V) to~~~~

~~~~a. I 2 0 • ^0 o -u~~~~

~~~~u A A >l **<* ^--vt u3 *y,-\*=l- i-u ^"3 A0l4 2^^\X^Wd x^onyr OJry (cJc^JK-ted)~~~~

~~~~OD0d°0dOba i~~~~

~~~~c00= (. o O in &*. = i.~~~~

~~~~cc UJ £ = O.ff~~~~

~~~~I i i i i ! i i i ) i "" i"* -! -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 W torV^CY-* —^~~~~

~~~~£**+ rv\»ti~~~~

~~~~^•0-1~~~~

~~~~/.o- >?N~~~~

~~~~CJ,~~~~

~~~~o. —~~~~

~~~~C*J0 v.£rivi*^ ^Vy-.^o-cy } 2T =ICl0U|~~~~

~~~~5*0 =. 3 1 -°0 ll?*\'~~~~

~~~~«0 .3" ^/(Svi^ajj: Cvii/ii^p") °~~~~

*vo-

~~~~^s r-p; •«^>v - T -o- >, •CO~~~~

~~~~n^a, Uq a Ha <— """^HllHUBBUU,, -- O'O *7~~~~

~~~~/£TOWARDAFULL3DHYBRIDTRANSPORTCODE; A REACTORSIMULATIONCODE sn Q LLL H~~~~

~~~~UJ r~~~~

~~~~o *t* S—~~~~

~~~~Low Field Solenoid~~~~

~~~~« P v o UI~~~~

~~~~Hinlmum|B| end cells 3 0~~~~

~~~~« .. | vn~~~~

~~~~3 J L5 PRECURSORS AND FOUNDATIONS~~~~

~~~~3D MHD Fully Implicit Fixed Grid. Finan & Killeen FLUID CODES {3D MHD Implicit MALICE. Barnes & Brackbill~~~~

~~~~3D MHD Semi Implicit Resistive MALICE. Brackbill~~~~

~~~~ID Hybrid. Crystal, Rathmann, Vomvoridis, & Denavit BIG At PARTICLE ID Hybrid. Byers, Cohen, Condit, & H?nson AND HYBRID CODES |2D Hybrid. Hewett & Nirlscp~~~~

~~~~2D Superaverage. Cohen, Brerolc, C-rrlev, & Freis~~~~

~~~~NUMERICAL (2D ^EM &alerkin# Gresho METHODSANALYTIC {ID1 (& 2D) Moving Node FEM Gelerkin. Gelinas IICCG. Meijerink & Vandervorst. Kershaw~~~~

~~~~THREE MODEL 3D TRANSPORT CODE~~~~

~~~~THREE MODELS: 1. Long Timestep Transport on Full Time Interval~~~~

~~~~2. Short Timestep 3D MHD Calculations at Occasional Sub-Intervals~~~~

~~~~3. Short Timestep 3D Particle Simulation at Occasional Sub-Intervals~~~~

~~~~ONE GRID: All Simulations on 3D Almost Lagrangian Grid~~~~

~~~~x> LS~~~~

~~~~REGIMES OF CODE OPERATION~~~~

~~~~1. Transient Start Up~~~~

~~~~2. Slow Evolution Near Steady State~~~~

~~~~3. Fast MHD Disruption~~~~

~~~~4. Fast Micro-Turbulent Disruption~~~~

~~~~ECONOMIC REGIMES~~~~

~~~~A, Only 2. is Advantageous For Computer Economics B. Fusion Reactors Won't Work Unless in Regime 2~~~~

~~~~SUMMARY~~~~

~~~~1. Need 3D Transport As In TMR. 2. Keep Track Of MHD Stability.~~~~

~~~~3. Get Transport Coefficients From Particle Simulation.~~~~

~~~~4. Calculate All Of These On Same Grid.~~~~

~~5. Most Economical For Reactor Confiourations. 6. Galerkin and ICCG Methods Developed. "**% 7. MALICE Exists. Its Companion Particle Code Does Not. 8. Currently Available Computers Barely Adequate.~~

~~~~a S6 iMHvfeJfef* CoJk *r~~~~

~~~~UCLA~~~~

~~~~(jriUUr+fo/Z ____ ._^_5_>Mr/r»/~~~~

~~~~Skilly /• -~-*f£+fpr*/ri+(i *..*... 0.*s~~~~

~~~~._.._- T-~- .. Xis _*#lr*"~«T* .t Cfc*r*»*y -fL*lJ ..§ Jurt*+t+«**> , e^HMPc^Vn~~~~

~~~~4i,-($*f**Ce* *rt Ottos Cocks* •>• ^ V~~~~

~~~~v*>~~~~

~~~~r I~~~~

~~~~O H o •*- b.~~~~

It •f P -**% K .**

~~~~o Si •T \> "1 \~~~~

~~~~>-H io (,1 VifuA* .k gJ«™» MH2> Codes~~~~

~~~~IX^X~~~~

iA% ft? (-J fe1 ^K 'hia/lJ '/, Klmoi/tJt of uiyicujff** if Tfose fe*/S(H4s 63 PUOS9 Cede.*

~~~~I ihfo—Sh Ifr*d-*tlrtA cfu*+-£pc*yfrc/esj - G^A-—4n4~~CAJLt*< (*fi*AS—C** t xiO—~~~~

~~~~OP Pixoa (A^'forhx msb •i .11'~ !! I! ..- i~~~~

~~~~More* *H%n*OY% JW Ctfr^fvClAjiiA^, 53 H 4h 4 tf<5 4 ^ © 1 v5~~~~

~~~~i -s 2~~~~

~~~~lb 3' 1 v£x v~~~~

~~~~v» A Sv *a •< i t $1 J V i o~~~~

~~~~.9 I nr I i i ^ >>~~~~

~~~~W l \j Ii ft i?i« '^%~~~~

~~~~r to~~~~

~~~~V3* 3~~~~

*4* S^

~~~~U t fI JP 1 J-~~~~

& ^ r^ r r> •*

~~~~(I U -A ** CO \ (* ^ •3 •+ I I ^ I "to X X rf~~~~

~~~~« ^ l?K "5* i /.~~~~

~~~~n O M s~~~~

~~~~7S * £~~~~

~~~~as X~~~~

~~~~1 I~~~~

I . ill A 3 A v 1} t ar^ us I* 3 $ *

~~~~ce CP <* *i*i3 I ! 4; (I ii~~~~

~~~~2r v 9 J* 2 +~~~~

t»

~~~~G> ^ J* *, JH -^ o *• o x o *• x x~~~~

~~~~mm •~~~~

+ *^% x* x £*£ *

~~~~9 — * \A I ;W~~~~

!•

~~~~3 — • 4 i ; *3 *1 IS* ^\ I "">* ^ & « .* S ^-\ ; kL 'Ci t \ I 4 x* f aw~~~~

* U. H* r <3>

~~~~n j ' w GO*fu CO fl 00 -3J PS~~~~

~~~~3 4 4^~~~~

~~~~ta « *» <*/ ^~~~~

~~~~o <0~~~~

~~~~i C8L If IK » > \ < ft. > T SL I ,sC 9» •> I ! ^ +~~~~

~~~~k •^ 5. as. St. i *~* 5 « U '-T r (i~~~~

~~~~A. I~~~~

~~~~0 <*£ r £ £ **$ fiUTQCORRELATlOH FUHCTION SPECTRUM _1S_ f~r— i i ' i i i i i i i i~~~~

~~~~' * " '->' *—»• *• - / Sr— N..O~NM*ia*'» ? 7 t f T f I I~~~~

~~~~MODE • 3~~~~

~~~~J2 Qrf'd- /]rjsK Wi*tL~~~~

~~~~A(f«-<-~~~~

~~~~^> .if D~~~~

~~~~aH \HJO*r^Z~~~~

*>•* j. ~* *•

~~~~51 OtAA. - A*4* toJ+?**( *" k&X -lAJj^Jrttf1 l-U;W'Cr~~~~

~~~~%e>Y\Jl - Ann VtjtftJ- Ux'^^ArciL~~~~

~~~~Bora r~~~~

~~~~• TFT . f AUTOCORt^LATlCN FUNCTION DEN KX IS ' •. 12 * ' 1~~~~

~~~~1 \.*M- to « i~~~~

•' i • 4 \hri 1 i • <•.••:•'£•;•-.• ' !:- : 1 - — •• / - \ « • i i 4 / • ; ;i*' $0u*jzt te&/*S4 • I' ''• j 2 '; .-.-. '-'i. ..•'/( o a <• * « a t* ia a m ' a b a o a o 6 6 «J

~~~~MODE • 4~~~~

~~~~00- Ft T- Lfy?yj*Jw&~~~~

~~~~tmtmtitNtn** m* miit^atmmwtmmtjii~~~~

~~~~t«^-/v,~~~~

~~~~r r /~~~~

~~~~1 ftUTDCOgRELflTlCN PUMCT1CN 6X KX~~~~

~~~~^aftybf™* C-~~~~

~~~~f t T? V~~~~

~~~~MODE - 4~~~~

~~~~SOD- Vt'f- La* (fiU&ff~~~~

~~~~••• - •.•.•.•41 ?~7~~~~

~~~~AlfiY*>s\ U?**M.~~~~

~~~~MODE • 4~~~~

~~~~SUD-TFT -.UfitiWi 'if-~~~~

~~~~./-^t~~~~

~~~~1 RUTDC0RRELRT1ON FUNCTION SPECTRUM~~~~

*—t—> V •? v Of « s

~~~~WS& f ♦~~~~

*rara

~~~~CS 4 r* ft 3 4 W !&~~~~

~~~~i 4 M 8 N v ••-> 4~~~~

~~~~•a ft * I~~~~

~~~~— ia^i~~~~

~~~~12 mu~~~~

~~~~-'A t * 20CM~~~~

~~~~,.-.• r4'~~~~

^s_ V..-,-^-**

~~~~b! Mnaiaads NainNfij NOtiBii38a3ainu J~~~~

~~~~W I— r~~~~

~~~~.1'~~~~

~~~~!3 O* 3UBU eo+na/sta-z Eo-a^tzars 88*efr -i X1B '~~~~

~~~~SSSsSsaBSSS^SBBSaAtBBBBsZsXs~~~~

~~~~-«• N~~~~

~~~~tl. " ...*• ~ . • " • / #»t < • ''.* It ' , .~~~~

~~~~« .•. * • :... -- ',,*... ,. '* ** -•• « -.• <• '• ' • . *.. .• .. .#* _*'* *-* , ijr-~~~~

~~~~n >S?'as^^^w~~~~

n P.I* ESS^•••ITT1 ••", ,.- •• *.*• i *. a ••-* '"' . •* •• .* • • t. ii M II •• . • 141 • • tl *.* ' -t—i—» •", 1, , i. Sjijs "« l .1 l l JMIM MMJ; ; :?f

~~~~BENSITY T- 188.88 3.a8271E-ai 8.7777!E+e8 ravfiw •*/ Numeric I Stud lea. oMhe- L i.aaa Theta, Finch ^«v I—T~~~~

~~~~dj s| s -a a •• .q a « a a •~~~~

~~~~rle> 2*.~~~~

The elapaad -prob|ea-tta . to aacb'ceaa. la. enough. Cor- tha. rarefaction to>acafar- gate about SO ca. •r.-. Tha aore difficult coaputattona.vlth cMgnetle alrroT» are* betnaj pe»teraaa*U * coaparlaoo of the aaaa dace* rata* fr***-Hwv*~eBletilatloo*, obtained) aa> a«|geeted.la (f), wilt B»»a tha crucial taat of tba abllltjp. of Idaal KHB t» deacrtbe andlsaa. Tha reeulta- Cor tha etralght caaa-are-vera eoceureajlag, however, not onlf baeauaa of their egreeaeat with ejrperlaeot bun alao hiiwi tha* euggeet that, aoae*eat-of- equation*), even* alt*?la* tkaej tba>'KHD>eqaatleaa daaerlba endtoeei. ">ferencea•

~~~~III .rraldbers.J f end-Uelttnar Hi. Noel, rualoa.1% 21>l»WJ>.~~~~

HI BracVblll J U. Numerical rlagaetohydrodvnealce. for Breh>»•«» Meeaea. Hatha, la Coap. rhye. 16t Acadcale freee, Hit*. (to>be. -galeae**). Ml Braettblll J 0 and Barnea.DC. Ttae-Depeadeat' CelcutaCtoaa at B4»IUhe*>aB and Stebltlt*>rroblaae.fa Scvllac and*Totcaaaa Ceoactxiea. rroc. of tha third Topical Ceaf. oa rulaadiHigh.lata, MaaaeevCttlbaat<»*7>>. m

1*1 freldberg J•» aod-UeeeonJ A. Fhyav Malda-1* MM*OvTOB. hi Winner It. (ptlveea-coaanalcatloaK Ffgure 7>$*3< DilTcrtntitagtiof \*ortcx rin*^ formed in water by allowing drops of culuurvd water to fall vertically from the end of a pipcttu I cm ahove a free lurfjcc. [I-torn OLjIh; and Inoue n>6i.)

~~~~Q* \ ' " \. -\ \ pr> s \~~~~

~~~~•I r r~~~~

~~~~oc <= —. a. - - = t£ • U *" "r £ *~ - — — ~"' U CJ — — K* ;. •~~~~

~~~~•O ll~~~~

~~~~— o * in «••? I CA J Y*\ ^ I. s 7; •M~~~~

~~~~» 7^ # f7~~~~

~~~~MVMROOtA»SrV^WTH-pBNE.TrVATIOr45V5TIMEr^OK12."SU&MEr\66NC£~~~~

~~~~9 — TEST I A — TEM 2 + — £AL6ULATI*N~~~~

>-£—*

~~~~t* .SO~~~~

~~~~|0. \oo 120 |40 |^ * ?T5 V-a~~~~

~~~~ul 5 o.qooi \00 \lo 140 ^ [Cop \bo 100 TIME (Mi»EO) • t'.HC J n~~~~

~~~~.*. e..^ c t MUSHROOM GROWTH-WIDTHS VS TIME FOR 12 SUBMERGENCE~~~~

~~~~o —TEST I 'J <3KJ^~rJ-^ a - TEST *2 * — CALCULATION~~~~

~~~~/• /*<• /• / < • *• « s S / S s s / / SSSS* "fs/f/ /~~~~

~~~~"V -» •> 3 9- ±~~~~

*t*ift-aaV**^*****-.' /I /•

~~~~J- 4/» -7_-~~~~

~~~~PT-*- 19 (*- ; \10 140 160 * TIME (MSEC.) -*/* *A~~~~

~~~~p/ij. ./• Sa*»t>k-Vjrh* Sheet- Pnl>le~%~~~~

~~~~f«mBne007 JIW'BI 1, .c~~~~

~~~~4e» »e> o t » b 5 •a 5 o •»• 1~~~~

~~~~V o ^ f >: 8 Hi 2 I* av t c 2 I 55 *»p»~~~~

* V)

~~~~'/P**\~~~~

~~~~1 I.J? i i ^'«~~~~

~~~~t Jl.l .L.i_i_ :.:..~~~~

~~~~r» .•?.^- ©~~~~

I2|1»«i>btll*lt VjI*"0*'*1* Y.->h>tMMA. nt»m*ti 2 i-iivsks Limns apiplltade , oncmutlb : 30- 30 tick nani|- tfumdctei tNptfimenl jrf\>lion o not gutian 7 80- the theoiei . 8 (tie hansp 4- 10- In fact, t l.o »-° tnomalous aaoitrt l«HI»l»* piesence ol IC. 1. *U«o»lrfti«&c«lopofce*^1*>tlbe»oealleaedB» •rl,***'*. •11 current Vibe* ml Important Ora eorlerea. Theeymbol f, . fluctuation •irueftbe »ep»r««rU. <,U tor «url»ee»llh theodalmam flC. >. Ul DreiltT a andflochuttloa amplitude eVej and(b> «s,-.s LeW boe. and t,•• the aiaMUiy bmSI auriaee. The acta 5 4 .» d,» r »*, etectrostali •alerylactore, aaa fttocHoa ol fitteoca above thetopMem*! l«*en the t ».* i.s.r.rtiy »»'•' left. ectopote beep. Fig. I. Theobterved oloeof tnh n meattned by Lanpneb mode.k0 k piobeib plottedita function of poailon, +. -4M130BD C19»Ame».«J» Innate* »*•»«» *34 toi. and tin aoutd gft* wavespiopagated upthedentil)' giadienl aswed at tin*»"»0. peipendkulai to Ittesultingln oblique propagation al aitempl wa •. roughly 45*lo Iheditection of the demity gtadient. fluctuation The parallel wavelength watmeauiied lo beabout a 0.1 the e equal to the minorcfacumfeience ofthe toitn or bifet than •bout 3.5 m tetulttngIna parallel phate velocity mtti- bire tiantf mediate belween Ion and election thermal speeds as |4.B)otti expected lot• drift wi»e. The observed values of 8-i/n weteapproximately equal lorb$ITtandweieob served lo glow fromabout0.1%Imide •>, to about 25% at1>t «* shown infig. I for • value olBp ~ 750 C. Tig. I.Theobieiveddtmltjp-ofee w«b Bp - 730C «kIu Shown in fig. 2isthe limeevolution of Ihedensity anWtblrondllloa atI• Oand e»ol»ed foiaai* l»tJma by profile observed when fl„ - 750 Ccomplied with the eemc-kal fetation tothedrTanoa caaatlon oilng only din* predicted evolution due toclassical diffusion using the cal dlffiitlon. ThliptetBettdA.pt Ittompattd alththa •»- 1*0 ptofileataninitial condition. The agieement pnlmental polali al I- $o«cand I-1 aw.thef'**T • notmattttd to1.0 arlth tha p»kWtul d.»tl*y rooJ toU watverygood bothinmagnitude andInprofile shape X10'»ian^^Katlti»tt«^»pntmentalponMib««ete t»i.tr i«w a withIheprofile relaxing to aneatnoimalmodeshape dWllitng coon b.the daiaacqeWrton tyUno and the en y IC. *• Fhrtuattoa amplitude aa a function ol aafery lactor f |S| in9 msecIt thendecay* withadecay timecom thrtettloat. The nomal n»4tdecay olthe planne •««*•*• *• lar at Situ* plaaoia If, « TJ. . parable lo Ihe 100 msec decay time of themagnetic , Okwagnetle field ?mtee aftat Injection it 10» miec field. Thb good apeement watobtainedboth Inthe FIG. "J. !•» riurnulloo ampbtoor manured atlb etnabove Ko. 21 hoop »»a fraction ol a. lb) PlMina uletlmt» a> a quiescent legion intlde f, and In thelegion with large amplitude fluctuations outside **,. Geailythen the Weconclude, iheiefore. lhal thediirosion due to . jV*ta^, ^jC***- ,0.0**"'* , Wovx-j function di. .' the fluctuations./^. mast be lets (han 10* ofthe ItanspotI due to these drift mode* we* much let* than magnitude ofthe classical diffusion or» significant d»- riattt, Efimi, OMaMt. and Wong 4 lhal dueto classical diffusionIn thiscase,although "feienee In the time evofetion ofthe density piolue In anomalous Itanspott due to Ihitmode ha» been ob side *, and outside *,would have been observed. served inother experiments |6J. Purthermote, over" This implies th.ti>r<0/250)DBohw for.Huctua- Ihe range inmagnetic field horn Bp —250 CtoBf — 1.25 kC ihe confinement lime increased by over an lion amplitude oftn/ti -20% In contradiction lothe olderol magnitude whfle Ihe value of on/itm Ihe avet- usual assumption ofBtfwi-like liantport due lolaige agemlnimum-fl regions actually Ineieased somewhat.

~~~~224~~~~

~~~~^eSJsoV&Sai^aiSrisWVB^ -i.7- -4 •- /05 SkP fi"»eJoL~~~~

Wnpt =1000 ne(r) (x|04) ^^U^V^^W *****

~~~~exp(ikzZ) expansion (upet=l960 * MjWrova*^Vrt^t~~~~

~~~~i 3w9.«l4-9pCO€>~~~~

iax^4b«j^ (*

~~~~762384 Pig. 13. Particlediffusionin the absenceof convectivecells. Fig. 11 is also plotted for comparison.~~~~

|| 772047 Fig.1. Asketchoft^.i^tie*^l-^^{^,^J*^ 9nexteralmagneticfield.ThemodelisPer.10?^ctions> >,letheconductingwallsareassumedinx.^rec ^^ ^^ •T as«0 '(T'S) • (u'w) apoui a*cqv-)sun •»8C*a ay.} ;o aan*j,3nj^s apoui i^Tpea au.3 30 *)uauidoxaAap auiTj *y 'Stj

~~~~0891- 49d (^ 0fr9 =T9dn 50*0~~~~

~~~~CO —1 -8- CD 3~~~~

~~~~- vo~~~~

~~~~OWI»48d"«~~~~

~~~~('»n>)-w*ty (j«»vrrre»irv c*»tt«A) (1*9") • (u*ui) CivS^rr^vvO «-w»iqa^ x^gr»/\ it>»rn.;v.\~~~~

*° 7-3 y p-«a

~~~~3 O 9) SI r» Q. * «- < h. -» 0 » n 1- 1- _i_ c II -Tit 3 H-0 O O. * 3 (0 (0 "S nS sj c a o » IT e 1— « • »~~~~

~~~~-?8~~~~

~~~~0 -si fO M I- ••* r o Jo uo~~~~

~~~~"o9.»t'~~~~

~~~~j^*\~~~~

~~~~W1 » 0 •iiii'— ~fl 11 •1 V( r It tl?~ It !'V u~~~~

~~~~1 It 7 •~~~~

~~~~- -Vt*" YAoeL*.~~~~

~~~~~le '« " VWV ht^^fJ*Vt;M •I i1 i~~~~

~~~~a 1 i i a~~~~

~~~~ti j •~~~~

~~~~a h' • i \t tl P « K i n (• i~~~~

~~~~• B) • .•' ! Ml II x x *-H •'I K K x » r* *T x-x S"x xli TTR K-Jt-S * V U L JUL . 1 if~~~~

32 B1 • t B e 9 n * a a a ' 8 POTiNTIf-L Pea PILS OPT)S 10-TH KRAKU te ii atlU^VV* Jx«>Vfi.l>fM 4 KTH no THSl, i-nt tnnrtoHiC IH •re 3 ii i • 'A a KI 3IK0UMI KM 3U 4XU Hi* HI pirfrOM Hi-CZ j\ HO mtiOJM 7IUK310a ^^1; >6 0 <• SB B = .' S 2 3 •!.-. 3

~~~~x x X !|~~~~

~~~~X X~~~~

~~~~V-*/* T+J4~~~~

~~~~oQcun * <^V~~~~

~~~~-4 -. / 9>~~~~

~~~~WB^eByBBgaWUJJLUia^jaai^^ . a. uj, „ , m M_j j^LBBUJBtUUIJBiiaaB ULI1I *aae«a\ * '/^"l iff* ^ -taas~~~~

~~~~' II~~~~

~~~~• | j I j~~~~

~~1 • • it X* X X X x V ^ X x X»j» * XX * » *|" X|X x XX *-*** "1 ... ^ on ass sas nrntrnt.norns ortk to-thmmne m mrnto tw j-th matnic w « .-*. aw. ok. Hut hiaukum w-it su jo310011. miuam~~

~~~~,$p**N~~~~

~~~~agates}~~~~

~~~~•• i~~~~

~~~~2f ti~~~~

~~~~z* hi annuavi hi-z >iHt aw him hi 9tKou»« hi-is au jo nidcw -utiroiod~~~~

~~~~a 1 a it a s S: a 1 1 {I Js ?'~~~~

X y *-* x x X x Xj *~* *-*

~~~~T ]U (•iS.+u-i...~~~~

~~~~e^pcnu 'VSV^A~~~~

~~~~(•tin\ *<3 *«. +,+l J) _2_9^ •=~~~~

~~~~9 ••8 r-e-o-** 9- ua~~~~

~~~~-Y "TD t*»7*>*3~*U_ P iX% *r^s b-^-^ 1 i \ i t' 1~~~~

~~~~! *i~~~~

~~~~•V* .. • i 1 / Oa •» •~~~~

~~~~!5 j • r V~~~~

~~~~i 1 it • u 4- •tsMa.60 :• n 1~~~~

~~~~N I I It I It '. I II Mt ! t. I it 1r 1 * 1 ;• <4 '<* at~~~~

pi r 1= i • |i 1 " ir 1 o; a — r >- a .i.V-i "J : 4- 1, -3.. i ; i K X X X y it »• X <*; . . X '!•( Ml » F X K <><*j yj "XT X1 i. wx - X x X «x '• f X K v v •cr^. •.«!••» : c if : r s.9 n a • i a a oc poromn. pnvtui op h *|»I-TH KSWHTC IH ICTH W 1KB S-TH rWJlOHlC til KZ . 'l *>«> ' • • • . t ' J r Si

~~~~2> i: .-',. i. li~~~~

~~~~\ ..si oast *umi * Hoiiraiiusia uauaa swairo-x~~~~

~~~~a a a 8 S D a a v 8 3 .< a a • |a 1* IV„ >•«, •an •»• UU ••~~~~

~~~~t ""a~~~~

•

~~~~1 •~~~~

~~~~^ • 1 0C9V' | t 1 1 1 1 • 1 1 1 1 1 1 i • ' ,._~~~~

~~~~TT-wf'T'lL ^••a'S V^»vs 5 • s2«6 aaaaaoasj: 3 s a S 3 = 3 «• =s B b a a a a a 8 A s s s : s s 3 3 3 - ie •41 X !~~~~

~~~~a * X e-ci : , 5 e * e X c a. < < 3 e > a X X eo X X * t-ta X : c X tt X X X \ :l *^ X~~~~

~~~~\^ ij »Pjl > X ia . X X u *• • X~~~~

~~~~n X X i S! X -J* ' e c I X X te vl-> X « X X X » X~~~~

* : X X 3 X X 0- X n * E

~~~~f« <♦// a hi awouajH m-c su emu ma hi :ihowmi m-ot au jo Titdtsu tiuimod a a a .a a u a .a s s a Js a » a 2a » .. Tx"7TT x x x xjH « '-.J J l_ 1 II 1 "—i**, i X~~~~

~~~~X X X X ** *l » X~~~~

~~~~1 1 1 ,. 1 i~~~~

~~~~1 l 1 1~~~~

~~~~— — 11 1 " Ml •a 1~~~~

~~~~at , 1 I~~~~

~~~~a '~~~~

• T* *~x jj~X r. x x * x ,r X at X • X Ml X *-# xl r—• x- «y|m I s*x|x« a a f a Q •r m t - w Sss»nsaaa'a mm* pwpjw cp tw xa-TH „«»»« wmw ;„, l-w (nNMie w^ ***-• CO • UJ UJ • I— (_> 2 .. oo «j? o UJ <_> 1— >- . ^- CO 5S o CO >- UJ < u_ CO LU _ UJ «—l __l UJ *~* UJ CO £ :J QC c UJ LL. -J _J j p— z <*c CD 1— CO ca LU LU •» «*c <_) > CD oo UJ o *= a*: CO p— 1— eo CO L_> _ OS UJ UJ **3 UJ =) p— CO 1 __i _l o z: 3= £ t_> - ca ca LL. LU >- SZ CD >- <_) oc ^^* o

~~~~.«* eB /~~~~

~~~~HEM LIBRIS NEW LIBRIS (CO NT.)~~~~

HIGHLIGHTS: L1BR1S REWRITE • NUMEROUS RETRIEVAL FIELDS, E.G. •TECHNICAL PROBLEMS IN CURRENT VERSION AUTHOR DIVISION INSTALLATION (FILE FORMATS, FILE STORAGE, LANGUAGE SUBJECT INFLEXIBLE OUTPUT) •INTERACTIVE SEARCHING THROUGH SUBJECT •MORE HUMAN ENGINEERING NEEDED CATEGORIES, E.G.

MATH ODE •AVAILABILITY ON CRAY-1 AMD 7600 PHYSICS (HANDOUT) (ULTIMATELY SHARED DATA BASE) ROOTS ENGINEERING FUHCTIOtIS •MUCH IMPROVED RETRIEVAL LINEAR SYSTEMS PDE GRAPHICS NEULIBRIS PROPOSAL IN DOCUMENT: •ADD ABSTRACTS EITHER INTERACTIVELY DOCUMENT PRINT NEULIBRIS device BOX tm id OR WITH TEXT FILES

~~~~COMMENTS TO CAROL TULL USER # 7 61 • EDIT AND REVISE YOUR ABSTRACTS WITH YOUR FAVORITE EDITOR «v ^y W~~~~

~~~~INFORMAL CONFERENCE GN PARTICLE LI BR IS STATUS AND HYBRID CODES FOR FUSION~~~~

~~~~PHYSICS + ENGINEERING CODES 3 3 ABSTRACTS December 10-11, 1979~~~~

~~~~MATHEMATICAL ALGORITHMS AND 18 ABSTRACTS CODE SYNOPSIS COMPUTER SCIENCE Code Name: MATHEMATICAL LIBRARIES 3 ABSTRACTS Contact(s):~~~~

~~~~UTILITY ROUTINES 11 ABSTRACTS~~~~

~~~~(SEE HANDOUTS FOR A BRIEF Machines: DESCRIPTION OF THE ABSTRACTS) Description:~~~~

~~~~FUTURE WORK PERIODIC MFECC PUBLICATION OF CODE LISTINGS TO ALL USERS~~~~

~~~~ADDITION OF MATHEMATICAL ROUTINES Other Comments: ( K. F 0 N G , E T. a L. )~~~~

~~~~ADDITION OF NESC ABSTRACTS~~~~

ONLINE PHYSICS CODE DOCUMENTATION??? Return to: Carol Tull National Magnetic Fusion Energy Computer Center P. 0. Box 5509 L-402 Lawrence Livermore Laboratory Livermore. California 94550 r i U-7 PHYSICS CODES angle.TheelectronsaretakenasHoawellion.Applications

~~~~includemirrordevices(bothstandardandtandom)andtokamaks.~~~~

~~~~Variousphysicalphenomenamay lieIncluded. Fokker-Planck code3 No.12TDHFP- A. A.Mirin,NHFBCC,Cray-1andCDC-7600~~~~

Thisisa twodimensionalmultispeciescodewhichmodelstwo No. 7 1SOTIONS - A. A. Mirln, NHFBCC CDC-7600 dimensionalelectronsandionsinthepresenceofa mirrorloss This is a one-dimensional multispecies code in which the cone.RFdiffusionisIncluded.Applicationsincludemodeling functions distribution depend on speed only. Applications with theeffectthatuntrappedparticleshaveontrappedparticlesin this code include 0 calculations for mirror reactors in which a tandom mirror device. there is neutral beam injection. NO.56FPPAC- M.G.McCoyandA.A.Hirin,NMFECCCray-1andCDC-7600

~~~~No. 8 ISOLEGEND - A. A. Mirin, NHFBCC Cray-1 and CDC-7600 ThisFokker-Planckpackageisdesignedtobeincorporatedinto~~~~

This is a two dimensional code in which the distribution largephysicscodedrivers.Itsolvesthefulltimedependent, functions depend on both speed and pitch angle (using a Legendre nonlinear,multispeciesequationintwodimensions,speedand expansion). The code includes an axial electric field and models pitchangle.ThepackageIsoptimizedforboththeCray-1and electrons and one ion species. The primary application for the the7600withtheCrayversionrunningup to 12timesfasterthan code is to study runaway electrons and ions. the 7600 version.

~~~~No. 9 THDSZ - A.v A. Mirin, NMFECC Cray-1 and CDC 7600~~~~

Thi3 is a three-dimensional multispecies code in which the Transport code3 particle distribution functions depend on speed, pitch angle, and one spatial co-ordinate. An electric field is computed No. 11 FPT - A. A. Mirin,NMFECCCray-1and CDC-7600 self-consistently using a Poisson solver. Thisisa transportcodeapplicabletobeaminjectedtokamaks. No. 10 IIYBRID-II A. A. Hirin, NHFBCC Cray-1 and CDC-7600 Theenergeticionsaredescribedbytwodimensionalvelocity

This is a two-dimensional multispecies Fokker-Planck code in spacedistributionfunctionswhicharetime-integratedusingthe which the ion distribution functions depend on speed and pitch nonlinearFokker-Planckoperator.ApplicationsincludePLT,

~~~~TFTR, PDX, and D1TB. I T '•>? No. 37 EPFI - Stevo Sackett, LLL Cray-1 and CDC-7600 Eijiti librium codes This code calculates magnetic flux lines, fields, forces, and~~~~

~~~~No. 18 ELLIPTI - J. C. Taylor, University of Clascow (Scotland) PDP-10 inductance for an arbitrary system of rectangular cross section~~~~

~~~~This code is designed to solve nonlinear elliptic equations in conductors.~~~~

~~~~two-dimensional regions.~~~~

~~~~No. 43 ORNLEO - B. Dory, ORNL CDC-7600 Dispersion equations and the plasma dispersion function~~~~

~~~~This is the conducting shell version of the two-dimensional~~~~

~~~~axisymmetric tokamak equilibrium code developed at ORNL for No. 21 ROOTS - Michael Gerver, Cornell CDC-7600~~~~

~~~~tokamak MHD calculations. ROOTS calculates and plots the linear normal modes for electrostatic perturbations of a Vlasov plasma.~~~~

~~~~Magnetic field calculation codes No. 36 ZETA - Bill Sharp, LLL CDC-7600 ZETA numerically computes the value of the Fried-Conte plasma~~~~

~~~~No. 13 MAFC076 - C. Finan, NMFECC CDC-7600 dispersion function using three different approximation methods~~~~

~~~~This code computes magnetic fields for an arbitrary set of point depending on the value of the argument of the function. conductors and calculates dl/B integrals.~~~~

~~~~No. 19 MSUPBR - Larry Miller, UCLA CDC-7600 Hybrid codes~~~~

~~~~This is an interactive code for designing magnetic fields using~~~~

Tektronix scopes. In this version, the code calculates magnetic No. 14 FLUPA - Bill Hobbs, NRL CDC-7600 fields due to point sources in systems with azimuthal symmetry. FLUPA is a particle-fluid hybrid code in which the thermal

~~~~No. 20 MFIELD - Larry Miller, UCLA CDC-7600 portion of the distribution function is represented as a fluid~~~~

~~~~MFIELD is an interactive code for designing and plotting (on and the high energy tail of the distribution is represented by~~~~

Tektronix scopes) magnetic fields for an arbitrary set of point particles. Applications with this code include studies of the current sources. evolution of ion-accoustic waves, landau damping, and nonlinear oscillations of the O'Neil analysis. r t$t

NO. 25 CUIDON77 - C. G. Tull, NHFECC COC-7600 No.5-iSPLASH- D. NielsenandJamesCreen,U.h andStanfordCDC-7600 This is a three-dimensional self-consistent guiding center SPLASHis a packageof 9 codesusedforthree-dimensional particle model coupled with a plasma equilibrium calculation. self-consistentelectromagneticparticlesimulations.Several

Some particle collisional effects are included. Applications geometriesand physicalsystemsmay be simulatedwiththiscode include neutral beam Injection into toroidal systems. includinguniformplasmas,columnarplasmas,RFplasmaheating,

~~~~mirrorplasmas,andneutralbeamInjectionintomirrorsystems.~~~~

~~~~Particle codes Reaction rate calculations~~~~

~~~~No. 16 TIBROX R. Stephen Devoto, LLL CDC-7600~~~~

TIBROX computes single particle orbits for charged particles in NO. 17 SIGMA*V- Kris Rothe,ORNL CDC-7600and IBM 360/91 magnetic fields. The fields may be computed analytically, input Thissubroutinepackagecalculatesreactionrates,,for

~~from other codes, or calculated internally using MAFC076 type 25 lightelementsfora specifiedtemperaturein therangefrom1 algorithms for a general set of point current sources. The code to 1000 keV.~~

~~~~also performs field curvature, d(lnB)/ds, and guiding center No. 65 SIGV - R. Stephen Devoto, LLL PDP-10~~~~

~~~~calculations. SIGVis a subroutinepackageforevaluatingreactionrates,,for~~~~

~~~~No. 26 ESI A. Bruce Langdon, LLL Cray-1 and CDC-7600 severalcommonlyoccurringdistributionfunctions:1).twoMaxwellianspecie*;~~~~

~~~~This is a one-dimensional self-consistent electrostatic particle 2). beam and Maxwellianspecies,3). a cold gas distributionand a Haxwelliar~~~~

~~~~code with the provision for an external magnetic field. ESJ.has species,and 4). a beam and a mirror confinedplasma. been used as a starting point in research applications as well as~~~~

~~~~an introductory and experimental code. Neutron and radiation transport codes~~~~

~~~~No. 34 RINGA - A. Mankofsky, Cornell CDC-7600~~~~

~~~~'RINGA is a self-consistent magnetostat 1c particle code (2.5 No. 5 ANISN-L - T. Wilcox and R. Herrick, LLL CDC-7600~~~~

~~~~dimensional) used to study the behavior of strong axisymmetric ANISN-L solves solves the one-dimensionalBoltzman transport~~~~

ion rings. equation for neutrons or gamma rays in slab, spherical,or cylindricalgeometries. ANISN-L was designed to solve 132 T •* No. 62 postaco - mi i M., deep-penetration problems in which angle dependent spectra are lU HaSOn' L'-LCDC-7600 Thlsi« araphicspos,processorf„r seal calculated in detail. * «••**elementcodes.It wa„ »~ *—1 No. 6 MORSE-t. - T. Wilcox and R. Ilerrick, LLL CDC-7600 d , "'"' "— boththesap4candLhe11112:^to——*- This is a Monte Carlo multigroup transport code for neutron-gamma penetratio] problemsin either a one-dimensionalsphericalgeometryor a generalized three-dimensional model using quadratic surfaces as the interface between CHAPEis agraphicsdisplayprogramforthreedime, andpolyhedralmodels „ Bafi, ^-nsional polygon adjoining material media. e*s. It was developed specifie»i !„ Procesor for finite al Pacifically as a post rinite element and finite HiFf. SAP4CandTACO). <*ifferencecodes^ ^ ! ENGINEERING CODES »- « SW- MichaelGerhard,LLLCDC-7600

Stress and thermal analysis codes No. 40 SAP4C - Steve Sackett, LLL CDC-7600 -scriptlonofthegeoraetry- ;— —hcompletethe This is a general structural analysis code for linear static and dynamicanalysisof complexelastic structures. A large variety - ™ - - 9L::;r::;ra and--- of both two dimensional and three dimensional structures may be £ilS^i£fi__engineering modeled with this code.

No. 61 TACO - Bill Mason, LLL Cray-1 and CDC-7600 No- 27 EMTP - TACO is an implicit finite element code for heat transfer WaldoMagnuson,LLLCDC-7600 Thisnetworkanalysiscodedevel analyses. It performs linear and nonlinear analyses and is used opedbytheBonnevillePower for both transient and steady state problems. ;i:::;:;:and/or——- -——«:: I3S N°. 28 MP - K p„„ No.66SCEPTRE- WaldoMagnuson.I.LLCDC-7600

"SCEPTREisageneral-purposeelectronicengineeringprogramdesignedto " U """"*»««•.«u,p„»„., ««•»«*•.Packa„vhu.„laQll,PCeC'5i<'""°»««*-I.' -isttheelectricalengineerindeterminingtheinitialconditions ••«.»l.r,<„„„,„„„ ' '"-tl°'s "»•.»«..*1„ and/ortransientresponseofelectroniccircuits. function,. """"CO°°°"*l'<•»*<>fecial No.67SPICE-2- WaldoHagnuson.LLLCDC-7600 »•»^syth,.MaIc0BondHojor ^ SPICEIsageneral-purposecircuitsimulationprogramfor pack,„e„f t.„ „„„, '* """• »«««CDC-7S0, nonlinearDC,nonlineartransient,andlinearACanalyses. *" I. a *•«•«*«-.»theautllors. mathe0a""* Circuitsmaycontainresistors,capacitors,inductors,mutual ..J' «-"*. '-•*. roUtlnMfor.„,„,;;;—"- -*-.. n•41 OENICand SOLIC- M . ., end °" •""• No.3 ETBFCTandPRBFCT- JayBoris,NRLCDC-7600 "» o-ic so,.c . ,*"Ci,Vlc' P01-'» Thesetworoutinesimplementaflux-correctedtransportalgorithm. andaredesignedforEulerian,slidingrezone,or Lagrangian finitedifferencecalculations. «•*« IC- - T.c„„ec,lu coc.76no•

No.23MOVBLES- K.Estrabrook,LLLCCDC-7600 »'• «•• -,..•, ,ptl„lMd80brootIne »l9orlth»„sl„,<,„,„-„,„, "' *"»,*"~«««lthe lece ThisisaCOMPASScoded,one-dimensionalelectrostaticparticlepushing ' """ Cor

These are vectorized CAL-coded routines for a 2:1 data No. 58 IHTPAN - Oscar Buneman, Stanford Cray-1 compression and expansion. The routines were used to pack This is a vectorized CAL-coded routine to generate an array of particle parameters in a three dimensional electromagnetic random integers. particle code. No. 63 SOLVER - II. S. AU-Yeung and A. Friedman, UCB Cray-1

~~~~SOLVER calculates roots of a user speciied function using a~~~~

~~~~simple version of Muller's method. r~ r '3?~~~~

~~~~/flft.fhemo.l-tctx.1 l_.brtxn.c-Ps £»f?(£l\)C£ L/iSL Subrou^ne Library ( tiAA/5 ABSTRACT #W %%^J NCAfc u.Wrxe Library ( LM^S *<3*7&Kr **#_ ) CDC. -760O~~~~

~~~~{from CHAL^ A»V£/e C0d-6fcoo <^C AK-* At^SOPy~~~~

-t\(\(Zu>CLL U *&flhfVt (LUltJ .fl^OrO AA>P ^^ 1W**,***- > * S 9 s 5 8 « ?» 6 $ * i *» I £: * s Ui u. d * C 3 V 2 ft VJ 2 \ \. u > S i •4 A CO a Ki t *\a. I tS U| I 2 VI •V Ar •s •t o i 5 b. *N 3 ii: u 3 •** 5 to K 4 s 8 u. 4 U hi ^ IA •v i *i ft U 0 5. ft. 9 ft* fi» At s $ £ a "*k a. 1 x VI XI ! a VI X X •* oc *!» v. CD o 0 5 2 I * Ol 3 a {J. & O U »• *? * a * VI < 00 Ml 0 w. M M 3 Or *• X *M *t 3 L 0, u. 14 <3 at Q § I 3 t 3 ♦» f* ft ft X •a u, ot •* 3 x A. S . &l «•<< no

~~~~"2 vV • •a: ^ ' tc* 0 -VJ *.;~~~~

~~~~l w \A) ol' < V- 9 2 p UJ W IA J r; 0 V. £ a- «J0 eft* £ 3 -J /* so p <. U- «$ 3 J* •*< 5 ^ >» 5 ^ ^ m 5 fc s~~~~

~~~~in •n 10 T *i 2 O~~~~

~~~~>n •Tl 0 JO ! 5 3 JO "< N » 8 w *»• **» 3 8 a f.~~~~

~~~~Jr. X! 2 i 5~~~~

~~~~i J *< 9 x o I ll «5 «l 1 ar 3 !~~~~

GO IS 39 O a 09 ^ o *> ? H7 3" I vs •Jfc ^ ~i 3 ? vi 1%. i " 1 0 *l 5 4a 5 kr o 3 *»> **» 00 C? >[ ft i* rO 0 s 2 c ' i* n * 0 H I I I r+ $ I** *9 •0 O 5 i ? I 09 8 9 sr 09 *N Cw to m n I 0 J >l 1 3

~~~~! 00 ^ i *i >. I S3 79 3 (S. j~~~~

~~~~\a§ 1a \~~~~

~~~~/*»*, r*~ y r* ^o-/»^o r*4 "•/*•« H.'**'* '.'lU~~~~

~~~~/Voo»zi ^•»X/r-»'»oyi oo/r"'-?^~~~~

££

~~~~'• :iiiM '{$?*% v:':i:y. *.*>•>*•"**•, tfrrx"*".. *"Q •~~~~

~~~~• ** *': " // ^ •••\- •"\ ' .' • ;I*V .'•I-.-'!-: I. -.."- :•',- '.* ' . -if" .~~~~

* U Ut M A C 8 00 X. B 6t •a: u (X M o >i * a o a A N 4 <* do M s la, s u. 2 K. I*. •«*» O 0 l» ** K X 5 o r» I s 0 5 » 8 •*. 3

v*t It CO u •5 s o vu * * o ft •4 •». o it s ill •ac **s> 0 /Vow Ca}* OPT/* tlAT/tsU UVAT y*<* ? . *

*l*»)tS to ifkf.

~~~~IT"~~~~

~~~~r 2>roo/»'** •/>»« eif/ e**/o»r •••; .,•#•.}.*•.,~~~~

~~~~0o€J.a ***»;* *• ***** *•«*/•'" *** S ©**r~~~~

~~~~•••• 0*» \ •*?••~~~~

~~~~>r rJiA*/v) f*%*t9jd~~~~

~~~~. '^-^ovji»;»^« -fl^8 JV*9*T '(**-*'") /9**1~~~~

~~~~•:'• ^/iViiV»^'* ''/*• *»»"*'V< (**>*0 *"•'* *b> •'•' ^?">k-"*/:...•.'•.'•, .. o<:: "' '~~~~

~~~~r^tfoa &4LL0 OX ^^'7~~~~

~~~~I .. , f&OQVd-LfOc/ re«J fiO4~»9~~1/f*<'9)0/J'V'9 *"}*P »9»-/r \%***i) ?y* t*t*9yf i. eT9p •*•"/**/***?***' we~~~~

~~~~!•!' J.VQJJ3 1Vr**t-L*Jjnd **0~~~~

~~~~/*i fr.-« -»''•«•» »A*»*"**« i^m/;, v*'**4U. "TKDJ"' j+*>.1* >"''MUy IA~~~~

* a

~~~~\ o ^*% .v.'- ^ >%, «**r.~~~~

~~~~fu«r<( ac •cufet+d *o;* ">.?*•'• -**-?ir*fm £#£**'* •".":.~~~~

~~~~»>•>* A»/*t.A*£~~~~

~~~~.-H**'*:ft*F *** )'•"*+ r~~~~

~~~~•H v"~~~~

~~~~\ £? ttiHF "^r /£3~~~~

TK* LXSP /lacklK*

~~~~• b*v«l*/>*d *+ +ka /i.r/r»i.x-i» • ^a-bi+ tag*4J awc^iWukA nrilkmam g. liroc&ssor • lVf.a*,.ct«f.ftJ Disk C6o»*Mti)j lor He LISP MeAm- r 7 M-brt~~~~

~~~~K, Ba4rmav\ * Co*vpUx FunctionaMy iw /ttcsococl<6~~~~

~~~~S. o^m TyP* l*%fi+\A\ CgAtvftHc ©/OS*) fliU *Y/U */>.S * \~~~~

•' S/i't**** 5©*f+«ook4 ( Window* S^ $+••*!•) C P/\ CCoâVf#•*»»( Pkysfcs /i**Uii+) * Nolfyfa piroe*s**VVTÔ U/ky LXSP/1 as «> Kost p«>casscH, Lompâ Vav * /AA^X-M •*• avoilati/;^

**^£|C S/5+«0KA Coat* • ; :i.= '•' • r • 9 < M

~~~~0 b "° " 40' I • T~~~~

~~~~>J 0 3: c 9 0"* O I l 9 to~~~~

*-• o-

~~~~. f *J • «l €~~~~

~~~~. * I A* c . I •c- Ik -~~~~

~~~~- » X Hi \ -lb -~~~~

* : F

~~~~•J ? 0 19 M I "7* /s* 1S7~~~~

~~~~4P Fjmf u^as Fo**ct#»v\ 0*rk is «S clock~~~~

~~~~Tc, ;(yigfopy lojic 57 >*lFtoP3 Chawi} 35 h« A//10S M4fcio>y c^iiic~~~~

~~~~//a«o CAD s^atifet BoJ«»*» ^ s*i /"f+j do* fc k j&r~~~~

~~~~A/otn~~~~

F««* x/o - smuH 4fi i.«ktt flmttril tntafft* *,/*/( Alu HtylKt* * * i dlUdU -*+V l douq Uj»r 4C . A|A>HOr|iftS Gi%o,b«s - IAt - J6/i w«*J* * Ql|o hi acco*$/cycl$ AdcV*SS Owt+i - Alu */S -KbUl tefiftytf AP M*«*o*y - *i*K - IM \jo*h bi*f Mw4»»r4 $-(4-3,-1 w»^ |n'f4H'«ai^C lop c^^.p^yd<*Vw^ ^rc>**wo,p^ ftA^Ji+4lrs - 3. bon|f8. 3*6 4*. o/o*(c ttowslti^

~~~~AdJ*J>«5 - 9l /. 1*6 lo 6,'j S-OGuftwC^tA •>- Condi 4~~~~

~~~~'^G btHi ) lo vtCj a-way c~~~~

~~~~/ST /i I'AKTICI.K 'SIMULA! MN ON THE VAP IS1~~~~

~~~~AP ftjtti-*fo»-mowcJt SgfiVwufat W. E. Druuimonct and B. N. Moore Austin Ri":i-arrh Anr.iul.il rs 1901 RuH.-ind Drive Atisl in, Texas 78/58~~~~

~~~~fWuU l>ufH.{%nM.*.*C4 infew^MtA.}~~~~

~~~~ABSTRACT~~~~

I6 eyeUs j .$"6 >**e. — -CullyOv4vftyM4.i..feCc^D***fti ArePortw,»b«givenofthestatusofhardwareandsoftwaredevelopmentonthe CHCktfct.ftX' VAP#™* VAP1S,,e8l*«M*as• HeatingpointvectorarrayprocessorofconsIderable powerandwillbeabletoexecutelargevectorprogramsathighspeedina stand-alone mode.Otherdesignfeaturesarcmultiplelargefastdatamemorieswithindependent datapathstoa pipelinedarithmeticunit.Fourofthesememoriesarevector(serial) memorieswitha maximumsizeof8 millionwords.Thereisalsoa sralar(ram) memorywitha totalof1 nflllonwords.Practicaloperationatspedsof1?mflnps f willbepossible.A softwarepackageisunderdevelopmentwhichwilleventuallymakeit possibletoprogramat theFortranlevel.Applicationsto plasmasimulationwillbe discussed. Ft>*-ho />/V*'l*'*3y »*««»Hi/tjr"Hjc"4-iwuijX a-

INTRODUCTION 2. It has multiple,fast, data memorieswith independentdata paths to AustinResearchAssociatesis develop the pipelinedarithmeticunits. Four of ing a floatingpointvectorarrayprocessor, thesememoriesare vector (serial) the VAP. Thisdevelopmentwas originally memoriesand one is a fast scalar (ram) motivatedby the need for sn inexpensive memory. The initialconfigurationwill high speedvectorprocessorfor large-scale have a total of 1.5 million words of fast Ocaq Co«yU* FFr C»*<*»*n±*): l.g n$ plasma simulations. However, the archi memory and can be easily expanded to a tecture of the VAP provides an extraordi much larger totalmemory. nary degree of flexibilityin vectorizing ^|r.**9 «*r*.\ tai*.

The principal attributes of the VAP 3. Programmingis carriedout using a subset of standard Fortran statements 91 WAc(tt*«45-fo*\&*i C>vsr^/tir) plusa fewadditionalstatementsunique V/ 1. It executeslargevector programs to vector programming. at high speed.

4. It is Inexpensive—less than For plasma simulation problems, it « percentof the cost of a Cray I or a is expected to be approximately three Star. times faster than a CDC-7600. In the followingsections,the orp.inlz.itJon of the VAP memories and data flow logic will ^escribed, togcthur standard Fortran processor .nid |>»m;i aws »>M Table 2. VAP features. «« with certain features of the VAP .software execute appioxiniaiely three limes faster and utility packages. than on a CDC-7600. *\ i~a | H tS!T I * All features of conventional AP MACHINE DESCRIPTION Tin' |>i iiiripal inndifi rat ions to the ava i1,-ibl e. AP-120B involve the addition of four The CPU of the VAP is a modified high-speed data and control paths to the * * (n x 6«K) serial memories con Floating Point Systems, Inc. AP-120B array CPU, together with the expanded hardware figurable under program control as processor. The AP-120B is a high-speed logic to facilitate the flexible use of inputs and outputs of arithmetic synchronous processor with a cycle Iline these additional data paths. The four units. of 167 nanoseconds and executes one instruc high-speed data paths are connected tion per cycle. The instruction word pro through controllers to four high-speed * Access rates of 6 million words/ vides the capability of overlapping 10 ram memories which are used as vector second to each memory giving total Independent operations in each instruction, (serial) memories. The use of ram memo data flow rate of 30 million i.e., 10 independent instructions per ries avoids the latency problems asso words/second. cycle. The floating point arithmetic ciated with CCD or bubble memories. units consist of a pipelined multiplier * Setup time for vector operations and a pipelined adder, each of which can The resulting VAP functional diagram s U microseconds. produce one result per cycle. Thus, the is shown in Figure 1 and more details of maximum execution rate for floating point the array processor section are given in operations is 12 megaflops. Figure 2. In summarizing machine features Fig. 2. Details of Array Processor CPU. The additional control logic incor and capabilities, it is convenient to porated in the VAP takes advantage of the To write and debug optimized code for distinguish between those characteristics flexibilities provided by a mixture of the AP-120B requires assembly language of the conventional AP mode of operation vector and scalar memories so that programming which is extremely tedious given in Table 1 and the extended VAP scatter/gather operations which, in the because of the overlapping of multiple mode given in Table 2. Under software Table 1. AP mode features. past, were not thought to be vectorizable, operations on each instruction. In control, the VAP can operate either as are, in fact, easily vectorizable. For addition, because of data path and memory a vector array processor or in a con plasma simulation problems, the scatter/ conflicts, even carefully written assembly * 167 nsec cycle time. ventional AP mode. gather operations of interpolation and language code does not make very efficient allocation are thus materially speeded up. use of the arithmetic pipelines. Finally, * Independent pipelined floating there is only one data memory which is adder and multiplier. SOFfWARE limited to a maximum of a million words. I tuui J I ,a:a I iuut I I mu | Even with these restrictions, however, we I -tcm s I ^JO j I «na j I I -son .I * Pipelined access to as much as 512K words of ram data memory at rates up Parallel development of software and have written plasma simulation codes on H- r Ljr__rJ hardware is being undertaken in order the AP-120B which execute at approximately to 6 million words/second. IlliK wctt r.iiuact that hardware design options may be the same speed as on the CDC-7600.. Thus, realistically evaluated as they become the AP-120B is a cost-effective processor * Sixteen 16-bit integer scratch registers with associated ALU. apparent. The earliest possible useful for many applications. Jj?lc production from the machine will also be * 2K table memory rom. obtained. VAP software items can be The VAP was designed to make use of -rJ1 ± I classified as support, e.g., compiler, the many attractive features of the assembler, linker, etc., or as utilities. AP-120B, while at the same time removing :izn Typical utilities include commonly used the memory size and data path conflict D For large vector programs, the execu vector arithmetic operations, as well as restrictions. The resulting hardware con tion speed is primarily limited by the specialized utilities for plasma simula figuration lends itself to vector process maximum rate from the data memories to the tion problems. Figure 3 is a block dia ing and the development of a vector ALU. The addition of the four fast data gram indicating the stages required to Fortran compiler removes the need for memories increases the data rate for the convert VAP Fortran source code into an assembly language coding. With this Fig. 1. VAP Machine Organization VAP to 30 million words per second as execution module. All of the support vector Fortran compiler, programming can compared to 6 million words per second in software items appear in Figure 3. All be carried out as easily as on a the AP mode. of the compiling, assembling, and linking isdoneon thehostcomputerandthe Fortran statement A rrllli.ilelementin any vector completebinaryexecutionmoduleis rapacity, and l

*'A?B *C 6Megaflops APPLICATIONS OUTLOOK 2'A" " Megaflops the is 3. 12Megaflops Although VAP beingdeveloped Fig. 3. Support software. A- B* C± D An operatingprototypeof theVAP becauseitisneededfora ratherspecific «Ithreduceddatapathwidthandskeleton «•A- (B+C)• D 12Megaflops problem,itwillbe capableof quite ProgrammingisdoneprincipallyIn serialmemoryshouldbeavailablebvthe generalapplication.Someproblemsfor termsof a subsetof standardFortran endof theyeartoperformtestsorthe Sca'.*.Iw/CaIherVectorOperations vhlchtheVAPwillbe usefularelistedin statementswitha fewmodificationsunique design.Ifa]]gocsr,.nsonablvwe|1flt Tablet,. Considerationof the2-1/2D to vectoroperations.Variablesare 1.A- B*C±M 12Megaflops thatpoint,it Isanticipatedthata plasmasimulationproblemwillIllustrate declaredaseithervectororscalarvaria rullyoperationalmachinewillbeworking 2.M.M+A*B «Megaflops someol thefeaturesof theVAP. A 2-1/20 blesandthenusedIntheusualFortran bythemiddleof nextyear. fully-electromagnetic,fully-relnlivistic syntax. v-e-c.t0Jj,.U.,..,.M.e8 c-bcarasimulationcodehasbeeninproduc Mostof thesoftwarewillbeavaila J-A- SQRT(B) !.5„eefflr,opg tionontheAPlorsometimeanditwillbe Forexample,ifB.C & D arevectors blebeforetheprototypeVAPIsopera thefirstmajorcodetobe implementedon ofthesamelength,locatedindifferent tional,andworkwillproceedonactual i'.*.".1/" 1-5Megaflops theVAP.Anestimatedperrormancccompari vectormemories,the Fortranstatement code development. son is givenin Table5. TheVAP can Inthistable.A,B,C andJ arevectors handlelargerHeldarraysandparticle A - B * C 4 D located(ndifferentvectormemory. tablesbecauseoftheserialmemory isthecontentsof theBcalar multiplieseachelementof B timesthe M\J/ memorylocationwhoseaddressis the correspondingelementof C andaddsthe correspondingelementof the vectorJ. thewritetothismemorylocationtrom the producttothecorrespondingelementof rirstor theseelementswouldnotbecom D,toproducetheresultingvectorA, * pletedberorethereadofthatsamememory whichIsstoredintheremainingvector Otherscatter/gatheroperations,e.g., forthe memory. location nextelement.Toguard No. inTable scatter/gatheroperation 2 3 againstthispossibility,theutilityhas may run more slowly f„r two reasons: beenslowed.Forplasmasimulationcodes, Scatter/gatheroperationsmakeuseor ThescalarmemoryisaccessedtwiceTor a specialutilityhas beenwrittenfor additionalsymbolsbutthesameFortran theoperationsoneachclement;andif thisoperationwhichexecutesat6 mega E.g.,If B. syntax. C A J arevectors adjacentelementsofJ havethesame flops. locatedindifferentserialmemories,the valuei.e.,ift|lesaBCmvmoryiocatlon isaddressedfortwoconsecutiveelements. "1 ICH Tablele 5.>. C..»i|>..C«.nip.. JPfjH of the t'.in.iltilitcaji.iltil i t iesof the AP mode and the VAP m.nle for a 2-1/2 D LAWRENCE UVERMORE L^, fully-electromagnetic, fully-relativistic, pariIcle push program.

~~~~AP VAP~~~~

~~~~Number of particles 32K 2II0K~~~~

~~~~Number of cells Z5K~~~~

~~~~a 1 Execution time per particle 81 /isec/particle 21 /isec/particle~~~~

~~~~Ease of programming Difficult Same as Comparing the Floating Point Systems, Inc. AP190L Standard Fortran Co Representative Scientific Computers: JJL. Some Benchmark Results~~~~

For purposes of comparison, standard Fortran programming of the CDC-7600 gives an Thomas A* Brengle execution lime of about 80 microseconds per particle and standard Fortran programming Lawrence Livermore Laboratory of the Cray I initially achieved an execution lime of approximately 12-1/2 microseconds December 10, 1979 per particle. However, more recent hand-optimized coding on the Cray I has led to a significant reduction in this push time. (Private communication - D. Fnrslund) b This result is not a measured result. However, since the processor is a synchronous - What is the AP190L? processor, it is believed to be accurate. 38 bit, 6 MHz clock, 64K Main Data Memory. Kicrocoded. 2 way Interleave memory. 2 cycle add pipe. 3 cycle multiply pipe. Driven as peripheral by DEC PDP-10 Model KI.

~~~~- AP FORTRAN~~~~

~~~~Cross-compiler, runs on DEC-10. First operable version. Rudimentary optimization (reglster-sllocatloa, code foldback).~~~~

~~~~- Benchmark~~~~

~~~~MAGIC2 (Cohen, Brengle) ID, cylindrical, magneto-inductive. Speed bound by Boris mover (about 85 floating point operations, Including SQRT).~~~~

~~~~UnwivtyolCuiluna POBox808 Uwrmon-Cablotna 94550 O TeleplK>i>ei415)447 1100O hvx 910 386 8339 UCUL LVMR (^~~~~

~~~~/tV7 LAWRENCE UVERMORE LABORATORY LAWRENCE UVERMORE LABORATORY~~~~

~~~~- Results~~~~

~~~~PDP-10 AP190L 7600 CRAT~~~~

MIPS 1 18 36 80 - Comments Theoretical HFLOPS .25 12 34-56 160-240 Effect of offloading code to API90L was to reduce Elapsed time 5640 564 375 271 utilization of PDP-10 CPU by fsctor or 50. Enlargement of Program Memory will reduce overlaying. AP FORTRAN will get additional enhancements. CPU 2559 276 99 50 New electron physics code shows problems with passing MFLOPS .14 1.3 3.5 7.0 dsts ss formal parameters instead of as COMMON data.

~~~~Realized HFLOPS .56 .11 .06 .04 Theoretical MFLOPS~~~~

~~~~Hegsbucks/MFLOP 2 '.08 1 1~~~~

~~~~NotesI 1. AP190L CPU time was inferred by counting cycles. 2. Priorities en 7600 and CRAT were set st 1.4.~~~~

UriversityotCaStomia POBox BOB Lncrmore CaKlorna 94550O Tchfjhonel4lS}447 1100 O Ttv* 910386 8339 UCLLL LVMR UnivenJlyotCaMcinia PO Box BOB Lnrfiooir Cahfcixa 94550 O TdcplKitelJI5'4J7 1100O T»v» 910 386 8339 UCLlt LVMR w M

~~~~\5 ft *cko cell CodU^ (o r tfttieM ?*rkcl~~~~

~~~~mutt c*(( i*tnpordry vttkrs h ntke force Were \*kr poMw and t,Urci*. <*£Stf>me/>4 1VAW« Dtsoripke Cell QescnpU* run f+skr> This is C/ih^tA^lf irekkJ~~~~

~~~~//»4fi«ie4r —Lift 4jr-*4**'+t* -l***sp*rU hoi <-* /> j^(0lif nMZ O, -5 ^ S ^ 5 °~&~~~~

~~~~ft lAg,** K X <\)Ck~~~~

~~~~> » ™» ^ • —~~~~

* £

~~~~o /^*%l~~~~

~~~~\) Kf t- 3 i? 5> * 8 « H 11 S~~~~

~~~~o 8 § *' i (* VI * ^ T 0 T 3 sf r» <» e-».M «1 I s* i (% 5~~~~

*.* p ftl*4 1 % ?T5 a* 9- 3 .1 r 1" 1-*-

~~~~i I n i~~~~

~~~~'« j.Hi~~~~

~~~~«M Cl~~~~

~~~~•u.~~~~

«0

~~~~Si r~~~~

~~~~/7y I7S~~~~

Somephysical problems that arc studied with particle-mesh simulation techniques Macroeetl Cotllng for Efficient Psrticle Movers on Arithmetic Processors demandefficient algorithms to allow complete investigations of the parameter spaces 15 that are encountered. Ourinterest in problems in nonlinear plasma physics leads us to formulate model problems withmillions of particles and meshes withhighspatial Robert II. Ilermsn* rcsoullion for long periods or timc(2,9|. The typical limesteploopin our electrostatic George C. Csrrelte* codes docs: i, a charge collection of the particles onto a mesh; 2,a potential solution; 3, integration or equations of motion. Thesestepsmay typically involvemultilinear Pfosma Fuwm Center interpolation on or off the mesh. Higher order interpolation may be necessary for Massachusetts Institute of Technology short-range accuracy. The particles move independently through the mesh in their Cambridge, Mass. 02139 self-consistent field and any external fields that are present. An array processor or pipelinemachine tikethe LISP Machine A-Dox[3,

The calculation precedes asMows. First, weassume that the electric potential alts been calculated andthe linked listL hasbeen prepared. We loop over the macrocclls. Thus, particlecoordinates and velocities, forces and geometrical interpolation factors •Thiswork wmBtipported, inpurl, byIhe Untied Stair* Department ofEnergy tinder Contract arecalculatedand gathered intotemproary vectors. Thisis facilitated by the macroccll Number BT785-02-10H2. '•-J '76 , 177

coordinate L. We usethe time-centered leap-frog scheme to integrate the equations of naive calculation. motion. This can beperformed inavectorized mode on theCray-1 using thelemproary vectors as well as calculating the linear momentum and kinetic energy. Next, the We have employed several of the techniquesadvocated by Dcrman[8] for efficient particle spatial coordinates have to be reset to lie in the periodic mini-incsli. To do FORTRAN coding in the particle push which accounts for the major difference in this, we need check only the macrocclls that lie on the perimeter (surface) ofthemesh. TPUSH for these two methods. We used initial coordinates randomly spread over Furthermore,weneedonlycheck the particles which havemovedinthe otward normal' the active mesh within a border. This accounts for in part for the difference in opera direction. Inlerm6 ofcells, wc need only check the 4M macrocclls onthe perimeter of tion counts presented below. Listingsin FORTRAN of this implementation are freely the mini-mesh for particles that moved outward, rather than the the N2 cells for all available from the authors. the particles. Finally, thecharge collection can also bedone using temprorary vectors Table 1 to calculateweighting factors. At the sametime, the link coordinate L is recalculated. It is easier to recalculate themacroccll coordinates rather then trying to dosome sort of garbage collection. Timings for Macrocell Coding We must caution herethat we do not advocate indescriminate use of macrocclls. Naive FORTRAN MC=4 MC=8 MC<=*16 The overhead in memory and computing may be too high a price to pay for small N per cell 1.5 25 100 400 rainimeshs and small numbers ofparticles. We bclcive thatmacrocell coding becomes TFORCE (sec) .9283 .4165 .3664 .3581 an efficient alternative when there are a significant number of particles and a large TPUSH (sec) .0320 .0095 .0094 .0094 numberof macrocclls. The efficiency of this scheme is optimized when there are an TRIIO (sec) .5379 .2747 .2746 .2742 even number of particles per macroccll. MFIops 3.827 10.743 11.529 11.678 InTable 1wepresent some timing studies ontheCray-1 using a two-dimensional . TOTAL (sec) 1.498 0.701 0.650 0.642 mesh with N=256 and M= 64, 32, 16. We arc using a macrocell thatcontains 4a, RATIO (time) 1.0 .468 .434 .429 8a and 16' minimesh cells. We call the linear dimension of macroccll measured in RATIO (Mflops) 1.0 2.8 3.0 3.1 minicclls MC. Wc present results for 102400 particles. The times seem to be linear withparticle numebr above 10240 particles. The critical issue in picking themacroccll We adovcate the use of macrocell coding especially in the following when large size MC is to ensure that the macroccll contains a substantial number or particles. particle numbers of high spatial resolution demand as much vectorization possible. We have computed the time (TFORCE) for bi-linear force interpolation to update The above table shows that even in naive FORTRAN, the extra coding necessary to the velocities; the lime (TPUSH) to update positions withthe new velocities in the perform amcroccll management iscompensated by increased vectorization asevidenced leap-frog lime-ccntcrcd integration algorithm and toreset theposition coordinates on by the Mflops. We would Euggest that this codebe writtenseriously in assembler for a spatially periodic mesh; thetime (TRIIO) to collect thenew charges with bilinear charge assignment. These operations include certain simulation overheads - zeroing production u6c. certain arrays, moving coordinates, etc. and in the the case ofthe macroccll methods, This macroccll method can be extended with the following considerations: recomputing themacroccll coordinates. To compare Ihc utility of macroccll coding, we also computed Mffops (Million floating point operations per second) bynoting the 1. The PPPM method can be used for short-rangeforcecorrection, which is what number N-f- and N# of multiplications and additions in these cases, excluding the originallymotivated our IhoughU this way.Thistechnique canbevectorized to provide arithmetic necessary to reset theperiodic mesh coordinates. The result shows that a large accuracy for simulations running many time steps. The PPPM method consists factor oftwo in raw cycle lime results over anaive method for coding the lime step. It of a Particle-Mesh calculated as wc have outlined above followed by a Particle-Particle should be pointed out that the naive method is not too naive in that it uses the Cray pair-wisecorrection forthe 6hort range force. The 6izeofthe macroccll then becomesa vector commands for the push and for theremap totheperiodic mesh. It is the non- natural distance over which to do this correction. One processes the current macroccll vector character of thecharge collection and force inelcrpolation thatslows down this applying a 6hort range correction to the particles in the current macrocell and the f"

~~~~m w~~~~

neighboring macrocclls. Further optimization is possible if the particles are partly length 04 in our example. ordered within a macroccll. 7. A schemeof buffering particles in macrocclls in main memcory for an array 2. Non-Cartesian coordinate systems can be treated with this technique. The processor can be envisioned. A large bufferof particles is read from disk, gathered macrocctln, then, arc more general than rectangles. It is still no more convenient to into macrocclls, processed, dispersed, andthen written out. This procedure could be calculategeometrical factors, but the opportunity to vectorize exists. overlapped with reading and writing other buffers. We believe that the buffer sizes need to be catfullymatched withthe mactocetl sizes and mintmesh sizes for efficiency. 3. Non-periodic boundary conditions ontheparticles can also betreated. Suppose we consider a bounded problem likea self-gravitating, isolated system or a bounded 8. A final observation about coding styles andpractices is appropriate here. We plasma. The particles'that wander offtheactive mesh need to be processed separlcly. have used a FORTRAN implementation of the macrocell method to illustrate the In the gravitating problem, a particle might move ina two-body approximalion|7,8). technique for this discussion. Production codes would undoubtedlywant this method *If a "wander-list" is created for these particles, they can be identified and processed •in assembler or microcode. There is a significant problem in balancing the gains in quite speedily. ' formulating analgorithm and theeaseof manipulating adata structure inaflexible high level langattge against the real-lime problem of gcllling the maximal raw computing 4. It is possible to extend theabstract structure of macrocclls assuggested in (3) power from a machine. Our observation is that ideas of data abstraction (message- to containmoreinformation. The particles within a macroccll arcallindependent or passing semantics,generalized data structures, data flow analysis) areconsistentwith any other macroccll, and grouped together in space. It may be convenient to have developing efficient vector codingbecausethey canuse informationabout the vector the electric field or potential grouped intothemacrocell configuration. 11 may also be structure of the data. Out, unlike other models of vectorization (DO loops, expan desirable to dothe charge collection In macroccll configuration as well. It needs only sions, temporary variablesor vectors), propcrply chosen dataabstractions can be used to guard the boundaries of themacrocclls and fix them uplater. This maybe highly throughout the formulation of a problem (program) to recognize suchactivitiesasdisk desirable if many independent parallel processors are present (eg. Illiac IV, el. (10]). buffering, multi-processing environments (each assigned to a macrocell), or parallel II may also be desirable to tag theidentity of particles by putting them in special machine processing (data movementover a network, another machinedoing pre- or macrocclls, asmight happen if it isnecessary to identify particles in different regions postprocessing). Any data structure that the computational physicist wishes to im of phase space that originally started close together. This is particular important in poseson the calculation should becompatible withthetechnique for computing as well measure phase space correlations in palsma turbulence or6tochasticily|2,9]. as the machinohe computeswith.We arefinding themacrocell coding is amenable for these reasons. In an two-dimensional electrostatic problem, say,the calculation for electric field proceeds bycomputing amesh ofpotential values and using finite difference techniques In conclusion, wc advocate the useof the macrocell asan efficient algoirthm Tor to caluculale Ihc force. This ismore efficient instorage than calculating extra meshs particle mesh simulations on array processors. Coding in naive FORTRAN produces for the electric field. A space centered fintic difference formula would cause 10 fps significant improvement in usage of the available resources..The macroccll concept, a (floating point operations) per particle for interpolation whereas it would cost 4/V* fps hierarchicalform of"divide- and - conquor", can beapplied tootherdifficult problems to calculate theelectric at all the points. Therefore, when the number of particles is that are studied on pipeline machines much larger than 0.25/V', it is desirable to prccalcttlate the electric field in a macroccll anddseit for interpolation rather than differencing for each particle. Thecostin extra RnrcncNccs storage in just2A/C*. A similar result obtains inthree dimesnions. Namely, when the number of particles ismuch larger than .11N*, theelectric field should beprccalculalcd 1. J.W Hntlwood, rt.W. Ifockney, and D.N. Lnwrence, "P3MJDP- The Three Dimensional In a macroccll at a costof JA/C memory locations! Periodic Pnrlirle-Parltrle - Porticlt-MethProgram," Internalmemo,Reading- Unlrtrtilj, Department of Computer Science(I97S). 8. Temporary vectors inmacrocclls can also beintroduced for currents, magnetic 2. D.J. Tetrrautt, il.tl. Herman, and T.II. Dupree, "Computer Simulation of Nonlinear fields, etc. for more general problems. We have typically used temporary vectors of Pha«e Space Structure," U.A.F.S. 14, (1979), 9«.

~~~~~J Ifo ttl~~~~

3. J.T. llolloway, "The LISP Machine ADOX,"inleraalmemo(Mov. 1979). 4. R.II. Dcrnion and J.L. Kulp, "A Pcrcpectiveon Environment*forComputational Physic* \{, 3P S/mJUfai, 4 T&M ~ 1Htm**ks — The CPM — An Integrated Pergonal Computer System," MafMchucclU Institute of TccbnologyPJofma Fusion Center Computational Phytic* Working Paper I (1979). 5. O. Duneman, "VECFFT," in LIU/IIS, MlFBCC(1979). 6. R.1T. Herman, "Vcctoriiing a Id Particle Push," Buffer3, (1979),12. «. 7. It. A. Jame* and J. A. Sellwood, "Computer Simulation of Galaxies," M.N.R.A.S. 182, (1978), 331. 8. R.W. Ilockney and D.R.K. Drownrigg, "Three Dimensional Computer Simulation of Galaxies," M.N.ILA.S 107,(1974), 331. 9. R.II. Herman, "Criteria for Transition to Slochasticity," D.A.P.S. 24„ (1979). 942,. 10. C.R. Jc**hopcand J.A.I. Craigie, "Some Principle* of PorallelUm in Particle and Me*h Modelling," internal memo,DepartmentofComputerScience, readingUniversity (1979). 'PW«eft6» pAY*kr» c c

~~~~J&*- in 1HENEEDFORNEW3D PARTICLEMODEL FWweM S £~~~~

~~~~A) REALISTICPLASMADEVICE(e.g.,TOKAHAK,0-MACHINE) v**8« «4T •»{ © AXIALLENGTH» CROSSSECTIONALLENGTH~~~~

~~~~FORLOHFREQUENCYHICROIHSTABILITIES<«*<£V4r,ft~~~~

~~~~k||«** /rw*v kJLriJ+***••~~~~

~~~~AXIAL LENGTH « S" %<9~~~~

~~~~V'i **oln~~~~

~~~~O-NACMNEPUtSW VOLUME~~~~

~~~~»« 1^-2 css ^ la j£ 100-~ 200 D IgSO** 100 ca >rsT M* _/*/-r / vT**#io,»,~~~~

~~~~FORTOKAMAKvT>vQ '~~~~

~~~~B) FEATUREOF NEW3D PARTICLEMODEL~~~~

~~~~KEEPONLYFEWLONG-WAVELENSTHHODESINTHEAXIALDIRECTION~~~~

~~~~MAKEEIGENFUKCTIONEXPANSIONINTHEAXIALDIRECTIONANDMULTIPLE~~~~

~~~~EXPANSIONONTHE2DSPATIALGRIDINTHECROSSSECTIONFOR~~~~

~~~~SOLVINGMAXWELL'SEQUATIONS~~~~

~~~~_L in Applied^ 4 V> M"^ t)*—tti*^*%-—rfr/~~~~

~~~~GUJa**v> •** f*it*/ -Vf 6r. C. drift J^to>*£*>*"*H.~~~~

~~~~-f»f. 2tv. Utt~~~~

~~~~X^, jLiKtmiYi Are «£**«*. c c~~~~

~~~~lib IV~~~~

~~~~i.) JevJjJl*. IfJilL*** «~Jt Them«J G*.Jm<&«.~~~~

~~~~i £ i~~~~

~~~~2j 'Bootstrap Current~~~~

~~~~EZ_.L 2T~~~~

~~~~5 "* * a £ t i 2 « %~~~~

~~~~M(\ (mjuui,j T& '*** f/*j -Wpt/3 *ta*tWJ?»d,s -»»<*WtA~~~~

~~~~fcfftoitint tfect~~~~

~~~~iO"~~~~

~~~~400 noo 1200 °0~ 0.5 1.0 "w''6" f/2 t 3^72 21 01POLOIOAL ANGLE) *.. jj.«r;w™***r ^ *»**« -•* .W^l**ail^^k^B^*>*BMBVBslBiB^»a>a^B*aV»«aW""~"*sw*u^j,r—'^ *,^poUitLUpo(o«^~~~~

~~~~0» i • r» »- M~~~~

'I .?§ 9 »*♦" t Or* • «H l» I MO irfutci •» M« t • •«ej »* ge a in *

~~~~5' «.(«•••) •A » o • ^ r»3 . A i i i i i n i~~~~

~~~~O 0)~~~~

* n 7T • "9 ft

~~~~• O. • f* n 3 S o a~~~~

t « »» *- «< w» o O •• **

~~~~I • • e o« n w s— i t • o f~~~~

~~~~- -e~~~~

~~~~fe—~~~~

~~~~h -J HX 1 I® J._~ _. y £(cdrm :pp" 1 ii »»{»•»,~~~~

~~~~'I i " r i ! I 1 1 '~~~~