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Signatureredacted 0/Department of Physics Msy 2, 1980 EXPERIMENTAL STUDY OF TOKAMAK PLASMAS WITH EXTERNAL ROTATIONAL TRANSFORM OF THE MAGNETIC FIELD by ALAN CHARLES JANOS Sc.M., Brown University (1976) B.S., Cornell University (1973) SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 1980 © Massachusetts Institute of Technology 1980 Signature of Author Signatureredacted 0/Department of Physics MSy 2, 1980 Certified by Signature redacted Bruno Coppi Th'esis Supervisor Accepted by -Signature redacted George Koster Chairman, Department Committee ARCHIVES OF TECHNLGY JUN .0 198E UBRANiES -2- EXPERIMENTAL STUDY OF TOKAMAK PLASMAS WITH EXTERNAL ROTATIONAL TRANSFORM OF THE MAGNETIC FIELD by ALAN CHARLES JANOS Submitted to the Department of Physics on May 2, 1980 in partial fulfillment of the requirements for the Degree of Doctor of Philosophy ABSTRACT This research was an experimental, numerical, and analytical study of the equilibrium and stability of a tokamak plasma with an externally applied rotational transform of the magnetic field produced by semi- stellarator field coils. The research utilized the Rector tokamak device, capable of 15-40 kA plasma current discharges at a toroidal field of BTO= 3 .5 kG at the geometric center of the vacuum chamber (Ro=57.5 cm, z=0). Typical plasma density and temperature were ne=2-10 1 3 cm- 3 and Teo=80-2 00 eV, respectively. Both positive and negative (elongated plasma) values of the vertical field index of curvature n= -(Ro/Bvo) 3Bv/3R were tested. Dependence of stability on amount of externally provided rotational transform was studied and compared to normal low safety factor q discharges in Rector. Improvement in gross positional stability was manifested in a tripling of the discharge time (limited only by ohmic heating volt-sec flux capacity), flat up-down position even for vertically unstable indices of curvature of the applied vertical field, limited in-out position excursions, and insensitivity to both device and inten- tionally applied error fields. The effect of the external rotational transform on the MHD activity in the tokamak plasma with and without semi- stellarator field was also studied. The observed magnetic fluctuations below 20 kHz in the high current, low q(a) (<2) phase of the discharges accompanied sawtooth horizontal plasma motion, changes in plasma minor radius, and increases in plasma current (minor disruptions). The semi- stellarator field did not inhibit or enhance MHD activity for the rela- tively low values of externally provided rotational transform XoO0.1. Results from numerical computer codes for field line trajectories and magnetic surfaces supported existence of closed magnetic surfaces, and provided added basis for interpretation of experimental observations. Extreme ultraviolet (XUV) photoelectric effect collimated vacuum diode detector arrays were used to monitor equilibrium and stability of plasma discharges. Their design, construction, and operation are described. Thesis Supervisor: Dr. Bruno Coppi Title: Professor of Physics -3- ACKNOWLEDGMENTS I wish to thank the following people: PROFESSOR BRUNO COPPI, for supporting this project, providing use- ful suggestions, and supervising my thesis; PROFESSOR RONALD R. PARKER, for his support, his valuable help, and his suggestions to develop the XUV detectors; DR. HIRO IKEZI, for his interest in this project, his significant scientific contributions to the semi-stellarator field research, and his encouragement; CHRISTIANE LUDESCHER, for providing useful computer codes; DR. FRAN OIS MARTIN, for his assistance at the start of my research work; and my wife and parents for their understanding and unfailing moral support. -4- TABLE OF CONTENTS ABSTRACT ........... ..... ....... 2 ACKNOWLEDGMENTS ....... ............ 3 LIST OF TABLES ...... .. ....... .... 8 LIST OF FIGURES ..... .. ............ 9 I. INTRODUCTION ... .. ....... ..... 1 5 1.1 PURPOSE ...... ....... ..... 16 1.2 SUPPORTIVE EVIDENCE ............ 8 1.3 EXPERIMENTAL DEVICE ............ 19 1.4 RELATION TO STELLARATOR . .... .... 23 1.5 PRINCIPAL RESULTS ... ....... .. 25 1.6 FORMAT ....... ........ 8 28 II. HELICAL FIELDS . .... ....... 29 2.1 TRANSFORM CONTROL . ........... 29 2.2 COMPARISON OF TOKAMAKS AND STELLARATORS . .. .. 31 2.3 EFFECTS OF STELLARATOR FIELDS ON TOKAMAK PLASMAS AND TOROIDAL CURRENTS IN STELLARATO R FIELDS .. 33 2.3.1 TOTAL TRANSFORM . ... .... .... 33 2.3.2 EXPERIMENTS........ ........ 37 III. EQUILIBRIUM THEORY .... .. ........ 38 IV. POSITIONAL STABILITY .. ..... 42 4.1 FORCE BALANCE .... .... .. ... .. 42 4.2 DISRUPTIVE INSTABILITY DESCRIPTION . .. .. 45 4.3 STABILITY CONTROL .. ... .. ... .. 46 4.4 HELICAL FIELD EFFECTS .. .. .. ... .. 49 4.4.1 STELLARATOR .. .. ... ... .. 49 4.4.2 SEMI-STELLARATOR .. ..... .. 51 -5- V. RECTOR TOKAMAK DEVICE DESCRIPT ION . .. .. .. 55 5.1 VACUUM SYSTEM.. .. 5 5 5.2 POWER AND COIL SYSTEMS . .. .. .. 5 6 5.2.1 VERTICAL FIELD . .. .. 5 6 5.2.2 OHMIC HEATING . .. 5 9 5.2.3 MAIN FIELD . .. 6 2 5.2.4 ACTIVE FEEDBACK . 6 3 5.3 DIAGNOSTICS. .. 6 4 VI. SEMI-STELLARATOR FIELD SYSTEM . .. .. 7 0 6.1 COIL . .. .. .. .. -- .. - . 70 6.2 PHASE OF HELICAL FIELD AND DIRECTION OF TRANSFORM . 78 6.3 CIRCUITS .. .. .. .. .. .. 80 6.3.1 POWER . .. .. .. .. .. 80 6.3.2 SWITCHING .. .. .. .. .. 84 6.3.3 MONITOR . .. .. .. .. 88 VII. SEMI-STELLARATOR FIELD DESCRIPTION - ANALYTIC 89 7.1 MAGNETIC FIELD FOR CURRENT ARRAY MODEL .. .. 92 7.2 WORKING EXPRESSION FOR MAGNETIC POTENTIAL .. 98 7.3 CURRENT ARRAY IN UNIFORM FIELD .. .. .. .100 7.4 APPLICATION TO TOROIDAL DEVICE - NUMBERS FOR RECTOR .104 7.5 FIELD LINE ORBIT EQUATIONS . ... .. ... .108 7.6 SOLUTIONS OF FIELD LINE ORBIT EQUATIONS .. .. .111 7.6.1 ZEROTH ORDER . .. .. .. .. 7.6.2 FIRST ORDER .. ... .. .. ... .112 7.6.3 SECOND ORDER ... ... ... ... .124 7.7 PROPERTIES OF SOLUTIONS ... ... ... .131 7.7.1 TRANSLATIONAL TRANSFORM ... ... .131 -6- 7.7.2 SHEAR . .. .. .. .. .. .132 7.7.3 AVERAGED VERTICAL FIELD . .. .. .. .132 7.7.4 INDEX OF CURVATURE .. .. .. .139 7.8 SEPARATRICES . .. .. .140 7.8.1 WITHOUT PLASMA CURRENT . .. .. 140 7.8.2 WITH PLASMA CURRENT . .. 146 VIII. FIELD LINES IN PLANAR GEOMETRY - NUMERICAL . .150 8.1 UNIFORM MAIN FIELD . .. .152 8.2 MAIN FIELD WITH -- DEPENDENCE . .. 157 R 8.3 HELICAL CONDUCTORS OUTSIDE TOROIDAL FIELD MAGNET .160 IX. FLUX SURFACES IN TOROIDAL GEOMETRY - NUMERICAL. .163 9.1 METHOD . .164 9.2 RECTOR DEVICE RESULTS . 166 9.2.1 VACUUM FIELDS . .. .. .166 9.2.2 TOKMAK OPERATION . .179 9.2.3 SEMI-STELLARATOR OPERATION . .179 9.2.4 REVERSAL OF EXTERNALLY APPLIED TRANSFORM. .180 9.3 IKEZI'S DEVICE RESULTS . .182 9.3.1 VACUUM FIELDS . .182 9.3.2 TOKAMAK OPERATION . .. .182 9.3.3 SEMI-STELLARATOR OPERATION . .. .182 9.4 MAGNETIC SURFACE DESTRUCTION AND RESONANCES . .195 X. TOKAMAK DISCHARGES IN RECTOR . .197 10.1 EVOLUTION OF DISCHARGES . .. .197 10.2 SCALING OF PARAMETERS . .205 10.3 DECAY INDEX OF VERTICAL FIELD . .. .. .209 -7- XI. SEMI-STELLARATOR OPERATION . ... .210 11.1 DISCHARGES ... .. .. ... 210 11.2 APPLIED VERTICAL FIELD .. .... 214 11.2.1 INDEX ..... 214 11.2.2 MAGNITUDE .... .. 215 11.2.2.1 OPTIMUM By .. .215 11.2.2.2 LOW By . 2 . 216 11.2.2.3 ZERO By . 225 11.2.2.4 HIGH By . ... .225 11.3 EFFECTIVE SEMI-STELLARATOR VERTICAL FIELD . ... 228 11.3.1 TRANSFORM ... ...... ... .228 11.3.2 MAGNITUDE . ........ ... .228 11.3.3 DEPENDENCE ON MAJOR RADIUS . ..... 231 11.3.4 DEPENDENCE ON HELICAL CURRENT . .. 233 11.4 PLASMA SHAPE ........... ... .237 11.5 LOW TOROIDAL FIELD OPERATION .... 3238 11.6 REVERSAL OF EXTERNALLY APPLIED TRANSFORM . ... 242 11.7 DISRUPTIVE BEHAVIOR ........ 4245 11.8 SAWTOOTH ACTIVITY ......... ..... 247 XII. CONCLUSIONS- AND RECOMMENDATIONS ... .. .255 APPENDICES ................. .... .257 A. XUV DETECTORS ............ 5257 B . USEFUL MATHEMATICAL EXPRESSIONS ..... .275 REFERENCES ................. 7276 BIOGRAPHICAL NOTE ............. .... .282 i -8- LIST OF TABLES 1. OHMICALLY HEATED STELLARATORS . 35 2. LOCATION (R,z) OF VERTICAL FIELD CONDUCTORS ON RECTOR . 60 3. CAPACITOR BANKS FOR SEMI-STELLARATOR FIELD COIL POWER SUPPLY. 80 -9- LIST OF FIGURES 1 Rector device with semi-stellarator windings - side view 20 2 Rector device with semi-stellarator windings - top of device showing yoke for semi-stellarator coils . 21 3 Co6rdinates R, z, p, ' . .. .. 40 4 Cross section of Rector vacuum chamber . .. .. 57 5 Vertical field for decay index n=+ 0.7 .. .. 58 6 Schematic of semi-stellarator coils on Rector . .. 71 7 Schematic of an m=3 semi-stellarator field coil system using one continuous length of conductor A. With IH= 4 iH amp-turns along a vertical leg, where iH is the current from the power supply to the semi- stellarator coil (this system was used) . .. 72 B. With IH= 2 iH amp-turns along a vertical leg (this alternate system was not used) . .. 73 8 Diagram of winding law for semi-stellarator coil for 75 A. Torus with small hole B. Torus with large hole, as in Rector 9 Semi-stellarator field coil power circuit and monitor circ uit. 81 10 Power supply HELI I (only one leg of full wave bridge rectifier is shown) - see Fig. 9 . .. 82 11 Trigger circuit HELI I - see Fig. 9 .. .. 85 12 Trigger circuit HELI II - see Fig. 9 . .. 86 13 Voltage comparator circuit HELI II - see Fig. 9 . 87 14 Current array • • -- - .. • . • - - .. .. 93 15 Magnetic potential function O(x,y=0+) at boundary for current array described in Fig. 14 . .. 97 16 Current array in uniform field . .. .. 101 17 Orientation of co6rdinate system and main field with respect to helical windings on toroidal
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