| Preface |
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xvii | |
| 1 Introduction |
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1 | (24) |
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1.1 Brief description of spheromaks |
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1 | (4) |
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1.2 History and time-line |
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5 | (20) |
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1.2.1 Pre-1970: Antecedents of the spheromak |
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6 | (5) |
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1.2.2 Advances in theory: Taylor relaxation and development of the theoretical model for the spheromak |
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11 | (1) |
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1.2.3 The 1980's: The spheromak investigated as a fusion confinement scheme |
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12 | (2) |
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1.2.4 The 1990's: Search for other applications and renaissance in confinement efforts |
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14 | (1) |
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1.2.5 2000 to time of writing: SSPX, HIT-SI, General Fusion, investigation of dynamics, inclusion of Hall term |
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15 | (10) |
| 2 Basic Concepts |
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25 | (30) |
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2.1 The different levels of description of a plasma |
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25 | (4) |
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2.1.1 Follow every particle |
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25 | (1) |
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26 | (1) |
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2.1.3 Two-fluid description |
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26 | (1) |
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27 | (1) |
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2.1.5 Dynamical and β assumptions in MHD |
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28 | (1) |
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2.2 Scaling between lab and space plasmas |
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29 | (4) |
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2.3 Vacuum (potential) magnetic fields |
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33 | (2) |
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2.4 Poloidal and toroidal fields |
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35 | (2) |
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2.5 Magnetic stress tensor |
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37 | (3) |
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40 | (1) |
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2.7 Magnetic flux and symmetry |
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41 | (1) |
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41 | (1) |
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2.9 Poloidal flux and particle confinement |
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42 | (1) |
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2.10 Relation between field, field lines, and flux |
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43 | (1) |
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44 | (5) |
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2.12 The plasma as a magnetic flux conserver |
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49 | (1) |
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2.13 The condition for frozen-in flux |
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50 | (2) |
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2.14 Tendency of the plasma to maximize its inductance |
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52 | (1) |
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52 | (3) |
| 3 Magnetic Helicity |
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55 | (40) |
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3.1 The issue of analyticity in Gauss's and Stokes's theorems |
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56 | (2) |
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3.2 Definition of magnetic helicity |
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58 | (2) |
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3.3 Helicity, safety factor, and twist of an isolated flux tube |
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60 | (3) |
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63 | (1) |
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64 | (4) |
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3.6 Simply connected volumes v. doubly connected volumes |
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68 | (2) |
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3.6.1 Relative helicity suitable for doubly connected volumes |
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68 | (2) |
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3.7 Helicity conservation equation |
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70 | (9) |
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3.7.1 Fixed volume, simply connected region, open field lines |
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71 | (5) |
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3.7.2 Fixed volume - Doubly connected region |
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76 | (1) |
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3.7.3 Comparison between helicity injection into simply and double connected volumes |
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77 | (1) |
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3.7.4 Time-dependent volume with no open field lines |
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77 | (1) |
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3.7.5 Comparison of helicity conservation to energy conservation |
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78 | (1) |
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3.8 Single species helicity and association with whistler wave physics |
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79 | (3) |
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3.9 Magnetic reconnection |
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82 | (1) |
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3.10 Geometric interpretation of magnetic helicity |
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82 | (7) |
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3.10.1 Twist, linking, and crossing |
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84 | (3) |
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87 | (2) |
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3.11 Magnetic reconnection and helicity conservation |
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89 | (3) |
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3.12 Reconnection and dissipation |
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92 | (3) |
| 4 Relaxation of an Isolated Configuration to the Taylor State |
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95 | (20) |
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95 | (2) |
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4.2 Helicity decay rate v. magnetic energy decay rate |
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97 | (1) |
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4.3 Derivation of the isolated Taylor state |
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97 | (3) |
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4.4 Relationship between helicity, energy, eigenvalue |
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100 | (2) |
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4.4.1 Equality of poloidal and toroidal field energies in an isolated axisymmetric spheromak |
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101 | (1) |
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4.5 Cylindrical force-free states |
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102 | (2) |
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4.6 Ohmic Decay of an Isolated Taylor State |
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104 | (3) |
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4.7 Comparison of minimum energy states in a long cylinder |
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107 | (2) |
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4.8 Spheromaks in spherical geometry |
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109 | (6) |
| 5 Relaxation in Driven Configurations |
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115 | (8) |
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5.1 Taylor relaxation in systems with open field lines |
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116 | (3) |
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5.1.1 Relation between energy and helicity for system with open field lines |
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119 | (1) |
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119 | (2) |
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5.2.1 Bounding surface is an equipotential |
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120 | (1) |
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5.2.2 Bounding surface is not an equipotential |
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120 | (1) |
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5.3 Impedance of the driven force-free configuration |
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121 | (2) |
| 6 The MHD Energy Principle, Helicity, and Taylor States |
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123 | (20) |
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6.1 Derivation of the MHD Energy Principle |
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123 | (11) |
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6.2 Relationship of the energy principle to Taylor states |
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134 | (1) |
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135 | (8) |
| 7 Survey of Spheromak Formation Schemes |
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143 | (18) |
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7.1 Magnetized coaxial gun |
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146 | (4) |
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7.2 Non-axisymmetric gun method |
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150 | (1) |
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150 | (5) |
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155 | (2) |
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7.5 Steady inductive injection using quadrature-phased oscillating inductive injectors |
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157 | (4) |
| 8 Classification of Regimes: An Imperfect Analogy to Thermodynamics |
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161 | (8) |
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8.1 Room full of springs analogy |
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161 | (1) |
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8.2 Analogy to thermodynamics |
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161 | (1) |
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8.3 Classification of thermodynamic problems |
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162 | (3) |
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8.4 Analogy between lambda and temperature |
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165 | (2) |
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8.5 Strong and weak coupling |
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167 | (1) |
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8.6 Overview of next five chapters |
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167 | (2) |
| 9 Analysis of Isolated Cylindrical Spheromaks |
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169 | (24) |
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9.1 Flux, current, magnetic field, helicity and energy |
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169 | (5) |
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9.2 Experimental measurements |
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174 | (2) |
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9.3 Safety factor for a zero-β spheromak |
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176 | (6) |
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9.3.1 Evaluation of q profile |
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178 | (2) |
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9.3.2 Effect of flux conserver shape on qwall |
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180 | (2) |
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182 | (11) |
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9.4.1 Derivation of the Grad-Shafranov Equation |
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183 | (1) |
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9.4.2 Finite-β spheromak solution |
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184 | (3) |
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187 | (1) |
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188 | (5) |
| 10 The Role of the Wall |
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193 | (14) |
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194 | (1) |
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194 | (5) |
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199 | (8) |
| 11 Analysis of Driven Spheromaks: Strong Coupling |
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207 | (30) |
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11.1 Force-free equilibria with open field lines |
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208 | (7) |
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215 | (7) |
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11.3 Safety factor variation with lambda |
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222 | (1) |
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222 | (1) |
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223 | (3) |
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226 | (3) |
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229 | (1) |
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11.8 Gun impedance and load line |
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230 | (7) |
| 12 Helicity Flow and Dynamos |
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237 | (30) |
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12.1 Downhill flow of helicity |
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237 | (4) |
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12.1.1 An isolated spheromak acting as a helicity source |
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239 | (2) |
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12.2 Dynamos and relaxation mechanisms |
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241 | (9) |
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12.2.1 Paradox of the driven axisymmetric spheromak |
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241 | (4) |
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12.2.2 Physical constraints on coefficients of Fourier expansions in cylindrical coordinates |
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245 | (5) |
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12.3 Observations of flux conversion dynamo behavior |
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250 | (9) |
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12.4 Lagrangian Description Showing no Dynamo in Plasma Frame |
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259 | (4) |
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12.5 Deviation from the Taylor state |
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263 | (4) |
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12.5.1 Force-free limit of Grad-Shafranov equation |
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263 | (4) |
| 13 Confinement and Transport in Spheromaks |
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267 | (22) |
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267 | (1) |
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13.2 Confinement on flux surfaces |
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267 | (2) |
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269 | (1) |
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13.4 Survey of transport mechanisms |
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270 | (9) |
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13.4.1 Diffusive processes |
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270 | (4) |
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13.4.2 Non-diffusive processes |
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274 | (1) |
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13.4.3 Magnetic energy and magnetic helicity |
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275 | (2) |
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13.4.4 Relationship between magnetic energy and thermal energy decay times |
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277 | (1) |
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13.4.5 Dissipation in a single flux tube |
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278 | (1) |
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13.5 Experiments on transport in spheromaks |
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279 | (6) |
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279 | (2) |
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13.5.2 S-1 particle confinement measurement |
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281 | (1) |
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281 | (1) |
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13.5.4 CTCC-I gettering experiments |
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281 | (2) |
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13.5.5 The CTX gettered, solid flux conserver experiment |
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283 | (1) |
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13.5.6 Evidence for good confinement: hard X-rays on CTX |
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283 | (1) |
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13.5.7 SSPX Conclusions: Conflict between good confinement and helicity injection |
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284 | (1) |
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13.6 Anomalous ion heating |
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285 | (4) |
| 14 Some Important Practical Issues |
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289 | (20) |
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14.1 Breakdown and Paschen curves |
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289 | (6) |
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295 | (2) |
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14.3 Wall desorption and contamination |
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297 | (3) |
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14.4 Impurity line radiation |
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300 | (2) |
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14.5 Refractory electrode materials |
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302 | (1) |
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14.6 Skin effect and the wall as a flux conserver |
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303 | (1) |
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304 | (1) |
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305 | (1) |
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14.9 Noise radiation from pulsed power supplies |
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305 | (1) |
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306 | (3) |
| 15 Basic Diagnostics for Spheromaks |
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309 | (22) |
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15.1 Magnetic field and electric current measurement |
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309 | (5) |
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15.2 Equilibrium reconstruction using measurements at the wall |
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314 | (1) |
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15.3 Voltage measurements |
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314 | (1) |
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315 | (7) |
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15.4.1 Langmuir probe (triple probe) |
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315 | (3) |
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318 | (4) |
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15.5 Ion temperature measurement |
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322 | (3) |
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15.5.1 Impurity Doppler shift |
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322 | (2) |
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15.5.2 Neutral particle analysis |
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324 | (1) |
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15.6 Electron temperature measurement |
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325 | (3) |
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325 | (1) |
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15.6.2 Thomson scattering |
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325 | (3) |
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15.7 Impurity radiation measurements |
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328 | (3) |
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15.7.1 Spectroscopic identification of ionization states |
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328 | (1) |
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328 | (3) |
| 16 Applications of Spheromaks |
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331 | (24) |
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16.1 The spheromak as a fusion reactor |
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331 | (8) |
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16.2 Accelerated spheromaks |
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339 | (4) |
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16.3 Magnetized Target Fusion |
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343 | (2) |
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16.4 Tokamak Fuel injection |
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345 | (2) |
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16.5 Helicity injection current drive in tokamaks |
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347 | (3) |
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16.6 Colliding spheromaks to investigate magnetic reconnection |
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350 | (2) |
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16.7 Proposed additional spheromak applications |
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352 | (3) |
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16.7.1 Pulsed high power X-ray radiation sources |
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352 | (1) |
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353 | (2) |
| 17 Initial dynamics leading to relaxation: MHD jets |
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355 | (22) |
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17.1 MHD-driven collimated jet |
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361 | (8) |
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17.1.1 Bernoulli relation: overview via garden hose analog |
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361 | (1) |
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17.1.2 Flux function derivation |
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362 | (7) |
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369 | (6) |
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17.2.1 Jet direction insensitive to current direction |
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375 | (1) |
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17.3 Association with helicity injection |
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375 | (2) |
| 18 Dynamics associated with relaxation: Kinks, Rayleigh-Taylor, Hard X-rays |
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377 | (20) |
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18.1 Kink Instability of the Jet |
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377 | (3) |
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18.2 The kink instability as an agent of relaxation |
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380 | (1) |
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18.3 The Rayleigh-Taylor instability as an agent of reconnection |
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381 | (3) |
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18.4 Production of Hard X-rays |
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384 | (11) |
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18.5 Comparison of actual experimental observations with predictions of Taylor relaxation model |
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395 | (2) |
| 19 Beyond MHD: Whistler Waves and Fast Magnetic Reconnection |
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397 | (40) |
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19.1 Quick review of resistive MHD reconnection |
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398 | (5) |
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19.2 Generalized Ohm's law and relative scaling of its terms |
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403 | (6) |
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19.3 Implications of Hall MHD |
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409 | (1) |
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19.3.1 Flux frozen into electron fluid rather than plasma center of mass |
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409 | (1) |
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19.4 Whistler waves in the limit of zero electron inertia |
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410 | (3) |
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19.4.1 Polarization and helicity content of whistler waves |
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411 | (2) |
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19.5 Hall reconnection with finite electron inertia |
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413 | (14) |
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19.5.1 Distinction between low and high β reconnection: guide field or not |
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414 | (1) |
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19.5.2 Hall reconnection analysis |
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414 | (11) |
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19.5.3 Discussion and implications |
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425 | (2) |
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19.6 Canonical circulation flux tube derivation of 2-fluid reconnection |
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427 | (10) |
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19.6.1 Non-linear equations for 2-fluid reconnection |
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427 | (1) |
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19.6.2 Using a Q flux tube instead of a magnetic field line |
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428 | (1) |
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19.6.3 Equilibrium properties of Q |
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429 | (1) |
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19.6.4 Perturbation from Q equilibrium |
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430 | (1) |
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19.6.5 Quadrupole magnetic field and quadrupole Q |
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430 | (4) |
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19.6.6 Proof that Q does not reconnect |
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434 | (1) |
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19.6.7 Amplification of Q |
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434 | (1) |
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19.6.8 Creation of torsion in Q flux tube |
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435 | (1) |
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19.6.9 Narrowing of current layer |
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436 | (1) |
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19.6.10 Growth rate scaling |
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436 | (1) |
| 20 Zero-β models for solar and space phenomena: Helicity, force-free equilibria |
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437 | (24) |
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437 | (11) |
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20.2 Sun-Earth connection viewed as helicity flux/relaxation |
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448 | (1) |
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20.3 A simple linear-force free model of coronal loops |
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449 | (9) |
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20.3.1 Setup of problem and derivation of solution |
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449 | (5) |
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454 | (1) |
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20.3.3 Flux tube bifurcation and breakup |
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455 | (2) |
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20.3.4 Comparison of magnetic field, field lines, flux tubes |
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457 | (1) |
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20.3.5 Relaxation and line tying |
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457 | (1) |
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20.4 Force-free state for arbitrary boundary conditions |
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458 | (3) |
| 21 Finite-β models and experiments for solar phenomena: collimation, flows, expansion |
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461 | (50) |
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461 | (17) |
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21.1.1 Solar surface boundary conditions |
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462 | (3) |
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21.1.2 Electrostatic potential drop associated with the twisting of a magnetic flux loop |
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465 | (8) |
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21.1.3 Relation to Alfven waves |
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473 | (2) |
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21.1.4 Motivation for the "gobble" finite beta model |
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475 | (3) |
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478 | (21) |
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21.2.1 Stage 1: current ramp-up |
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484 | (4) |
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21.2.2 Stage 2: ramped-up, constant current but no force balance |
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488 | (3) |
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21.2.3 Stage 3: Stagnation and collimation |
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491 | (3) |
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21.2.4 Hoop force and the gobble model |
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494 | (5) |
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21.3 Difference between finite and zero β models and consequences |
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499 | (1) |
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499 | (1) |
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21.3.2 Co- and counter-helicity merging |
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500 | (1) |
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21.4 Solar loop simulation experiment |
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500 | (11) |
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21.4.1 Loop expansion and collimation |
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503 | (3) |
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21.4.2 Plasma upflow from footpoints |
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506 | (2) |
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21.4.3 Magnetic measurements supporting the gobble model |
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508 | (1) |
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21.4.4 Hoop force explanation for the dip seen in the bottom of Fig. 21.12 |
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508 | (3) |
| 22 Beyond MHD: Extreme particle orbits in helical magnetic fields |
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511 | (12) |
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22.1 Radial Unstable Motion (RUM) in a helical magnetic field |
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512 | (5) |
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22.1.1 Lorentz equation derivation of RUM instability |
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512 | (2) |
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22.1.2 Hamiltonian-Lagrangian derivation of the RUM instability |
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514 | (3) |
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22.2 Stochastic ion heating in a rapidly twisting or untwisting magnetic field |
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517 | (6) |
| 23 Finite-β toroidal magnetic cloud model |
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523 | (8) |
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23.1 Introduction to magnetic clouds |
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523 | (2) |
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23.2 Magnetic cloud solution to Grad-Shafranov equation in toroidal geometry |
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525 | (6) |
| 24 Astrophysical Jets, Accretion, Angular Momentum Removal, and Space Dynamos |
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531 | (30) |
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531 | (1) |
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24.2 Field and current topology in astrophysical jets |
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532 | (8) |
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24.2.1 Inadequacy of Ideal MHD Ohm's Law to Model Accretion |
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538 | (2) |
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24.3 Kepler v. Cyclotron Orbits |
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540 | (2) |
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24.4 Charged dust grains as a method for having a cyclotron frequency comparable to the Kepler frequency |
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542 | (1) |
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24.5 Weak ionization as a method for having an effective cyclotron frequency comparable to the Kepler frequency |
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543 | (3) |
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24.6 Inward Spiral Orbits of Zero-Canonical Angular Momentum Particles |
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546 | (3) |
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24.7 Meta-Particle Inward Spiral from the Point of View of Hall Ohm's Law |
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549 | (1) |
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24.8 Accretion and removal of angular momentum via magnetic braking |
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550 | (7) |
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24.8.1 Braking torque interpreted using conservation of canonical angular momentum |
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550 | (6) |
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24.8.2 Braking torque from the MHD equation of motion |
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556 | (1) |
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24.9 Dynamo to generate poloidal magnetic field |
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557 | (4) |
| Appendix A Vector Identities and Operators |
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561 | (2) |
| Appendix B Bessel Orthogonality Relations |
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563 | (4) |
| Appendix C Capacitor Banks |
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567 | (4) |
| Appendix D Transmission lines, pulse forming networks, and transformers |
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571 | (18) |
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571 | (6) |
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D.1.1 Wave propagation along a transmission line |
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571 | (3) |
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D.1.2 Propagation velocity |
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574 | (1) |
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D.1.3 Characteristic impedance |
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574 | (1) |
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D.1.4 Matching, mismatching and reflected waves |
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575 | (2) |
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D.2 Pulse forming networks |
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577 | (1) |
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578 | (11) |
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579 | (1) |
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D.3.2 Transformer equations (resistive loads) |
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580 | (2) |
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D.3.3 Non-ideal transformers |
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582 | (2) |
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D.3.4 Pulse rise-time (high frequency response) |
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584 | (1) |
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D.3.5 Low-frequency response |
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584 | (1) |
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D.3.6 Air-core and iron-core transformers |
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585 | (1) |
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D.3.7 Calculating inductance of iron core transformers |
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586 | (1) |
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D.3.8 Measuring self and mutual inductances |
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586 | (3) |
| Appendix E Selected Formulae |
|
589 | (10) |
| Bibliography |
|
599 | (24) |
| Index |
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623 | |
Lisainfo e-raamatute kohta