| Preface |
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xiii | |
| Introduction |
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xv | |
| PART 1 Equilibrium of a Plasma in Toroidal Confinement Systems |
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1 | (64) |
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General Results of Equilibrium Theory |
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3 | (15) |
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Starting equilibrium equations |
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5 | (1) |
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Equilibrium of a circular cylinder |
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5 | (3) |
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Curvilinear coordinates associated with the magnetic surfaces |
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8 | (4) |
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General description of an isotropic-plasma equilibrium |
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12 | (4) |
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General expressions for a magnetic well and shear |
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16 | (2) |
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Equilibrium of a Plasma in Axisymmetric Systems |
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18 | (21) |
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Starting equations for the metric tensor |
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20 | (1) |
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General equations of equilibrium in axisymmetric systems |
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21 | (1) |
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Large-aspect-ratio approximation for axisymmetric systems |
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22 | (2) |
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Equilibrium of a low- and a finite-pressure plasma in a circular tokamak |
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24 | (4) |
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28 | (3) |
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Tokamak of a circular or almost circular cross-section with a high βp |
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31 | (8) |
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Appendix: Effect of current gradient on ellipticity and triangularity of magnetic surfaces in tokamaks |
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37 | (2) |
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Equilibrium of a Plasma in Systems without Axial Symmetry |
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39 | (26) |
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Equilibrium in helical systems with circular cross-sections |
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41 | (5) |
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Helical systems with elliptic cross-sections |
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46 | (3) |
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Systems with an arbitrary form of the magnetic axis |
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49 | (2) |
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Systems with an axially inhomogeneous magnetic field |
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51 | (3) |
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54 | (2) |
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Stellarators with helical windings |
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56 | (9) |
| PART 2 Internal Magnetohydrodynamic Modes in the Cylindrical Approximation |
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65 | (44) |
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Description of Magnetohydrodynamic Perturbations in a Cylindrical Plasma |
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67 | (9) |
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68 | (1) |
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Ideal perturbations at the stability boundary |
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69 | (3) |
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72 | (1) |
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Allowance for finite resistivity |
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73 | (3) |
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Ideal Magnetohydrodynamic Internal Modes in a Cylindrical Plasma |
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76 | (20) |
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79 | (6) |
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General dispersion relation for a plasma with parabolic profiles of the pressure and the longitudinal current |
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85 | (2) |
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Non-local modes with m > 1 |
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87 | (3) |
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The m = 1 ideal internal kink mode |
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90 | (3) |
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Appendix: Ideal asymptotics of precise solutions for kink modes |
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93 | (3) |
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Resistive Magnetohydrodynamic Modes in a Cylindrical Plasma |
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96 | (13) |
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Solving the magnetohydrodynamic equations in the inertial-resistive layer |
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97 | (3) |
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100 | (1) |
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General dispersion relation for resistive kink modes |
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101 | (1) |
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102 | (4) |
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106 | (3) |
| PART 3 Small-Scale Magnetohydrodynamic Instabilities in Toroidal Confinement Systems |
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109 | (84) |
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Description of Marginally Stable Small-Scale Ideal Perturbations |
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111 | (17) |
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Starting equations for small-scale ideal perturbations |
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114 | (4) |
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Perturbations localized near a rational magnetic surface |
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118 | (3) |
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121 | (2) |
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Local modes with finite n/kx |
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123 | (2) |
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Ideal ballooning modes in the weak-ballooning approximation |
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125 | (3) |
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Small-Scale Magnetohydrodynamic Stability of a Plasma in Tokamaks |
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128 | (18) |
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Circular tokamak with a low- and a finite-pressure plasma |
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131 | (2) |
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133 | (2) |
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Stabilization of the Mercier modes in a circular tokamak at β > (a/R)4/3 |
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135 | (1) |
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Self-stabilization of a plasma in the central region of a tokamak |
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136 | (1) |
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Shear-driven instability of local and ballooning modes |
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137 | (9) |
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Appendix A: Starting equations of localized and ballooning modes in a circular tokamak with β *** (a/R)4/3 |
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141 | (4) |
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Appendix B: Qualitative stability criterion of ideal ballooning modes |
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145 | (1) |
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Ideal Small-Scale Magnetohydrodynamic Stability of a Plasma in Complex Magnetic Systems |
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146 | (15) |
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Systems homogeneous along the magnetic axis with a low- and a finite-pressure plasma |
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149 | (2) |
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Self-stabilization of a high-pressure plasma in helical systems |
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151 | (2) |
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Stability of localized and ballooning modes in the helical column without a longitudinal current |
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153 | (1) |
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Systems with a magnetic field inhomogeneous along the magnetic axis |
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153 | (4) |
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Stellarator with helical windings |
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157 | (4) |
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Growth Rates of Ideal Small-Scale Modes |
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161 | (11) |
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Description of ideal small-scale modes in the inertial region |
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162 | (5) |
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General dispersion relations for the ballooning modes and the Mercier modes |
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167 | (1) |
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Ballooning modes in a plasma stable against the Mercier modes |
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167 | (2) |
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The role of the ballooning character of perturbations in a plasma unstable against the Mercier modes |
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169 | (3) |
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Resistive Small-Scale Magnetohydrodynamic Modes in a Toroidal Geometry |
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172 | (21) |
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Description of perturbations in the inertial-resistive layer |
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176 | (4) |
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180 | (1) |
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Resistive-interchange instabilities in toroidal geometry |
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181 | (3) |
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Resistive instabilities driven by external (ideal) ballooning |
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184 | (1) |
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Stabilizing effect of the averaged plasma compressibility on resistive-interchange modes |
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185 | (1) |
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Resistive instabilities with a finite and with a large (γqR/cs)2 |
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186 | (7) |
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Appendix: Allowance for the averaged plasma compressibility |
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190 | (3) |
| PART 4 Magnetohydrodynamic Internal Kink Modes in Toroidal Geometry |
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193 | (62) |
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Description of Ideal Kink Modes in Toroidal Geometry |
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195 | (11) |
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Starting magnetohydrodynamic equations for the tokamak geometry |
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197 | (3) |
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Equations for the ideal kink modes with γ = 0 |
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200 | (3) |
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Equations for the ideal kink modes in the inertial layer |
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203 | (1) |
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Equations for kink modes in a helical column and stellarators with helical windings |
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204 | (2) |
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Ideal Internal Kink Modes in Toroidal Confinement Systems with a Low- and a Finite-Pressure Plasma |
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206 | (25) |
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The m = 1 mode in a circular tokamak with a weakly inhomogeneous longitudinal current |
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212 | (5) |
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General stability criterion of the m = 1 mode in a circular tokamak |
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217 | (1) |
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The growth rate of the m = 1 mode |
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218 | (1) |
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Modes with m > 1 in a circular tokamak |
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219 | (3) |
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Internal kink modes in a non-circular tokamak |
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222 | (2) |
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The m = 1 mode in a helical column with a longitudinal current |
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224 | (1) |
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Internal kink modes in stellarators with helical windings |
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225 | (6) |
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Appendix: Simplification of the potential-energy functional for the m = 1 mode |
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227 | (4) |
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Ideal Internal Kink Modes in Toroidal Systems with a High-Pressure Plasma |
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231 | (15) |
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Equations for kink modes at a high plasma pressure |
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233 | (1) |
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The m = n = 1 mode in a tokamak with a high-pressure plasma |
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234 | (1) |
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The m > 1 modes in a tokamak with a high-pressure plasma |
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235 | (2) |
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The m = 1 mode in a helical column with a high-pressure plasma |
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237 | (1) |
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Appendix A: Description of the m = 1 mode in a circular tokamak with a high-pressure plasma |
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238 | (3) |
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Appendix B: Description of the modes with m > 1 in a circular tokamak with a high-pressure plasma |
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241 | (5) |
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Resistive Internal Kink Modes in Toroidal Systems |
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246 | (9) |
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Equations for kink modes in the inertial-resistive layer |
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247 | (4) |
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Resistive internal kink instabilities with m = 1 |
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251 | (2) |
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Tearing modes in tokamaks |
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253 | (1) |
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Tearing modes in stellarators with helical windings |
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253 | (2) |
| PART 5 Magnetohydrodynamic Modes in Collisionless and Neoclassical Regimes |
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255 | (62) |
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Description of Equilibrium and Perturbations in Collisionless and Neoclassical Plasmas |
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257 | (15) |
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Equilibrium particle motion in a collisionless plasma in a curvilinear magnetic field |
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259 | (2) |
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Equilibrium trapped-particle motion in a tokamak |
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261 | (2) |
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Equilibrium distribution function and macroscopic parameters of a collisionless plasma |
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263 | (3) |
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266 | (2) |
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Bounce-averaged perturbed distribution function of trapped particles in the ideal region |
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268 | (2) |
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Starting form of perturbation equations in the ideal region |
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270 | (2) |
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Effect of the plasma pressure anisotropy on magnetohydrodynamic stability |
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272 | (10) |
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Equilibrium of an anisotropic plasma in axisymmetric systems |
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273 | (2) |
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Description of ideal magnetohydrodynamic modes in an anisotropic plasma |
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275 | (2) |
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Qualitative analysis of the role of the plasma pressure anisotropy in the magnetohydrodynamic modes |
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277 | (1) |
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The Mercier and ballooning modes |
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278 | (2) |
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The m = 1 mode in an anisotropic-pressure plasma |
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280 | (2) |
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Effect of Trapped Particles on the Magnetohydrodynamic Stability |
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282 | (7) |
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Perturbed pressure of collisionless trapped particles |
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283 | (1) |
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Description of small-scale modes |
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284 | (2) |
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Stability of small-scale modes |
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286 | (1) |
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Potential energy of kink modes allowing for the trapped-particle contribution |
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287 | (1) |
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Stability of the ideal m = 1 mode in the banana regime |
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287 | (2) |
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Description of the Longitudinal Viscosity in the Magnetohydrodynamic Modes |
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289 | (12) |
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Qualitative notions of viscous effects |
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290 | (3) |
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Allowance method for viscosity effects |
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293 | (1) |
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Calculation of the viscosity coefficient in the Pfirsch-Schluter regime |
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294 | (1) |
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Starting equations for calculation of the viscosity coefficient in the kinetic regimes |
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295 | (2) |
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Viscosity in the plateau regime |
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297 | (1) |
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Viscosity in the banana regime |
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297 | (4) |
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Magnetohydrodynamic Modes in a Viscous Plasma |
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301 | (16) |
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The averaged ballooning equation in the inertial-resistive layer allowing for the viscosity |
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303 | (2) |
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Precise solution of the averaged ballooning equation in the inertial-resistive layer allowing for the viscosity |
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305 | (1) |
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Description of kink modes in the inertial-resistive layer allowing for the viscosity |
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306 | (1) |
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Dispersion relations in the presence of the viscosity |
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307 | (1) |
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Ideal-viscous ballooning and the Mercier modes |
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307 | (1) |
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The ideal-viscous m = 1 kink mode |
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308 | (1) |
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Influence of the viscosity on resistive-interchange modes |
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309 | (4) |
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Viscous-resistive ballooning instability |
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313 | (1) |
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Viscous-resistive kink modes |
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314 | (3) |
| PART 6 Drift-Magnetohydrodynamic Modes |
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317 | (64) |
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Ideal Drift-Magnetohydrodynamic Modes |
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319 | (24) |
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Canonical form of the averaged ballooning equation in the inertial layer allowing for drift effects |
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321 | (3) |
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Drift-magnetohydrodynamic modes in the two-fluid non-viscous approximation |
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324 | (4) |
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Viscous drift-magnetohydrodynamic modes in the Pfirsch-Schluter regime |
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328 | (1) |
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Collisional drift-magnetohydrodynamic modes in the plateau regime and related collisionless modes |
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329 | (1) |
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Collisional drift-magnetohydrodynamic modes in the banana regime |
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330 | (2) |
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Collisionless drift-magnetohydrodynamic modes and their resonant and collisional excitation |
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332 | (3) |
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Two-fluid modes with ω + vti/qr |
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335 | (3) |
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Drift modification of the m = 1 mode |
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338 | (5) |
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Appendix: Collisionless inertia renormalization for ω + ωb |
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339 | (4) |
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Resistive Drift-Magnetohydrodynamic Modes |
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343 | (9) |
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Description of perturbations in the inertial-resistive layer allowing for drift effects |
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345 | (2) |
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Dispersion relations for resistive ballooning and kink modes with ω << cs/qr allowing for drift effects |
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347 | (1) |
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Drift resistive-interchange instability |
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348 | (1) |
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Resistive drift-ballooning modes with ωqr/cs → 0 |
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348 | (1) |
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Resistive drift-kink instabilities with m = 1 |
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348 | (1) |
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349 | (1) |
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Resistive drift-ballooning modes with a finite and with a large (ωqr/cs)2 |
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350 | (2) |
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Neoclassical Resistive Drift-Magnetohydrodynamic Modes |
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352 | (11) |
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Averaged ballooning equation in the inertial-resistive layer allowing for drift and viscosity effects |
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353 | (4) |
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Effect of the longitudinal ion viscosity on resistive drift-magnetohydrodynamic modes |
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357 | (2) |
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Description of modes dependent on both the electron viscosity and the ion viscosity |
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359 | (1) |
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Drift resistive-interchange instability due to the electron viscosity and the ion viscosity |
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359 | (2) |
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Resistive drift-ballooning instability due to the electron viscosity and the ion viscosity |
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361 | (1) |
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Resistive drift-kink modes due to the electron viscosity and the ion viscosity |
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362 | (1) |
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Semicollisional Drift-Magnetohydrodynamic Modes |
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363 | (7) |
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The problem statement and starting equations |
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364 | (3) |
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Derivation of the dispersion relation |
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367 | (1) |
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Low-frequency perturbations |
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368 | (2) |
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Interaction of High-Energy Trapped Particles with Magnetohydrodynamic Modes |
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370 | (11) |
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Description of high-energy trapped particles |
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371 | (3) |
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Interaction of high-energy trapped particles with ideal ballooning modes |
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374 | (4) |
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Interaction of high-energy trapped particles with the ideal internal m = 1 kink mode |
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378 | (2) |
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Influence of high-energy trapped particles on the m = 1 drift-kink mode |
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380 | (1) |
| PART 7 External Kink Modes |
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381 | (32) |
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Ideal External Kink Modes |
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383 | (4) |
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Description of ideal external kink modes in a cylindrical plasma |
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383 | (2) |
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Ideal external kink modes in a cylindrical plasma with a homogeneous longitudinal current and density |
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385 | (1) |
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Ideal external kink modes in a cylindrical plasma with a parabolic distribution of the longitudinal current |
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385 | (2) |
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Resistive-Wall Mode Instability |
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387 | (16) |
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Resistive-wall mode instability in a cylindrical plasma |
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389 | (3) |
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Problem statement and starting equations in the case of toroidal geometry |
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392 | (4) |
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Integral expression for the growth rate |
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396 | (1) |
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Contribution to the growth rate from non-singular side-band harmonics |
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397 | (1) |
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Contribution to the growth rate from singular side-band harmonics |
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398 | (3) |
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The case of the main harmonic with a singular point lying near the plasma boundary |
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401 | (2) |
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Stabilization of Resistive-Wall Mode Instability in a Tokamak with a Rotating Plasma |
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403 | (10) |
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Allowing for modified inertia in equations for singular side-band harmonics |
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404 | (1) |
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Solution of the equation for side-band harmonics in the inertial layer |
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405 | (3) |
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Asymptotics of the ideal solution for singular side-band harmonics |
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408 | (1) |
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Matching the asymptotics of the inertial and ideal solutions |
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409 | (1) |
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Contribution of singular side-band harmonics to the growth rate |
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410 | (1) |
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Analysis of the role of inertia effects |
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411 | (2) |
| PART 8 Alfven Eigenmodes and Their Interaction with High-Energy Particles |
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413 | (32) |
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Toroidicity-Induced Alfven Eigenmodes |
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415 | (8) |
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Description of Alfven modes allowing for metric oscillations |
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416 | (2) |
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Toroidicity-induced Alfven eigenmodes |
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418 | (4) |
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Energy of toroidicity-induced Alfven eigenmodes |
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422 | (1) |
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Kinetic Toroidicity-Induced Alfven Eigenmodes |
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423 | (11) |
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Magnetohydrodynamic approach to studying kinetic Alfven modes |
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423 | (3) |
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Starting equations for kinetic toroidicity-induced Alfven' eigenmodes |
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426 | (3) |
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Inner eigenfunctions and dispersion relations of kinetic toroidicity-induced Alfven eigenmodes |
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429 | (3) |
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Energy of kinetic toroidicity-induced Alfven' eigenmodes |
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432 | (2) |
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Interaction of High-Energy Particles with Toroidicity-Induced Alfven Eigenmodes and Kinetic Toroidicity-Induced Alfven Eigenmodes |
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434 | (11) |
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General expression for the growth rate of toroidicity-induced Alfven eigenmodes and kinetic toroidicity-induced Alfven eigenmodes due to high-energy particles |
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435 | (3) |
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Growth rate of toroidicity-induced Alfven eigenmodes and kinetic toroidicity-induced Alfven eigenmodes due to strongly circulating high-energy particles |
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438 | (2) |
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Growth rate of toroidicity-induced Alfven eigenmodes and kinetic toroidicity-induced Alfven eigenmodes due to trapped high-energy particles |
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440 | (5) |
| References |
|
445 | (11) |
| Index |
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456 | |
