| Preface |
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xvii | |
| Part I Basic Theory And Observations |
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1 | (142) |
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3 | (17) |
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1.1 What is dynamo theory? |
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3 | (1) |
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1.2 Historical background |
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4 | (6) |
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4 | (4) |
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8 | (2) |
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1.3 The homopolar disc dynamo |
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10 | (2) |
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1.4 Axisymmetric and non-axisymmetric systems |
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12 | (8) |
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2 Magnetokinematic Preliminaries |
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20 | (39) |
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2.1 Structural properties of the B-field |
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20 | (5) |
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20 | (1) |
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2.1.2 The Biot-Savart integral |
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21 | (1) |
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2.1.3 Lines of force ('B-lines') |
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21 | (1) |
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2.1.4 Helicity and flux tube linkage |
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22 | (3) |
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25 | (5) |
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2.2.1 The rattleback: a prototype of dynamic chirality |
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26 | (2) |
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2.2.2 Mean response provoked by chiral excitation |
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28 | (2) |
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2.3 Magnetic field representations |
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30 | (6) |
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2.3.1 Spherical polar coordinates |
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30 | (2) |
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2.3.2 Toroidal/poloidal decomposition |
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32 | (2) |
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2.3.3 Axisymmetric fields |
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34 | (1) |
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2.3.4 Two-dimensional fields |
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34 | (2) |
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2.4 Relations between electric current and magnetic field |
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36 | (3) |
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36 | (1) |
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2.4.2 Multipole expansion of the magnetic field |
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37 | (1) |
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2.4.3 Axisymmetric fields |
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38 | (1) |
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39 | (4) |
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2.5.1 Force-free fields in spherical geometry |
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41 | (2) |
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2.6 Lagrangian variables and magnetic field evolution |
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43 | (4) |
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2.6.1 Change of flux through a moving circuit |
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44 | (1) |
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2.6.2 Faraday's law of induction |
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45 | (1) |
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2.6.3 Galilean invariance of the pre-Maxwell equations |
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45 | (1) |
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2.6.4 Ohm's law in a moving conductor |
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46 | (1) |
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2.7 Kinematically possible velocity fields |
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47 | (1) |
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48 | (5) |
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2.8.1 Toroidal decay modes |
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49 | (1) |
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2.8.2 Poloidal decay modes |
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50 | (1) |
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2.8.3 Behaviour of the dipole moment |
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51 | (2) |
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2.9 Fields exhibiting Lagrangian chaos |
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53 | (1) |
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54 | (5) |
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54 | (2) |
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2.10.2 Helicity of a knotted flux tube |
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56 | (3) |
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3 Advection, Distortion and Diffusion |
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59 | (40) |
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3.1 Alfven's theorem and related results |
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59 | (3) |
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3.1.1 Conservation of magnetic helicity |
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60 | (2) |
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3.2 The analogy with vorticity |
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62 | (2) |
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3.3 The analogy with scalar transport |
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64 | (1) |
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3.4 Maintenance of a flux rope by uniform irrotational strain |
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64 | (2) |
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3.5 A stretched flux tube with helicity |
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66 | (1) |
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3.6 An example of accelerated ohmic diffusion |
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67 | (1) |
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3.7 Equation for vector potential and flux-function under particular symmetries |
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68 | (2) |
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3.7.1 Two-dimensional case |
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69 | (1) |
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69 | (1) |
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3.8 Shearing of a space-periodic magnetic field |
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70 | (3) |
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3.9 Oscillating shear flow |
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73 | (3) |
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3.9.1 The case of steady rotation of the shearing direction |
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75 | (1) |
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3.10 Field distortion by differential rotation |
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76 | (1) |
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3.11 Effect of plane differential rotation on an initially uniform field: flux expulsion |
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77 | (9) |
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78 | (1) |
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3.11.2 The ultimate steady state |
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79 | (2) |
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3.11.3 Flow distortion by the flow due to a line vortex |
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81 | (1) |
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3.11.4 The intermediate phase |
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82 | (2) |
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3.11.5 Flux expulsion with dynamic back-reaction |
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84 | (1) |
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3.11.6 Flux expulsion by Gaussian angular velocity distribution |
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84 | (2) |
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3.12 Flux expulsion for general flows with closed streamlines |
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86 | (2) |
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3.13 Expulsion of poloidal field by meridional circulation |
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88 | (1) |
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3.14 Generation of toroidal field by differential rotation |
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89 | (4) |
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90 | (1) |
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3.14.2 The ultimate steady state |
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90 | (3) |
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3.15 Topological pumping of magnetic flux |
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93 | (6) |
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4 The Magnetic Field of the Earth and Planets |
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99 | (22) |
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4.1 Planetary magnetic fields in general |
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99 | (5) |
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4.2 Satellite magnetic fields |
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104 | (2) |
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4.3 Spherical harmonic analysis of the Earth's field |
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106 | (7) |
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4.4 Variation of the dipole field over long time-scales |
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113 | (3) |
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4.5 Parameters and physical state of the lower mantle and core |
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116 | (1) |
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4.6 The need for a dynamo theory for the Earth |
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117 | (1) |
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4.7 The core-mantle boundary and interactions |
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118 | (1) |
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4.8 Precession of the Earth's angular velocity |
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119 | (2) |
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5 Astrophysical Magnetic Fields |
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121 | (22) |
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5.1 The solar magnetic field |
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121 | (1) |
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5.2 Velocity field in the Sun |
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122 | (4) |
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5.2.1 Surface observations |
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122 | (2) |
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124 | (2) |
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5.3 Sunspots and the solar cycle |
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126 | (5) |
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5.4 The general poloidal magnetic field of the Sun |
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131 | (1) |
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132 | (2) |
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5.6 Magnetic interaction between stars and planets |
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134 | (2) |
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5.7 Galactic magnetic fields |
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136 | (4) |
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140 | (3) |
| Part II Foundations Of Dynamo Theory |
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143 | (154) |
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145 | (40) |
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6.1 Formal statement of the kinematic dynamo problem |
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145 | (1) |
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6.2 Rate-of-strain criterion |
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146 | (2) |
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6.3 Rate of change of dipole moment |
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148 | (1) |
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6.4 The impossibility of axisymmetric dynamo action |
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149 | (2) |
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6.4.1 Ultimate decay of the toroidal field |
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150 | (1) |
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6.5 Cowling's neutral point argument |
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151 | (2) |
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6.6 Some comments on the situation B . (Nabla B) equivalence 0 |
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153 | (1) |
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6.7 The impossibility of dynamo action with purely toroidal motion |
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153 | (3) |
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6.8 The impossibility of dynamo action with plane two-dimensional motion |
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156 | (1) |
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156 | (9) |
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6.9.1 The 3-sphere dynamo |
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158 | (3) |
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6.9.2 The 2-sphere dynamo |
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161 | (2) |
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6.9.3 Numerical treatment of the Herzenberg configuration |
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163 | (1) |
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6.9.4 The rotor dynamo of Lowes and Wilkinson |
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164 | (1) |
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6.10 Dynamo action associated with a pair of ring vortices |
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165 | (4) |
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6.11 Dynamo action with purely meridional circulation |
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169 | (2) |
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6.12 The Ponomarenko dynamo |
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171 | (5) |
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6.13 The Riga dynamo experiment |
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176 | (1) |
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6.14 The Bullard-Gellman formalism |
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176 | (7) |
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183 | (2) |
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7 Mean-Field Electrodynamics |
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185 | (31) |
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7.1 Turbulence and random waves |
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185 | (3) |
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7.2 The linear relation between epsilon and B0 |
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188 | (1) |
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189 | (4) |
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7.4 Effects associated with the coefficient βijk |
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193 | (2) |
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7.5 First-order smoothing |
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195 | (1) |
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7.6 Spectrum tensor of a stationary random vector field |
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196 | (4) |
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7.7 Determination of αij for a helical wave motion |
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200 | (2) |
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7.8 Determination of αij for a random u-field under first-order smoothing |
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202 | (3) |
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7.9 Determination of βijk under first-order smoothing |
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205 | (1) |
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7.10 Lagrangian approach to the weak diffusion limit |
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206 | (3) |
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206 | (2) |
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7.10.2 Evaluation of βijk |
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208 | (1) |
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7.10.3 The isotropic situation |
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209 | (1) |
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7.11 Effect of helicity fluctuations on effective turbulent diffusivity |
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209 | (3) |
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7.12 Renormalisation approach to the zero-diffusivity limit |
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212 | (4) |
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8 Nearly Axisymmetric Dynamos |
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216 | (15) |
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216 | (3) |
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8.2 Lagrangian transformation of the induction equation when η = 0 |
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219 | (2) |
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8.3 Effective variables in a Cartesian geometry |
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221 | (1) |
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8.4 Lagrangian transformation including weak diffusion effects |
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222 | (1) |
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8.5 Dynamo equations for nearly rectilinear flow |
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223 | (2) |
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8.6 Corresponding results for nearly axisymmetric flows |
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225 | (2) |
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8.7 A limitation of the pseudo-Lagrangian approach |
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227 | (1) |
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8.8 Matching conditions and the external field |
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228 | (2) |
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230 | (1) |
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9 Solution of the Mean-Field Equations |
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231 | (48) |
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9.1 Dynamo models of α2- and αω-type |
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231 | (2) |
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9.1.1 Axisymmetric systems |
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232 | (1) |
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9.2 Free modes of the α2-dynamo |
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233 | (3) |
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9.2.1 Weakly helical situation |
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235 | (1) |
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9.2.2 Influence of higher-order contributions to epsilon |
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235 | (1) |
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9.3 Free modes when αij is anisotropic |
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236 | (8) |
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9.3.1 Space-periodic velocity fields |
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237 | (1) |
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9.3.2 The α2-dynamo in a spherical geometry |
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238 | (3) |
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9.3.3 The α2-dynamo with antisymmetric α |
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241 | (3) |
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9.4 Free modes of the αω-dynamo |
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244 | (3) |
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9.5 Concentrated generation and shear |
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247 | (4) |
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9.5.1 Symmetric U(z) and antisymmetric α(z) |
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249 | (2) |
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9.6 A model of the galactic dynamo |
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251 | (7) |
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254 | (1) |
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255 | (2) |
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9.6.3 Oscillatory dipole and quadrupole modes |
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257 | (1) |
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9.6.4 Oblate-spheroidal galactic model |
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258 | (1) |
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9.7 Generation of poloidal fields by the α-effect |
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258 | (2) |
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9.8 The αω-dynamo with periods of stasis |
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260 | (1) |
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9.9 Numerical investigations of the αω-dynamo |
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261 | (8) |
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9.10 More realistic modelling of the solar dynamo |
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269 | (3) |
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9.11 The Karlsruhe experiment as an α2-dynamo |
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272 | (1) |
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9.12 The VKS experiment as an αω-dynamo |
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273 | (3) |
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9.12.1 Field reversals in the VKS experiment |
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275 | (1) |
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9.13 Dynamo action associated with the Taylor-Green vortex |
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276 | (3) |
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279 | (18) |
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10.1 The stretch-twist-fold mechanism |
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279 | (4) |
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10.1.1 Writhe and twist generated by the STF cycle |
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279 | (2) |
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10.1.2 Existence of a velocity field in R3 that generates the STF cycle |
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281 | (1) |
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10.1.3 Tube reconnection and helicity cascade |
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282 | (1) |
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10.2 Fast and slow dynamos |
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283 | (1) |
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10.3 Non-existence of smooth fast dynamos |
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284 | (1) |
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10.4 The homopolar disc dynamo revisited |
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285 | (2) |
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10.5 The Ponomarenko dynamo in the limit η right arrow 0 |
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287 | (1) |
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10.6 Fast dynamo with smooth space-periodic flows |
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288 | (5) |
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10.6.1 The symmetric case A = B = C = 1 |
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289 | (2) |
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10.6.2 The Galloway-Proctor fast dynamo |
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291 | (2) |
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10.7 Large-scale or small-scale fast dynamo? |
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293 | (1) |
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10.8 Non-filamentary fast dynamo |
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294 | (3) |
| Part III Dynamic Aspects Of Dynamo Action |
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297 | (185) |
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11 Low-Dimensional Models of the Geodynamo |
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299 | (16) |
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11.1 Dynamic characteristics of the segmented disc dynamo |
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299 | (3) |
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11.2 Disc dynamo driven by thermal convection |
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302 | (4) |
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303 | (2) |
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11.2.2 Coupling of Welander loop and Bullard disc |
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305 | (1) |
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306 | (2) |
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11.4 Symmetry-mode coupling |
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308 | (2) |
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11.5 Reversals induced by turbulent fluctuations |
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310 | (5) |
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11.5.1 Dipole-quadrupole model |
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311 | (4) |
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315 | (41) |
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12.1 The momentum equation and some elementary consequences |
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315 | (4) |
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316 | (1) |
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12.1.2 Alfven wave invariants and cross-helicity |
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317 | (2) |
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319 | (4) |
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12.2.1 Dispersion relation and up-down symmetry breaking |
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320 | (2) |
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12.2.2 Inertial and magnetostrophic wave limits |
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322 | (1) |
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12.3 Generation of a fossil field by decaying Lehnert waves |
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323 | (1) |
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12.4 Quenching of the α-effect by the Lorentz force |
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324 | (3) |
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12.4.1 A simple model based on weak forcing |
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324 | (3) |
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12.4.2 Quenching of the β-effect |
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327 | (1) |
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12.5 Magnetic equilibration due to α-quenching |
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327 | (6) |
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12.5.1 The case of steady forcing |
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328 | (1) |
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12.5.2 The case of unsteady forcing with ω2η&nukappa4 |
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329 | (2) |
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12.5.3 Cattaneo-Hughes saturation |
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331 | (2) |
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12.6 Quenching of the α-effect in a field of forced Lehnert waves |
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333 | (3) |
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12.7 Equilibration due to α-quenching in the Lehnert wave field |
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336 | (3) |
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12.7.1 Energies at resonance |
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338 | (1) |
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12.8 Forcing from the boundary |
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339 | (3) |
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12.9 Helicity generation due to interaction of buoyancy and Coriolis forces |
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342 | (1) |
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12.10 Excitation of magnetostrophic waves by unstable stratification |
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343 | (5) |
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12.11 Instability due to magnetic buoyancy |
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348 | (5) |
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350 | (3) |
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12.12 Helicity generation due to flow over a bumpy surface |
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353 | (3) |
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13 The Geodynamo: Instabilities and Bifurcations |
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356 | (40) |
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13.1 Models for convection in the core of the Earth |
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356 | (1) |
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13.2 Onset of thermal convection in a rotating spherical shell |
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357 | (9) |
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13.2.1 The Roberts-Busse localised asymptotic theory for small epsilon |
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360 | (1) |
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13.2.2 The Soward-Jones global theory for the onset of spherical convection |
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361 | (2) |
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13.2.3 Localised mode of instability in a spherical shell |
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363 | (2) |
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13.2.4 Dynamic equilibration |
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365 | (1) |
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13.3 Onset of dynamo action: bifurcation diagrams and numerical models |
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366 | (10) |
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369 | (2) |
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13.3.2 Model equations for super- and subcritical bifurcations |
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371 | (1) |
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13.3.3 Three regimes, WD, FM and SD; numerical detection |
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372 | (1) |
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373 | (2) |
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13.3.5 The WD / SD dichotomy |
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375 | (1) |
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13.4 The Childress-Soward convection-driven dynamo |
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376 | (5) |
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13.4.1 Mixed asymptotic and numerical models |
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380 | (1) |
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13.5 Busse's model of the geodynamo |
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381 | (3) |
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13.6 The Taylor constraint and torsional oscillations |
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384 | (6) |
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13.6.1 Necessary condition for a steady solution U(x) |
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384 | (1) |
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13.6.2 Sufficiency of the Taylor constraint for the existence of a steady U(x) |
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385 | (2) |
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13.6.3 The arbitrary geostrophic flow upsilon(s) |
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387 | (1) |
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13.6.4 Deviations from the Taylor constraint |
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388 | (1) |
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13.6.5 Torsional oscillations when the Taylor constraint is violated |
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388 | (1) |
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13.6.6 Effect of mantle conductivity |
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389 | (1) |
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390 | (6) |
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14 Astrophysical dynamic models |
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396 | (21) |
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14.1 A range of numerical approaches |
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396 | (6) |
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396 | (2) |
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398 | (2) |
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14.1.3 Direct numerical simulations |
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400 | (2) |
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14.2 From planets to stars |
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402 | (1) |
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14.3 Extracting dynamo mechanisms |
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403 | (2) |
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14.4 Dipole breakdown and bistability |
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405 | (1) |
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14.5 Kinematically unstable saturated dynamos |
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406 | (2) |
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408 | (1) |
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14.7 Accretion discs and the magnetorotational instability (MRI) |
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409 | (8) |
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14.7.1 Rayleigh stability criterion |
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410 | (1) |
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14.7.2 Magnetorotational instability |
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411 | (1) |
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14.7.3 Shearing-box analysis |
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412 | (2) |
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14.7.4 Dynamo action associated with the magnetorotational instability |
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414 | (1) |
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14.7.5 Experimental realisation of the magnetorotational instability |
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415 | (2) |
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417 | (24) |
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15.1 Effects of helicity on homogeneous turbulence |
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417 | (9) |
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15.1.1 Energy cascade in non-helical turbulence |
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418 | (1) |
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419 | (3) |
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15.1.3 Effect of helicity on energy cascade |
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422 | (4) |
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15.2 Influence of magnetic helicity conservation in energy transfer processes |
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426 | (6) |
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15.3 Modification of inertial range due to large-scale magnetic field |
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432 | (1) |
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15.4 Non-helical turbulent dynamo action |
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433 | (2) |
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15.5 Dynamo action incorporating mean flow effects |
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435 | (3) |
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15.6 Chiral and magnetostrophic turbulence |
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438 | (3) |
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16 Magnetic Relaxation under Topological Constraints |
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441 | (22) |
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16.1 Lower bound on magnetic energy |
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441 | (2) |
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16.2 Topological accessibility |
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443 | (1) |
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16.3 Relaxation to a minimum energy state |
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443 | (3) |
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16.3.1 Alternative 'Darcy' relaxation procedure |
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445 | (1) |
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16.4 Two-dimensional relaxation |
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446 | (2) |
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16.5 The relaxation of knotted flux tubes |
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448 | (3) |
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16.6 Properties of relaxed state |
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451 | (2) |
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453 | (1) |
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16.8 Structure of magnetostatic fields |
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453 | (1) |
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16.9 Stability of magnetostatic equilibria |
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454 | (3) |
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16.9.1 The two-dimensional situation |
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456 | (1) |
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16.10 Analogous Euler flows |
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457 | (1) |
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16.11 Cross-helicity and relaxation to steady MHD flows |
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458 | (5) |
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16.11.1 Structure of steady states |
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459 | (1) |
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16.11.2 The isomagnetovortical foliation |
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460 | (1) |
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16.11.3 Relaxation to steady MHD states |
|
|
461 | (2) |
|
17 Magnetic Relaxation in a Low-β Plasma |
|
|
463 | (19) |
|
17.1 Relaxation in a pressureless plasma |
|
|
463 | (2) |
|
17.2 Numerical relaxation |
|
|
465 | (2) |
|
|
|
467 | (1) |
|
17.4 Current collapse in an unbounded fluid |
|
|
468 | (3) |
|
17.4.1 Similarity solution when η = 0 |
|
|
470 | (1) |
|
17.5 The Taylor conjecture |
|
|
471 | (4) |
|
17.6 Relaxation of a helical field |
|
|
475 | (2) |
|
17.7 Effect of plasma turbulence |
|
|
477 | (2) |
|
17.8 Erupting flux in the solar corona |
|
|
479 | (2) |
|
|
|
481 | (1) |
| Appendix Orthogonal Curvilinear Coordinates |
|
482 | (3) |
| References |
|
485 | (26) |
| Author index |
|
511 | (4) |
| Subject index |
|
515 | |
