Magneto-Fluid Dynamics: Fundamentals and Case Studies of Natural Phenomena (Astronomy and Astrophysics Library)

The part "Fundamentals" is novel, first in that it stresses the use of electric currents in Magneto-Fluid-Dynamics. As a rule, authors discuss magnetic field lines without ever referring to the required electric currents. Second, the book stresses the importance of electric space charges inside conductors that move in magnetic fields. It is the custom to disregard both the required electric currents and the field of the space charges. The "Case Studies" concern solar phenomena and the EarthВґs magnetic field, stressing electric currents and electric space charges. Each case study is based on a published, or soon to be published, paper. Looking Ahead Xxi -- Part I The Early History -- 1 The Early History 3 -- 1.1 Magnetism In China 4 -- 1.2 Magnetism In Europe 5 -- 1.2.1 Navigation By Compass In Europe 5 -- 1.2.2 Pierre De Maricourt (petrus Peregrinus) 6 -- 1.2.3 From Gilbert On 7 -- 1.3 Faraday 8 -- 1.4 The Story Of The Maxwell Equations 9 -- 1.5 Maxwell 10 -- 1.6 Heaviside 11 -- 1.7 Zeeman, Hale, Gouy, And Larmor 13 -- 2 The Maxwell Equations 19 -- 2.2 The Equations In Differential Form 20 -- 2.2.1 Example: The Differential Form Of Gauss's Law For Electric Fields 23 -- 2.2.2 Example: The Differential Form Of Gauss's Law For Magnetic Fields 23 -- 2.2.3 Example: The Differential Form Of The Law Of Faraday 23 -- 2.2.4 Example: The Differential Form Of Ampere's Circuital Law 23 -- 2.3 The Equations In Integral Form 24 -- 2.3.1 Example: The Integral Form Of Gauss's Law For Electric Fields 25 -- 2.3.2 Example: The Integral Form Of Gauss's Law For Magnetic Fields 26 -- 2.3.3 Example: The Integral Form Of The Law Of Faraday 26 -- 2.3.4 Example: The Integral Form Of Ampere's Circuital Law 27 -- 2.4 The Displacement Current Density 27 -- 2.5 The Convection Current Density 28 -- 3 Electric Fields 31 -- 3.1 Electric Fields And Forces 31 -- 3.2 Electric Currents 33 -- 3.2.1 Example: The Drift Speeds Of Charge Carriers 34 -- 3.3 The Conservation Of Electric Charge 34 -- 3.3.1 Example: The Relaxation Time Of A Conductor 35 -- 3.4 The Electric Potential V 36 -- 3.5 The Equations Of Poisson And Of Laplace 36 -- 3.6 Joule Losses 37 -- 3.7 Electric Energy 37 -- 3.8 Example: A Proton Beam I 37 -- 3.9 Case Study: V At The Surface Of The Sun 41 -- 4 Constant Magnetic Fields 45 -- 4.1 Magnetic Field Lines 46 -- 4.2 The Magnetic Flux Density B At A Point 46 -- 4.2.1 Example: Calculating The Field Of A Coil 47 -- 4.3 Magnetic Flux [phi] 48 -- 4.4 Ampere's Circuital Law 48 -- 4.4.1 Example: The Thin Solenoid 48 -- 4.4.2 Example: The Thick Solenoid 49 -- 4.5 The Vector Potential 50 -- 4.5.1 Example: The Vector Potential Near A Circuit 51 -- 4.5.2 Example: The Thin Solenoid 51 -- 4.6 Magnetic Multipoles 51 -- 4.7 Case Study: The Earth's Magnetic Field 52 -- 4.8 The Magnetic Force 56 -- 4.8.1 Example: The Electromagnetic Pump 57 -- 4.8.2 Magnetic Field Lines Again 57 -- 4.9 Magnetic Pressure 59 -- 4.10 Magnetic Energy 59 -- 4.11 Example: A Proton Beam Ii 60 -- 5 Time-dependent Magnetic Fields: The Law Of Faraday 63 -- 5.1 The Electric Field Strength 63 -- 5.2 The Law Of Faraday 64 -- 5.2.1 Lenz's Law 64 -- 5.2.2 Example: The Thin Solenoid 65 -- 5.2.3 Example: A Secondary On A Long Solenoid 65 -- 5.3 Self-inductance 65 -- 5.3.1 Magnetic Energy In A Self-inductance 66 -- 5.4 Mutual Inductance 66 -- 5.5 Electromagnetic Waves 67 -- 5.5.1 Case Study: The Earth's Magnetic Field 67 -- Part Iii Moving Conductors -- 6 Ohm's Law For Moving Conductors 73 -- 6.1 Reference Frames 74 -- 6.2 Special Relativity 74 -- 6.2.1 The Kinematic Transformation Equations 75 -- 6.2.2 The Transformation Equations Of Electromagnetic Quantities 75 -- 6.2.3 Example: Straight Wire Carrying A Current 79 -- 6.3 The Lorentz Force 79 -- 6.4 Ohm's Law For Moving Conductors 79 -- 6.5 The Induction Equation 82 -- 6.6 The Magnetic Reynolds Number 83 -- 6.7 Magnetic Forces On Moving Conductors 85 -- 7 Charges Inside Moving Conductors 87 -- 7.2 Electric Charges In Moving Conductors 88 -- 7.2.1 Example: Rotating Solid Conductor 90 -- 7.2.2 Time-dependent Situations 91 -- 7.2.3 The Magnetic Field Of The Convection Current 92 -- 7.3 Example: The Faraday Disk 92 -- 7.3.1 The Faraday Disk With The Switch Sw Open 93 -- 7.3.2 The Faraday Disk As A Generator 94 -- 7.3.3 The Faraday Disk As A Motor 95 -- 7.3.4 The Currents In The Axle And In The Disk 96 -- 7.4 Example: The Rotating Sphere 97 -- 7.4.1 The Surface Charge 99 -- 8 Nine Examples: Magnetic Fields In Moving Conductors 103 -- 8.2 The Induced Currents 104 -- 8.3 Magnetic Field Lines Once More 105 -- 8.4 The Net Magnetic Field 106 -- 8.5 The Case Of Rotating Conductors 107 -- 8.6 The Newcomb Criterion 109 -- 8.7 Nine Examples 110 -- 8.7.2 Examples 1, 2, 3: The Three Shercliff Cases 111 -- 8.7.3 Example 4: Rigid Conductor 113 -- 8.7.4 Example 5: Rotating Cylinder 114 -- 8.7.5 Example 6: Rotating Solid Sphere 116 -- 8.7.6 Example 7: Moving Plate 117 -- 8.7.7 Example 8: Rotating Fluid Sphere 120 -- 8.7.8 Example 9: The Faraday Disk 122 -- 9 Case Study: The Azimuthal Magnetic Field In The Earth's Core 125 -- 9.2 The Ratio B[subscript T]/b[subscript P] 127 -- 9.3 The Reference Frames S And S' 130 -- 9.4 Does The Earth's Magnetic Field Rotate? 131 -- 9.4.1 Does The Other Magnetic Field Rotate? 132 -- 9.4.2 Does The Axisymmetric Field B[subscript Axi] Rotate? 132 -- 9.4.3 Example: The One-piece Faraday Generator 133 -- 9.5 Solid Core 133 -- 9.5.1 The Values Of V X B And Of Q 134 -- 9.6 Differential Rotation 135 -- 9.6.1 Ferraro's Law Of Isorotation 136 -- 9.6.2 B[subscript T] With V[subscript [phi]] A Function Of Z 137 -- 9.6.3 B[subscript T] With V[subscript [phi]] A Function Of P 141 -- Part Iv Natural Dynamos -- 10 Case Study: The Disk Dynamo Model For Natural Dynamos 147 -- 10.2 The Self-excited Disk Dynamo 149 -- 10.2.1 The Kinematic Self-excited Disk Dynamo 149 -- 10.2.2 The Dynamic Self-excited Disk Dynamo 153 -- 10.3 A Laboratory-sized Dynamo? 156 -- 11 Three Case Studies: Magnetic Flux Tubes, Flux Ropes, And Flux Coils 159 -- 11.2 Convecting, Conducting Fluids 161 -- 11.3 Magnetic Flux Tubes (mft's) 163 -- 11.3.1 The Magnetic Force Density In Mft's 167 -- 11.3.2 The Gas Pressure Inside Mft's 167 -- 11.3.3 [phi] B[subscript Z], And J[subscript [phi]] 169 -- 11.3.4 The Magnetic Energy In Mft's 170 -- 11.3.5 Power Dissipation In Mft's 170 -- 11.3.6 The Resistance Per Meter In Mft's 170 -- 11.3.7 The Inductance Per Meter In Mft's 171 -- 11.3.8 The Time Constant Of An Mft 171 -- 11.3.9 Fluctuating Mft's 172 -- 11.4 Magnetic Flux Ropes (mfr's) 174 -- 11.4.1 The Axial Current In An Mfr 176 -- 11.4.2 Magnetic Forces On Mfr's 178 -- 11.4.3 Power Dissipation And Time Constants In Mfr's 179 -- 11.5 Are Tubes And Ropes Light Guides? 179 -- 11.5.1 Electromagnetic Waves 180 -- 11.5.2 Guided Electromagnetic Waves 180 -- 11.5.3 Electromagnetic Waves In Plasmas 181 -- 11.5.4 Guiding Light In Magnetic Flux Tubes And Ropes 181 -- 11.6 Magnetic Flux Coils As Particle Accelerators 184 -- 12 Case Study: Solar Magnetic Elements 189 -- 12.2 Local Currents In Magnetic Elements 191 -- 12.3 A Simple Model 192 -- 12.3.1 The Magnetic Flux Density B 194 -- 12.3.2 The Current Density J And The Conductivity [sigma] 194 -- 12.3.3 The Stored Magnetic Energy 196 -- 12.3.4 The Gas Pressure Inside The Element 196 -- 12.3.5 The Dissipated Power 197 -- 12.3.6 The Time Constant 198 -- 12.3.7 Magnetic Element Dipoles 200 -- 12.3.8 Anchoring 200 -- 12.3.9 A More Realistic Approximation 200 -- 13 Case Study: Sunspots 205 -- 13.2 Our Model 206 -- 13.3 Plasma Flows Above And Below Sunspots 207 -- 13.4 The Magnetic-flux-tube Dynamo 208 -- 13.5 Why Are Sunspots Dark? 210 -- 13.6 The Radial Distribution Of J[subscript [phi]] 210 -- 13.6.1 The Magnetic Field Configuration 211 -- 13.7 Magnetic Pressure And Gas Pressure 215 -- 13.8 The Flux Tube Radius As A Function Of Z 216 -- 13.8.1 The Radius B Below -36 Megameters 216 -- 13.8.2 The Flux Tube Radius B Above -36 Megameters 219 -- 13.9 The Wilson Depression 220 -- 14 Case Study: Solar Spicules 223 -- 14.2 Devising A Model 226 -- 14.3 What Type Of Particle? 226 -- 14.4 Repulsion And Pinching 227 -- 14.5 Channeling 228 -- 14.6 Self-excited Dynamos 230 -- 14.7 A Self-excited Accelerator For Spicules 231 -- 14.7.1 J[subscript A] In The Accelerator, Self-excitation 233 -- 14.7.2 The Magnetic Force Density In The Accelerator 235 -- 14.8 The Proton Energy 236 -- 14.9 The Beam Head 236 -- 14.9.1 The Beam Head Speed 237 -- 14.9.2 The Return Current At The Beam Head 238 -- 15 Case Study: Solar Coronal Loops As Self-channeled Proton Beams I 239 -- 15.2 Observations 241 -- 15.3 What Are They? 242 -- 15.4 Loops As Charged-particle Beams 245 -- 15.5 The Beam Current And The Beam Power 246 -- 15.6 The Magnetic Energy 247 -- 15.7 The Proton Energy 247 -- 15.8 Our Reference Loop 250 -- 16 Case Study: Solar Coronal Loops As Self-channeled Proton Beams Ii 253 -- 16.2 Channeling A Broad Proton Beam 254 -- 16.2.1 Electric And Magnetic Forces On A Proton Beam, Without Rotation 255 -- 16.2.2 Magnetic Forces On A Beam Proton, With Rotation 259 -- 16.3 Diamagnetism Of The Proton Beam 261 -- 16.4 At The Beam Head 263 -- 16.5 Guiding The Proton Beam 264 -- 16.5.1 Near-axial Ambient Magnetic Field 264 -- 16.5.2 Diverging Or Converging Ambient Magnetic Field 265 -- 16.6 What Determines The Beam Diameter? 266 -- 16.7 The Forces Outside The Beam 267 -- 16.8 No Interaction Within A Given Family? 268 -- 16.9 Two Families Of Loops Repel? 270 -- A Characteristic Lengths And Times, A Justification 273 -- A.2 Amplitude And Norm Of A Function 275 -- A.3 L And T Of A Sinusoidal Function 277 -- A.4 Fourier Expansions 279 -- A.5 Characteristic Time T Of A Function 281 -- A.6 Characteristic Length L Of A Function 282 -- A.6.1 The Norm Of [down Triangle] F (r) 283 -- A.6.2 Vectors With Complex Components 283 -- A.6.3 The Norms Of [down Triangle] . A And Of [down Triangle] X A 283. Paul Lorrain, Francois Lorrain, Stephane Houle. Includes Bibliographical References (p. [297]-308) And Index. Magnetohydrodynamics (MHD) concerns the interaction between magnetic fields and conducting fluids. We are concerned here with macroscopic inter­ actions and, when the conducting fluid is a plasma, time scales are very much longer than the plasma period. Plasma periods vary widely, but are short, say 10~^^ second. We prefer the term Magneto-F/i^Z(i-Dynamics (MFD) because the disci­ pline concerns mostly plasmas, various liquid conductors, and the liquid part of the Earth's core. It seems that the only applications of MFD to water are the induction of electric currents in the oceans by the Earth's magnetic field, and ship propulsion. But even MFD is not quite appropriate because that term also includes solid conductors that move in magnetic fields. This book is meant for graduate and upper-division undergraduate stu­ dents in Physics, Geophysics, and Astrophysics, as well as for practicing sci­ entists in these fields. This book is no more than a brief introduction to MFD because this vast subject is closely related to many others, namely Astrophysics, Elec­ trodynamics, Fluid Dynamics, Geophysics, Oceanography, Plasma Physics, Thermonuclear Fusion, etc. We sketch the fundamentals, and provide many Examples, as well as Case Studies related to natural phenomena. MFD sorely needs a rethink: it must of course be totally compatible with Physics. On the contrary, it is the custom to discuss the shapes of imaginary magnetic field lines, without ever referring to the required electric currents. Front Matter....Pages I-XXX Front Matter....Pages 1-1 The Early History....Pages 3-15 Front Matter....Pages 17-17 The Maxwell Equations....Pages 19-30 Electric Fields....Pages 31-44 Constant Magnetic Fields....Pages 45-62 Time-dependent Magnetic Fields: The Law of Faraday....Pages 63-69 Front Matter....Pages 71-71 Ohm’s Law for Moving Conductors....Pages 73-86 Charges Inside Moving Conductors....Pages 87-101 Nine Examples: Magnetic Fields in Moving Conductors....Pages 103-123 Case Study: The Azimuthal Magnetic Field in the Earth’s Core....Pages 125-144 Front Matter....Pages 145-145 Case Study: The Disk Dynamo Model for Natural Dynamos....Pages 147-157 Three Case Studies: Magnetic Flux Tubes, Flux Ropes, and Flux Coils....Pages 159-187 Case Study: Solar Magnetic Elements....Pages 189-204 Case Study: Sunspots....Pages 205-222 Case Study: Solar Spicules....Pages 223-238 Case Study: Solar Coronal Loops as Self-Channeled Proton Beams I....Pages 239-251 Case Study: Solar Coronal Loops as Self-Channeled Proton Beams II....Pages 253-270 Front Matter....Pages I-XXX Characteristic Lengths and Times, a Justification 1 ....Pages 273-293 SI prefixes....Pages 295-295 Back Matter....Pages 271-319 This book concerns the generation of electric currents and of electric space charges inside conducting media that move in magnetic fields. The authors postulate nothing but the Maxwell equations. They discuss at length the disk dynamo, which serves as a model for the natural self-excited dynamos that generate magnetic fields such as that of sunspots. There are 36 Examples and 13 Case Studies. The Case Studies concern solar phenomena -- magnetic elements, sunspots, spicules, coronal loops -- and the Earth's magnetic field.