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DYNAMO THEORY
Chris A. Jones
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK

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Contents
1. Kinematic dynamo theory
1.1. Maxwell and Pre-Maxwell equations
1.2. Integral form of the MHD equations
1.2.1. Stokes’ theorem
1.2.2. Potential fields
1.2.3. Faraday’s law
1.3. Electromagnetic theory in a moving frame
1.4. Ohm’s law, induction equation and boundary conditions
1.4.1. Lorentz force
1.4.2. Induction equation
1.4.3. Boundary conditions
1.5. Nature of the induction equation: Magnetic Reynolds number
1.6. The kinematic dynamo problem
1.7. Vector potential, Toroidal and Poloidal decomposition.
1.7.1. Vector Potential
1.7.2. Toroidal-Poloidal decomposition
1.7.3. Axisymmetric field decomposition
1.7.4. Symmetry
1.7.5. Free decay modes
1.8. The Anti-Dynamo theorems
2. Working kinematic dynamos
2.1. Minimum Rm for dynamo action
2.1.1. Childress bound
2.1.2. Backus bound
2.2. Faraday disc dynamos
2.2.1. Original Faraday disc dynamo
2.2.2. Homopolar self-excited dynamo
2.2.3. Moffatt’s segmented homopolar dynamo
2.2.4. Hompolar disc equations
2.3. Ponomarenko dynamo
2.3.1. Ponomarenko dynamo results
2.3.2. Smooth Ponomarenko dynamo
2.4. G.O. Roberts dynamo
2.4.1. Large Rm G.O. Roberts dynamo
2.4.2. Other periodic dynamos
2.5. Spherical Dynamos
2.5.1. Dudley and James dynamos
2.5.2. Braginsky limit
2.6. More specimens from the dynamo zoo!
2.6.1. Gailitis Dynamo
2.6.2. Herzenberg Dynamo
2.6.3. Lowes-Wilkinson Dynamo Experiment

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3. Mean field dynamo theory
3.1. Averaging the Dynamo Equations
3.1.1. Mean Field Induction equation.
3.1.2. Evaluation of (u × B )
3.2. Validity of MFDT.
3.2.1. The averaging process.
3.2.2. Evaluation of (u × B ), a closer look.
3.3. Tensor representation of E

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