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Topics: Magnetic field, Maxwell's equations, Electrical generator Pages: 97 (14208 words) Published: November 10, 2013
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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C. A. Jones

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
3.4. First order smoothing
3.4.1. Connection with helicity
3.4.2. Connection with G.O. Roberts dynamo
3.5. Parker loop mechanism
3.5.1. Joy’s law
3.6. Axisymmetric mean field dynamos
3.6.1. The Omega-effect
3.6.2. Dynamo waves
3.6.3. α2 dynamos
3.7. Spherical αω dynamos
4. Fast and slow dynamos
4.1. Magnetic helicity
4.2. The Stretch Twist Fold dynamo
4.2.1. Stretching and folding in 2D
4.3. Baker’s maps and stretch, fold, shear
4.3.1. Stretch Fold Shear, SFS
4.3.2. Stretch Fold Shear in G.O. Roberts dynamo
4.4. ABC dynamos
4.5. Stretching properties
4.5.1. Line Stretching
4.5.2. Flux growth rate
4.6. Time dependent flow fields
5. Nonlinear dynamos
5.1. Basic ideas in nonlinear dynamos
5.1.1. Dynamical regimes
5.2. Stellar dynamo saturation mechanisms
5.2.1. Modelling saturation mechanisms
5.2.2. A truncated system
5.3. α-quenching
5.3.1. α-quenching: Small or Large scale?
5.3.2. α-quenching: magnetic helicity
5.3.3. β-quenching
5.4. Saturation in rapidly rotating systems
5.4.1. Busse rolls
5.4.2. J.B. Taylor’s constraint
5.4.3. Elsasser number
5.5. Dynamo models and Taylor’s constraint
5.5.1. Equipartition in rapid rotation?
5.5.2. Dissipation time
5.6. Dynamo saturation in experiments
6. Numerical methods for dynamos
6.1. The pseudo-spectral method for a convection-driven plane layer dynamo

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Dynamo Theory
6.1.1. Dimensionless plane layer equations
6.1.2. Toroidal-Poloidal expansion
6.1.3. Toroidal Poloidal equations
6.1.4. Fourier decomposition
6.1.5. Boundary conditions
6.1.6. Collocation points
6.1.7. Pseudo-spectral method
6.2. Methods for kinematic dynamos
6.3. Hyperdiffusion
6.4. LES models
6.4.1. Similarity model
6.4.2. Dynamical Similarity model
6.5. Finite Volume methods
6.6. Spherical Geometry: spectral...

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