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food security bill in hindi
DYNAMO THEORY
Chris A. Jones
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK

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jonesphoto.jpg

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



References: [1] G. Backus, A class of self-sustaining dissipative spherical dynamos, Ann. Phys., 4, 372-447, (1958) [2] J.P. Boyd, Chebyshev and Fourier Spectral Methods, Dover, 2001. [3] S.I. Braginsky, Nearly axisymmetric model of the hydromagnetic dynamo of the Earth, Geomag. Aeron., 15, 122-128, (1975). R. Soc. Lond. A, 247, 213-278, (1954). [5] F.H. Busse, Thermal instabilities in rapidly rotating systems, J. Fluid Mech., 44, 441-460, (1970). [6] F. Cattaneo and D.W. Hughes, Dynamo action in a rotating convective layer. J. Fluid Mech., 553, 401-418, (2006). [8] S. Childress and A.M. Soward, Convection-driven hydromagnetic dynamo. Phys. Rev. Lett., 29, 837-839, (1972). [10] U.R. Christensen and A.Tilgner, Power requirement of the geodynamo from ohmic losses in numerical and laboratory dynamos, Nature, 429, 169-171, (2004). [11] T.G. Cowling, The magnetic field of sunspots, Mon. Not. R. Astr. Soc., 94, 39-48, (1934). (2001). R. Soc. Lond., A 425, 407-429, (1989). Dormy and A.M. Soward, CRC Press, (2007). [17] D.R. Fearn, Hydromagnetic flow in planetary cores, Rep. Prog. Phys., 61, 175-235, (1998). Galdi and B. Straughan. Longmans. (1988) [19] A [21] D.J. Galloway and M.R.E. Proctor, Numerical calculations of fast dynamos for smooth velocity fields with realistic diffusion, Nature, 356, 691-693, (1992). [22] A.D. Gilbert, Fast dynamo action in the Ponomarenko dynamo, Geophys. Astrophys. Fluid Dynam., 44, 214-258, (1988). [25] A. Herzenberg, Geomagnetic dynamos, Phil. Trans R. Soc. Lond., A 250, 543-583, (1958). [26] D. Gubbins, C.N. Barber, S. Gibbons and J.J. Love, Proc. Roy. Soc. Lond. A., 456, 1333-1353, (1999). [29] C.A. Jones, Thermal and Compositional Convection in the Outer Core, in the Treatise of Geophysics, Core dynamics volume, pp 131-185, Volume editor, Peter Olson, Editor-in-Chief, Gerald Schubert, Elsevier (2007) [30] C.A Mech., 404, 311-343, (2000). [31] A. Iskakov and E Dormy, On the magnetic boundary conditions for non-spectral dynamo simulations, Geophys. Astrophys. Fluid Dynam., 99, 481-492, (2005). [32] C.A. Jones, N.O. Weiss, and F. Cattaneo, Nonlinear dynamos: a complex generalization of the Lorenz equations, Physica D, 14, 161-176, (1985). [33] F. Krause and K-H Rädler, Mean field magnetohydrodynamics and dynamo theory, Pergamon Press, New-York, (1980). [34] F.J. Lowes and I. Wilkinson, Geomagnetic dynamo : an improved laboratory model, Nature, 219, 717-718, (1968). [36] M. Meneguzzi and A. Pouquet, Turbulent dynamos driven by convection, J. Fluid Mech., 205, 297-318, (1989). [37] R.T. Merrill, M.W. McElhinny and P.L. McFadden, The Magnetic Field of the Earth, Academic Press (1996). [38] H.K. Moffatt, Magnetic field generation in electrically conducting fluids, Cambridge University Press (1978). [39] H.K. Moffatt, A self-consistent treatment of simple dynamo systems, Geophys. Astrophys. Fluid Dynam., 14, 147-166, (1979). [40] N.F. Otani, A fast kinematic dynamo in two-dimensional time-dependent flows, J. Fluid Mech., 253, 327-340, (1993). [41] E.N. Parker, Hydromagnetic dynamo models, Astrophys. J., 122, 293-314, (1955).%0

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