Introduction to Magnetohydrodynamics

P. A. Davidson

doi:10.1017/9781316672853

Magnetohydrodynamics (MHD) plays a crucial role in astrophysics, planetary magnetism, engineering and controlled nuclear fusion. This comprehensive textbook emphasizes physical ideas, rather than mathematical detail, making it accessible to a broad audience. Starting from elementary chapters on fluid mechanics and electromagnetism, it takes the reader all the way through to the latest ideas in more advanced topics, including planetary dynamos, stellar magnetism, fusion plasmas and engineering applications. With the new edition, readers will benefit from additional material on MHD instabilities, planetary dynamos and applications in astrophysics, as well as a whole new chapter on fusion plasma MHD. The development of the material from first principles and its pedagogical style makes this an ideal companion for both undergraduate students and postgraduate students in physics, applied mathematics and engineering. Elementary knowledge of vector calculus is the only prerequisite.

Review of previous edition:‘… an excellent book, which provides a refreshing introduction and a welcome addition to the MHD literature.'

A. M. Soward Source: Journal of Fluid Mechanics

Review of previous edition:'The language of this book is simple, vivid, yet fully scientific. It is a real pleasure to read … worth recommending, not only to students, but also to everyone who is interested in MHD, particularly to theoreticians who, as a rule, know almost nothing about metallurgical applications of MHD.'

Source: Applied Mechanics Review

Review of previous edition:‘Like other texts in the series, the typography is easy on the eyes and the price easy on the purse. All in all, a wonderful introduction to the subject and more!'

Stanley A. Berger Source: Physics Today

Review of previous edition:‘… a thorough introduction to conducting fluid mechanics … an excellent and informative book that can be well recommended.'

S. W. H. Cowley Source: Contemporary Physics

Review of previous edition:‘The author writes lucidly and maintains the reader's interest in several ways: he formulates arguments provocatively, sometimes as paradoxes; he provides apt quotations; he points to exciting applications; and he enlivens his text with historical snippets … It is written with love, and in a completely consistent style.'

Paul H. Roberts Source: SIAM Review

Review of previous edition:‘The book is unique in bringing together a number of diverse topics … [It] makes for rewarding reading, and I recommend it to all students of MHD, no matter what their persuasion. It would be an excellent textbook for students with interest in the engineering applications, but also will serve as a perfect complementary text for an introductory plasma MHD course.'

Elena V. Belova Source: American Journal of Physics

'The rich scholarship presented in this monograph is a result of the author’s ongoing study of these concepts … This careful documentation, also provided for modern technology, gives the reader an exceptional glimpse into this field.'

J. W. Jerome Source: MathSciNet

'This book is at once a useful basic textbook introducing the elements of electromagnetism and fluid dynamics and, at the same time, an informative research monograph targeting professional researchers in industry and academia.'

K. Alan Shore Source: Optics & Photonics News

References

Part I:

Part I: From Maxwell’s Equations to Magnetohydrodynamics

Chapter 1

Chapter 1 A Qualitative Overview of MHD

Davidson, P.A., 1999, Magnetohydrodynamics in material processing. Annual Reviews Fluid Mech. 31, 273–300.

Chapter 2

Chapter 2 The Governing Equations of Electrodynamics

Feynman, R.P., Leighton, R.B. and Sands, M., 1964, The Feynman lectures on physics, Addison-Wesley.

Jackson, J.D., 1999, Classical electrodynamics, 3rd ed., Wiley.

Lorrain, P. and Corson, D., 1970, Electromagnetic fields and waves, 2nd ed., W.H. Freeman & Co.

Chapter 3

Chapter 3 A First Course in Fluid Dynamics

Acheson, D.J., 1990, Elementary fluid dynamics, Clarendon Press.

Batchelor, G.K., 1967, An introduction to fluid mechanics, Cambridge University Press.

Davidson, P.A., 2004, Turbulence: an introduction for scientists and engineers, Oxford University Press.

Davidson, P.A., 2013, Turbulence in rotating, stratified and electrically conducting fluids, Cambridge University Press.

Davidson, P.A., Staplehurst, P.J. and Dalziel, S.B., 2006, On the evolution of eddies in a rapidly rotating system. J. Fluid Mech., 557, 135–144.

Feynman, R.P., Leighton, R.B. and Sands, M., 1964, The Feynman lectures on physics, Vol. II. Addison-Wesley.

Ranjan, A. and Davidson, P.A., 2014, Evolution of a turbulent cloud under rotation. J. Fluid Mech., 756, 488–509.

Tennekes, H. and Lumley, J.L., 1972, A first course in turbulence, The MIT Press.

Part II:

Part II: The Fundamentals of Incompressible MHD

Chapter 5

Chapter 5 Kinematics: Advection, Diffusion, and Intensification of Magnetic Fields

Biskamp, D., 1993, Nonlinear magnetohydrodynamics, Cambridge University Press.

Galloway, D.J. and Weiss, N.O., 1981, Convection and magnetic fields is stars., Astrophys. J., 243, 309–316.

Moffatt, H.K., 1978, Magnetic field generation in electrically conducting fluids, Cambridge University Press.

Priest, E., 2014, Magnetohydrodynamics of the sun, Cambridge University Press.

Shercliff, J.A., 1965, A textbook of magnetohydrodynamics, Pergamon Press.

Chapter 6

Chapter 6 Dynamics at Low Magnetic Reynolds Numbers

Chandrasekhar, S., 1961, Hydrodynamic stability, Dover.

Davidson, P.A., 1997, The role of angular momentum in the magnetic damping of turbulence, J. Fluid Mech., 336, 123–150.

Moreau, R., 1990, Magnetohydrodynamics, Kluwer Acad. Pub.

Müller, U. and Bühler, L., 2001, Magnetofluiddynamics in channels and containers, Springer.

Shercliff, J.A., 1965, A textbook of magnetohydrodynamics, Pergamon Press.

Chapter 7

Chapter 7 Dynamics at High Magnetic Reynolds Numbers

Arnold, V.I., 1966, Sur un principe variationnel pour les écoulements stationaires des liquides parfaits et ses applications aux problèmes de stabilité non-linéaires. J. Méc., 5, 9–15.

Balbus, S.A. and Hawley, J. F., 1998, Instability, turbulence and enhanced transport in accretion disks. Rev. Modern Phys., 70 (1), 1–53.

Bernstein, I.B., et al., 1958, An energy principle for hydromagnetic stability problems. Proc. Roy. Soc. Lond. A., 244.

Biskamp, D., 1993, Non-linear magnetohydrodynamics, Cambridge University Press.

Chandrasekhar, S., 1960, The stability of non-dissipative Couette flow in hydromagnetics. Proc. Nat. Acad. Sci., 46, 253–257.

Chandrasekhar, S., 1961, Hydrodynamic and hydromagnetic stability, Oxford University Press.

Davidson, P.A., 2000, An energy criterion for the linear stability of conservative flows. J. Fluid Mech., 402, 329–348.

Davidson, P.A., 2013, Turbulence in rotating, stratified and electrically conducting fluids, Cambridge University Press.

Frieman, E. and Rotenberg, M., 1960, On hydromagnetic stability of stationary equilibria. Rev. Mod. Phys., 32(4), 898–939.

Kelvin, Lord, 1887, On the stability of steady and of periodic fluid motion. – Maximum and minimum energy in vortex motion. Phil. Mag., 23, 529.

Moffatt, H.K., 1978, Magnetic field generation in electrically conducting fluids, Cambridge University Press.

Moffatt, H.K., 1986, Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part 2. Stability considerations. J. Fluid Mech., 166, 359–378.

Velikhov, E.P., 1959, Stability of an ideally conducting liquid flowing between cylinders rotating in a magnetic field. Soviet Physics JETP, 36, 1398–1404.

Chapter 8

Chapter 8 An Introduction to Turbulence

Batchelor, G.K., 1953, The theory of homogeneous turbulence, Cambridge University Press.

Batchelor, G.K. and Proudman, I., 1956, The large-scale structure of homogeneous turbulence. Phil. Trans. R. Soc. Lond. A 248, 369–405.

Biskamp, D., 2003, Magnetohydrodynamic turbulence, Cambridge University Press.

Davidson, P.A., 2009, The role of angular momentum conservation in homogeneous turbulence. J. Fluid Mech., 632, 329–358.

Davidson, P.A., 2010, On the decay of saffman turbulence subject to rotation, stratification or an imposed magnetic field. J. Fluid Mech., 663, 268–292.

Davidson, P.A., 2013, Turbulence in rotating, stratified and electrically conducting fluids, Cambridge University Press.

Davidson, P.A., 2015, Turbulence: an introduction for scientists and engineers, 2nd ed., Oxford University Press.

Davidson, P.A., Okamoto, N. and Kaneda, Y., 2012, On freely decaying, anisotropic, axisymmetric, Saffman turbulence. J. Fluid Mech., 706, 150–172.

Gödecke, K., 1935, Messungen der atmospharischen Turbulenz in Bodennähe mit einer Hitzdrahtmethode. Ann. Hydrogr., 10, 400–410.

Ishida, T., Davidson, P.A. and Kaneda, Y., 2006, On the decay of isotropic turbulence. J. Fluid Mech., 564, 455–475.

Kolmogorov, A.N., 1941a, Local structure of turbulence in an incompressible viscous fluid at very large Reynolds numbers. Dokl. Akad. Nauk SSSR, 30(4), 299–303.

Kolmogorov, A.N., 1941b, Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR, 32(1), 19–21.

Kolmogorov, A.N., 1941c, On the degeneration of isotropic turbulence in an incompressible viscous fluid. Dokl. Akad. Nauk. SSSR, 31(6), 538–541.

Kolmogorov, A.N., 1962, A refinement of the concept of the local structure of turbulence in an incompressible viscous fluid at large Reynolds number. J. Fluid Mech., 13 (1), 82.

Krogstad, P.-A. and Davidson, P.A., 2010, Is grid turbulence Saffman turbulence? J. Fluid Mech. 642, 373–394.

Landau, L.D. and Lifshitz, E.M., 1959, Fluid mechanics, 1st ed., Pergamon.

Proudman, I. and Reid, W.H., 1954, On the decay of a normally distributed and homogeneous turbulent velocity field. Phil. Trans. R. Soc. Lond. A, 247, 163–189.

Saffman, P.G., 1967, The large-scale structure of homogeneous turbulence. J. Fluid Mech. 27(3), 581–593.

Tennekes, H. and Lumley, J.L., 1972, A first course in turbulence, MIT Press.

Chapter 9

Chapter 9 MHD Turbulence at Low and High Magnetic Reynolds Numbers

Batchelor, G.K., 1950, On the spontaneous magnetic field in a conducting liquid in turbulent motion. Proc. Roy. Soc. London, A201, 405–416.

Batchelor, G.K., 1953. The theory of homogeneous turbulence, Cambridge University Press.

Davidson, P.A., 1997, The role of angular momentum in the magnetic damping of turbulence. J. Fluid Mech., 336, 123–150.

Davidson, P.A., 2009, The role of angular momentum conservation in homogeneous turbulence. J. Fluid Mech., 632, 329–358.

Davidson, P.A., 2010, On the decay of Saffman turbulence subject to rotation, stratification or an imposed magnetic field. J. Fluid Mech., 663, 268–292.

Davidson, P.A., 2013, Turbulence in rotating, stratified and electrically conducting fluids, Cambridge University Press.

Davidson, P.A., 2015, Turbulence: an introduction for scientists and engineers, 2nd ed., Oxford University Press.

Federrath, C., Schober, J., Bovino, S. and Schleicher, D.R.G., 2014, The turbulent dynamo in highly compressible supersonic plasmas. Astro. Phys. Lett., 797, L19.

Ishida, T., Davidson, P.A. and Kaneda, Y., 2006, On the decay of isotropic turbulence. J. Fluid Mech., 564, 455–475.

Ohkitani, K., 2002, Numerical study of comparison of vorticity and passive vectors in turbulence and inviscid flows. Phys. Rev. E., 65, 046304.

Okamoto, N., Davidson, P.A. and Kaneda, Y., 2010, On the decay of low magnetic Reynolds number turbulence in an imposed magnetic field. J. Fluid Mech., 651, 295–318.

Saffman, P.G., 1967, The large-scale structure of homogeneous turbulence. J. Fluid Mech. 27(3),581–593.

Schekochihin, A.A., Iskakov, A.B., Cowley, S.C., McWilliams, J.C., Proctor, M.R.E. and Yousef, T.A., 2007, Fluctuation dynamo and turbulent induction at low magnetic Prandtl numbers. New J. Phys., 9, 300.

Stribling, T. and Matthaeus, W.H., 1991, Relaxation processes in a low-order three-dimensional magnetohydrodynamic model. Phys. Fluids B 3, 1848–1864.

Taylor, J.B., 1974, Relaxation of toroidal plasma and generation of reversed magnetic fields. Phys. Rev. Lett., 33, 1139–1141.

Tobias, S.M., Cattaneo, F. and Boldyrev, S., 2013, MHD turbulence: Field guided, dynamo driven and magneto-rotational. In Ten chapters in turbulence, Davidson, P.A., Kaneda, Y. and Sreenivasan, K.R., eds, Cambridge University Press.

Zhdankin, V., Boldyrev, S., Perez, J. C. and Tobias, S.M., 2014, Energy dissipation in MHD turbulence: Coherent structures or nanoflares? Astrophys. J., 795, 127–135.

Part III:

Part III: Applications in Engineering and Materials

Chapter 10

Chapter 10 The World of Metallurgical MHD

Moffatt, H.K. and Proctor, M.R.E., 1984, Proceedings of the 1982 IUTAM Symposium on Metallurgical Applications of Magnetohydodynamics, The Metals Society, London.

Chapter 11

Chapter 11 The Generation and Suppression of Motion in Castings

Birat, J. and Chone, J., 1982, 4th International Iron & Steel Congress, London.

Davidson, P.A., 1992, Swirling flow in an axisymmetric cavity or arbitrary profile driven by a rotating magnetic field. J. Fluid Mech. 245: 660–99.

Davidson, P.A.,1995, Magnetic damping of jets and vortices. J. Fluid Mech., 299: 153.

Davidson, P.A., 1997, The role of angular momentum in MHD turbulence. J. Fluid Mech., 336: 123–50.

Davidson, P.A. and Hunt, J.C.R., 1987, Swirling, recirculating flow in a liquid metal column generated by a rotating magnetic field. J. Fluid Mech. 185: 67–106.

Marr, H.S., 1982, Electromagnetic stirring in continuous casting of steel. In Moffatt, H.K. and Proctor, M.R.E., Proc. metallurgical applications of MHD, The Metals Society.

Takeuchi, E., Masafumi, Z., Takehiko, T. and Mizoguchi, S., 1992, Applied MHD in the process of continuous casting. In Magnetohydrodynamics in process metallurgy, Szekely, J., Evans, J.W., Blazek, K. and El-Kaddah, N., The Minerals, Metals and Materials Soc. of USA.

Chapter 12

Chapter 12 Axisymmetric Flows Driven by the Injection of Current

Bojarevics, V., Freidbergs, Y., Shilova, E. I., and Shcherbinin, E.V., 1989, Electrically induced vortical flows. Kluwer.

Davidson, P.A. and Flood, S.C., 1994, Natural convection in an aluminium ingot: a mathematical model Metallurgical and materials trans. B., 25B, 293.

Davidson, P.A., Kinnear, D., Lingwood, R.J., Short, D.J. and He, X., 1999, The role of Ekman pumping and the dominance of swirl in confined flows driven by Lorentz forces. European J. Mech. B, 18, 693–711.

Kinnear, D. and Davidson, P.A. 1998. Forced recirculating flow J. Fluid Mech., 375, 319–344.

Shercliff, J.A., 1970, Fluid motion due to an electric current., J. Fluid Mech., 40, 241–249.

Chapter 13

Chapter 13 MHD Instabilities in Aluminium Reduction Cells

Bojarevics, V. and Romerio, M.V., 1994, Long wave instability of liquid metal-electrolyte interface in an aluminium electrolysis cells: A generalisation of Sele’s criterion. Eur. J. Mech. B, 13: 33–56.

Davidson, P.A., 2000, Overcoming instabilities in aluminium reduction cells: A route to cheaper aluminium. Materials Science and Technology, 16, 475–479.

Davidson, P.A and Lindsay, R.I., 1998, Stability of interfacial waves in aluminium reduction cells. J. Fluid Mech. 362, 273–295.

Sneyd, A.D. and Wang, A., 1994, Interfacial instability due to MHD mode coupling in aluminium reduction cells. J. Fluid Mech. 263, 343–359.

Part IV:

Part IV: Applications in Physics

Chapter 14

Chapter 14 The Geodynamo

Bin Baqui, Y. and Davidson, P.A., 2015, A phenomenological theory of rapidly rotating turbulence. Phys. Fluids, 27(2), 025107.

Christensen, U.R., 2010, Dynamo scaling laws and application to the planets. Space Sci. Rev. 152, 565–590.

Christensen, U.R., 2011, Geodynamo models: Tools for understanding properties of Earth’s magnetic field. Phys. of Earth and Planetary Interiors, 187, 157–169.

Christensen, U.R. and Wicht, J., 2007, Numerical dynamo simulations, In Treatise on geophysics, Olson, P., ed., Elsevier.

Cowling, T.G., 1934, The magnetic field of sunspots. Mon. Not. Roy. Astro. Soc., 94, 39–48.

Davidson, P.A., 2013, Turbulence in rotating, stratified and electrically conducting fluids, Cambridge University Press.

Davidson, P.A., 2014, The dynamics and scaling laws of planetary dynamos driven by inertial waves. Geophys. J. Int., 198(3), 1832–1847.

Davidson, P.A. and Ranjan, A., 2015, Planetary dynamos driven by helical waves: Part 2. Geophys. J. Int., 202, 1646–1662.

Gailitis, A., Lielausis, O., Platacis, E., Gerbeth, G. and Stefani, F., 2002, Laboratory experiments on hydromagnetic dynamos. Rev. Mod. Phys., 74, 973–990.

Jackson, J.D., 1998, Classical electrodynamics, 3rd ed., Wiley.

Jones, C.A., 2011, Planetary magnetic fields and fluid dynamos. Ann. Rev. Fluid Mech., 43, 583.

Lowes, F.J. and Wilkinson, I., 1963, Geomagnetic dynamo: A laboratory model. Nature, 198.

Moffatt, H.K., 1978, Magnetic field generation in electrically conducting fluids, Cambridge University Press.

Olson, P., Christensen, U.R. and Glatzmaier, G.A., 1999, Numerical modelling of the geodynamo: Mechanisms of field generation and equilibration. J. Geophys. Res., 104 (B5), 10383.

Parker, E.N., 1955, Hydromagnetic dynamo models. Astrophys. J., 122, 293–314.

Roberts, P.H. and King, E.M., 2013, On the genesis of the Earth’s magnetism. Rep. Prog. Phys., 76(9).

Sakuraba, A. and Roberts, P.H., 2009, Generation of a strong magnetic field using uniform heat flux at the surface of the core. Nature Geoscience, 2, 802–805.

Sreenivasan, B., 2010, Modelling the geodynamo: Progress and challenges. Perspectives in Earth Sciences, 99(12), 1739–1750.

Taylor, J.B., 1963, The magnetohydrodynamics of a rotating fluid and the Earth’s dynamo problem. Proc. Roy. Soc. A274, 274–283.

Veronis, G., 1959, Cellular convection with finite amplitude in a rotating fluid. J Fluid Mech., 5, 401–435.

Chapter 15

Chapter 15 Stellar Magnetism

Armitage, P.J., 2011, Dynamics of protoplanetary disks. Ann. Rev. Astron. Astrophys., 49, 195–236.

Balbus, S.A. and Hawley, J.F., 1998, Instability, turbulence and enhanced transport in accretion disks. Rev. Modern Phys., 70, 1–53.

Carroll, B.W. and Ostlie, D.A., 1996, Introduction to modern astrophysics, Addison Wesley.

Cravens, T.E., 1997, Physics of solar system plasmas, Cambridge University Press.

Davidson, P.A., 2013, Turbulence in rotating, stratified and electrically conducting fluids, Cambridge University Press.

Frank, J., King, A. and Raine, D., 2002, Accretion power in astrophysics, 3rd ed., Cambridge University Press.

Hughes, D.W., Rosner, W. and Weiss, N.O., eds, 2007, The solar tachocline, Cambridge University Press.

Mestel, L., 1999, Stellar magnetism, Oxford University Press.

Parker, E.N., 1993, A solar dynamo surface wave at the interface between convection and nonuniform rotation. Astrophys. J., 408, 707–719.

Priest, E., 2014, Magnetohydrodynamics of the Sun, Cambridge University Press.

Pringle, J.E. and King, A.R., 2007, Astrophysical flows, Cambridge University Press.

Shakura, N.I. and Sunyaev, R.A., 1973, Black holes in binary systems. Observational appearance. Astron. Astrophys., 24, 337–355.

Chapter 16

Chapter 16 Plasma Containment in Fusion Reactors

Biskamp, D., 1993, Nonlinear magnetohydrodynamics, Cambridge University Press.

Boyd, T.J.M. and Sanderson, J.J., 2003, The physics of plasmas, Cambridge University Press.

Bühler, L., 2007, Liquid metal magnetohydrodynamics for fusion blankets. In Magnetohydrodynamics, Molokov, S., Moreau, R., Moffatt, K., Springer.

Davidson, P.A., 1994, Global stability of two-dimensional and axisymmetric Euler flows. J. Fluid Mech., 276, 273–305.

Freidberg, J.P., 2014, Ideal MHD, Cambridge University Press.

Khan, R., Mizuguchi, N., Nakajima, N., Hayashi, T., 2007, Dynamics of the ballooning mode and the relationship to edge-localised modes in a spherical tokamak. Phys. Plasmas, 14, 062302.

Müller, U. and Bühler, L., 2001, Magnetofluiddynamics in channels and containers, Springer.

Introduction to Magnetohydrodynamics

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Contents

References

Part I: From Maxwell’s Equations to Magnetohydrodynamics

Chapter 1 A Qualitative Overview of MHD

Chapter 2 The Governing Equations of Electrodynamics

Chapter 3 A First Course in Fluid Dynamics

Part II: The Fundamentals of Incompressible MHD

Chapter 5 Kinematics: Advection, Diffusion, and Intensification of Magnetic Fields

Chapter 6 Dynamics at Low Magnetic Reynolds Numbers

Chapter 7 Dynamics at High Magnetic Reynolds Numbers

Chapter 8 An Introduction to Turbulence

Chapter 9 MHD Turbulence at Low and High Magnetic Reynolds Numbers

Part III: Applications in Engineering and Materials

Chapter 10 The World of Metallurgical MHD

Chapter 11 The Generation and Suppression of Motion in Castings

Chapter 12 Axisymmetric Flows Driven by the Injection of Current

Chapter 13 MHD Instabilities in Aluminium Reduction Cells

Part IV: Applications in Physics

Chapter 14 The Geodynamo

Chapter 15 Stellar Magnetism

Chapter 16 Plasma Containment in Fusion Reactors

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