Contents18 Magnetohydrodynamics 118.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118.2 Basic Equations of MHD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.2.1 Maxwell’s Equations in MHD Approximation . . . .......... 418.2.2 Momentum and Energy Conservation . . . . . . . . . . . . . . . . . . 718.2.3 Boundary Conditions ........................... 1018.2.4 Magnetic field and vorticity . . . . . . . . . . . . . . . . . . . . . . . 1118.3 Magnetostatic Equilibria ............................. 1318.3.1 Controlled thermonuclear fusion . . . . . . . . . . . . . . . . . . . . . 1318.3.2 Z - Pinch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518.3.3 Θ Pinch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.3.4 Tokamak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718.4 Hydromagnetic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1818.5 Stability of Hydromagnetic Equilibria . ..................... 2218.5.1 Linear Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . 2218.5.2 Z-Pinch; Sausage and Kink Instabilities ................. 2518.5.3 Energy Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2718.6 Dynamos and Reconnection of Magnetic Field Lines . . . . . . . . . . . . . . 2918.6.1 Cowling’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2918.6.2 Kinematic dynamos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3018.6.3 Magnetic Reconnection . . . . . . . . . . . . . . . . . . . . . . . . . . 3118.7 Magnetosonic Waves and the Scattering of Cosmic Rays . . . . . . . . . . . 3318.7.1 Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3318.7.2 Magnetosonic Dispersion Relation . . . . . . . . . . . . . . . . . . . . 3418.7.3 Scattering of Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . 360Chapter 18MagnetohydrodynamicsVersion 1018.1.K.p df, 11 March 2009.Please send comments, suggestions, and errata via email to [email protected] or on paper toKip Thorne, 350-17 Caltech, Pasadena CA 91125Box 18.1Reader’s Guide• This c hapter relies heavily on Chap. 12 and somewhat on the treatmen t of v orticitytransport in Sec. 13.2• Part V, Plasma Physics (Chaps. 18-21) relies heavily on this chapter.18.1 OverviewIn preceding chapters, we have described the consequences of incorporating viscosity andthermal conductivity into the description of a fluid. We now turn to our final embellishmentof fluid mechanics, in which the fluid is electrically conducting and moves in a magneticfield. The study of flows of this type is known as Magnetohydrodynamics or MHD for short.In our discussion, we eschew full generality and with one exception just use the basic Eulerequation augmented by magnetic terms. This suffices to highlight peculiarly magnetic effectsand is adequate for many applications.The simplest example of an electrically conducting fluid is a liquid metal, for example,mercury or liquid sodium. However, the major use of MHD is in plasma physics. (A plasma isahot,ionizedgascontainingfreeelectronsandions.) Itisbynomeansobviousthatplasmascan be regarded as fluids since the mean free paths for collisions between the electrons andions are macroscopically long. However, as we shall learn in Part V (Sec. 19.5 and Chap. 21),collective interactions between large numbers of plasma particles can isotropize the particles’velocity distributions in some local mean reference frame, thereby making it sensible todescribe the plasma macroscopically by a mean density, velocity, and pressure. These mean12quantities can then be shown to obey the same conservation la ws of mass, momentum andenergy, as we derived for fluids in Chap. 12. As a result, a fluid description of a plasmais often reasonably accurate. We defer to Part V further discussion of this point, askingthe reader to take this on trust for the moment. We are also, implicitly, assuming thatthe average velocity of the ions is nearly the same as the average velocity of the electrons.This is usually a good approximation; if it were not so, then the plasma would carry anunreasonably large current density.There are two serious technological applications of MHD that may become very importantin the future. In the first, strong magnetic fields are used to confine rings or columns of hotplasma that (it is hoped) will be held in place long enough for thermonuclear fusion to occurand for net power to be generated. In the second, which is directed toward a similar goal,liquid metals are driven through a magnetic field in order to generate electricity. The studyof magnetohydrodynamics is also motivated by its widespread application to the descriptionof space (within the solar system) and astrophysical plasmas (beyond the solar system). Weshall illustrate the principles of MHD usingexamplesdrawnfromeachoftheseareas.After deriving the basic equations of MHD (Sec. 18.2), we shall elucidate hydromagneticequilibria by describing a Tokamak (Sec. 18.3). This is currently the most popular schemefor magnetic confinement of hot plasma. In oursecondapplication(Sec.18.4)weshalldescribe the flow of conducting liquid metals or plasma along magnetized ducts and outlineits potential as a practical means of electrical power generation and spacecraft propulsion.We shall then return to the question of hydromagnetic confinement of hot plasma and focuson the stability of equilibria (Sec. 18.5). This issue of stability has occupied a central placein our development of fluid mechanics and it will not come as a surprise to learn thatit has dominated research into plasma fusion. When a magnetic field plays a role in theequilibrium (e.g. for magnetic confinement of aplasma),thefieldalsomakespossiblenewmodes of oscillation and some of these modes can be unstable to exponential growth. Manymagnetic confinement geometries are unstabletoMHDmodes. Weshalldemonstratethisqualitatively by considering the physical actionofthemagneticfield,andalsoformallyusingvariational methods.In Sec. 18.6 …
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