Contents18 Magnetohydrodynamics 118.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118.2 Basic Equations of MHD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.2.1 Maxwell’s Equations in the MHD Approximation . .......... 418.2.2 Momentum and Energy Conservation . . . . . . . . . . . . . . . . . . 818.2.3 Boundary Conditions ........................... 1018.2.4 Magnetic field and vorticity . . . . . . . . . . . . . . . . . . . . . . . 1218.3 Magnetostatic Equilibria ............................. 1418.3.1 Controlled thermonuclear fusion . . . . . . . . . . . . . . . . . . . . . 1418.3.2 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2818.6 Dynamos and Reconnection of Magnetic Field Lines . . . . . . . . . . . . . . 3018.6.1 Cowling’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3018.6.2 Kinematic dynamos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3118.6.3 Magnetic Reconnection . . . . . . . . . . . . . . . . . . . . . . . . . . 3118.7 Magnetosonic Waves and the Scattering of Cosmic Rays . . . . . . . . . . . 3318.7.1 Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3418.7.2 Magnetosonic Dispersion Relation . . . . . . . . . . . . . . . . . . . . 3418.7.3 Scattering of Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . 370Chapter 18MagnetohydrodynamicsVersion 1118.1.K.p df, 4 March 2012.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. 13 and somewhat on the treatmen t of v orticitytransport in Sec. 14.2• Part VI, Plasma Physics (Chaps. 19-22) 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 examplemercury or liquid sodium. However, the major application of MHD is in plasma physics.(A plasma is a hot, ionized gas containing free electrons and ions.) It is by no meansobvious that plasmas can be regarded as fluids, since the mean free paths for Coulomb-forcecollisions between the electrons and ions are macroscopically long. However, as we shall learnin Sec. 19.5, collective interactions between large numbers of plasma particles can isotropizethe particles’ velocity distributions in some local mean reference frame, thereby making itsensible to describe the plasma macroscopically by a mean density, velocity, and pressure.12These mean quantities can then be shown to obey the same conservation laws of mass,momentum and energy, as we derived for fluids in Chap. 13. As a result, a fluid descriptionof a plasma is often reasonably accurate. We defer to Part VI further discussion of thispoint, asking the reader to take it on trust for the moment. In MHD, we also, implicitly,assume that the average velo city of the ions is nearly the same as the average velo city of theelectrons. This is usually a good approximation; if it were not so, then the plasma wouldcarry an unreasonably 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 or plasmas are driven through a magnetic field in order to generate electricity.The study of magnetoh ydrodynamics is also motivated b y its widespread application to thedescription of space (within the solar system) and astrophysical plasmas (beyond the solarsystem). We shall illustrate the principles ofMHDusingexamplesdrawnfromeachoftheseareas.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 MHD modes can be unstable to exponential …
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