2 - Magnetohydrodynamics  pp. 10-32

By Dieter Biskamp

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Magnetohydrodynamics, or MHD in short, describes the macroscopic behavior of an electrically conducting fluid – usually an ionized gas called a plasma –, which forms the basis of this book. By macroscopic we mean spatial scales larger than the intrinsic scale lengths of the plasma, such as the Debye length λD and the Larmor radii ρj of the charged particles. In this chapter we first derive, in a heuristic way, the dynamic equations of MHD and discuss the local thermodynamics (Section 2.1). Since most astrophysical systems rotate more or less rapidly, it is useful to write the momentum equation also in a rotating reference frame, where inertial forces appear (Section 2.2). Then some convenient approximations are introduced, in particular incompressiblity and, for a stratified system, the Boussinesq approximation (Section 2.3). In MHD theory the ideal invariants, i.e., integral quantities that are conserved in an ideal (i.e., nondissipative) system, play a crucial role in turbulence theory; these are the energy, the magnetic helicity, and the cross-helicity (Section 2.4). Though this book deals with turbulence, it is useful to obtain an quick overview of magnetostatic equilibrium configurations, which are more important in plasmas than stationary flows are in hydrodynamics (Section 2.5). Also the zoology of linear modes, the small-amplitude oscillations about an equilibrium, is richer than that in hydrodynamics (Section 2.6). Finally, in Section 2.7 we introduce the Elsässer fields, which constitute the basic dynamic quantities in MHD turbulence. In this chapter we write the equations in dimensional form, using Gaussian units, to emphasize the physical meaning of the various terms.