Contents19 Waves in Cold Plasmas: Two-Fluid Formalism 119.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119.2 D ielectric Tensor, Wave Equation, and General D ispersion Relation . . . . . 319.3 Wave Modes in an Unmagnetized Plasma . . . . . . . . . . . . . . . . . . . . 519.3.1 Two-Fluid Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 519.3.2 Dielectric Tensor and Disp ersion Relation for a Cold Plasma . . . . . 619.3.3 Electromagnetic Plasma Waves . . . . . . . . . . . . . . . . . . . . . 819.3.4 Langmuir Waves and Ion Acoustic Waves in Warm Plasmas . . . . . 919.3.5 Cutoffs and Resonances . . . . . . . . . . . . . . . . . . . . . . . . . 1319.4 Wave Modes in a Cold, Magnetized Plasma . . . . . . . . . . . . . . . . . . 1619.4.1 Dielectric Tensor and Disp ersion Relation . . . . . . . . . . . . . . . 1619.4.2 Parallel Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1719.4.3 Perpendicular Propagation . . . . . . . . . . . . . . . . . . . . . . . . 2019.5 Propagat io n of Radio Waves in the Ionosphere . . . . . . . . . . . . . . . . . 2219.6 CMA Diagram for Wave Modes in Cold, Magnetized Plasma . . . . . . . . . 2519.7 Two Stream Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280Chapter 19Waves in Cold Pl a s mas: Two-FluidFormalismVersion 08 19.1.K.pdf, 0 8 April 2009.Please send comments, suggestion s, and errata via emai l to ki [email protected]. edu or on paperto Kip Thorne, 130-3 3 Caltech, Pasadena CA 91125Box 19.1Reader’s Guide• This chapter relies significantly on:– Chapter 19 on the particle kinetics of plasmas.– The basic concepts of fluid mechanics, Secs. 12.4 and 12.5.– Magnetosonic waves, Sec. 1 7.7.– The basic concepts of geometric optics, Secs. 6.2 and 6.3• The remaining chapters 21 and 22 of Part V, Plasma Physics, rely heavily on thischapter.19.1 Ove rviewThe g rowth of plasma physics came about historically through the studies of oscillations inelectric discharges a nd the contemporaneous development of the means to broadcast radiowaves over increasing distances by reflecting them off a layer of partially ionized gas in theupper atmosphere known as the ionosphere. It is therefore not surprising that most earlyresearch was devoted to describing the different modes of wave propagation. Even in thesimplest, linear approximation, we will see that the variety of possible modes is immense. Inthe previous section, we have introduced several length and time scales, e.g. the gyro radius12and Debye length, the plasma period, the gyro period and the collision frequency. To thesemust now be added the wavelength a nd perio d of the wave under study. The relative orderingof these different scales controls the characteristics of the waves, and it is already apparentthat there are a bewildering number of possibilities. If we further recognize that plasmasare collisionless and that there is no guarantee that the particle distribution functions canbe characterized by a single temperature, then the possibilities multiply.Fortunately, the techniques needed to describe the propagation of linear wave pertur-bations in a particular equilibrium configuration of a plasma are straightforward and canbe amply illustrated by studying a few simple cases. In this section, we shall follow thiscourse by restricting our attention to one class of modes, those where we can either ignorecompletely the thermal motions of the ions a nd electrons that comprise the plasma (in otherwords treat these species as co l d ) or include them using just a velocity dispersion or temper-ature. We can then a pply our knowledge of fluid dynamics by treating the ions and electronsseparately as fluids, upon which act electromagnetic for ces. This is called the two-fluid for-malism for plasmas. In the next chapter, we shall show when and how waves are sensitive tothe actual distribution of particle speeds by developing the more sophisticated kinetic-theoryformalism and using it to study waves in warm plasmas.We begin our two-fluid study of plasma waves in Sec. 19.2 by deriving a very general waveequation, which governs weak waves in a homogeneous plasma that may or may not havea magnetic field, and also governs electromagnetic waves in any other dielectric medium.That wave equation and the associated disp ersion relation for the wave modes depend on adielectric tensor, which must be derived from an examination of the motion of the electronsand proto ns (or other charge carriers) inside the wave.In Sec. 19.3 we sp ecialize to wave modes in a uniform, unmagnetized plasma. Usinga two-fluid (electron-fluid and proton-fluid) description of the charge-carriers’ motio ns, wederive the dielectric tensor and thence the dispersion relation for the wave modes. Themodes fall into two classes: (i) Transve rs e or electromagn e tic modes. These are modifiedversions of electromagnetic waves in vacuum. As we shall see, they can propagate only atfrequencies above the plasma frequency; at lower frequencies t hey become evanescent. (ii)Longitudinal waves, which come in two species: Langmuir waves and ion acoustic waves.Longitudinal waves are a melded combination of sound waves in a fluid and electrostaticplasma oscillations; their restoring force is a mixture o f thermal pressure and electrostaticforces.In Sec. 19.4, we explore how a uniform magnetic field changes the character of thesewaves. The B field makes the plasma anisotropic but axially symmetric. As a r esult, thedielectric tensor, dispersion relation, and wave modes have much in common with those inan anisotropic but axially symmetric dielectric crystal, which we studied in the context ofnonlinear optics in Chap. 9. A plasma, however, has a much richer set of characteristicfrequencies than does a crystal (electron plasma frequency, electron cyclotron frequency, ioncyclotron frequency, ...). As a result, even in the regime of weak linear waves and a coldplasma (no thermal pressure), the plasma has a far greater richness of modes than does acrystal.In Sec. 19.4 we derive the general dispersion relation that encompasses all of these cold-magnetized-plasma modes, and explore the special cases of modes that propagate parallel to3and perpendicular to the magnetic field. Then in Sec. 19.5 we examine a practical example:the propagation of radio waves in the Earth’s ionosphere, where it is a good approximation toignore the ion motion and work with a one-fluid (i.e.
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