Contents21 Nonlinear Dynamics of Plasmas 121.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2 Quasilinear Theory in Classical Language . . . . . . . . . . . . . . . . . . . . 221.2.1 Classical Derivation of the Theory . . . . . . . . . . . . . . . . . . . . 221.2.2 Summary of Quasilinear Theory . . . . . . . . . . . . . . . . . . . . . 921.2.3 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 921.2.4 Generalization to Three Dimensions . . . . . . . . . . . . . . . . . . . 1021.3 Quasilinear Theory in Quantum Mechanical Language . . . . . . . . . . . . 1221.3.1 Wave-Particle Interactions . . . . . . . . . . . . . . . . . . . . . . . . 1221.3.2 The relationship between classical and quantum mechanical formalismsin plasma physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1821.3.3 Three-Wave Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1921.4 Quasilinear Evolution of Unstable Distribution Functions: The Bump in Tail 2221.4.1 Instability of Streaming Cosmic R ays . . . . . . . . . . . . . . . . . . 2421.5 Parametric Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2621.6 Solitons and Collisionless Shock Waves . . . . . . . . . . . . . . . . . . . . . 280Chapter 21Nonlinear Dynamics of PlasmasVersion 0 821.2.K.pdf, 18 April 2009. Changes from 0 821.2.K.pdf a re only correction of a fewminor typographical errors.Box 21.1Reader’s Guide• This chapter relies significantly on:– Portions of Chap. 2 on kinetic theory: Secs. 2.2.1 and 2.2.2 on the distributionfunction, Sec. 2.2.5 on the mean occupation number, and Sec. 2.6 on Liouville’stheorem and the collisionless Boltzmann equation.– Section 1 8.3 on Debye shielding, collective behavior of plasmas and plasmaoscillations.– Sections 20.1–20.5 on kinetic theory of warm plasmas.• This chapter also relies to some extent but not greatly on:– The concept of spectral density as developed in Sec. 5.3.– Section 5.7 o n the Fokker-Planck equation.No subsequent material in this book relies significantly on this chapter.Please send comm ents, sugge s tion s, and errata via email to [email protected], o r onpaper to Kip Thorne, 130-33 Caltech, Pasadena CA 9112521.1 OverviewIn Chap. 20 we met o ur first example of a velocity space instability, the two stream instabil-ity, which illustrated the general principle that departures from Maxwellian equilibrium in12velocity space in a collisionless plasma might be unstable and lead to the exponential growthof small amplitude waves, just as we found can happen for departures from spatial unifor-mity in a fluid. In Chap. 21, where we analyzed warm plasmas, we derived the dispersionrelation for electrostatic waves in an unmagnetized plasma, we showed how Landau dampingcan damp the waves when the phase space density of the resonant particles diminishes withincreasing speed, and we showed that in the opposite case of an increasing phase space den-sity the waves can grow at the expense of the energies of near-resonant particles (providedthe Penrose criterion is satisfied). In this chapter, we shall explore the back-reaction of thewaves on the near-resonant particles. This back-reaction is a (weakly) nonlinear process, sowe shall have to extend our analysis of the wave-particle interactions to include the leadingnonlinearity.This extension is called quasi l i near theory or weak turbulence theory, and it allows us tofollow the time development of the waves and the near-resonant particles simultaneously.We develop this formalism in Sec. 21.2 and verify that it enforces the laws of particle con-servation, energy conservation, and momentum conservation. Our original development ofthe formalism is entirely in classical language and meshes nicely with the theory of elec-trostatic waves as presented in Chap. 21. In Sec. 21.3, we reformulate the theory in termsof the emission, absorption and scattering of wave quanta. Although waves in plasmas al-most always entail large quantum occupation numbers and thus are highly classical, thisquantum formulation of the classical theory has great computational and heuristic power(and as one would expect, despite the presence of Planck’s constant ~ in the formalism, ~nowhere appears in the final answers to problems). Our initial derivatio n and developmentof the formalism is restricted to the interaction of electrons with electrostatic waves, but wealso describe how the formalism can be generalized to describe a multitude of wave modesand particle species interacting with each other. We also describe circumstances in whichthis formalism can fail, and the resonant particles can couple strongly, not to a broad-banddistribution of incoherent waves (as the formalism presumes) but instead to o ne or a fewindividual, coherent modes. In Sec. 22.6 we explore an example.In Sec. 21.4 we turn to our first illustrative application of quasilinear theory: to a warmelectron beam propagating through a stable plasma. We show how the particle distributionfunction evolves so as to shut down the growth of the waves and we illustrate this by de-scribing the problem of the isotropization of Galactic cosmic rays. Next, in Sec. 2 1.5, weconsider parametric instabilities which are very important in the absorption of laser lightin experimental studies of the compression of small deuterium-tritium pellets - a possibleforerunner of a commercial nuclear f usion reactor. Finally, in Sec. 21.6 we return to ionacoustic solitons and explain how the introduction of dissipation can create a collisionlessshock, similar to that found where the earth’s bow shock meets the solar wind.21.2 Quasilinear Theory in Classical Language21.2.1 Classical Derivation of the TheoryIn Chap. 21 we discovered that a distribution of hot electrons or ions can Landau damp awave mode. We also showed that some distributions lead to exponential growth of the waves3in time. Either way there is energy transfer between the waves and the particles. We nowturn to the back-reaction of the waves o n the near-resonant particles that damp or amplifythem. For simplicity, we shall derive the back-reaction equations (“quasilinear theory”) inthe special case of electrons interacting with electrostatic Langmuir waves, and then shallassert the (rather obvious) generalization to protons or other ions and to interaction withother types of wave modes.We begin with the electrons’ one-dimensional
View Full Document
Unlocking...