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Plasma Physics

by Abhishek Apoorv
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Contributors

Abhishek Apoorv
Abhishek Apoorv
Lecture Notes in Physics Introduction to Plasma Physics Michael Gedalin
Contents 1 Basic definitions and parameters 1.1 What is plasma . . . . . . . . . 1.2 Debye shielding . . . . . . . . . 1.3 Plasma parameter . . . . . . . . 1.4 Plasma oscillations . . . . . . . 1.5 Ionization degree . . . . . . . . 1.6 Coulomb collisions . . . . . . . 1.7 Summary . . . . . . . . . . . . 1.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 4 4 5 6 8 8 2 Plasma description 2.1 Hierarchy of descriptions . . . . . . . . . . . . . . . 2.2 Fluid description . . . . . . . . . . . . . . . . . . . 2.3 Mass conservation - continuity equation . . . . . . . 2.4 Momentum conservation - motion (Euler) equation 2.5 State equation . . . . . . . . . . . . . . . . . . . . . 2.6 Energy conservation . . . . . . . . . . . . . . . . . 2.7 MHD . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Order-of-magnitude estimates . . . . . . . . . . . . 2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . 2.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 9 9 10 11 12 12 13 14 15 15 . . . . . . . 17 17 18 19 22 22 23 23 . . . . . . 25 25 26 26 27 27 31 . . . . . . . . . . . . . . . . . . . . . . . . 3 MHD equilibria and waves 3.1 Magnetic field diffusion and dragging 3.2 Equilibrium conditions . . . . . . . . 3.3 MHD waves . . . . . . . . . . . . . . 3.4 Alfven and magnetosonic modes . . . 3.5 Wave energy . . . . . . . . . . . . . . 3.6 Summary . . . . . . . . . . . . . . . 3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 MHD discontinuities 4.1 Stationary structures . . . . . . . . . . 4.2 Discontinuities . . . . . . . . . . . . . . 4.2.1 No-flow discontinuities . . . . . 4.2.2 Alfven (rotational) discontinuity 4.3 Shocks . . . . . . . . . . . . . . . . . . 4.4 Why shocks ? . . . . . . . . . . . . . . iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CONTENTS 4.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5 Two(multi)-fluid description 5.1 Basic equations . . . . . . . . . . . . . 5.2 No magnetic field case . . . . . . . . . 5.2.1 Small-amplitude (linear) waves 5.3 Nonlinear ion-acoustic waves . . . . . . 5.3.1 Stationary waves . . . . . . . . 5.4 Time-dependent nonlinear waves . . . 5.5 Reduction to MHD . . . . . . . . . . . 5.6 Generalized Ohm’s law . . . . . . . . . 5.7 Nonlinear magnetosonic waves . . . . . 5.7.1 Time-dependent waves . . . . . 5.7.2 Magnetosonic soliton . . . . . . 5.8 Problems . . . . . . . . . . . . . . . . . 6 Waves in dispersive media 6.1 Maxwell equations for waves . 6.2 Wave amplitude, velocity etc. 6.3 Wave energy . . . . . . . . . . 6.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . 7 Waves in two-fluid hydrodynamics 7.1 Dispersion relation . . . . . . . . . 7.2 Unmagnetized plasma . . . . . . . 7.3 Parallel propagation . . . . . . . . 7.4 Perpendicular propagation . . . . . 7.5 General properties of the dispersion 7.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Kinetic theory 8.1 Distribution function . . . . . . . . . . . . . . . 8.2 Kinetic equation . . . . . . . . . . . . . . . . . 8.3 Relation to hydrodynamics . . . . . . . . . . . . 8.4 Dielectric tensor without external magnetic field 8.5 Waves . . . . . . . . . . . . . . . . . . . . . . . 8.6 Landau damping . . . . . . . . . . . . . . . . . 8.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 33 34 34 36 36 37 37 38 39 39 40 41 . . . . 43 43 44 46 49 . . . . . . 51 51 53 53 55 56 56 . . . . . . . 59 59 60 61 61 63 63 66 9 Micro-instabilities 67 9.1 Beam (two-stream) instability . . . . . . . . . . . . . . . . . . . . . . 67 9.2 More on the beam instability . . . . . . . . . . . . . . . . . . . . . . 68 9.3 Bump-on-tail instability . . . . . . . . . . . . . . . . . . . . . . . . . 68 10 ∗ Nonlinear phenomena∗ 71 A Plasma parameters 73 iv
Chapter 1 Basic definitions and parameters In this chapter we learn in what conditions a new state of matter - plasma - appears. 1.1 What is plasma Plasma is usually said to be a gas of charged particles. Taken as it is, this definition is not especially useful and, in many cases, proves to be wrong. Yet, two basic necessary (but not sufficient) properties of the plasma are: a) presence of freely moving charged particles, and b) large number of these particles. Plasma does not have to consists of charged particles only, neutrals may be present as well, and their relative number would affect the features of the system. For the time being, we, however, shall concentrate on the charged component only. Large number of charged particles means that we expect that statistical behavior of the system is essential to warrant assigning it a new name. How large should it be ? Typical concentrations of ideal gases at normal conditions are n ∼ 1019 cm−3 . Typical concentrations of protons in the near Earth space are n ∼ 1 − 10cm−3 . Thus, ionizing only a tiny fraction of the air we should get a charged particle gas, which is more dense than what we have in space (which is by every lab standard a perfect vacuum). Yet we say that the whole space in the solar system is filled with a plasma. So how come that so low density still justifies using a new name, which apparently implies new features ? A part of the answer is the properties of the interaction. Neutrals as well as charged particles interact by means of electromagnetic interactions. However, the forces between neutrals are short-range force, so that in most cases we can consider two neutral atoms not affecting one another until they collide. On the other hand, each charged particle produces a long-range field (like Coulomb field), which can affect many particles at a distance. In order to get a slightly deeper insight into the significance of the long-range fields, let us consider a gas of immobile (for simplicity) electrons, uniformly distributed inside an infinite cone, and try to answer the question: which electrons affect more the one which is in the cone vertex ? Roughly speaking, the Coulomb force acting on the chosen electron from another one which is at a distance r, is inversely proportional to the distance squared, fr ∼ 1/r2 . Since the number of electrons which are at this distance, Nr ∝ r2 , the total force, Nr fr ∼ r0 , is distance independent, which means that that electrons which are very far away are of equal importance as the electrons which are very close. In other 1

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