J TRITA-EPP-72-23 BASIC FEATURES OF PLASMA INSTABILITIES B. Lehnert Stockholm, October 22, 1972 y ics and Fusion RoyaRov^Tnc^l Institute ?of Technolog^ l y Research 100 Ht* Stockholm 70, Sweden 1. BASIC FEATURES OF PLASMA INSTABILITIES B. Lehnert Royal Institute of Technology, S-10044 Stockholm 70, Sweden ABSTRACT A review is given on the present stat^i of research on plasma instabilities: (1) The basic characteristics of various instability phenomena .v own so far are outlined, including the methods of the underlying analysis, the general physical properties of unstable modes, the corresponding energy sources and coupling mechanisms, as well as applied stabilization methods. (2) A classification scheme is developed for a systematic subdivision of existing types of plasma disturbances, including both stable and unstable modes. In addition, a general description is given of each instability mode being detected so far, (3) The present state of research on plasma instabilities is summa- rized, including discussions both on linear and nonlinear modes and on associated loss mechanisms. The established classification scheme is applied in a systematic search for instability phenomena which so far have remained undiscovered. CONTENTS 1. Introduction 1 2. Basic theoretical means 2 2.1. Basic equations 2 2.1.1. Particle orbit theory 2 2.1.2. Kinetic theory 2.1.3. Macroscopic fluid theory 4 2.2. The unperturbed state 4 2.2.1. Stationary states 5 2.2.1.1. Static states 5 2.2.1.2. Non-static states 5 2.2.2. Non-stationary states 6 2.2.2.1. Laminar states 6 2.2.2.2. Non-laminar states 6 2.3. The perturbed state 6 2.3.1. Energy sources 6 2.3.1.1. Intrinsic sources of the plasma 7 2.3.1.2. Externally imposed force fields 8 2.3.1.3. External injection of energy 8 2.3.2. Coupling mechanisms 8 2.3.3. The stages of linear and nonlinear 10 analysis 2.3.4. Methods of theoretical analysis of 10 plasma disturbances 2.3.4.1. Particle orbit analysis 11 2.3.4.2. Linear normal mode analysis 11 2 .3.4.3. Nonlinear mode analysis 12 2.3.4.4. The energy principle 13 2.3.4.5. Computer analysis 13 3. General properties of plasma instabilities 14 3.1. Definitions of instability 14 3.1.1. Singe-stage processes 14 3.1.1.1. Onset of instabilities at small 14 ampl5.tudes 3.1.1.2. No onset of instabilities at small 15 amplitudes 3.1.2. Multi-stage processes 15 3.2. External effects on plasma instabilities 15 3.3. Relativistic effects 16 3.4. Linear instability properties 16 3.4.1. The restoring forces 16 3.4.2. Unstable oscillations and waves 17 3.4.2.1. Absolute and convective 17 instabilities 3.4.2.2. Negative energy waves 18 3.4.2.3. Plasma oscillations and Bernstein 19 waves 3.4.2.4. Velocity space effects 19 3.4.2.5. Parametric instabilities 19 3.4.3. Particle-wave interaction 19 3.4.3.1. Trapping instabilities 20 3.4.3.2. Drift instabilities 20 3.4.3.3. Resonance phenomena in general 21 Ill 3.4.4. Collective modes 21 3.U.U.I. Untrapped particle instabilities 21 3.4.4.2. Trapped particle instabilities 21 3.4.4.3. Resonance phenomena of collective 22 modes 3.5. Nonlinear instability properties 22 3.5.1. Nonlinear mode interaction 22 3.5.1.1. Resonant interaction 23 3.5.1.2. Non-resonant interaction 24 3.5.2. Plasma turbulence 24 Stabilization mechanisms 26 4.1. Internal plasma mechanisms 27 4.1.1. Linear mechanisms 27 4.1.1.1. "Hard" mechanisms 27 4.1.1.2. "Soft" mechanisms 28 4.1.2. Nonlinear mechanisms 29 4.2. Effects due to the boundary conditions 29 4.3. Externally applied mechanisms 30 4.3.1. Stationary mechanisms 30 4.3.2. Non-stationary mechanisms 30 4.3.2.1. Dynamic stabilization 30 31 4.3.2.2. Feedback stabilization 32 Classification of plasma disturbances 32 5.1. The rotation 32 5.2. The classification scheme External space effects 33 Relativistic modes 33 Collisional phenomena 34 The macroscopic motion in the 34 unperturbed state The unperturbed state in velocity 35 space The externally imposed magnetic field 36 Electromagnetic induction effects 37 Collective effects 37 37 5.2.10. Resonance effects 38 5.2.11. The disturbance amplitudes 39 5.2.12. The growth of the disturbances 39 6. A survey of instability subclasses 40 7. Suggestions for discoveries of new phenomena 41 7.1. Instabilities 41 7.1.1. Feedback destabilization 41 7.1.2. Relativistic instabilities 41 7.1.3. Collisional non-resonant effects 42 7.1.4. Non-static state phenomena 42 IV. 7.1.5. Magnetic induction effects 42 7.1.5.1. High-beta phenomena 43 7.1.5.2. Low-bera phenomena 43 7.1.6. Collective effects 43 7.1.7. Resonance phenomena 44 7.1.7.1. Localized phenomena 44 7.1.7.2. Collective phenomena 45 7.1.8. Nonlinear effects 45 7.1.9. Plasma interaction with other 45 immersed media 7.1.10. Ambiplasma effects 45 7.2. Stabilization effects 46 8. Conclusions 47 9. References 48 9.1. General references on plasma disturbance 48 phenomena 9.1.1. Plasma instabilities 48 9.1.2. Plasma waves 49 9.2. References to special classes and 49 properties of plasma instabilities 9.2.1. Absolute and convective instabilities 49 9.2.2. Relativistic instabilities 50 9.2.3. Resistive instabilities 50 9.2.4. Drift instabilities 50 9.2.5. Two-stream and beam-plasma 51 instabilities 9.2.6. Collective instabilities 51 9.2.7. Trapped particle instabilities 51 9.2.8. Cyclotron instabilities 51 9.2.9. Parametric instabilities 51 9.2.10. Nonlinear phenomena and turbulence 51 9.2.11. Moving striations 52 9.2.12. Numerical simulation by means of 52 computers 9.3. Stabilization methods 53 9.3.1. Dynamic stabilization 53 9.3.2. Feedback stabilization 53 9.4. Special references 53 10. Index of instabilities 55 10.1, Classification index 55 10.2. Name index 62 Tables on instability subclasses T1-T20 Figure 1. Simple mechanical analogies of Fl stability problem. Figure 2. Classification scheme of plasma F2 instabilities. Figure 3. General division in subclasses. F3 Appendix. Basic properties of individual A0-A127 instability modes and subclasses 1. 1. INTRODUCTION A magnetized plasma constitutes a complex system with many degrees of freedom, within which various energy forms interact with each other. Such an interaction is provided by the numerous types of coupling mechanisms being involved in the dynamics of plasma distur- bances. As a result, the latter may develop either into instabilities or into wave phenomena and other stable perturbations. Since the end of the 1950:s, the number of detected basic types of plasma instabilities has been steadily increasing, up to the present date where most elementary modes appear to have been explored. This development has earlier been summarized in a number of general reviews [jL - SFJ of which the contribution by F. Cap and collaborators dl»?Zl represents the most extensive and updated one. The latter in- cludes abstracts of 5112 papers as well as extensive lists of references. Reviews on specific classes of instabilities have further been given by a number of investigators Q-J> - B9^\ . The large efforts made in the study of plasma instabilities are mainly due to their importance to plasma confinement in fusion research. Nevertheless there is also a widespread general interest in such phenomena within basic plasma physics, as well as in other fields of application such as cosmical and space physics, magnetohydrodynamic power generation, and in the fields represented by some special tech- nical problems. Thus, in addition to the confinement problem, the instability analysis plays a central role in the investigations of collisionless heating mechanisms, collisionless shock transitions, anomalous transport phenomena, nonlinear plasma phenomena, and turbu- lence. Within the various fields of application just mentioned, there is further a considerable interest in plasma wave phenomena, as des- cribed in several contexts J£lO - 1*C] • This review is mainly con- centrated on the basic features of plasma instabilities. Plasma wave phenomena will only be treated as far as they are connected with growing unstable plasma disturbances. We shall further restrict ourselves to pure plasma *phenomena; i,e. the semiconductor effects earlier reviewed by Hartnagel C5~j are not included in the present discussions. The review serves four purposes, namely to outline the basic physical features of plasma instabilities, to make an attempt to systematize the rich flora of various types of existing modes, to summarize the present state of research on instabilities, and to give hints to the possible detection of new modes and plasma disturbance phenomena. 2. BASIC THEORETICAL MEANS I i Starting with the basic means of plasma theory, we shall now j attempt a systematic study of plasma disturbances and their associated I physical effects. i i 2.1. Basic Equations The electromagnetic field is governed by Maxwell's equations, where curlE = - 3B/3t ; divB = 0 (1) represents the law of electromagnetic induction of the electric and magnetic fields £ and JB, and curlB/v = j + £ 3E/3t (2) — o **- o — expresses the magnetic field in terms of its sources. Here j[ denotes the electric current density and p and e the magnetic permeability and dielectric constant in vacuo, given in MKS units. Conservation of the electric charge density a further yields divi = - 3a/3t ; divE = a/eQ (3) The Equations (1) - (3) of the electromagnetic field have to be coupled with those determining the motion of the charged particles in the plasma.