Basic Features of Plasma Instabilities

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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 summarized, 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 disturbances. 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 includes 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 technical 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 turbulence. Within the various fields of application just mentioned, there is further a considerable interest in plasma wave phenomena, as described 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. We restrict ourselves in this Section to the non- relativistic case, keeping in mind that the present equations can easily be recast in an equivalent relativistic form. There are in principle three approaches leading to closed sets of equations, as described in the following paragraphs (see also Table 1).

2.1.1. Particle Orbit Theory The first approach consists of using directly the exact equation

of motion of single particles of each species (v> as given by v

nyiw/dt = ^ + qv(E + w^ x B)

where m , q , and w are the mass, charge, and velocity of a particle, and ? is the corresponding external force of non-electromagnetic 3. origin. This approach has the advantage of being exact, but is at the same time encumbered with the drawback of leading to extremely complicated orbits and calculations which in most cases cannot be mastered even by excessive amounts of work. A significant simplification is achieved by means of Alfven's first order orbit theory of the guiding centre motion. This leads to a better insight in the physical behaviour of the plasma, and sometimes ' also to final solutions of sufficiently good accuracy. However, care is necessary when orbit theory is applied to a quasi-neutral plasma. Even the smallest differential motions between ions and electrons, due to terms which might appear to be of orders high enough to be neglected, can give rise to charge separation effects and electric fields of crucial importance to the dynamical plasma behaviour.

2.1.2. Kinetic Theory The second approach is represented by kinetic theory in which a density distribution function f(£,w,t) is defined in phase space and 0_ denotes the position vector. For the particle species ( ) this leads to the Boltzmann-Vlasov equation

3f —- + w -Vf + i-l F + q (EE +• w , x B)) JI «.V Vfv ==1 -^r— I ,, (5) 3t \ where 7 indicates the gradient in velocity space, and the right-hand member is due-to collisions. Further, the electric space charge is given by

• « > Q.. W VWvxdwvydwvz and the electric current density by

f f dw dw dw %JjJJjJ vvHH v vx vy v2 This approach makes possible the treatment of quasi-neutral plasmas under rather general conditions and in a much more rigorous way than by first order orbit theory. Still there are assumptions underlying Eq. (5) which sometimes may restrict its applicability. These are partly due to the fact that the N-particle distribution in 6N-dimensional phase space cannot always be factored into a product of single particle distribution functions f in 6-dimensional (£,w) space 2.1.3. Macroscopic Fluid Theory The third approach consists of taking the moments in velocity space of the Boltzmann-Vlasov equation, and "cutting off" this pro- cedure at some stage to form a closed system of equations. If the cut-off is performed to give the first three conservation equations, we obtain

n m i- v • = n r ^ v v —v v;v v l_~ v ^v — —V

-div TT + M. - m P v (9) = V ~v v v-v 3 3Pv + liv(p v } + div £ H 2 3t 2* v -V for *:he conservation of matter, momentum, and heat of a nearly isotropic plasma. In eqs. (8)-(10), n is the particle density of the vth species, v the corresponding macroscopic fluid velocity, 3 the pressure tensor with p as its scalar part, (£ the heat-flow vector, and P , M and H represent rates of gain of particles, momentum, and heat due to various collisional processes. When the anisotropy of the plasma cannot be neglected, part of the terms in Eqs. (9) and (10) have to be modified accordingly. In the particular case of a collisionless plasma, the Chew-Goldberger-Low approximation can sometimes be used, in which the pressure tensor is represented by two scalar pressures, p, and p , parallel and perpendicular to the magnetic field B. The macroscopic fluid model of Eqs. (8)-(10) has the advantage of being simpler than that of the kinetic theory. It still becomes more rigorous than first order orbit theory, especially when finite Larmor radius effects are included in the pressure tensor. Its main drawbacks are due to the fact that most of the information on the motions in velocity space has been eliminated by taking the moment integrals of Eq. (5).

2.2. The Unperturbed State

Before looking into the dynamics of the plasma disturbances, the corresponding unperturbed state has to be specified. Without clear definitions of such a state, the investigations oi plasma disturbances may even lose their physical meaning. This is e.g. the case when 5. attempts are made to discuss instabilities in a system where there does not exist any defined plasma equilibrium. In the extreme and trivial sense, the only equilibrium state which finally establishes itself after an infinitely long time, is that of a plasma within which all motions produced by imposed driving forces neve been relaxed. The plasma then fills the entire available coordinate space, and has reached a Maxwellian distribution in velocity space. However, in practice there are unperturbed states of a plasma within a finite volume of phase space, having prescribed patterns of motion and velocity distributions lasting long enough to represent a well-defined "equilibrium". The possible types of such states are described in the following paragraphs.

2.2.1. Stationary States A large group of unperturbed states discussed so far is characterized by the condition of explicit time-independence, i.e. 3/3t=0. These stationary states consist of two subclasses, namely:

2.2.1.1. §tatic_states There are static states, including no macroscopic fluid motions of ions and electrons, except those being necessary to produce electric currents for balancing the pressure gradients and other forces.

2.2.1.2. Non-static_states

There are non-static states, including macroscopic motions in excess of those required to balance the forces just mentioned. Here there is again a subdivision, namely into those states having a centre-of-mass velocity v = (n.m.v.+nÄmAv )/(n-m.+n m ), and those J —o i l—i e e—e i i e e where there is no such velocity but an extra imposed electric current j=e(n.v.-n y). In this connection should be stressed that it is not sure that a "local static equilibrium1' defined by vsO always exists. To make a plasma equilibrium possible in non-uniform magnetic configurations, it may namely become necessary for st to differ locally from zero C6O • Ohm's generalized low further shows that y^ and ^ cannot vanish simultaneously in the unperturbed state, unless the lowest order approximation E^+vxB=0is adopted and the pressure gradients and other higher order effects are neglected. In addition, it is not sure that putting the unperturbed electric field E^ = 0 always corresponds to a physically relevant equilibrium state. Even small space charges within the plasma may produce local fields E which cannot be short-circuited, and which may contribute to or cancel part of the fluid motion in the unperturbed state, by means of the corresponding E- x B drift.

2.2.2. Non-Stationary States There are time-dependent states defined by &/at^0 in the laboratory frame. They consist of two subclasses, namely:

2.2.2.1. Laminar_States The first class is defined by a time-dependent motion which is laminar and regular. It may consist of a travelling wave, of a regular oscillation such as that of a standing wave, or of some other time- dependent laminar pattern of motion.

2.2.2.2. Non-laminar States The second class is represented by a non-laminar motion, such as in a turbulent state. This may still be considered as a form of unperturbed state with certain specified properties, upon which an additional disturbance can be superimposed.

2.3. The Perturbed State

2.3.1. Energy Sources The dynamics of growing, steadily oscillating or decaying disturbances is due to the processes by which various energy forms are transferred into each other. Some of the most important energy sources which represent driving forces of instabilities and also become associated with wave phenomena are summarized in the following paragraphs (see also Table 2). In this connection it should be observed that the existence of an energy source does not necessarily lead to instability, but only to the energetic possibility of it. Thus, the dynamical possibility of making the transition from a higher to a lower energy state by an efficient coupling mechanism also enters the i.ability problem as a subsidiary condition. 7.

2.3.1.1. Intrinsic §ources_of tha Plasma The most frequently discussed group of energy sources is directly associated with the "intrinsic" properties of the plasma, as given by its particle densities, velocities, energies, space charges, and electric currents:

(1) Expansion energy. The internal energy of the plasma can be lowered and relaxed by an expansion towards spatial uniformity. This is always a large source of energy. (2) Deviations from local thermal equilibrium. Free energy is always gained by relaxing an arbitrary velocity distribution towards the Maxwellian in velocity space, somewhat in analogy with the relaxation of the expansion energy to uniformity in coordinate space. The available energy depends here on the magnitude of the deviation from the Maxwellian distribution, e.g. by the degree of anisotropy. (3) Kinetic drifts. Kinetic energy is provided by the plasma drift motion. This source arises from such phenomena as diamagnetic and other currents balancing the plasma pressure gradient and various other types of forces, from motions in the unperturbed state due to beams, unsheared and sheared stationary or non-stationary fluid motions, electric currents not being associated with the balance of forces, and even from motions due to turbulence in some special cases. The kinetic drifts usually represent a small energy source, but they are still able to generate instabilities which affect the balance of the plasma. (k) Electrostatic energy. The electrostatic energy usually plays an unimportant role in all but the most dilute plasmas. However, there are some special cases in which a release of such energy from space charges in the plasma is of some interest. (5) Magnetic energy. The magnetic energy stored in fields being induced by plasma currents can be released by changes in the field distribution within the plasma. This source is small at low ratios (beta values) between the energy density of the plasma and of the 8.

totally imposed magnetic field. Even when this is the case, however, instabilities driven by magnetic field fluctuations can have a marked influence on the energy balance.

2.3.1.?. Externally Imposed Force Fields In addition to the intrinsic energy sources of the plasma, there are externally imposed force fields from which energy can be extracted:

(1) Static fields such as imposed gravitation and electric fields are able to produce plasma motions and growing deviations from a Maxwellian unperturbed state under certain conditions.

(2) Non-static fields, e.g. in the form of regular oscillatory components , may feed energy into the plasma and produce growing disturbances. This is the case especially when the corresponding frequency is tuned to some of the plasma eigenfrequencies. Most of the so called parametric instabilities can be generated in this way. In this connection, care is necessary when the intrinsic sources of the plasma are replaced by equivalent external forces of some simplified theoretical models. As an example, the driving force of instabilities arising from the magnetic field curvature and the thermal particle motions in a confined plasma is not always completely equivalent to an imposed gravitation force

2.3.1.3. External Injection of Energy A further way of feeding energy into a plasma disturbance is due to the "external" injection (or sometimes extraction) of energy by mechanisms which are coupled to the disturbance itself. Examples are given by the heat released or removed by reactions which depend on the local plasma density and temperature, and which also may become coupled to the radiation field.

2.3.2. Coupling Mechanisms The transfer of energy from the previously described sources to the plasma disturbances can take place by means of a manifold of coupling mechanisms. Under certain conditions these mechanisms may also become involved in the stabilization of the plasma, as described in Section H. Here the following types should be mentioned 9. (see also Table 3):

(1) Electric charge separation is produced by a manifold of differential drift mechanisms of ions and electrons such as by guiding centre drifts including finite Larmor radius contributions, by drag phenomena due to differential forces acting on the ion and electron gases, by differential relativistic contraction, and by bunching effects due to velocity dispersion in a collisionless plasma.

(2) Changes in the local ion and electron density and temperature can be produced by the effects of convective motions of the ion and electron gases in a non-uniform plasma.

(3) Changes in the magnetic-field arise from the induced plasma currents.

(U) Changes in the momentum balance are produced by variations in the p B force.

(5) A transfer of momentum and energy is caused by resonant particle- wave or wave-wave couplings.

(6) Collisional effects provide a number of coupling mechanisms. Firstly, there is one due to resistive diffusion across a magnetic field which connects the energy states in a way not being possible in a collisionless plasma. Secondly, ion-ion collisions produce viscous effects which influence the plasma balance. Thirdly, the collisions put the disturbed field quantities such as the density and potential fluctuations out of phase, thus leading to a "feedback" and negative damping under certain circumstances. Fourthly, there is a coupling by collisions between charged and neutral particles affecting both the centre-of-mass motion and the charge separation processes in the plasma.Fifthly, processes such as ionization, recombination, attachment and nuclear reactions provide special coupling mechanisms.

(7) Heat conduction connects the temperatures of neighbouring plasma fluid elements in space. 10.

(8) Magnetic line-tying connects the plasma fluid elements along a magnetic flux tube.

(9) Feedback by external circuits and fields coupled to the plasma can affect the dynamics of the disturbances, both in the case of passive circuits and of imposed external stationary or non- stationary fields.

2.3.3. The Stages of Linear and Nonlinear Analysis The behaviour of plasma disturbances is usually analysed in terms of a combination of the basic equations of Section 2.1 for the perturbed and unperturbed states, by subtracting the former from the latter. The theoretical analysis can be performed by the following steps (see also Table

(1) A linear analysis is developed by studying disturbances of small amplitudes and neglecting all terms of second and higher order in respact to the amplitudes. In the case of growing disturbances, such an analysis will only provide information about the onset of an instability and its initial growth rate.

(2) The analysis is extended such as to include finite amplitude effects. There are several more or less restrictive simplifications which can be imposed on such an analysis, to make the difficult non-linear deductions possible. In its most complete form, the analysis then gives information on the growth to large amplitudes, as well as on the balance and the limitations of the fully developed disturbances. It also provides quasi-linear and non-linear methods for tackling the important problems of associated transport effects, loss mechanisms, and turbulent plasma states. This is a crucial question from the practical point of view because not all instability modes, including some of the rapidly growing ones, will necessarily be able to reach large amplitudes, or to produce large losses which affect the plasma balance noticeably.

2.3.^. Methods of Theoretical Analysis of Plasma Disturbances There are a number of different methods by which the dynamics of plasma disturbances can be analysed, each having its advantages 11.

and drawbacks, and all being in a way complementary to each other (see also Table 5).

2.3. U.I. P§£ticle_Orbit_Analysis The direct study of the exact particle orbits is usually a hopeless task. In many cases first order orbit theory provides a simple and surveyable mean to get a physical insight into the behaviour of the disturbances. As already pointed out in Section 2.1.1., however, it should be handled with care when charge separation phenomena are being considered.

2.3.4.2. When treating a linearized perturbation as described in Section 2.3.3. (1), an analysis can be applied in terms of kinetic or fluid theory, where the perturbation is dissolved into a sum of normal modes being extended over the entire spectrum of possible frequencies and wave numbers. Such an analysis is subject to three basic assumptions, namely that the normal modes exist, that they form a complete set describing an arbitrary disturbance, and that their sum is bounded when each mode itself becomes bounded |^6Z1# In some situations such an approach may fail, e.g. when there are singularities in the force driving an instability, and when a given initial field quantity can grow and e-fold many times locally before it becomes bounded or dies out | 6 ^f. The analysis can be performed mainly in three ways:

(1) Dispersion relations connecting the frequency u and the wave number k can be obtained for small regions within the plasma body by means of a localized analysis. This is the simpliest method by which all unperturbed quantities are treated as constants having their local values, and no boundary conditions are taken into account. Sometimes this method can be used in the form of a rough dimensional analysis of the perturbations in a limited plasma, to estimate the order of magnitude of instability growth rates and wave frequencies. On the other hand, it has to be kept in mind that the stability criteria obtained from the localized analysis sometimes deviate noticeably from those obtained from a more accurate theory P"60 ~J. ^ This is especially the case when the wave-lengths of the if 12.

disturbances are no longer small compared to the characteristic lengths of the unperturbed field quantities.

(2) In a non-localized analysis the variations of the unperturbed quantities in space and time are explicitly taken into account, as well as the boundary conditions. This analysis is more rigorous than the localized, but it often leads to complex dispersion relations and eigen-value problems. Full information is obtained by this method on the linear growth rates and frequencies of oscillation.

(3) When the non-localized analysis leads to complicated and un- surveyable dispersion equations, stability criteria may still be achieved by the Nyquist diagram technique. This yields stability conditions, but does not determine the growth rates.

2.3.4.3. Nonlinear_Mode_AnalYsis Some of the most important nonlinear features of the plasma disturbances can be treated by means of a mode analysis of weakly nonlinear plasmas Cjil»1*1*^]" Arbitrary perturbations are then still expressed as a superposition of linear eigenmodes, but the nonlinearity provides a weak coupling between the modes. The theory is based on three main types of interaction:

(1) The nonlinear wave-wave interaction, which can be treated in terms of macroscopic fluid equations.

(2) The quasi-linear particle-wave interaction, which has to be developed in terms of kinetic theory. Here the effects of the disturbance amplitudes on the distribution function are taken into account.

(3) The nonlinear particle-wave interaction, which also has to be developed in terms of kinetic theory, takes mode coupling into account.

An analysis of strong mode coupling and of the general nonlinear state of motion is, of course, still outside of the frame of these approaches. 13.

2.3.4.H. The_Energy_Princigle The stability of an arbitrary system can be investigated by studying virtual displacements in respect to the unperturbed state, and by determining the corresponding change in potential energy. The displacements should be made under the constraints of the equations of motion which then are to be treated as subsidiary conditions. Thus, it is not certain that a system should become unstable when there are neighbouring states of lower energy, because a transition to these states also has to become dynamically possible, according to the equations of motion.

The advantages of this energy principle are that it can be applied to complex systems and that it is not restricted to small disturbance amplitudes. On the other hand, it only gives an answer to the question of stability, but cannot be used to determine the growth rates.

2.3. h.5. Comguter_Analysis During recent years, stability analysis by computers has attracted a strongly increasing interest. Such an analysis can be applied to any of the approaches described in Sections 2.1.1.-2.1.3, and is particularly useful when nonlinear problems have to be treated. Its limitations are due to the fact that each run on the computer has to be performed at fixed values of part of the parameters involved, and that many runs are usually required to obtain a general and surveyable picture of the system behaviour within the whole parameter range. 3. GENERAL PROPERTIES OF PLASMA INSTABILITIES

We now leave the general subject of plasma disturbances and limit ourselves to plasma instabilities, the properties of which will be briefly discussed in this Section.

3.1. Definitions of Instability

To make sense, the discussion of a particular instability has to be based on an unperturbed state which represents some type of equilibrium. This is illustrated in Fig. 1 by the elementary analogy of a ball-shaped particle sliding on a smooth wall surface of various shapes, under the action of gravity. Here equilibrium demands that the external forces on the ball balance each other in a static state. Such a state becomes possible in Figs. 1.2-1.6, but not in Fig. 1.1. Further, the situation of Fig. 1.2 represents a marginal state of neutral stability, having zero growth rate and oscillation frequency of the ball motion. Finally, Figs. 1.3-1.6 represent stable or unstabe states corresponding to a number of various types of plasma behaviour, as described in the coming paragraphs.

3.1.1. Single-stage Processes In its simpliest form the growth of an elementary plasma instability mode can be considered as a single-stage process, not being influenced by the presence of other existing modes. There are a number of ways in which such an instability may develop.

3.1.1.1. Onset of Instabilities at Small Amplitudes One possible situation becomes equivalent to Figs. 1.3 and 1.5. Here small displacements of the ball from the equilibrium position at the bottom of the through formed by the wall lead to restoring forces. The system then becomes linearly stable. On the other hand, situations equivalent to Figs. I-1* and 1.6 lead to the onset of small-amplitude instabilities. During a further development of these, there are two alternatives:

(1) The instability grows "indefinitely" also at large amplitudes, and may finally be limited by some additional constraints, such as those due to the extensions of the available space and to certain boundary conditions. This corresponds to a potential through given by the combination of the full and broken wall contours of Fig. l.H. 15.

(2) The instability grows to a finite amplitude at which it becomes limited(saturated)on account of nonlinear restoring forces. This corresponds to a well-shaped wall curvature at large displacements from the equilibrium position, as represented by the model of Fig. 1.6.

3.1.1.2. No Onset of Instabilities at Small Systems being stable in respect to small-amplitude disturbances, such as in Figs. 1.3 and 1.5, can behave in two ways when the imposed initial disturbance amplitudes become increased: (1) The system remains stable also in the nonlinear case, such as in the potential through given by the full and broken wall contours of Fig. 1.3.

(2) The system becomes unstable at a certain critical amplitude, as represented by the top points of the wall surface in Fig. 1.5.

3.1.2. Multi-Stage Processes It is not always sure that a certain plasma instability should not affect the conditions of growth of other instabilities, especially if the amplitude of the former is allowed to become large enough to change the average "equilibrium" properties of the plasma. In the simple model of Fig. 1 this would be represented by a wall shape which is charged under the influence of finite amplitude ball motions. Conditions for the growth of other instabilities may then become realized, which cannot be satisfied as long as the original unperturbed state remains unaffected. As a consequence, one instability mode may first set in, then being followed by other modes at later stages. An example of such multi-stage instability processes has been given by Nezlin

3.2. External Effects on Plasma Instabilities The effects governing a plasma instability are not always localized only to the plasma body itself. Thus there are effects of external origin being coupled to the instability:

(1) Wall and boundary phenomena as well as Debye sheath effects in the immediate neighbourhood of the plasma body may influence the stability of the latter. 16.

(2) Feedback effects may arise due to the coupling between the plasma and external space, sometimes including external circuits.

3.3. Relativistic Effects

The extension of the stability analysis to include relavistic effects often leads to modifications of the stability conditions and the growth rates. Only in a few cases treated so far, however, the inclusion of relativistic effects has resulted in the discovery of new modes. In some cases these effects even have a stabilizing influence. Some references on recent investigations on such relativistic instabilities are given in the papers by McKee £ 2(T] and Santini and Szamosi

3.4. Linear Instability Properties We now turn to some of the general properties of instabilities existing already at small disturbance amplitudes. Even in this case the elementary models of Fig. I can only be taken as a rough description of the plasma, the mechanisms of which do not always become equivalent to those of a purely mechanical system.

3.4.1. The Restoring Forces In the linear normal-mode analysis of instabilities, a set of associated eigenfrequencies OJ is obtained from an expansion of the form exp (iwt). Three physically different situations correspond the solutions obtained from such an analysis [I14»

(1) If all eigenfrequencies are real or have positive imaginary parts, the system becomes stable. This corresponds to Fig. 1.3 in which the restoring force tends to push the system back into its equilibrium state.

(2) If io is pure imaginary and negative for certain values of the parameters involved, there is a purely growing instability. This corresponds to Fig. 1.4 in which the restoring force tends to push the system away from the equilibrium state. 17.

(3) If u) is complex with a negative imaginary part, there is an instability in the form of an overstable mode. This alternative has no correspondence to the simple mechanical analogies in Fig. 1. The underlying physical mechanism is due to restoring forces directed towards The equilibrium state, but being so strong that they overshoot the corresponding position on the other side of the equilibrium, i.e. they produce oscillations of growing amplitude. Such overstable situations are quite frequent in magnetized plasmas.

3.4.2. Unstable Oscillations and Waves Plasma instabilities often consist of growing oscillations and waves which do not have the localized character of the simple models in Fig. 1 but also become extended in coordinate and velocity space. In fact, the distinction between growing (amplifying) and decaying (evanescent) oscillations and waves in a plasma is by no means straight-forward and has to be interpreted within the framework of wave kinematics as discussed by several investigators H~15-19^]. Here we summarize some of the more important questions in this connection.

3.4.2.1. Absolute and_Convective Instabilities When a plasma disturbance travels in space, the definitions of instability becomes less straight-forward than in the models of Fig. 1. Such definitions have been extensively discussed in the papers by Sturrock E"lQ , Bers and Briggs C15T1 > Briggs O°H» McCune and Callen Cl7^] , and Ketterer and Melcher Ql*O among others. Considering an initially localized disturbance in a system of large spatial extensions, the physical picture of a growing (unstable) disturbance leads to two alternatives:

(1) The disturbance grows with time locally at all points in space within the system. It is then denoted as an absolute (non-convective) instability. The plasma thus operates somewhat like an oscillator.

(2) The disturbance propagates in space at the same as it grows. It is then observed to increase in time at all points within a frame of reference moving at the corresponding propagation velocity. In the laboratory frame, however, it propagates away from its original locations, and there it is seen to fade away. This type of mode is denoted as a convective instability. In systems of finite spatial extensions it depends critically on their size and on the boundary conditions. In the case of a convective instability the plasma thus operates somewhat like a travelling wave amplifier, rather than as an oscillator.

In reality it is often not a straight-forward task to judge from the dispersion relation or from other analytical expressions whether an instability is of the absolute or convective type. A rigorous criterion for this has been developed by Bers and Briggs • According to x*he criterion, absolute instability becomes possible only when there is a simultaneous solution (u , k ) of the dispersion relation D(u),k) = 0 and of its derivative 3D(d),k)/3k = 0 in the "unstable half" of the complex u-plane. This is equivalent of requiring a mode with zero complex group velocity (du/dk) = 0, i.e. a saddle point on one of the Riemann sheets representing D = 0. Physically it implies that the disturbance should grow at all points in space, being essentially a non- propagating wave packet. For further details of the criterion reference is made to the original papers QL5-193].

3.4.2.2. Negative Energy Waves When analysing the various types of waves in plasmas, modes are sometimes found for which the total energy of the perturbed state does not become larger but smaller than that of the unperturbed state. An example of this is the slow space-charge wave of a perturbed monoenergetic stream of charged particles, where electrons travel slower than the average in regions of increased density and faster in regions of decreased density [^3(f^\ . To increase the amplitude of such a negative energy wave one therefore has to remove energy from it. This type of waves naturally provides a source for instabilities both in the linear and nonlinear cases. They can be made to grow by means of coupling mechanisms which remove the energy and thus increase the amplitude:

(1) The energy can be transferred to a growing positive energy wave or oscillation.

(2) The energy can be removed by means of Landau damping.

(3) The energy can be absorbed in dissipative processes through collisions. 19.

3.4.2.3. Plasma Oscillations and Bernstein Waves In the study of certain problems it is sufficient to treat a collisionless magnetized plasma in which the velocity of light is considered to be infinite. Apart from such effects as Landau damping, it then becomes possible to describe small-amplitude plasma disturbances under quite general conditions, in the form of plasma oscillations and Bernstein wave modes ClO^] where the latter can be considered as a kind of cyclotron harmonics. These modes are obtained from a Fourier analysis in space and a Laplace transform in time of the Vlasov equation (collisionless Boltzmann-Vlasov equation).

3.4.2.4. VelocitY._Space_Effects When the particle distributions deviate from the Maxwellian, or become bi-Maxwellian with different equivalent temperatures along and across an immersed magnetic field, there is a driving force in velocity space which sometimes turns the plasma disturbances into velocity space instabilities. In particular, it has been shown by Grad CT63"] , that sufficiently steep gradients in this space always lead to such modes in the form of velocity gradient instabilities of an arbitrary contained plasma.

3.4.2.5. Parametric Instabilities When an externally imposed oscillating field has a frequency being chosen in the vicinity of some of the eigenfrequencies of otherwise stable plasma oscillations and wave modes, the latter may be turned into unstable growing disturbances. Such parametric instabilities are subject to an increasing interest at the present stage.

3.4.3. Particle-Wave Interaction The linear particle-wave interaction is an important mechanism which can only be treated in terms of kinetic theory, and which gives rise to large classes of instability phenomena. To discuss these, the concept of Landau damping becomes an important starting point. When a plane electromagnetic wave propagates through a plasma, it will retard the particles which slightly lead it, and will accelerate those which lay slightly behind it. For velocity distributions with a negative 20.

slope 3f/3w<0 such as the Maxwellian, there will always be more particles accelerated than retarded by the wave. This results in Landau damping. However, there are several conditions under which this situation may become changed such as to turn the wave into an unstable mode, as described in the following paragraphs.

3.4.3.1. Trapping Instabilities When there are deviations from the Maxwellian distribution in the form of a "hump" in velocity space, there will exi^t a corresponding region for which 3f/3w>0. At phase velocities of a plasma wave situated in this region of velocity space there is negative (inverse) Landau damping producing a growth of the wave amplitude under certain conditions, in the form of a trapping instability. The two-stream instabilities arising from two interpenetrating beams are due to conditions being rather similar to those of the trapping instabilities. However, there is an essential difference due to the fact that the latter depend on the specific plasma properties, whereas the former are independent of the specific type of coupling mechanism being involved f3 8 ZH -

3. k. 3.2. Drif ^Instabilities In the unperturbed state of an inhomogeneous plasma, there are diamagnetic drifts generated by the spatial gradients of density and temperature. These drifts are "universal" in the sense that they exist in any confined plasma of limited spatial extensions. The result of these drifts is to increase the effective number of retarded particles, and to increase the number of those being accelerated. This effect is opposite to that of Landau damping, and may sometimes ^produce drift instabilities Q283• These can also be considered as the result of a resonance between the drift motion and a wave . A particular class of modes is due to the drift in crossed electric and magnetic fields. These E_ x 13 drift instabilities have recently been reviewed by Sanderson £^29J. In presence of collisions, the drift modes are further converted into drift-dissipative instabilities the growth rate of which becomes more related to collisions than to resonant particles. The instability is then caused by collisions which put the potential and density fluctuations out of phase. 21.

3.4.3.3. Resonance Phenomena in General The particle-wave interactions described in Sections 3.4.3.1 and 3.4.3.2 can both be considered as special types of particle-wave resonance phenomena. There are other similar types of interactions leading to instability, whenever some kind of fluctuations or motions in the plasma is in resonance with some possible wave type. This can even be the case when the absolute values of the corresponding phase velocities differ considerably, provided that the directions of propagation are chosen nearly at right angles, such as to satisfy the resonance condition. Among these possibilities are the resonances with the electrostatic and acoustic electron and ion oscillations, the electron and ion cyclotron motion, and the various types of Alfven, whistler and Bernstein waves. Further discussions on resonance phenomena are postponed to Section 7.1.7. It should finally be pointed out that linear wave-wave interactions may even occur in some special cases, such as when an imposed oscillating field is tuned to a plasma wave resonance.

3.4.4. Collective Modes There is a subclass of instabilities which does not only depend on the local properties of the plasma, but also on the properties of the unperturbed state being integrated over a large plasma volume. Such collective instabilities have been described by Coppi £ 333 and Kadomtsev and Pogutse C35 3 among others.

3.4.4.1. Untragped Particle Instabilities A large number of the collective modes are merely a generalization of the localized ones, where the driving forces result from an average taken over parts of the plasma volume. One example is given by the collective flute instability driven by the net effect of magnetic field curvature obtained from integration along a magnetic flux tube, under the constraint of magnetic line-tying. These instabilities concern all particles within the flux tube.

3.4.4.2. Tragged^Particle^Instabilities There are also collective modes associated w"th certain groups °^ trapped particles. These modes are developed in a "background11 of transit particles which do not contribute to the driving forces. 22. j They therefore become less violent than the corresponding untrapped I particle instabilities. In principle there exist two types: i I j (1) Modes produced by trapping in electrostatic field or magnetic j mirror configurations, being static in space. i

(2) Modes produced by trapping in the potential throughs of a travelling wave.

3.4.4.3. Resonance_Phenomena of_Collective Modes All the resonance phenomena mentioned in Section 3.4.3.3. also become important to the collective modes. In addition, there exist resonance effects which are characteristic only of these modes. Such i effects are associated with the oscillation and bounce periods of i transit and trapped ions and electrons moving across macroscopic parts of the plasma body.

3.5. Nonlinear Instability Properties

Most of the properties already discussed in connection with the linear instabilities are also important in the nonlinear case. This applies especially to the particle-wave and wave»wave phenomena, the negative energy waves, the parametric instabilities, and to the resonance phenomena in general. Here we summarize some of the properties which are specific to the nonlinear modes.

3.5.1. Nonlinear Mode Interaction In linear theory arbitrary perturbations can be expressed as a superposition of independent eigenmodes, just like small-amplitude waves on a water surface. However, as the amplitudes increase, nonlinear mode interaction becomes important, in the same way as large-amplitude waves interact on a water surface. Especially in nonlinear plasma dynamics it is sometimes useful to introduce an approach being equivalent to quantum mechanics. Thus, the plasma can be considered to contain a mixture of particles and quasi-particles (plasma wave packets), all interacting with each other C^Z) • With this approach there are two possible types of interaction: 23.

(1) There is a resonant interaction in which the particles emit or absorb quasi-particles. The particles do not return to their original states after such an interaction.

(2) There is a non-resonant interaction during which the particles oscillate in the wave fields. If the interaction is due to a wave of finite duration which passes by, the particles return to their original states after the interaction has taken place

3.5.1.1. Resonant Interaction The resonant interactions between particles and quasi-particles (par tide-wave) and between quasi-particles among themselves (wave- wave) have to take place under the constraints of total momentum and energy conservation. These conservation laws can be expressed in terms of the wave number vectors k and frequencies w of the plasma waves, and of the frequencies k°w corresponding to the velocities w of the individual plasma particles. The conditions for momentum and energy conservation are necessary, but not always sufficient, because the changes in k and w produced by the interaction also have to satisfy the dispersion relation D (w,k)=0 of the plasma. This leads to an additional constraint on the transitions produced by resonant interaction \241,443] • In this connection should further be mentioned that the presence of negative energy waves makes the resonant interactions particularly violent, because these waves grow when energy is removed from them. In the nonlinear case this leads to explosive instabilities, the amplitude of which goes to infinity after a finite time t=t, , and not in the limit t= » as for most other unstable modes. The nonlinear interactions are of several types, partly depending on the degree of nonlinearity [1

(1) In the first approximation of moderately large amplitudes, quasi-linear particle-wave interaction is taken into account. Here the Landau damping or growth of the waves and the associated changes in the particle distribution are taken into account, whereas the non-linear wave-wave interaction is neglected. At this level of approximation it is therefore possible to study the effects of quasilinear diffusion of particles in phase space, as produced by the finite wave amplitudes. 24.

(2) In the next approximation the nonlinear throe-wave interaction is being considered. Here two waves beat together to form and match the wave number and frequency of a third wave, or a wave decays into two other waves by an analogous process. This leads to the so called decay instabilities, as well as to a sub-class of nonlinear parametric instabilities in which the initial wave plays the role of an "external" field, exciting other wave modes in the plasma.

(3) In the same approximation as (2), there is further a nonlinear particle-wave interaction. Its basic mechanism is similar to the three-wave interaction, with the exception that there are instead particles which maintain a constant phase with the beats of two waves.

The interactions given by (2) and (3) are to be considered as the lowest approximations to nonlinear mode coupling. In the fully developed nonlinear case there are finally four-wave and higher approximations.

3.5.1.2. Non-Resonant Interaction Many instabilities are algebraic in nature also in a nonlinear- state, and have nothing to do with resonant interactions. In such cases non-resonant diffusion in phase space describes the relaxation of the instability. Some examples of this are given by Sagdeev and Galeev [7^^Z] anc* by "the numerous calculations being performed by means of computers during recent years £"47-55^] . An additional type of nonlinear phenomena of interest in this connection are the echoes due to phase-mixing and unmixing, as reviewed by Lerch

3.5.2. Plasma Turbulence When the nonlinear plasma instabilities are allowed to grow sufficiently, the various individual modes can no longer be disting- uished from each other, and a turbulent state is being developed. The features of plasma turbulence have been summarized by Kadomtsev [J^ among others. The turbulent state leads to the following "anomalous" transport phenomena in the plasma: 25.

(1) Anomalous diffusion of the plasma across a confining magnetic field.

(2) Anomalous resistivity and inomalous heatirg power produced by electric currents.

(3) Anomalous heat conduction across a magnetic field. 26.

STABILIZATION MECHANISMS

There are at least three basic conditions which have to be taken into account when considering the stability of a plasma, namely:

(1) The spatial distributions and states of imposed magnetic, electric and other force fields.

(2) The plasma distribution in phase space, including both the velocity distribution and the macroscopic plasma properties in coordinate space.

(3) The boundary conditions.

One or two of these three conditions alone are not always sufficient to specify the stability properties uniquely. Thus, knowledge of the magnetic field geometry only is not sufficient for a judgement of the stability of a magnetically confined plasma. In every stability problem all the conditions just mentioned have to be examined in detail. The effects of nonlinear phenomena also become important in this connection. A .linearly unstable mode need not become "dangerous" to the practical problem of magnetic confinement if its amplitude becomes saturated at a low level in a. situation similar to that demonstrated in Fig. 1,6, Also the reverse situation demonstrated by Fig. 1.5 is not quite unimaginable. Stabilization of a particular mode is achieved by choosing the plasma parameters as well as the imposed fields and boundary conditions within the stable ranges of the corresponding dispersion relation or its equivalent. Throughout this review, the term "stabilization" should have the meaning "tending to stabilize", i.e. we include both the mechanisms which lead to a stable state and those which only produce a reduction in the growth rate of a mode still remaining unstable. The stabilization methods can be divided into two categories:

(1) The "hard" method by which configurations and conditions are chosen where no free energy becomes available from the sources listed in Section 2.3.1.

(2) The "soft" method by which configurations and conditions are chosen such as to weaken or remove the coupling mechanisms of 27.

Section 2.3.2 for the transition from higher to lower energy states.

U.I. Internal Plasma Mechanisms

Within the plasma body itself there are a number of intrinsic stabilization mechanisms.

4.1.1. Linear Mechanisms A large sub-class of intrinsic stabilization mechanisms exists already in the linear case, most of which also become effective at large amplitudes.

U.I.1.1. "Hard" Mechanisms The possibility of removing the free energy has frequently been discussed in connection with some specific modes and configurations of fully ionized collisionless plasmas:

(1) A pinched plasma column can be stabilized against sausage and kink instabilities by applying a sufficiently strong axial magnetic field. The perturbed states then have a larger magnetic energy than the unperturbed one, by "compressing" and "bending" the externally imposed field.

(2) The flute type instability arises under the constraint of the induction law given by Eq. (1), in a plasma with perfect magnetic line-tying. Any perturbation in terms of this mode leads to an increase in the expansion energy, provided that minimum-B or minimum-average-B configurations are being chosen. In the former configurations, the magnetic field is increasing in the direction away from the core of the plasma, and the centre of curvature of the field lines is on the plasma side of the boundary region. In the latter configurations this becomes an average property, the "good" regions having the same magnetic field gradient and curvature as in the minimum-B case, and dominating the "bad" regions where the reverse situation prevails. 28.

(3) In a sheared magnetic field 15 which changes its direction with some coordinate x, the quantity k«IS of perturbations with the wave number k cannot become zero at every point within the plasma. The magnetic field then becomes disturbed by the perturbations in a way to increase the magnetic energy.

In certain types of impermeable plasmas confined in a magnetic field and surrounded by neutral gas, the driving forces of flute and ballooning modes can be removed from the plasma interior and become localized only to a partially ionized boundary region DO-

(5) A convective instability which would have time and space to grow while propagating in an infinitely extended medium, can sometimes be suppressed by choosing configurations with sufficiently small extensions in the direction of propagation.

4.1.1.2. ^Soft"^Mechanisms Among the specific stabilization effects associated with the coupling mechanisms in the plasma, the following should be mentioned:

(1) Relativistic effects are sometimes stabilizing, leading to extra dispersion effects in the relations between u> and k.

(2) Higher-order terms in Ohm's law sometimes have a stabilizing effect. A particular example is given by the finite-Larmor- frequency effects which produce an overlapping of ion and electron flute disturbances and are represented by the Hull term.

(3) Finite-Larmor-radius effects give rise to extra charge separation phenomena, and to collisionless contributions to the pressure tensor, having the form equivalent to an anisotropic viscosity.

The Coriolis force sometimes has a stabilizing effect in rotating plasmas.

(5) High-beta plasmas sometimes become more stable than low-beta plasmas. 29. (6) Dissipation and losses often produce a damping effect on instabilities. On the other hand, it also has to be kept in mind that collisions may provide instabilities by making transitions possible which otherwise become blocked in a dissipation-free plasma. Examples on dissipative damping mechanisms on the growth of instabilities are given by finite resistivity, ion-ion viscosity,] plasma-neutral gas friction, and by losses of particles, momentum and heat by diffusion, conduction and convection.

(7) An important mechanism originating from kinetic theory is due to Landau damping.

(8) The thermal spread of particles in velocity space sometimes stabi- lizes microinstabilities such as the two-stream modes.

U.I.2. Nonlinear Mechanisms In addition to the mechanisms already operating in the linear case, the following nonlinear ones should be mentioned:

(1) A large group of nonlinear saturation mechanisms limit the amplitudes, somewhat as demonstrated in Fig. 1.6. The available energy then becomes exhausted.

(2) For the resonant transitions described in Section 3.5.1.1, there are subsidiary conditions due to the dispersion relations which sometimes block the development of a nonlinear instability. Thus the necessary coupling is missing in this case.

(3) In the particular situation of flute instabilities, it has earlier been suggested that nonlinear stabilization should occur under the joint action of finite-Larmor-frequency, ion-ion viscosity, and neutral gas friction effects under special conditions C"68^] • This also provided a kind of blocking effect due to the coupling mechanisms being involved.

U.2. Effects due to the Boundary Conditions

The effects due to externally imposed boundary conditions may sometimes produce important stabilizing effects in the form of "soft" mechanisms:

V 'yr-. \ ; -•;, T 30.

(1) The presence of limiters and metal walls being in contact with the outermost layers of a confined plasma puts a heavy constraint on the electric potential distribution. It also influences the particle, momentum, and heat balance such as to affect the stability properties.

(2) Metal plates being placed all the way across a magnetic flux tube, and being in contact with a considerable part of the plasma body through magnetic line-tying, may have similar and sometimes stronger effects on the electric state than limiters. Given potentials can further be applied to the plates, such as to contribute to stability.

(3) The presence of a cool plasma being adjacent to a hot plasma, sometimes has a stabilizing influence on the latter.

4.3. Externally Applied Mechanisms

The stability of the plasma can be further affected by applying additional external mechanisms of the "hard" type, including stationary or time-dependent fields, or imposed beams.

4.3.1. Stationary Mechanisms These mechanisms are so far rather unfrequent. They include the application of additional dc electric fields.

4.3.2. Non-Stationary Mechanisms There is a large class of non-stationary mechanisms which have attracted considerable interest during the last years.

4.3.2.1. The work on dynamic stabilization by the application of ac electric and magnetic fields has been reviewed by Berge C^6]]. By "this*method extra electric currents and fields are induced in the plasma which on the average counteract the destabilizing forces during a period of oscillation. Stability is achieved, provided that the amplitude of the imposed oscillation is chosen large enough. One of the difficulties of this method is to limit the corresponding required power input. 31.

Dynamic stabilization can be considered as the inversion of the process leading to parametric instabilities. For the latter the ac fields give instead rise to average destabilizing forces.

4.3.2.2. £eedback_Stabilization The various feedback stabilization methods have been reviewed by Thomassen £[ 57 ] . Feedback control of a number of instabilities can be achieved by picking up signals from the instabilities. The signals are then fed back to the plasma in the form of impulses on arrays of electrodes, probes, magnetic coils, microwave signals, beams, or of any other type of means by which the driving forces of the instabilities can be counteracted. One of the limitations of this method is due to the finite number of elements in the arrays which can be technically realized. 32. s- CLASSIFICATION OF PLASMA DISTURBANCES

With the previous sections as a starting point, an attempt shall now be made to classify the basic features of plasma disturbances in a systematic way. The scheme presented in this section should be considered as preliminary, possibly being subject to future modifications and extensions. This classification should apply both to instabilities and stable disturbances such as waves. Neither mixtures of various elementary modes, nor modifications of already existing modes will be explicitly taken into account.

5.1. The Notation To develop a classification scheme, the following notations and rules are introduced:

(1) For a certain property Q of the plasma disturbances, there are two classes specified by

(Q) which denotes all disturbances having the property Q; (Q') which denotes all disturbances not having the property Q, i.e. Q'=non-Q.

(2) The class of disturbances having both the properties P and Q is denoted by (PQ).

(3) The sum of the subclasses having the properties P,Q and P,Q* is simply denoted as (P), i.e. (PQ)+(PQf)=(P).

5.2. The Classification Scheme

We now develop a classification scheme closely related to the way in which the plasma disturbances arc analysed theoretically and interpreted physically. The results of this development are demonstrated by Fig. 2. The division into classes and subclasses is obtained from Fig.2 by following the centre line from top to bottom, and branching off at each point where there is a choice between two mutually exclusive alternatives. 33.

It should further be mentioned that only such properties Q are included in the classification scheme which contribute in an essential way to the characteristics of a given disturbance. To take a concrete example, this implies that an instability, all the essential features of which are given by a non-relativistic treatment, will be classified as non-relativistic even in cases where there exists a corresponding relativistic and slightly modified mode.

5.2.1. External Space Effects The mechanisms which determine the dynamics of the disturbances are not always localized to the plasma body only. They may also become coupled to plasma sheaths and boundary layers, circuits, and field structures being localized in or originating from regions outside of the plasma body. This leads to a division of the disturbances into two classes, namely:

(P) Disturbances depending only on the conditions within the plasma body and on its boundary conditions. (Pf) Disturbances the properties of which are also in an essential way dependent on the conditions and effects of external space.

Among the disturbances of the class (P1) there are such modes as those being essentially affected by external electric circuit feedback.

5.2.2. Relativistic Modes As a next step, it has to be decided whether the velocities involved lead to relativistic effects of essential importance to the dynamics of the disturbances. This leads to a division into the classes:

(V) The velocities are large enough for relativistic effects to be included, and these effects are essential to the characteristic behaviour of the disturbances.

(Vf) The main features of the disturbance are given by a non- relativistic treatment.

Relativistic instabilities are defined as those modes belonging to the class (V). They may be produced by such mechanisms as those due to relativistic velocity distributions and phase bunching, 3U. relativistic frequency shifts and resonance effects due to deviations from the rest mass, Serenkov waves, and charge separation by differential Lorentz contractions.

5.2.3. Collisional Phenomena Collisions in the sense of particle-particle interactions affect the balance of matter, momentum, and energy by introducing extra source and sink terms in the corresponding equations. The collisional couplings produce a transfer both between particles of different species such as ions, electrons and neutrals, and between groups within the velocity spectrum of the same species. The following classes and subclasses are defined:

(C) Particle-particle collisions are important. (Cn) Collisions with neutral particles are included. (CnT) Collisions with neutral particles are not included. (C) Particle-particle collisions are unimportant.

Among the instabilities belonging to the class (Cn) may be mentioned those produced by ionization, recombination, attachment, momentum and heat exchange by collisions between charged and neutral particles, and by the neutral drag and its charge separation effect on a moving partially ionized plasma. The instabilities belonging to the class (Cn1) include finite resistivity modes due to electron-ion collisions and related modes due to "banana" effects of trapped particles in toroidal fields, finite heat conductivity modes, drift-dissipative modes, and instabilities caused by the energy released in fusion and other possible reactions.

5.2.4. The Macroscopic Motion in the Unperturbed State The macroscopic motion in the unperturbed state leads to basic conditions for the development of plasma disturbances, and to the following classes and subclasses: 35.

(S) There is a stationary state in which all macroscopic quantities are time-independent. (Ss) The state is static, only with electric currents left to balance the pressure gradients and other forces. (Ssf) The state is non-static. There are macroscopic motions among the various species, in addition to those required to balance the pressure gradients and other forces. (Ssfv) There is a centre-of-mass motion having a velocity

(Ss'v1) There is no centre-of-mass motion, i.e. v =0, —o but electric currents i are imposed in addition to those required for the force balance. (Sf) There is a non-stationary state being explicitly dependent on time. (S'l) The time-dependent motion is laminar. (Sfl!) The time-dependent motion is non-laminar.

Among the class (Ssfv) are many of the drift instabilities for which there are drift and centre-of-mass motions being uncompensated by the unperturbed electric field and its corresponding E xB drift. The class (Ssfvs) contains a number of current-convective and two-stream instabilities in which there is an imposed unperturbed electric current, in absence of a centre-of-mass motion. Further, the class (Sfl) includes regular time-dependent motions, such as those producing parametric instabilities when an external oscillating field is being imposed. Finally, the class (S'l1) includes states such as those which are turbulent already from the beginning.

5.2.5. The Unperturbed State in Velocity Space The unperturbed state in velocity space may be approximated by a localized Maxwellian distribution or not, leading to the following division: 36.

(T) The state is described by a local thermal and isotropic velocity distribution upon which are superimposed local everage macroscopic velocities of various species. The limiting case of zero temperature is included in this class.

(Tf) The state cannot be described by a local thermal and isotropic distribution with superimposed macroscopic motions. States with anisotropic bi-Maxwellian distributions of unequal temperatures T and T are included in this class.

The class (T) represents states in which all "driving forces" in velocity space have been relaxed and a Maxwellian isotropic equilibrium has established itself. Within the class (T*) there are on the other hand driving forces left which act as energy sources for velocity space instabilities.

5.2.6. The Externally Imposed Magnetic Field Concerning the existence of an externally imposed magnetic field, there are the following classes and subclasses:

(B) There is an imposed magnetic field in the unperturbed state. (Bb) The unperturbed state is a high-beta case, in which the energy density of the plasma is comparable to that of the imposed magnetic field. (Bb1) The unperturbed state is a low-beta case, in which the energy density of the plasma is negligible compared to that of the imposed magnetic field. (Bf) The disturbances are of flute-type, in the sense that their spatial distribution along the magnetic field lines does not give rise to electromagnetic induction effects. In particular, such disturbances become independent of the coordinate in the direction of a homogeneous unperturbed field B . (Bf1) The disturbances are of ballooning (wiggly) type, i.e. they are non-flutes which vary along the magnetic field direction and often give rise to electromagnetic induction effects. (Bf) There is no imposed magnetic field in the unperturbed state. 37.

In J:his connection should be pointed out that the class (Bb*) also includes small-amplitude magnetic field disturbances, that such disturbances also exist within the claso CBf) of flutes for perturbations of the magnetic field in the directions perpendicular to the field lines, and that there are certain disturbances of the class (Bf1) which do not lead to electromagnetic induction effects.

5.2.7. Electromagnetic Induction Effects According to Eq. (1), the disturbances can further be divided into the following two classes:

(E) Electromagnetic perturbations for which the electric field cannot be derived from a. potential. (Ef) Electrostatic perturbations for which the electric field 5. ~ ~ Z$ ^3 derived from the potential $.

5.2.8. Collective Effects In a plasma there are some-cim

Among the collective modes (L') there are those which arise from the integrated effects taken along a magnetic flux tube, over the potential through of a given initial wave, or over the orbits of particles oscillating back and forth across the plasma body.

5.2.9. The Perturbed State in Velocity Space To describe the dynamics of the disturbances there are various choices of the theoretical approach, such as between those represented by Eqs. (4)-(10). A possible subdivision which will be adopted here leads to the following classes and subclasses; 38.

(M) Macroscopic disturbances which can be described by simple macroscopic fluid equations containing an isotropic and scalar pressure. (M1) Microscopic (non-macroscopic) disturbances for which a description in terms of fluid equations with an isctropic and scalar pressure becomes insufficient. (M*t) Microscopic disturbances which include groups of trapped particles. (M'tf) Microscopic disturbances which do net include trapped particles.

The macroscopic instabilities (macroinstabilities, hydromagnetic, or hydrodynamic instabilities) defined here thus represent a situation where the motions in velocity space have been approximated by isotropic Maxwellian distributions, and where the dynamics of the instabilities is restricted to motions in coordinate space only. The present microscopic instabilities (microinstabilities) are on the other hand due to such phenomena which also depend on the deviations from the isotropic Maxwellian in velocity space. In most cases the full kinetic theory has to be applied here, with the exception of such special cases as those of bi-Maxwellian distributions. The distinction between what is usually denoted as macro- and microinstabilities is not always as sharp as might appear from the present classification. There are namely more advanced fluid models in which anisotropy in pressure and finite-Larmor-radius effects can be included. However, speaking in general terms and more frcjn the physical point of view, one may state that macroinstabilities are mainly associated with rapid large-scale motions of the whole plasma body. Most of the microinstabilities, on the other hand, concern small groups of resonant and trapped particles which give rise to disturbances of short wave length, and to a rather slow "anomalous" diffusion of microscopic nature. There are also other somewhat different ways in which the distinction between macro- and microinstabilities can be defined

5.2,10. Resonance Effects The possible resonance effects by particle-wave and wave-wave interaction can be looked upon as an additional type of collision processes in which the plasma wave "quanta" (quasi-particles) are involved. We have the classes and subclasses given by: (R) Resonant interaction. (Rp) Particle-wave interaction. (Rp1) Wave-wave interaction. (Rf) Non-resonant interaction.

The particle-wave interaction of (Rp) is important both in the linear and nonlinear cases, whereas the class (Rp1) of wave-wave interaction mainly concerns nonlinear disturbances. However, there are some specific cases in which even the linear analysis leads to modes belonging to (Rpf), such as when parametric excitation of certain plasma resonances is taking place.

5.2.11. The Disturbance Amplitudes The possible influence of nonlinear effects leads to the following classes:

(A) The disturbance amplitudes are large, and nonlinear theory has to be applied. (A1) The disturbance amplitudes are small enough to justify the use of a linearized theory.

5.2.12. The Growth of the Disturbances The final purpose of the analysis is usually to decide whether there is a growing disturbance or not. This leads to the classes:

(I) The disturbance is growing and has the form of an absolute or convective instability. (II) The disturbance is decaying or remains at its initial amplitude, having the form of a stable wave or some other type of stable motion.

Consequently, the present classification scheme should specify some of the main features of all possible unstable and stable disturbances. A number of examples of specific instability modes are given in Section 6 and in the Appendix. As an application of this scheme to wave motions we finally mention one example here, namely that of the ncn-relativistic large-amplitude Alfvén wave in a collisionless plasma being at rest in the unperturbed state. Consequently, this mode re- ceives the code: (PV'C'SsTBb'f»ELMR'AI1). 40.

6. A SURVEY OF INSTABILITY SUBCLASSES

The specific modes listed in the Appendix are now rearranged in subclasses as outlined in Fig. 3, and demonstrated in detail by Tables 7-19, where the corresponding codes have been introduced. In the latter Tables a further subdivision has been undertaken, as given by the notation in the right hand columns. Finally, the essential mechanisms of a specific instability have been roughly indicated by the corresponding underlined letter symbols in the codes. 7. SUGGESTIONS FOR DISCOVERIES OF MEW PHENOMENA

By means of the present review, a closer examination may be made of instabilities and other disturbance phenomena which so far have remained undiscovered.

7.1. Instabilities

Most of the individual instability modes described in Section 3 and in the Appendix have been studied both theoretically and experimen- tally. However, it is neither certain that all these modes have a decisive influence on the plasma balance, nor that they correspond to physically realistic situations under all circumstances. In particular, the treatment of drift instabilities based on a vanishing unperturbed electric field E leads to modes which might not always exist in a real laboratory plasma where the assumption E =0 becomes questionable for any frame of reference. This question is closely Telated to the earlier discussion in Section 2.2.1. On the other hand, there might also exist nev? important instabilities, not being discussed so far. In this section we shall give some hints in terms of the present classification scheme, under what conditions such instabilities might be searched for.

7.1.1. Feedback Destabilization The feedback stabilization methods described in Section 4.3.2.2 can in principle be "inverted" to produce unstable modes in ?r otherwise stable plasma. Thus, within the class (P'D there may exist instabilities due to an externally applied "feedback destabilization". Important technical applications may be found within this class of modes. The few ones listed in Table 7 further suggest that there is still a large unexplored field to tackle here.

7.1.2. Relativistic Instabilities The possibilities within the field of relativistic instabilities does not yet seem to be exhausted. In particular, among the class (PVI) the following mechanisms should be important:

(1) Even a small difference between small Lorentz contractions of ions and electrons in a slightly relativistic quasi-neutral plasma can lead to significant charge separation effects ,652) . This may, in its turn, produce stabilizing or destabilizing mechanisms, depending upon the specific problem to be considered.

(2) The relativistic dispersion in gyro frequency due to the deviations from the rest mass produces resonance or off- -resonance effects which sometimes destabilize or stabilize a given system.

7.1.3. Collisional Non-Resonant Effects The "reaction rate" terms P , M , and H in Eqs. (8)-(10) can all provide sources driving instabilities, as already seen from a number of cases listed in Tables 11, 13, and I1*. Among the class (PV^R'I) there might exist more unstable non-resonant modes than those discussed so far. In particular, this applies to charge-exchange collisions and to collisions leadi.ig to a loss or gain of energy by other processes than ionization and thermonuclear reactions.

7 .1.4. Non-Static State Phenomena The instability analysis performed so far has often been restricted to unperturbed states in which either the centre-off-mass velocity —vo or the electric field —Eo are assumed to vanish identically all over the plasma volume. There still remains a number of cases to be treated in which v and E cannot vanish identically for any frame of reference In particular, the class (PV^s'vI) with plasma motions should still offer some new possibilities. An example is given by the critical velocity phenomenon earlier postulated by Alfvén [J 58 ~] . This phenomenon has been partly discussed in terms of a mechanism by which spatially non-uniform electric fields and plasma density distributions are built up in combination with finite-Larmor-radius effects, non-Maxwellian ion distributions, and charge-exchange and ionization effects ["67 I! • A corresponding "critical velocity instability" is imaginable, probably having the form (PV'CnSs'vT'Bb'E'M't'R'A'I).

7,1.5. Magnetic Induction Effects The instabilities produced by induced plasma currents still require further investigations, at least in some special cases to be mentioned in the coming paragraphs. 7.1.5.1. High-Beta_Phenomena There is a large class (PV'BbEI) of phenomena which both includes destabilizing and stabilizing high-beta effects, As one concrete example we may here suggest a mode somewhat analogous to the mirror instability, but existing in a rotating plasma with large mirror and radial ratios. This instability should be due to the centrifugal force which pushes out the magnetic field lines in the mid-plane, thereby increasing the radial ratio and the centrifugal confinement. This, in its turn, leads to a further concentration of plasma in the mid-plane, and to a reinforcement of the centrifugal expansion of the field lines. A corresponding "centrifugal expansion instability" is likely to exist in the form (PV'CfSs'vTBbf'EL'MR'A1I).

7.1.5.2. Low-Beta Phenomena In some types of configurations, such as in Tokamaks and Stella- rators, there is a rotational transform. Even for small perturbations of the plasma currents, within the limits of the class (PVfBbfEI), the field topology may become changed in a way to produce instabilities, or even to destroy the equilibrium. In particular, it is not sure that the induced plasma currents in Tokamaks always become perfectly axisymmetric. As a consequence, growing axial asymmetries, superbananas, and various associated instability phenomena may arise. Further, since the magnetic field in Tokamaks and Stellarators mainly consists of a toroidal part, asymmetric perturbations in its poloidal part may destroy the magnetic surfaces and make the field lines go to the walls.

7.1.6. Collective Effects An extensive survey has been given by Coppi £ 33]] on collective, ccllisionless, electrostatic, low-beta instabilities in a two- -dimenslonal magnetic field. It is likely that there are more collective modes to be discovered, where collisions and electromagnetic effects play a role in high-beta plasmas and more general types of field geometries. 7.1.7. Resonance Phenomena As seen from many examples in this review, the particle-wave and wave-wave resonances play an important role as instability coupling mechanisms. It is quite likely that the richness in combinations of various characteristic frequencies and corresponding phase and signal velocities lead to more resonant interactions and associated modes than those discovered so far. These possibilities arc listed in the coming paragraphs, as well as in Table 20.

7.1.7.1. Localized_Phenomena There is a large group of characteristic frequencies depending on the local state at a given point in the plasma. These can, in their turn, be divided into the following cate^orios;

(1) The individual particle motion is directly associated with collisional phenomena, as given by the characteristic frequencies and relaxation times of ionization, excitation, recombination, attachment, charge exchange, elastic collisions, fusion reactions, and other types of elementary processes. The particle motion is further associated with collisionless phenomena, such as those represented by the ion and electron gyro frequencies. (2) The various wave motions and oscillations of the plasma are associated with certain wave lengths and frequencies. These motions are due to sound waves, electrostatic plasma oscillations of ions and electrons, and to Alfvén, whistler and other electromagnetic waves. (3) Drift waves due to pressure gradients and equivalent gravitation fields, to associated transverse electric fields and magnetic gradients, and to other applied force fields are characterized by certain wave lengths and frequencies. In particulars the "overlapping" frequencies of the differential drift motions of ions and electrons may lead to new types of resonance phenomena. Externally imposed oscillating fields with piven tunable frequencies lead to a large variety of parametric resonance phenomena of which only a relatively small number has so far been investigated. 45.

7.1.7.2. Collectiye_Phenomena In the case of small collision rates and weak non-adiabatic effects there will exist well-defined frequencies associated with the transit particle motion over macroscopic distances, as well as with the periodic trapped particle motion between magnetic mirrors and other types of turning points.

7.1.8 Nonlinear Effects Research on the strongly nonlinear mode coupling and on other problems of a fully developed nonlinear state is still in its earliest phase. It is possible that new instability modes will be discovered within this field.

7.1.9. Plasma Interaction with Other Immersed "Media" In addition to the interaction with an immersed neutral gas, the plasma balance and stability may also become affected by other immersed "media", especially in situations of interest in cosmical physics:

(1) The plasma interaction with a cosmic ray background.

(2) The plasma interaction with an energetic photon gas background.

7.1.10 Ambiplasma Effects The possible existence of an ambiplasma consisting of a mixture of matter and antimatter has earlier been discussed by Alfvén [1^83 • In such a plasma there are lijht and heavy particles of both polarities present. This should among other things, introduce new types of inertia and charge separation mechanisms, thus leading to additional forms of instability and wave phenomena. 46.

7.2. Stabilization Effects

The stabilization effects discussed so far are mainly associated with small-amplitude disturbances. Only rather recently, when "mathematical experiments'5 have become possible by means of computers, nonlinear saturation phenomena and other large-amplitude effects have attracted an increasing interest H37-5Q • Tt ^s likely that much more can be done in the field of nonlinear stabilization3 which also gets much closer to the real physical situation of plasma experiments and associated loss phenomena. As one particular example should finally be mentioned the nonlinear effects on a gravitational flute instability in a dissipative plasma. Here the damping effects of viscosity by ion-ion collisions reduce the growth rate such as to make it smaller than the overlapping frequency due to the differential drift motions of ions and electrons. As a consequence, a flute of finite amplitude should become "'smeared out" and damped by the corresponding viscous, diffusion, and mixing effects C68ZI • 8. CONCLUSIONS

The present state of research on plasma instabilities can be shortly summarized as follows:

(1) In linear theory, a r^reat number of modes have been extensively investigated, and the conditions for their onset and stabilization are mostly known in great detail. New important contributions to the development of the linear analysis have recently been provided by the studies of collective and parametric instabilities.

(2) In nonlinear theory, considerable progress has been made during recent years in the approach to weakly nonlinear plasmas, i.e. in terms of the nonlinear wave-wave interaction, and the quasi- -linear and nonlinear particle-wave interactions. The rapid development and increasing use of computers has further provided a new and powerful method in studying nonlinear plasma phenomena. Still the exploration of the vast field of nonlinear plasma physics is only in its first stage, and much work remains to be done in the study of fully developed nonlinear mode coupling, of the loss mechanisms and loss rates in a fully developed large-amplitude state, and of plasma turbulence in general.

(3) The classification scheme developed in this review may be used in a systematic search for so far undiscovered instability phenomena. It is likely that new modes are to be found especially within the classes of the feedback destabilized modes, relativistic modes, collisional modes, non-static states, magnetic induction modes, collective modes, resonance phenomena of various kinds, and in ambiplasmas.

Stockholm, October 22, 1972 48.

9. REFERENCES

The present list of references does not make any claim of being complete or always to be ordered in respect to first discovereies. It merely serves the purpose of giving information where review articles can be found on the various subclasses of plasma disturbance phenomena For further details the reader is recommended to study the general references and descriptions of the elementary modes given in this context and in the Appendix.

9.1. General References on Plasma Disturbance Phenomena

9.1.1. Plasma Instabilities

L_1_J Auer, G., Cap, F., Floriani, D., and Grat? , 11., Inst. for Theor. Phys., Univ. of Innsbr'- ., Sci. Rep. NR. 75 (1970).

l_2__J Auer, G. , Cap, F., Floriani, D.- ratzl-, H., and Weil, J., Inst. for Theor. Phy - , Univ. of Innsbruck, Sci. Rep. NR. 81-89 (1971-19^2).

L3j Cap, F., Einfiihrung in die Plasmaphysik, I-III, Akademie-Verlag, Berlin; Pergamon Press, Oxford; Vieweg and Son, Braunschweig (1970-1972).

Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, Oxford, Clarendon Press (1961).

L5J Hartnagel, H., Semiconductor Plasma Instabilities, Heinemann Educational Books, London (1969).

L_6j Kulsrud, R.M., Proc, of the Int. School of Physics "Enrico Fermi", Course XXXIX (Ed. by P.A. Sturrock), Plasma Astrophysics, Academic Press, Mew York, and London (1967), p.64.

Lehnert, B., Plasma Physics £ (1967) 301. L 8-Tl Rostoker, N. , Plasma Physics in Theory and Application (VA. by W.3. Kunkel), McGraw-Hill Book Comp., Mew York, St. Louis, San Francisco, Toronto, London, Sydney (1966), Ch. 5.

Sato, M., Inst. of Plasma Physics, Nagoya Univ., Rep. IPPJ-7 (1963).

9.1.2. Plasma Waves

Bernstein, I.B., Phys. Rev. 109 (1958) 10.

Bernstein, I.B., and Trehan, S.K., Nuclear Fusion 1^ (1960) 3.

| 12^1 Denisse, J.F., and Dolcroix, J.L., Plasma Waves, Interscience Publ. and Wiley, --Tew York, London, and Sydney (1963).

Ll3_J Kadomtsev, B.B., and Karpman, V.I., Soviet Physics Uspekhi, 14_ (1971) 40.

Stix, T.H., The Theory of Plasma Waves, McGraw-Hill Book Comp., New York, San Francisco, Toronto, and London (1962).

9.2. References on Special Classes and Properties of Plasma Instabilities

9.2.1. Absolute and Convective Instabilities

| 15 | Bers, A., and Briggs, R.J., Mass. Inst. of Technology, Res. Lab. of Electronics, Quarterly Progress Report No. 71, Oct. 15 (1963), p. 122.

Bijlsma, S.J., Plasma Physics 1^ (1972) 420.

L_17_J McCuce, J.E., and Callen, J.D., Plasma Physics 12_ (1970) 305.

Q.8J Ketterer, F.D., and Melcher, J.R., Phys. Fluids 11 (1968)2179.

Ö.9H Sturrock, P.A., Phys. Rev. 112 (1958) 1488. 50. 9*2.2. Relativistic Instabilities

McKee, C.F., Phys. Fluids 14. (1971) 2164.

£227] Santini, F., and Szamosi, G., Nuovo Cimento 3_7 (1965)685

9.2.3. Resistive Instabilities

Baldwin, P. and Roberts, P.H., Phil. Trans of Roy.Soc. A, 272 (1972) 303.

C233 Coppi, B., and Rosenbluth, M.N., Plasma Physics and Controlled Nuclear Fusion Research, I.A.E.A., Vienna, Vol. £ (1966)617.

2«T] Frieman, E., Weimer, K., and Rutherford, P., Plasma Physics and Controlled Nuclear Fusion Research, I.A.E.A., Vienna, Vol. I. (1966) 595.

Furth, H.P., Advanced Plasma Theory, Proceedings of the International School of Physics, Course XXV, Varenna; Academic Press, New York (1964).

Furth, H.P., Killeen, J., and Rosenbluth, M.N., Phys. Fluids S_ (1963) 459.

Stringer, T.E., Plasma Physics and Controlled Nuclear Fusion Research, I.A.E.A., Vienna, Vol. X (1966) 571.

9.2.4. Drift Instabilities

Krall, N.A., Advances in Plasna Physics (Ed. by A. Simon and W.B. Thompson), Interscience Publ. and Wiley, New York, London, Sydney, Toronto, Vol. i (1968) 153.

29^| Sanderson, J.J., Max-Planck-Institut for Plasmaphysik, Rep. IPP 1/128 (1972). 51. 9.2,5. Two-stream and Beam-plasma Instabilities

30Q Briggs, R.J., Advances in Plasma Physics (Ed. by A. Simon and W.B. Thompson), Interscience Publ. and Wiley, Mew York, London, Sydney, Toronto, Vol. 4 (1971) 43.

31^] Gould, R., Symposium of Plasma Dynamics (Ed. by F.H. Clauser), Addison-Wesley Publ. Comp., Reading,Mass. (I960), Ch. 4.

£[323 Fainberg, Ya., Plasma Physics (Journ. of Nucl. En., Part C), 4 (1962) 203.

9.2.6. Collective Instabilities

CT^dl Coppi, B., Advances in Plasma Physics (Ed. by A. Simon and W.B. Thompson), Interscience Publ. ?.nd Wiley, New York, London, Sydney, Toronto, Vol. £ (1971) 173.

9.2.7. Trapped Particle Instabilities

£* 34^) Kadomtsev, B.B., JETP Letters 4, (1966) 15.

Kadomtsev, B.B., and Pogutse, O.P., Nuclear Fusion, (1971) 67.

Pogutse, O.P., Physics Letters 27A (1968) 63.

9.2.8. Cyclotron Instabilities

L37Z1 Hall, L.S., Heckrotte, W., and Kammash, T., Phys. Rev. A_l,39 (1965) 1117.

9.2.9, Parametric Instabilities

Perkins, F.W., and Flick, J., Phys. Fluids 14, (1971) 2012.

9.2.10. Nonlinear Phenomena and Turbulence

[I3O Bernstein, I.B., Greene, J.M., and Kruekal, M.D., Phys. Rev. 108 (1957) 546. 52.

Harris, E.G., 'vances in Plasma Physics (Ed. by A. Simon and W,B. Th- ipson), Interscience Publ. and Wiley, New York,

London, Syca ty9 Toronto, Vol. 3_ (1959) 157.

Kadomtsev, B,:>., Plasma Turbulence , Academic Press, London, New York (5 165).

Kalman, G. and Feix, M., Editors, Nonlinear Effects in Plasmas, Gordon and Breach, New York, London, Paris )1969)

Lerch, L., Nonlinear Plasma Phenomena: Echoes, Inst. for Theor. Phys., Univ. of Innsbruck, Sci.Rep. No. 91 (1972)

Sagdeev, R.Z. and Galeev, A.A., Nonlinear Plasma Theory, W.A. Benjamin Inc. Publ., New York, Amsterdam (1969).

9.2.11. Moving Striations

_] Lee, D.A., Bletzinger, P., and Garscadden, A., Journ. Appl. Phys. 21 (1965) 377.

Swain, D.W. and Brown, S.C., Phys. Fluids 14 (1971) 1383.

9.2.12. Numerical Simulations by Means of Computers

3vers> J.A., Phys. Fluids 10, (1967) 2235.

Byers, J.A., Rensink, M.E., Smith, J.L., and Walters, G.M., Phys. Fluids !U (1971) 826.

0**3 Crume» E.C., Oak Ridge National Lab., 0NRL-TM-3812 (1972).

Crume, E.G., Meier, H.K., and Elridge, O.C., Oak Ridge National Lab., ONRL-TM-3675 (1972).

Elridge, 0., Oak Ridge National Lab., ONRL-TM-3843 (1972).

Elridge, 0., Crume, E.C., and Meier, H.K., Oak Ridge National Lab., ONRL-TM-3642 (1972). v 53. C533 Finzi, U., Plasma Physics 14 (1972) 327.

Meier, H.K., Oak Ridge National Lab., ONRL-TM-3447 (1972).

C55ZI Morse, R.L., anc Nielson, C.W., Phys. Fluids 14_ (1971) 830. )

9.3. Stabilization Methods

9.3.1. Dynamic Stabilization

Berge, G., Nuclear Fusion r2 (1972) 99.

9.3.2. Feedback Stabilization

Thomassen, K.I., Nuclear Fusion 11^ (1971) 175.

9.4. Specific References

Q 583 Alfven, H., On the Origin of the Solar System, Clarendon Press, Oxford (1954).

Q59] Alfvjn, H., Scientific American, .m (1967) 106.

Croci, R., and Saison, R., J. Plasma Physics £ (1971) 375

Gartenhaus, S., Elements of Plasma Physics, Holt, Rinehart and Winston, New York, Chicago, San Francisco, Toronto, London (1964), Ch. 4.

62] Fälthammar, C.G., J. Geophysics Res. 6£ (1962) 1791.

H.63I] Grad, H., Phys. Fluids £ (1966) 498.

Jungwirth, K., J. Plasma Physics 2 (1969) 155. 54.

1365*3 Lehnert, B., Dynamics of Charged Particles, North-Holland Publ. Comp. Amsterdam (1964), p. 221.

Lehnert, B., Nuclear Fusion 1£ (1970) 283.

'Lehnert, B., Phys. Fluids 1£ (1967) 2216.

Lehnert, B., Fifth European Conference on Controlled Fusion and Plasma Physics, Grenoble, 21-25 August (1972), p.32, EURATOM-CEA, Centre d1Etudes Nucléaires de Grenoble; Electron and Plasma Physics, Royal Inst. of Tech., Stockholm, TRITA-EPP-72-06 (1972) and TRITA-EPP-72-21 (1972).

[36931 Nezlin, M.V., Soviet Physics JETP, £6 (1968) 693. 55.

10. INDEX OF INSTABILITIES

The various instability modes and classes of modes described in this review and in the Appendix are listed in this Section, both in respect to the developed classification scheme and to their names.

10.1. Classification Index

1. (PVC'SsT'Bb'fE'LM'tRpA1!) Cyclotron instability; relativistic mode 2. (PVC'Ss'vTBb'f'ELMRpA'I) fierenkov instability

f f f Whistler instability; 3. (PVC'Ss'v'T'Bb'f ELM tRpA I) current-driven relativistic mode Negative radiation absorption H. (PVC'Ss'v'T'LM'tRpA'I) instabilities; relativistic modes

? Amplitude dispersion instabilities; 5. (PVC'S 1 TBb'ELMR'AI) relativistic modes Resonant diffusion instabilities 6. (PVC'S'l'T'Bb'fE'LM'tRpA'I) of a relativistic plasma

7. (PV'CnSsTBb'f E'LMR'A'I) Ionization (electrothermal instability) 8. (PV'CnSsTB'E'LMR'A'I) Acoustic wave instability in a partially ionized plasma 9. (PV'CnSsTB'E'LMR'A'I) Electroconvective instability 10. (PV'CnSsTB'E'LMR'A'I) Ionization-recombination ion sound instability 11. (PV'CnSsTB'E'LMR'A1!) Recombination instability 12. (PV'CnSsTB'E'L'M't'R'A'I) Continuity equation instability 13. (FV'CnSsTfBbff?ELM't'RfAfI) Whistler instability; collisional mode (PV'CnSs'vTB^fE'LMR'A'I) Drift instability by neutral collisions 15. (PV^nSs^TBb'fE'LMR'A'I) Magneto-acoustic wave instability 16. (PV'CnSs'vTBb'fE'LMR'A'I) Neutral drag (crossed field) instability 56. 17. (FV'CnSs'vB'E'R'A'I) Corona discharge instabilities » 18. (PVCnSs'v'TBb'f'E'LMR'A'I) Screw instability in a partially ] ionized plasna 19. (PVtCnSslvlTBlEtLKR'AlI) Gunn instability i i i 20. (PV'CnSs'v'TB'E'LMR'A'I) Striations in the positive ; column j

t t l l s f l t Collision-induced electromagnetic 21. (PV»CnSs v T Bb f ELM t R A'I) instability in partially ionized gas Negative radiation absorption ; 22. (PV'CnSs'v'T'LM'tRpA'I) instabilities; non-relativistic modes

23. (PV'Cn'SsTBb'f'ELMR'A'I) Resistive ballooning instability , i 24. (PV'Cn'SsTBb'f'ELMR'A1]:) Resistive gravitational insta- l bility 25. (PV'Cn'SsTBb'f'ELMR'A'D Resistive surface instability 26. (PV'Cn'SsTBb'f ELMR'A»]!) Resistive tearing instability 27. (PV'Cn'SsTBb'f'E'LMR'A'I) Finite heat conductivity instability

28. (PV'On'SsTBb'f'E'LM't'R'A'I) Collision-induced electrostatic 4 instability in fully ionized p?.asma 29. (PV'Cn'SsTBb'f'E'LM't'R'A'I) Entropy wave instability 30. (PV'Cn'SsTB'E'LMR'A'I) Ion sound wave instability in fully ionized plasma 31. (PV'Cn'SsTBLMR'A'I) Fusion reaction instabilities 32. (PV'Cn'Ss'vTBbfELM't'R'A'I) Gradient instabilities in high-pressure plasmas 33. (PV'Cn'Ss'vTBb'f'E'LMR'A'I) Hydrodynamic electrostatic instabilities in a collision- dominated plasma . (PV'Cn'Ss'vTBb'f'E'LM't'R'A'I) Drift-dissipative electrostatic instabilities 35. (PVfCnfSsfvTBbff?E'L'M't1R'A'I) Finite orbit instability 36. Resistive rippling instability 37. (PV'Cn'Ss'v'TBb^'E'LMR'A1!) Screw instability in fully ionized plasma

38. (PV'Cn'Ss'v'TB'E'LM't'R'A'I) Run-away instability 57.

39. (PV'Cn'S'l'TB'E'LMR'A1!) Acoustic instabilities 40. (PV'CSsTBb'f'ELMR'A'I) Thermal convection (Rayleigh- -Taylor) instabilities 41. (PV'CSBb'L'A1!) Trapped-particle dissipative instabilities in a magnetized plasma

42. (PV'C'SsTBbf'ELMR'A'I) Ballooning instability 43. (PV'C'SsTBbf'ELMR'A'I) Buckling instability 44. (PV'C^SsTBbf'FLMR'A'I) Sausage instability 45. (PV1 C' SsTBbf'ELMR'A'I) Surface (Suydam) instability 46. (PV'C'SsTBbf'EL'MR'A1!) Flip instability 47. (PV'C'SsTBbf^.L'MR'A1!) Haas-Wesson instability 48. (PV'C^SsTBb'f Flute(electrostatic inter- charge or Rayleigh-Taylor) instabilities 49. (PV'C'SsTBb'fE'L'MR'A'I) Flute instability; collective mode 50. (PV^'SsTBb'f'ELMR'A1!) Gravitation instability; collisionless macroscopic mode 51. (PV'C'SsTBb'fELMRfAfI) Hall instability 52. (PV'C'SsTBb'f'ELMR'A'I) Kink and Kruskal-Shafranov instabilities 53. (PV'C^sTBb'f ELMR?A»I) Tearing instability; collisionless macroscopic mode 54. (PV^'SsTBb'f ELMftRpAfI) Gravitati.on instability; collisionless microscopic mode 55. (PV'C^SsTB'E'LMR'A'I) Radiation instability 56. (PVfCfSsTfBbfELMftRpAfI) Loss-cone instability; finite beta electromagnetic mode 57. (PV^'SsT'Bbf'ELM't'R'A1!) Alfvén wave (fire-hose) instability 58. (PV'C'SsT'Bbf'ELM't'R'A'I) Mirror instability 59. (PV'C'SsT'BbfELM't'R»AfI) Self-gravitational instability in anisotropic magnetized plasmas 58.

60. (PV'C'SsT'Bb'f'ELM'tRpA'I) Temperature-anisotropy transverse wave instability 61. (PV'C'SsT'Bb'f'ELM'tRpA1!) Whistler instability; pressure-driven mode 62. (PV'C'SsT'Bb'fE'LM'tRpA'I) Harris (ion cyclotron resonance) instability 63. (PV'C'SsT'Bb* fE'LM'tRpA'I) Negative-energy instability in inhomoaeneous mirror geometry 64. (PV'C'SsT'Bb'E'LM'tRpA'I) Loss-cone (maser) instabilities; electrostatic modes 65. (PV'C'SsT'B^LM't'R'A'I) Transverse wave (Weibel) instability 66. (PV'C'Ss'vTBbfE'LM'tRA1]:) Negative energy wave instabilities of shock wave 67. (PV'C'Ss'vTBbf'ELM'tRpA'I) Ion drift wave instability 68. (PV'C'Ss'vTBbf'ELM'tRpA'I) Tearing and sheet pinch instability; collisionless microscopic mode 69. (PVlCfSslvTBblfEfLMRpfAlI) Diocotron (slipping-stream) instability 70. (PVTCfSs'vTBb'fLM?A?I) Drift instabilities; flute-type collisionless modes 71. (PV'C'Ss'vTBb'f'ELMR'A1]:) Kelvin-Helmholtz instability; macroscopic electromagnetic mode 72. (PV'C'Ss'vTBb'f'ELM'tRpA'I) Universal Alfvén wave instability 73. (PV'C'Ss'vTBb'f'E'LMRpA1!) Beam-centrifugal instability . (PV'C'Ss'vTBb'f' E'LMRfAfI) Kelvin-Helmholtz instability; macroscopic electrostatic mode 75. (PV'C'Ss'vTBb'f'E'LM'tRpA'I) Drift-beam instability 76. (PV'C'Ss'vTBb'f'E'LM'tRpA1!) Drift cyclotron resonance (Mikhailovskii-Timofeev)instability 77. Kelvin-Helmholtz instability; microscopic electrostatic mode 78. (PVfC?Ss?vTBb'f'EaMftRpAfI) Temperature drift instability 79. (FV'CfSs?vTBb7f'EfLMTtRpA?I) Universal low-frequency drift instability 80. (PV'C'Ss'vTBb^'E'LM^1!) Impurity ion drift instabilities

i 59. 81. (PV^'Ss'vTBb'f'LM'A'I) Drift instabilities; ballooning- -type collisionless low- -frequency modes 82. (PV'C'Ss'vTB'E'LM't'RpAI) Ion beam instability; nonlinear modes 83. (PV'C'Ss'vTB'E'L'MR'A'I) Electromechanical co-and counterstreaming instability 84. (PV'C'Ss'vT'Bb'fE'LrrtRpA'I) Drift-velocity space instability; electrostatic collisionless mode 85. (PV'C'Ss'vT'Bb'fE'LM'T'R'A'I) Negative mass instability 86. (PV'C'Ss'vT'Bb'f'ELM1t'R1A1I) Kelvin-Helmholtz instability in anisotropic plasmas; electromagnetic mode 87. (PV'C'Ss'vBf'LM'tRpA7!) Hybrid instabilities 88. (PV'C'Ss'vB'E'LM't'R'A'I) Plasma diode instabilities 89. (PV'C'Ss'v'TBb'f'ELMR'A'I) Cross-stream instability in a magnetized plasma 90. (PV'C'Ss'v'TBb'f'E'LMR'A'I) Rippling instability; collisionless macroscopic mode 91. (PV'C'Ss'v'TBb'rE'LM'tRpA'I) Universal drift instability; current-driven mode 92. (PV'C'Ss'v'TBb'f'E'LM't'RpA»!) Ion-wave instability in a magnetized plasma; current-driven mode 93. (PV'C'Ss'v'TB'ELMR'A'I) Cross-stream instability in an unmagnetized plasma 94. (PV'C'Ss'v'TB'E'LM't'RpA'r» Two-stream and beam-plasms, instabilities; basic modes 95. (PV'C'Ss'v'TB'E'LM't'R'A'I) Current chopping instability 96. (PV'C'Ss^'TB'E'L'MRpA'I) Finite length instabilities 97. (PV'C'SsW'T'Bb'fE'L'M'tRpA'I) Ion resonance instability in a non-neutral plasma 98. (PVfCfSs'TfEfLM'tRpAfI) Trapping and t:bump" instabilities; basic modes 99. (PV'C'STBb'E'L'M'A'I) Collective electrostatic instabilities in a two-dimensional field 100. (PV'C'ST'Bb^LM^RpA1!) Cyclotron instabilities in a plasma with anisotropic temperature 60.

101. (PV'C'SBb'L'tA'I) Trapped-particle collisionless instabilities in a nrapnetized plasma

102. (PV'C'S'lTBbfE'LM'tRDA'I) Ion sound instability in a collisionless shock wave

103. (PV'C'S'lTBbf'ELKF'AI) Whistler-dominated laminar shock instability

104. (PV'C'S'lTBb'f'ELMRp'A'I) Parametric Alfven wave instability 105. (PV'C'S'lTBb'f'L'LMR'AI) Amplitude dispersion instability of whistlers 106. (PV'C'S'lTBb'f^LM't'Rp'AI) Parametric instabilities of ion cyclotron waves 107. (PV'C'S'lTBb'f E'LM't'RpA'I) Flectron-ion streaming instabili- lity; electrostatic mode

108. (PV'C'S'lTBh'f'E'LM't'Rp'A1!) Parametric electrostatic plasma oscillation instability 109. (PV'C'S'lTB'E'LMRp'A'I) Parametric ion acoustic instability

110. (PV'C'S'lTB'E'LM'tRpAI) Ion sound wave instability; non-linear current-driven mode 111. (PV'C'S'lTB'E'L'M'tRp'A1!) Bernstein-Greene-Kruskal wave instability 112. (PV'C'S'lTLRp'AI) Decay instabilities by three-wave interaction 113. IPV'C'S'lTLRp'AI) Explosive (negative energy) wave-wave interaction instabilities . (PV'C'S'lTLRp'AI) Four-wave interaction instabilities 115. (PV'C'S'ITLRAI) Sideband instabilities 116. (PV'C'S'rrB'E'LM'tRpA'I) Trapped-particle instability in a strong electrostatic wave 117. (PV'C'S'rPB'E'L'M'tRpAI) Stochastic instability 118. (PV'C'S'lLM'RpAI) Wave-particle nonlinear interaction instabilities 119. (PV'C'S'l'T^'ELM't'R'AI) Turbulence instability 120. (PVfCfLMfRpAI) Wave-particle quasi-linear interaction instabilities 61.

121. (PV'Ss'vTBb'f'E'LMA'I) Hydrodynamic drift modes 122. (PV'S'lTBbf ELM't'RpA'I) Electron-ion streaming instabilities ; transverse electromagnetic modes 123. (PST'LM'tRpA'I) Trapping instabilities in moving waves-, basic modes

124. (P'V'CnSsTB'E'L'MR'A'I) Emission instability 125. (P'V'CnSsTB'E'L'MR'A'I) Negative characteristic instability 126. (P;VfCfSs'vTB'EfL'M'tJRfAI) Plasma diode instability; non-linear mode 127. (P'V'C'Ss'v'TB'E'LMR'A'I) Pierce (space charge) instability 62. 10.2. Name Index

In this index the number preceding the class symbol refers to the corresponding page in the Appendix.

Absolute instability: see Section 3.4.2.1. (1) Acoustic instabilities 39. (PV'CnlStl'TBfEtLMRlAtI) Acoustic wave instability 8. (PV'CnSsTB'E'LMR'A1!) in a partially ionized plasma Alfven wave (fire hose) 57. (PV'C'SsT'Bbf'ELM't'R'A'I) instability Amplitude dispersion in- 105. (PV'C'S'lTBb'f'ELMR'AI) stability of whistlers Amplitude dispersion insta- 5. (PVC'S'lTBb'ELMR'AI) bilities; relativistic modes Anisotropic temperature see velocity space instability instability: Ballooning instability 42. (PVfCTSsTBbf'ELMR'A'I) Beam-centrifugal insta- 73. (PV'C'Ss'vTBb'f'E'LMRpA'I) bility Beam-plasma instability: see two-stream instability Bernstein-Greene-Kruskal 111. (PV'C'S'lTB'E'L'M'tRp'A'I) wave instability Buckling instability 43. (PV'C'SsTBbf'ELMR'A1!) Bulge instability: see sausage instability Bunching instability: see two-stream instability Buneman instability: see two-stream instability Centrifugal instability: see flute, Rayleigh-Taylor, and gravitation instabilities Serenkov instability 2. (PVC'SsWTBb'f'ELMRpA'I) Collective instability: see Sector 3.4.4. Collective electrostatic 99. (PV'C'STBb'E'L'M'A'I) instabilities in a two- -dimensional field i Collisional drift insta- see drift-dissipative instability: bility Collisionless gravitation see gravitation instability; instability: collisionless modes Collisionless tearing see tearing instability; instability: collisionless modes 63. Collision-induced electro- 28. (PV^n'SsTBb'f'E'LM't'R'A1!) static instability in fully ionized plasma Collision-induced electro- 21. (PV'CnSs'v'T'Bb'f'ELM't'R'A1!) magnetic instability in partially ionized gas Continuity equation in- 12. (PV'CnSsTB'E'L'M't'R'A'I) stability Convective instability: see Section 3.4.2.1. (2) Coriolis instability: see centrifugal and gravitation instabilities Corkscrew instability: see screw and kink instabilities Corona discharge instabilities 17. (PV'CnSs'vB'E'R'A'I) Crossed field instability see neutral drag instability Cross-stream instability in 89. (PV'C'Ss'v'TBb'f»ELMR'A1!) a magnetized plasma Cross-stream instability 93. (PV'C'Ss'v'TB'ELMR'A1!) in an unmagnetized plasma Current chopping instability 95. (PV'C^s'v'TB'E'LM't'R'A'I) Current instability: see ion sound wave instability; non-linear current-driven mode Current-convective insta- see screw, Kadomtsev, and bilities: rippling instabilities. Cyclotron instabilities in 100. (PV'C'S'T'Bb'LM'tRpA1!) a plasma with anisotropic temperature Cyclotron instability; 1. (PVC'SsT'Bb'f'E'LM'tRpA'I) relativistic mode Decay instabilities: see Section 3.5.1.1. (2). Decay instabilities by 112. (PV'C'S'lTLRp'AI) three-wave interaction Density gradient drift in- see flute instability; stabilities : electrostatic mode Diocotron (slipping-stream) 69. (PV'C'Ss'vTBb'fE'LMRp'A'I) instability Double-stream instability: see two-stream instability Drift instabilities: see Section 3.H.3.2. 64.

Drift instabilities; 81. (PV'C'Ss'vTBb'f'LM'A'I) ballooning-type collisionless low-frequency modes Drift instability by neutral 14. (PV^nSs'vTBb'fE'LMR'A'I) collisions Drift instabilities; flute- 70. (PV'C'Ss'vTBb'fLM'A1!) -type collisionless modes Drift-ballooning instability: see ballooning instability Drift-beam instability 75. (PV'C'Ss'vTBb'f 'Ef LMftRpAfI) Drift-cone instability: see drift-velocity space instability Drift cyclotron-resonance 76. (PV'C'Ss'vTBb'f»E'LM1tRpA1I) (Mikhailovskii-Timofeev) instability Drift-dissipative instabi- see Section 3.1.3.2. lities: Drift-dissipative electro- 34. (PV'Cn'Ss'vTBb'f•E'LM't'R'A'I) static instabilities Drift-temperature instability: see temperature drift instability Drift-velocity space insta- 84. (PV'C'Ss'vT'Bb'fE'LM'tRpA'I) bility; electrostatic collisionless mode E x I* drift instabilities: see Section 3.4.3.2. E-layer instability: see diocotron and two-stream instabilities Electroconvective instability 9. (PV'CnSsTB'E'LMR'A'I) Electromechanical co-and 83. (PV'C'Ss'vTB'E'L'MR'A1!) counterstreaming instability Electron-beam instability: see two-stream instability Electron cyclotron instability: see cyclotron instabilities Electron drift wave instability: see drift instabilities Electron-electron instability: see two-stream instability Electron-ion instability: see two-stream and ion sound wave instabilities Electron-ion streaming 107. (PV'C'S'lTBb'f'E'LM't'RpA'I) instability;electrostatic mode Electron-neutral atom colli- see ionization and recom- sion instability: bination instabilities Electron wave instability: see decay instability 65.

Electron-ion streaming 122. (PV'S'lTBbf'ELM't'RpA'I) instabilities; transverse electromagnetic modes Electrothermal instability: see ionization instability Emission instability 124. (P'V'Cn? ^TB'E'liMR'A'I) Entropy wave instability 29. (PV'Cn'SsTBb'f'E'LM'tfRfAfI) Explosive instability: see Section 3.5.1. Explosive (negative energy) 113. (PV'C'S'lTLRp'AI) wave-wave interaction instabilities Finite heat conductivity 27. (PV'Cn'SeTBb'f'E'LMR'A'I) instability Finite length instabilities 96. (PV'C'Ss'v'TB^'L'MRpA'I) Finite orbit instability 35. (PV'CnfSsfvTBb!f'E'L'M't'R'A'I) Fire-hose instability: see Alfvén wave instability Flip instability 46. (PV'C'SsTBbf'EL'MR'A'I) Flute Electrostatic inter- 48. (PV'C'SsTBb'fE'LMR'A'I) charge or Rayleigh-Taylor) instabilities Flute instability; 49. (PV'C'SsTBb'fE'L'MR'A'I) collective mode Fluttering instability: see surface instability Four-wave interaction 114. (PVfCfSflTLRpfAI) instabilities Fusion reaction instabilities 31. (PV'Cn'SsTBLMR'A'I) G-mode instability: see gravitation instability Garden-hose instability: see Alfvén wave instability Gradient instabilities in 32. (PV'Cn'Ss'vTBbfELM't'R'A'I) high-pressure plasmas Gravitation instability; 50. (pV'C'SsTBb'f'ELMR'A'I) collisionless macroscopic mode

Gravitation instability: see also flute, Rayleigh- -Taylor and Kruskal- -Schwarzschild instabilities 66.

Gravitation instability; 54. (PV'C'SsTBb'f'ELM'tRpA'I) collisionless microscopic mode Gunn instability 19. (PV'CnSsTB'E'LMR'A'I) Haas-Wesson instability 47. (PV'CtSsTBbftELfMRtAfI) Hall instability 51. (PV'C'SsTBb'ffELMRfAfI) Harris (ion cyclotron 62. (PV'C'SsT'Bb'fE'LM'tRpA'I) resonance) instability Harrison instability: see diocotron instability Helical instability: see screw and kink instabilities Helicon wave instability: see whistler instability Hirschfield instability: see trapping and negative radiation absorption instabilities Hose instability: see Alfven wave instability Hybrid instabilities 87. (PV'C'Ss'vBf»LM'tRpA'I) Hydrodynamic c^rift modes 121 (PV'Ss'vTBb'f'E'LMA'I) Hydrodynamic electrostatic 33. (PV'Cn'Ss'vTBb'f'E'LMR'A'I) instabilities in a collision- dominated plasma Impurity ion drift insta- 80. (PV'C'Ss'vTBb'f'E'LM'A1!) bilities Inertial instability: see collisionless gravitation and tearing instabilities; macroscopic modes Inertial drift instability: see collisionless gravitation and tearing instabilities; microscopic modes Interchange instability: see flute and Rayleigh-Taylor instabilities Ion acoustic instability: see ion sound and ion sound wave instability Ion beam instability; nonlinear 82. (PV'C'Ss'vTB'E'LM't'RpAI) mode Ion cyclotron resonance insta- see also cyclotron and Harris bility instabilities Ion drift wave instability 67. (PV'C'Ss'vTBbf'ELM'tRpA'I) Ion-ion instability: see ion sound wave, cyclotron, and anisotropic temperature instabilities 67.

Ion resonance instability in a 97. (PV'C'Ss;v'T7Bb'fE'L:MstRpAfI) non-neutral plasma Ion sound instability in a 10?. (PVfC"S'lTBbfEfLM'tRDA'I) collisionless shock wave Ion sound wave instability 30. (PV'Cn'SsTB'E'LMP'A'I) in fully ionized plasma Ion sound wave instability; 110. (PV'C'S'lTB'E'LM'tRpAI) nonlinear current-driven mode Ion-wave instability in a 92. (PV'C'Ss'v'TBb'f'E'LM't'RpA'D magnetized plasma; current- -driven mode Ionization (electrothermal) 7. (PV'CnSsTBb'fE'LMR'A1!) instability Ionization-recombination ion 10. (PV'CnSsTB'E'LMR'A1!) sound instability Jeans instability: see self-gravitational instability Kadomtsev instability: see screw instabilities Kaufman instability: see negative characteristic instability Kelvin-Helmholtz instability; 71. (PVfC'Ss'vTBb?f 'ELMR'A'D macroscopic electromagnetic mode Kelvin-Helmholtz instability; (PV'C'Ss'vTBb'f'E'LMR'A'I) macroscopic electrostatic mode Kelvin-Helmholtz instability; 77. (PV'C'Ss'vTBb'f?E?LMftRpA'I) microscopic electrostatic mode Kelvin-Helmholtz instability in 86. (PV'C'Ss'vT'Bb'f'ELM't'R'A'I) anisotropic plasmas; electromagnetic mode Kink and Kruskal-Shafranov 52. (PV'CfSsTBbfffELMRfAfI) instabilities Kruskal-Schwarzschild see gravitation instability instability: Langmuir wave instability: see decay instability Longitudinal instability: see two-stream instability Loss-cone (maser) instabilities; 64. (PV'C'SsT'Bb'E'LM'tRpA'I) electrostatic modes Loss-cone instability; 56. (PV'C'SsT'BbfELM'tRpA1!) finite beta electromagnetic mode 68.

Mikhailovskii-Timofeev see drift-cyclotron instability: resonance instability Mirror instability 58. (PV'C'SsT'Bbf'ELM't'R'A'I) Magneto-acoustic wave instability 15. (PV'CnSs'vTBb'fE'LMR'A'I) Magneto-gravitational insta- see gravitation instability bility: Maser instability: see loss-cone instability Mass-conjugate instability: see negative mass and two- -stream instabilities Multi-stage instability: see Section 3.1.2. Necking-off instability: see sausage instability Negative characteristic 125. (P'V'CnSsTB'E'L'MR'A'I) instability

Negative-energy instability in 63. (PV'C'SsT'Bb'fE'LM'tRpA'I) inhomogeneous mirror geometry Negative energy wave insta- see Section 3.4.2.2. bility: Negative energy wave-wave see also explosive instability instability: Negative energy wave insta- 66. (PV'C'Ss'vTBbfE'LM'tRA'I) bilities of shock wave Negative mass instability 85. (PV'C'Ss'vT'Bb'fE'LM't'R'A1!) Negative radiation absorption 22. (PV'CnSs'v'T'LM'tRpA1!) instabilities; non-relativistic modes Negative radiation absorption 4. (PVC'Ss'v'T'LM'tRpA1!) instabilities; relativistic modes Neutral drag (crossed field) 16. (PV'CnSs'vTBb'fE'LMR'A'I) instability Neutral point instability: see resistive tearing instability Non-convective instability: see Section 3.4.2.1. (1) Overstability: see Section 3.4.1. (3) Parametric instability: see Section 3.4.2.5. Parametric instability; see Section 3.5.1.1. (2) non-linear modes: 69.

Parametric Alfven wave 10U. (PV'C'S'lTBb'f'ELMRp'A'I) instability Parametric electrostatic 108. (PV'C'S'lTBb'f »E'LMH'Rp'A'I) plasma oscillation instability Parametric instabilities of 106. (PV'C'S'lTBb'f'ELM't'Rp'AI) ion cyclotron waves Parametric ion acoustic 109. (PV'C'S'lTB'E'LMRp'A1!) instability Particle-wave interaction see Section 3.5. instability: Pierce (space-charge) 127. (PlVtClSstvlTBfEfLMRtAfI) instability Plasma diode instabilities 88. (PV'C'Ss'vB'E'LM't'R'A1!) Plasma diode instability; 126. (P'V'C'Ss'vTB'E'L'M't'R'AI) non-linear mode Radiation instability 55. (PV'C'SsTB'E'LMR'A'I) Rayleigh-Taylor instability: see also flute, interchange, o£.thermal convection insta- ilities Rayleigh-Taylor bulk see electroconvective instability: instability Recombination instability 11. (PV'CnSsTB'E'LMR'A'I) Relativistic instabilities: see Sections 3.3 and 5.2.2. Resistive ballooning 23. (PV'Cn'SsTBb'f'ELMR^'I) instability Resistive drift instabilities see drift-dissipative instabilities Resistive gravitational 2k. (PV'Cn'SsTBb'f'ELMR'A1!) instability Resistive rippling insta- 36. (PV'Cn'Ss'v'TBb'f'E'LMR'A'I) bility Resistive surface instability 25. (PV'Cn'Ss'rBb'f'ELMR'A'I) Resistive tearing instability 26. (PV'Cn'SsTBb'f'ELMR'A'I) Resonant diffusion instabi- 6. (PVCtSflfT'BblfEtLMftRpAlI) lities of a relativistic plasma Resonant particle instability: see trapping instabilities Rippling instability; collision- 90. (PV'C'Ss'v'TBb'f'E'LMR'A'I) less macroscopic mode Rotational instability: see gravitation and centrifugal instabilities 70.

Run-away instability 38. (PV'Cn'Ss'v'TB'E'LM't'R'A'I) Sausage instability UU. (PV'C'SsTBbf »ELMR'A'D Shear instability: see Kelvin-Helmholtz instability Screw instability in a partially 18. (PV'CnSsWTBb'f 'E'LMR'A'D ionized plasma Screw instability in fully 37. (PVfCnlSsfvfTBbfffEtLMRlAtI) ionized plasma Self-gravitational insta- 59. (PV'C'SsT'Bbf'ELM't'R'A1!) bility in anisotropic magnetized plasmas Sheet pinch instability: see surface and tearing instabilities Sideband instabilities 115. (PV'C'S'ITLRAI) Simon-Hoh instability: see neutral drag instability Sinous instability: see Kink instability Slip instabilities: see resistive instabilities Slipping-stream instability: see dlocotron instability Space-charge instability: see Pierce instability Soper-Harris instability: see Harris instability Spiral instability: see screw instability Stochastic instability 117. (PV'C'S'lT'B'E'L'M'tRpAI) Striations in the positive 20. (PV'CnSs'v'TB'E'LMR'A'I) column Surface (Siiydam) instability 1*5. (PV'C'SsTBbf'ELMR'A'I) Suydam instability: see surface instability Synchrotron instability: see cyclotron instability Tearing instability; collion- 53 (PV'C'SsTBb'f'ELMR'A'I) less macroscopic mode Tearing and sheet pinch insta- 68. (PVfCfSsfvTBbf »ELM'tRpA'D biltiy; collisionless microscopic mode Temperature-anisotropic 60. (PVfCfSsTfBbfffELMftRpAfI) transverse wave instability Temperature drift instability 78. (PV'C'Ss'vTBb'f'E'LM'tRpA1!) Thermal convection (Rayleigh- (PVfCSsTBbfffELMRfAfI) -Taylor) instabilities 71.

Three-stage instability: see Section 3.1.2. Timofeev instability: see ion cyclotron and drift- -cyclctron resonance instabilities Transverse wave (Weibel) 65. (PV^'SsT'B'ELM't^'A1!) instability Trapped particle instability: see Section 3.4 Trapped-particle dissipative 41. (PV'CSBb'L'A1!) instabilities in a magnetized plasma Trapped-particle collisionless 101. (FV'C'SBb'L'tA'I) instabilities in a magnetized plasma Trapped-particle instability 116. (PV'C'S'lT'B'E'LM'tRpA1!) in a strong electrostatic wave Trapping instability: see Section 3.4.3.1. Trapping and "bump" insta- 98. (PV'C^Ss'T'E'LM'tRpA'I) bilities; basic modes Trapping instabilities in 123. (PST'LM'tRpA'I) moving waves; basic modes Turbulence instability 119. (PV'C'S'l'T'B'ELM't'R'AI) Two-stream and beam-plasma 94. (PV'C'Ss'v'TB'E'LM't'RpA'I) instabilities; basic modes Universal Alfven wave 72. (PVfCfSsfv TBb'f'ELM'tRpA'I) instability Universal drift instability; 91. (PV'C'Ss'v'TBb'f'E'LM'tRpA'I) current-driven mode Universal low-frequency 79. (PV'C'Ss'vTBb'f'E'LM'tRpA1!) drift instability Velocity anisotropic see trapping and Harris instability: instabilities Velocity gradient instability: see Section 3.4.2.4. Velocity shear instability: see Kelvin-Helmholtz instability Velocity space instability: see Section 3.4.2.4. Wave-particle interaction see Section 3.5. instabilities: 72. Wave-particle nonlinear 118. (PV'C'S'lLM'RpAI) interaction instabilities Wave-particle quasi-linear 120. (PV'C'LM'RpAI) interaction instabilities Wave-wave interaction see Section 3.5 instabilities: Weibel instability: see transverse wave insta bility Whistler-dominated laminar 103. (PV'C'S'lTBbffELMRfAI) shock instability Whistler instability; 13. (PV'CnSsT'Bb'f'ELM^'R'A'I) collisional mode Whistler instability; current- 3. (PVC'Ss'v'T'Bb'f'ELM'tRpA'I) -driven relativistic mode Whistler instability; 61. (PV'C'SsT'Bb'jrELM'tRpA'I) pressure-driven mode Wiggly instability: see Kink and ballooning instabilities Wriggle instability: see Kink and ballooning instabilities T 1

Table 1. Basic theoretical approaches

. . . Approach Main properties Comments

Orbit Theory 1.Exact orbit 1.Direct study of the l.No approximations are in- theory. exact euqtion of volved. Has the drawback of motion. leading to extremely compli-• cated calculations which usually cannot be mastered. 2.First order 2.Particle orbit appro- 2.Much simpler than exact orbit ximated by guiding orbit theory. Leads to phy- theory. centre motion of sical insight in particle equivalent magnetic behaviour. Difficulties dipole. arise when charge separatioi phenomena due to differential motions are considered in a quasi-neutral plasma.

Kinetic The Boltzman-Vlasov Makes possible the treatment Theory equation "of motion" of quasi-neutral plasmas in a is considered for a more rigorous way than by density distribution first order orbit theory.Is in phase space. based on the restriction that the distribution in 6N- dimensional phase space can be factored in a product of single particle distribution functions.

Fluid Moments are taken of Is simpler than the kinetic Theory Bcltzman-Vlasov equa- theory,and still more rigo- tion, and a "cut-off" rous than first order orbit is introduced to theory. Main drawback is that obtain a closed set information on the behaviour of equations. The lat- in velocity space is lost ter yields conserv- when the moments are formed. ation laws of matter, momentum, and energy. T 2

Table 2. Energy sources driving plasma instabilities.

Class of Sources Special features sources

Intrinsic 1.Expansion energy l.The internal energy of the sources of plasma. the plasma 2.Deviations from 2.The free energy due to devia- local thermal tions from the Maxwellian equilibrium distribution in velocity 3.Kinetic drifts space. 3.Driftlances o arisinf an inhomogeneoug from the bas - spatial plasma state,and from imposed motions. 4.Electrostatic '*.The energy of immersed space energy charges. 5.Magnetic 5.The energy stored in the energy magnetic fields being induced by the plasma currents.

Externally 1.Static fields 1.Imposed gravitation,electro- imposed static,or other fields. force fields 2.Non-static 2.Imposed high-frequency fields fields

External Energy released Energy can be injected injection (or absorbed)ins id 2 (or extracted) by imposed beams (or extract- the plasma,origina- and by locally released or ab- ion) of ting from "external" sorbed heat due to reactions energy sources or sinks. between the plasma particles, sometimes with a radiation field. T 3

Table 3. Some coupling mechanisms.

Mechanisms Produced effects

Collisionless differential Electric charge separation, and drifts of ions and elec- electric fields with associated J trons and bunching effects drift motions. due to velocity dispersion.

Convective motions in a Changes in the local plasma non-uniform plasma. properties.

Induced plasma currents Changes in the magnetic field.

Variations of the j_x B Changes in the momentum balance force.

Resonant particle-wave and Changes in the plasma state as wave-wave coupling. a whole.

Collisional effects: 1. Resistive effects in a 1. Resistive diffusion across a fully ionized plasma. magnetic field. 2. Viscosity effects in a 2. Ion-ion collisions influence fully ionized plasma. the plasma balance. 3. Collisional "feedback" 3. Collisions put fluctuations out of phase, such as to create negative damping in some cases. 4. Collisional coupling U. Influence centre-of-mass between charged and neutral motion and charge separation particles. process, by the neutral drag. 5. Reactions produced by 5. Extra source terms arise in collisions between charged balance equations due to and neutral particles. ionization, recombination, attachment, nuclear reactions etc.

Heat conduction A thermal coupling takes place between neighbouring plasma elements•

Magnetic line-tying. The plasma elements become connected in the direction along a magnetic flux tube.

Feedback by external circuits Influences the dynamics and sta- and fields. bility of the plasma disturbances T k

Table 4. Stages of the instability analysis.

Stage Type of analysis Important points

1. Onset of an Linear Conditions for the onset of small- instability -amplitude instabilities and the magnitudes of their initial growth rates.

2. Growth of an Nonlinear The magnitude of the growth rate instability at large amplitudes, and the to large possible nonlinear saturation of amplitudes the amplitude.

3. The state of Quasi-linear The transport effects arising in fully deve- and non- a state of fully developed insta- loped insta- -linear bility amplitudes; associated bility loss mechanisms and the charac- amplitudes teristics of plasma turbulence.

4. Comparisons Nonlinear The determination of the modes between exis- which are in practice the most ting fully "dangerous" ones, in respect to developed the plasma balance and the losses, instability The questions about mode inter- modes action, multi-stage phenomena, and of the influence of the boundary conditions on the fully developed amplitude modes. i T 5 Table 5. Methods of analysis of plasma disturbances.

Method Special features Comments

Orbit analysis Direct study of the Difficult with exact equation by analytic particle orbits. of motion.First order orbit methods. theory often not rigorous enough for a quasi-neutral plasma analysis. Linear normal A disturbance is dis- Rased on the assumption that mode analysis solved into a spec- normal modes exist,form a trum of normal modes, complete set, and that their treated in terms of sum is bounded. kinetic or fluid theory. 1.Localized 1.Unperturbed quan- 1.Boundary conditions not tities treated as taken explicitly into constants.Yields account.Theory fails when full dispersion disturbance wave lengths relations and cannot be treated as small growth rates. quantities. 2.Non-localized 2.Variation of unper- 2.Boundary conditions inclu- turbed quantities ded. Valid for arbitrary taken into account. large wave lengths.May lead Yields full disper- to complicated analysis. sion relations and growth rates. 3.Nyquist 3.Applicable to any 3.Applicable to complicated diagram dispersion relation, analytical expressions for Yields stability which only stability con- condition,but not ditions are to be deduced. growth rates.

Nonlinear Weakly nonlinear ana- Arbitrary disturbances ex- mode analysis lysis developed so pressed as a superposition of far. eigenmodes. 1.Nonlinear l.In terms of macros- 1.Three-wave as well as wave-wave copic fluid equa- multi-wave interactions. interaction tions . 2.Quasi-linear 2.In terms of kinetic 2.The effect of the disturbance particle-wave theory. amplitudes on the distri- interaction. bution considered. 3.Nonlinear 3.In terms of kinetic 3.Only weak mode-coupling particle-wave theory,with mode included so far. interaction coupling. The energy Energy changes due to Can be applied to complex principle virtual displacements systems.Yields stability studied,under the sub- conditions but not growth sidiary condition of rates. the equations of motion. Computer Direct computation of Eliminates many analytical analysis the disturbance beha- difficulties,but usually viour, in terms of any yields only answers to speci- available theory. fic problems for restricted parameter values. T 6 Table 6. Stabilization mechanisms.

Localization General classes Examples on specific of mechanisms mechanisms

Internal plasma mechanisms Linear mecha- "Hard" mecha- 1.Applied axial magnetic field in pinch. nisms. nisms ,removing 2.Minimum-B and minimum-average-B. free energy. 3. Shear, 4.Adjustment of spatial plasma distribution . 5.Limitation of plasma extensions. "Soft" mecha- l.Relativistic dispersion effects. nisms ;affecting 2.Higher-order terms in Ohms law, the coupling. such as those producing finite- -Larmor-frequency effects. 3.Finite-Larmor-radius effects. ^.The Coriolis force of rotating plasmas. 5.High-beta effects. 6.The damping effects of dissipation. 7.Landau damping. 8.The thermal spread in velocity space.

Nonlinear Nonlinear sa- Amplitudes limited by saturation mechanisms turation effects ("hard"). Dispersion Resonant transitions made impossible relation by dispersion relation conditions. conditions ("soft"). Dissipative and Dissipation of large-amplitude dis- "mixing" ef- turbances, in combination with other fects ("soft"). stabilizing effects. Boundary "Soft" mecha- 1.Metal limiter effects,in combination condition nisms affecting with magnetic line-tying. effects the coupling. 2.Cool plasma boundary effects.

Externally "Hard" mediae : applied nisms removing mechanisms free energy- Stationary Stationary sta- The application of additional dc mechanisms bilization electric fields. mechanisms

Non- Dynamic stabi- The application of additional ac stationary lization electric and magnetic fields. mechanisms Feedback sta- The feedback of plasma disturbance bilization signals. I i Ibö/e 8. (fl/l) Rcfcrtmst/c

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Table 20. Characteristic frequencies in a plasma.

Phenomena Effects Frequencies

Localized Single l.Collisional 1.Various reaction rate and particle relaxation frequencies due to collisions. 2.Collisionless 2.Gyro frequencies.

Macros- 1.Waves and 1.Frequencies corresponding to copic oscillations sound,electrostatic,and electromagnetic oscillations. 2.Drift waves 2.Frequencies corresponding to drift waves due to pressure and magnetic field gradients and associated,equivalent,or applied force fields. 3.Externally 3„Frequencies of the applied fields. applied fields

Collective Transit and Transit and bounce frequencies. trapped particle effects. Fl

P2.g»xm oiinpiB ötxi«tioy,y ox pxyunjffi t-iû sliding on a smooth wall surface. Its positions given in the figures represent the unper.turbed state» 1,1* Ko equilibrium' exists, 1*2« Marginal state with zero growth rate* 1,5» 3table equilibrium for both small and large disturbances» 1,4» Unstable equilibrium for both small and larê disturbe.ri.ces. l.% Stable equilibrium for small disturbances, but unstable equilibrium for larê disturbances. 1,6» Unstable equilibrium for small disturbances, but large disturbance amplitudes being limited (saturated). Fig.2. CLASSIFICATION SCHEME OF PLASMA DISTURBANCES

(P) In plasma only (P1) External space involved

(V) Relativistic (V) Non-relativistic GENERAL | CONDlTlONSj (C) Collisional (C) Collisionless

(n) Neu u als j(n')No neutrals I (S) Stationary (S*) Non-stationary

(s) Static j(s')Non-static (I) Laminar (l')Non-laminar \M {(V7} I [Mass {No mass UNPERTUBED; (motion j mot ion 1 STATE i (T) Thermal-isotropic (D Not thermal-isotropic CONDITIONS I

(B) Magnetized (B1) Unmagnetized

(b)High-beta J(bf)Low-beta

(f)Flutelike j(f) Ballooning I (E) Electromagnetic (E') Eletrostatic

(L) Localized (L*) Collective

(M) Macroscopic (M1) Microscopic PERTURBED STATE (t) Trapped |(t')Untrapped CONDITIONS i

(R) Resonant (Rf) Non-resonant

(p)Particle- Up1) Wave- wave ! wave

(A) Large-amplitude (A') Smalhamplitude

(I) Unstable disturbance (D Stable disturbance RESULTS F3 Fig.3. GENERAL DIVISION IN SUBCLASSES Numbers refer to corresponding Tables

Instabilities

(PI) In plasma only Externally coupled(7)

(P\ri) Non-relativistic Relativistic (8)

(PV'LI) (PVLII) Localized Collective (9)

(PVLA1D (PV'LAI) Linear Non-linear(IO)

(PV'BLA'I) (PVB:LAI) Magnetized Unmagnetized

(PV'CBLA'I) (PV'CBLA'I) (PVCB'LA'I) (PV'CB'LA'I) Collisional Collisionless Collisional 01) Collisicnless(12)

(PV'CnBLA'I) (PVCrVBLA'I) (PV'C'XBLA'I) (PVCTBLA'D Neutrals (13) No neutrals Thermal-isotropic Velocity space (15)

(PVC'TBLMA'I) (PVC'TBLNrA'D Macroscopic Microscopic

(PV'C'TBELMA'I) (PVC'TBE;MA'I) (PV'C'TBELM'A'I) (PV'C'TBrLM'A'I) Etec tromagnet ic(16] Electrostatic (17) Electromagnetic (18) Electrostatic (19) AO

APPENDIX. BASIC PROPERTIES OF INDIVIDUAL INSTABILITY MODES AND SUBCLASSES

In this Appendix are given the basic properties of a number of individual instability modes and subclasses, including their energy sources, coupling mechanisms, and stabilization mechanisms. The references piven are by no means complete, and should merely serve as a hint for further studies of the literature.

The modes are ordered according to the classification scheme and index of Sections 5 and 10.1 in which a class of property Q is listed before that of the property Q1. Further, in cases where two modes belong to the same class defined by the present rotation, the modes are put in alphabetic order in respect to their names. Al

(PVC'SsT'Bb'f'E'LM'tRpA'I)

Cyclotron instability; relativistic mode R.A. Blänken» T.H. Stix, and A.F. Kuckes, Plasma Physics 11 (1969)91+5.

: The Larmor motion of relativistic electrons

Main_features: A resonant exchange of energy takes place between relativistic electrons and electrostatic waves. The latter are associated with the finite temperature electrostatic resonance slightly above 2 w where oi e e is the non-relativistic electron cyclotron frequency in the magnetic field B>. Here relativistic cyclotron phase burching occurs for k'B=0, where k is the wave number. This is contrary to the corresponding non- relativistic case which requires k*_B^0 for instability to occur. A loss-cone nature of the hot electron distribution is further necessary for this mode. The instability is convective. §£§kilization: A decrease in the reflection coefficient of a wave packet at the boundaries decreases the growth rate. A2

(PVC'Ss'vTBb'f!ELMRpAfI)

Cerenkov instability C.F. Knox, J. Plasma Physics 1 (1967)1

source: ^-e motion of a plasma streaming through another "background" plasma.

A "background" static plasma is placed in a homogeneous magnetic field. Another plasma component is streaming at a relativistic velocity along the magnetic field, through the former plasma. This generates unstable Cerenkov and cyclotron waves. This represents a "cold" model and a special case of the "bump" instability produced by a double-humped velocity distribution.

Stabilization:At certain values of the involved parameters. A3

(PVC^s'v'T'Bb'f •ELM'tRpA'I)

Whistler instability; current-driven relativistic mode D.B. Chang, Astrophys. Journ. 138 (1963) 1231 and General Atomic, Report GA-6696 (1965).

Energy Source: Current due to electrons moving along magnetic field.

This instability exists in the frequency range w.<

possible frequency range for instability is reduced when the plasma density is decreased.

,v (PVC'Ss'v'T'LM'tRpA'I)

Negative radiation absorption instabilities; relativistic modes R.Q. Twiss, Australian Journ. Phys. _11 (1958) G. Bekefi, J.L. Hirschfield, and S.C. Brown, Phys. Fluids M (1961) 173.

: Deviations from Maxwellian velocity distribution

Main^features: These modes are similar to the corresponding non-relativistic ones, and belong to the class of trapping instabilities. In the relativistic case, the conditions for negative absorption of synchrotron radiation can also be satisfied in a fully ionized plasma, in a way not having its non-relativistic correspondence.

suppressing the deviations from the Maxwellian distribution, or by choosing certain stable ranges of the radiation parameters. A5

(PVC'S'lTBb'ELMR'AI)

Amplitude dispersion instabilities; relativistic modes T. Tang and A. Sivasubramanian, Phys. Fluids 14 (1971)

source: Energy of main wave.

£y^§i These modes represent a relativistic correspondence to the non-relativistic amplitude dispersion instability of whistlers. A second-order frequency shift of an electromagnetic wave is not only produced by a finite amplitude, but also by relativistic effects. As a result, unstable non-linear modes arise from the decay of an initial wave train. Contrary to the non-relativistic case, however, unstable waves are produced by the relativistic effects, also when the wave number vector k is parallel to the immersed magnetic field B .

§£äbilization: Ev the effects of a finite temperature and by electron-ion collisions. A6

(PVC'S'l'T'Bb'fE'LM'tRpA'I)

Resonant diffusion instabilities of a relativistic plasma I. Lerche, Phys. Fluids 11 (1968) 1720.

EnergY.source: The kinetic drift motions of the relativistic part of the particle velocity spectrum.

!^§iO-fi§iiiE^§i A collisionless relativistic plasma is immersed in a homogeneous magnetic field, in presence of a cold non-relativistic background plasma. A weakly turbulent unperturbed state is assumed, and the quasi- linear behaviour of the plasma is considered. In particular, it is found that unstable electrostatic flute modes become coupled with selective groups of particles. A resonance occurs with the associated waves, since the particle "mass" is no longer the rest mass in a relativistic case. Thus, there is a resonant diffusion of relativistic particles and a transfer of energy from the particles to the waves.

Stabilization: By decreasing the number of relativistic particles compared to that of the background plasma. A7

(PVlCnSsTBblfElLMRlAlI)

Ionization (electrothermal) instability J. Kerrebroc* AIAA Journ. 7_ (196U) 6; A.H. Nelson and M.G. Haines, Plasma Physics 11 (1969) 811.

I The Power input due to an imposed electric current.

a partially ionized plasma the ionization rate increases steeply with the electron temperature. The Ohmic heating due to an imposed electric current density is at the same time enhanced in regions of high electron density, provided that the heat losses of the electron gas by radiation and by collisions with heavy particles are sufficiently small. Then, perturbations in the local electron density become unstable due to the corresponding strong fluctuations in the ionization rate, provided that the imposed magnetic field strength and the Hall parametor are large enough to reduce the heat losses below a certain critical level.

losses of the electron gas due to collisions, conduction, convection, and radiation. A8

(PV'CnSsTB'E'LMR'A'I)

Acoustic wave instability in a partially ionized plasma U. Ingard and K.W. Gentle, Phys. Fluids Q_ (1965) 1396; U. Ingard, Phys. Rev. 1415 (1966) 41.

source: ^he thermal energy of the electron gas.

Main_features: In a weakly ionized gas with hot electrons, the heat transfer to the neutral gas by electron- neutral collisions leads to an excess heating of the neutrals, in places where the neutral gas density has its maxima. This produces growing sound waves in the neutral gas.

Stabilization: By collisional damping, viscosity and heat conduction in the neutral gas. A9

(PV'CnSsTB'E'LMR'A'I)

Electroconvective instability R.J. Turnbull, Phys. Fluids 11 (1968) 2588, 12_ (1969) 1160.

: Electrostatic energy

25A fluid of high and stronpn., temperature- dependent resistivity is placed bet^ren two horisontal highly conducting plates. A verti J electric field and temperature gradient are applir •. The electric rield acts on the free electric charges ^used by the gradient in resistivity, such as to counteract the effect of viscosity and to produce an instability in the form of convection.

Stabilization: By viscosity. Al O

(PV'CnSsTB'E'LMR'A'I)

Ionization-recombination ion sound instability A.I. Akhiezer, I. A. Akhiezer, and V.V. Angeleiko, Sov. Phys. JETP, 3£ (1970) 476.

£0§r.SY_§2Hr.9§: thermal energy of the electron gas.

S§ a partially ionized plasma the ionization- recombination rate becomes coupled to the fluctuations in plasma density which, in their turn, are associated with ion-acoustic oscillations. Under certain conditions these oscillations become growing perturbations. This mode is somewhat related to the icnization instability.

collisional damping and other loss mechanisms. All

(PV'CnSsTB'E'LMR'A'I)

Recombination instability N. D'Angelo, Phys. Fluids 1£ (1967) 719; 0. Kofced-Hansen, Phys. Fluids, 1£ (1969) 212»f.

Energy^ source: The internal energy of the plasma.

: This instability arises in systems where ion-electron recombinations is the dominant loss process, with a recombination coefficient being a strongly decreasing function of the electron temperature. It is further necessary that the density and temperature fluctuations of the electron gas can bo treated as adiabatic, yielding low temperature in regions of low density, and vice-versa. Recombination is then speeded up in the low-density regions, thus resulting in an enhanced perturbation, provided that the temperature dependence of the recombination rate j is stronger than its density dependence. 1 i Bv "t^le density dependence on the recombina tion rate, and by deviations from adiabaticity due to heat conduction, collisions and other loss processes. Al 2

(PV'CnSsTB'E'L'M't'R'A'I)

Continuity equation instability J.R. Rooth, Phys. Fluids 1£ (1967) 2712 and LI (1969) 763

: The thermal energy of the electrons, being coupled to the ionization rate.

continuity equation of a partially ionized plasma, the ion production is usually represented by a term containing ion and electron densities n "localized" in space and time, i.e. there are dn particles within a velocity interval between w and w+dw at all locations and for all times. In general, particles are constantly appearing and disappearing in the gas, partly on account of transit time effects. This leads to extra contributions to the divergence term in the continuity equation, not being present in a simple fluid model. These contributions may lead to growing low-frequency oscillations in the plasma density.

Stabilization: By reducing the transit time effects. A13

(PV'CnSsT'Bb'f»ELI'

Whistler instabili*y, collisional mode H. Derfler and J.Q. Howell, Ninth International Conference on Phenomena in Ionized Gases (G. Musa et al., Editions), Institute of Physics, Academy of the Socialist Republic of Romania, Bucharest, Sept. 1-6 (1969), page 478.

Eö§r.2y._SQu.rQe: The deviation from the Maxwellian distribution due to an electron peak.

a magnetized partially ionized collisional plasma electrons are released, e.g. by photoionization. The corresponding sharply peaked isotropic and monoenergetic electron distribution is assumed to survive until it is destroyed by an instability. In presence of the phase shift and the corresponding feedback due to collisions, the whistler mode then splits into two branches, one of which becomes unstable under certain conditions.

When collisional phenomena destroy the non-Maxwellian part of the distribution before the instability has had time to grow, and when the plasma parameters are chosen within certain ranges. A1U

(PV'CnSs'vTBb'fE'LMR'A'I)

Drift instability by neutral collisions A.v. Timofeev, Sov. Phys. Tech. Phys. S_ (1964) B82.

.: The transverse density gradient of the plasma and its associated macroscopic flow of matter,

A weakly ionized inhomogneous plasma with ^ is assumed to be confined in a strong homogeneous magnetic field. On account of the collisions between changed and neutral particles, the drifts of ions and electrons across the magnetic field will become unequal and lead to charge separation and growing electrostatic perturbations. This mode is closely related to the neutral drag instability.

decreasing the density gradient and magnetic field strength. Al 5

(PV^nSs'vTBb'fE'LMR'A'I)

Magneto-acoustic wave instability E.P. Velikhov, Symp. MPD Elec. Power Gen., Newcastle-upon -Tyne (1962); J.E. Me Cune, First Int. Symp. on Magnetic- Kydrodynamical Electric Power Generation, Paris, Paper 33 (196U).

Energy source: An imposed electric field and its associated current.

Main^features: In presence of the Hall effect and an electric current i perpendicular to a homogeneous magnetic field B , there will exist an electric field E forming a certain angle with the current vector in the plane perpendicular to B . At small magnetic Reynolds numbers in a cool, partially ionized plasma, the magnetic field will further remain nearly unaffected by a plane acoustic wave traveling along the positive or negative direction of j. On the other hand, the density fluctuations in this wave will produce local changes in the Hall parameter which, in their turn, give rise to a corresponding perturbation current. The latter produces an additional force with the field B which amplifies part of the waves propagating along i .

Tne growth rate decreases with the Hall effect.

' *' if Al 6

(PV'CnSs'vTBb'fE'LMR'A'I)

Neutral drag (crossed field) instability A. Simon, Phys. Fluids £ (1963) 382; F.C. Hoh, Phys. Fluids £ (1963) 1184.

source: fluid motion of a plasma through a neutral gas.

ö-£S§HE!:§ An imposed radial electric field produces a fluid motion in a partially ionized plasma. The resulting drag forces on ions and electrons due to neutral gas friciton are unequal and lead to charge separation and to an azimuthal electric field. In its turn, the latter produces a drift motion of the plasma. With proper signs of the radial electric field and the density gradient, growing perturbations are then generated.

Tne instability is suppressed by changing the polarity of the radial electric field, and by the effects of diffusion.

U. A17

t '4t (PV^nSs'vE'E'R'A'I)

Corona discharge instabilities G. Buchet and M. Goldman, Ninth International Conference on Phenomena in Ionized Gases (G. Musa et al., Editors), Inst. of Physics, Acad. of Socialist Republic of Romania, Bucharest, Sept. 1-6 (1969), pages 291-292.

Energy_source: An applied electric field.

ff^ures£ Streamers are observed in corona discharge experiments at atmospheric pressure when the applied voltage exceeds a certain critical level. No theory is given on this phenomenon.

^ili5^Si22: At sufficiently low voltages and by special choices of the gas components. Al 8

(PV'CnSs'v'TBb'f»E'LMR'A1!)

Screw instability in a partially ionized plasma B.B. Kadomtsev and A.V. Nedospasov, Plasma Physics, Journ. Nucl. En. Part C, 1. (1960) 230; B. Lehnert, Proc. Sec. Int. Conf. on the Peaceful Uses of Atomic En., Geneva, 32 (1958) 319, United Nations, New York; F.C. Hoh and B. Lehnert, Phys. Rev. Letters 1_ (1961) 75.

imposed axial electric current.

Ma?.n features: An electric current is flowing along a partially ionized axially symmetric plasma column situated in an axial magnetic field. Since ions and electrons are drifting in opposite directions along the magnetic fiedd, screw-shaped perturbations of the plasma column lead to charge separation and to an induced azimuthal electric field. The latter produces, in its turn, radial drift motions which enhance the perturbations,by moving parts of the dense plasma radially outwards into regions of lower density.

The neutral drag forces on ions and electrons drifting in the radial ambipolar electric field produce an additional charge separation effect which in certain cases counteracts that from the axial current. Further stabilization arises from diffusion and conduction.

*0 " A19

\-k (PV'CnSs'v'TB^'LMR'A'I)

Gunn instability H. Sabadil, Beiträge z. PI. Physik 8; (1968) 299.

applied electric field and the corresponding current.

Main_features: An instability was detected by Gunn in semi- conductors, being caused by the negative differential resistivity of its current characteristics. This behaviour was caused by an increase of the number of heavy charged particles at increasing electric fields and temperatures, causing in its turn a decrease in mobility. A similar situation may occur in partially ionized gaseous discharges in certain gases, on account of the recombination, attachment, and ionization processes being involved.

Instability arises only in certain parameter ranges, and not in halogene gases. A20

(PV'CnSs'v'TB'E'LMR'A'I)

Striations in the positive column G. Francis» Handbuch der Physik, Springer-Verlag, Berlin 1956, Vol. 22-> p. 1^0; D.A. Lee, P. Bletzinger, and A. Garscadden, Journ. Appl. Phys. 37. (1966) 377; D.W. Swain and S.C. Brown, Phys. Fluids 1M_ (1971) 1383.

The axial drifts of charged particles produced by an imposed electric field.

Main^features: The mobility of electrons in the positive column of a partially ionized plasma is much larger than that of the ions. Thus, a disturbance in the plasma density will after a short time produce corresponding perturbations in the space charges, electric fields, electron temperature, ionization rate, and the properties of the radiated light of the column. The self-excited striations arising from this process can consist both of stationary and of wave-type moving disturbances. They can further be coupled with other oscillations in the plasma.

choosing the discharge parameters within certain ranges. A21

(PV'CnSs'v'T'Bb'f»ELMlt'R'A'D

Collision-induced electromagnetic instability in partially ionized gas K. Suzuki, Journ. of Phys. Soc. Japan Z2. (1967)

A peak in the unperturbed velocity distribution of the electrons.

Main_features: A partially ionized gas is considered where the electron-neutral collision cross section depends strongly on the velocity. A superimposed peak in the electron velocity distribution then leads to anisotropic scattering conditions in velocity space. This corresponds to an equivalent net electric current being 180 degrees out of phase with the applied peak, thus giving rise to a positive feedback and to growing electromagnetic oscillations •

Stabilization: At large wave numbers. A22

(PV'CnSs'v'T'LM'tRpA'I)

Negative radiation absorption instabilities; non-relativistic modes R.Q. Twiss, Australian Journ. Phys. 3A, (1958) 564; G. Bekefi, J.K. Hirschfield, and S.C. Brown, Phys. Fluids 4 (1961) 173.

: The deviations from a Maxwellian velocity distribution.

S-£!=ä£ Tne condition 3f/8w>0 for the distribution function f of the plasma particles is not sufficient for a plasma to become unstable, such as by inverse Landau damping from a corresponding "hump" in the distribution. To arrive at a situation of growing waves where the plasma has a negative total absorption and temperature, it is further necessary to satisfy the condition 9[R(w)w j/3w>0 where w is the particle velocity and R(w) the absorption cross section of the radiation process in question. This leads to a class of trapping instabilities, by which radiation takes place in the form of bremsstrahlung and cyclotron radiation in a partially ionized gas, partly by pressure broadening.

ili5äti2i suppressing the deviation from the Max- wellian distribution, or by choosing certain stable ranges of the radiation parameters. A23

(PVfCnfSsTBb'f 'ELMR'A'D

Resistive ballooning instability R.M. Kulsrud, Plasma Instabilities, Proc» of the Int, School of Physics, Course XXXIX, Varenna (Ed. by P.A. Sturrock) July 1966, Academic Press, New York and London.

52e.r.SY._§°.urce: Gravity, or equivalent forces due to the magnetic field curvature.

B-£.§:H.. A plasma is supported against gravity by a horizontal magnetic field. A ballooning disturbance of a given wave-length, which would be stabilized in a dissipation-free case, becomes unstable even at low beta values, due to finite resistivity and to the corresponding diffusion across the field. This mode is closely related to the resistive gravitational instability, its name usually refers to comparatively long disturbance wave lengths.

Growth rate is reduced at increasing temperatures and field strengths.

¥4\

/ 4 }i J A24

(PV'Cn'SsTBb'f»ELMR'A'I)

Resisitve gravitational instability H.P. Furth, J. Killeen, and M.N. Rosenbluth, Phys. Fluids £(1963) 4 54; H.P. Furth, Advanced Plasma Theory, Proceedings of the Int. School of Physics, Course XXV, Varenna, Academic Press, New York 1964.

Energy^source: Gravity, or its equivalent due to the magnetic field curvature.

Main_features: Like the resistive rippling mode this instability develops in a limited region where k#_B is small and finite resistivity plays an important role in Ohm's law, i.e. plasma diffusion takes place across the magnetic field. A counter-circulatory motion is set up by the bouyancy forces due to gravity, leading to an interchange of heavy and light plasma elements. This mode is closely related to the resistive ballooning mode. The latter is not restricted to limited regions and small values of k*J3.

When the gravitation field, or its equivalent, points in the stabilizing direction, such as in minimum-B geometry. A25

(PV'Cn'SsTBb'f 'ELMR'A'I)

Resistive surface instability J.D. Jukes, Phys. Fluids ± (1961) 1527.

: The plasma pressure gradient.

Main_features: This mode is closely related to the surface (Suydam) instability, with the exception that finite resistivity is taken into account. This leads to a type of resistive ballooning mode, localized to a small region around the surface in a plasma column where k»B=0. This mode can occur even at low beta values.

: Growth rate is reduced at strong magnetic fields and high temperatures.

1 xLf4ef S4,^ A26

(PV'Cn'SsTBb'f' ELMR'A'I)

Resistive tearing instability J.W. Dungey, Phil. Mag. UM_ (1953) 725; H.P. Furth, J. Killeen, and M.H. Rosenbluth, Phys. Fluids 6 (1963)

e.: The magnetic energy.

iB.g^ Like the resistive rippling mode this instability develops in a limited region where k«I3 is small and finite resistivity plays an important role in Ohm's law, i.e. plasma diffusion takes place across the magnetic field. In the case of the present mode, the perturbations are of a long-wave type, being coupled to the plasma outside of the region where k*I3 is small. The field line diffusion makes the plasma break up into a set of individual pinches. This releases magnetic energy-

Stabilization: Growth rate is reduced by increasing the plasma temperature. A27

(PV'Cn'SsTBb'f'E'LMR'A'I)

Finite heat conductivity instability A.A. Galeev, V.N. Oraevskij and R.Z. Sa^deev, Sov. Phys. JETP 17 (1963) 615.

The thermal plasma pressure.

Main_features: A plasma with antiparallel density and temperature gradients is immersed in a homogeneous magnetic field being perpendicular to the gradients. Spoke-shaped disturbances extended along these gradients are considered, having phase velocities w/k_ in the magnetic field direction satisfying the condition w.«w/k <

When the heat conductivity along the magnetic field is reduced, the disturbances tend to a state of stable adiabatic oscillations.

är. i A28

(PV'Cn'SsTBb'f»E'LM1tfRfA'l)

Collision-induced electrostatic instability in fully ionized plasma K. Jungwirth, J. Plasma Physics 3_ (1969) 155.

i The expansion energy of the plasma

^iD-f^É^ a collisionless, fully ionized plasma with large inn temperature gradients, stability can be achieved by means of sufficiently large minimum-B effects. However, in presence of ion-ion collisions a new branch of unstable oscillations arises. This mode does not exist in gravity-simulated curvature models, because it depends directly on the geometric properties of the magnetic field.

Stabilization: By a sufficiently curved magnetic field.

K;,, f A29

(PV'Cn'SsTBb'f E'LM't»R'A1!)

Entropy wave instability B.B. Kadomtsev, Sov. Phys. JETP 37_ (1960) 780 i Shin-Tung Tsai, F.W. Perkins, and T.H. Stix, Phys. Fluids 13: (1970) 2108; A.B. Mikhailovskii and V.S. Tsypin, Zh. Eksp. Teor. Fiz. 5£ (1970) 524.

The thermal energy of the plasma.

G-Stnr§A low-beta plasma is immersed in a static uniform magnetic field IJ. The density and temperature vary in a direction perpendicular to .B, with the relative temperature gradient being much smaller than the relative density gradient. Electrostatic low-frequency waves with a wave number almost perpendicular to B are considered. The combined effects of ion-ion.collisional viscosity and heat transport then gives rise to a new entropy-wave overstability mode. The entropy waves are characterized by having no magnetic field and overall pressure disturbances, with the temperature perturbations T-=-T .

5ili5åti2 Within certain ranges of the wave numbers and plasma parameters. The instability vanishes at low plasma densities.

pltvV. i. Sfwr A30

(PV'Cn'SsTB'E'LMR'A'I)

Ion sound wave instability in f\:lly ionized plasma M. Schultz, Phys. Fluids 3J) (1967) 243.

'thermal energy of the electron gas.

Main features: In a plasma with hot electrons, there is a heat transfer to the ions by Coulomb collisions. When T >Y.T. with Y. being the specific heat ratio of the e i i ' i ° c ion gas, this leads to heating of the ions in places where the ion density perturbations have their maxima, thus giving rise to growing ion sound waves. This instability resembles the acoustic wave instability in a partially ionized plasma.

£ii§i collisional damping, viscosity and heat conduction in the plasma. A31

(PV'Cn'SsTBLMR'A1!)

Fusion reaction instabilities A.L. Fuller and R.A. Gross, Phys. Fluids IJ^ (1968) 53U; M.S. Chu, Phys. Fluids L5 (1972) 413; M. Ohta, H. Yamato, and S. Mori, Nippon Genshiryoku Gakkaishi 13^ (1971) 259 and United States AEC-tr-7291, Techincal Information Center, Washington D.C.

: Thermonuclear reactions and injected heat.

Main_features^ The balance of energy production and energy losses of a thermonuclear plasma is involved in this mode. In the case of magnetically confined plasmas various types of energy loss mechanisms such as classical and Bohm diffusion, affect this balance. Temperature perturbations then lead to an unbalance between the energy production and losses and to instability within certain ranges of the plasma parameters. Thermonuclear reaction waves are found to differ from ordinary chemical reaction waves in several ways.

Stabilization: By external feedback control.

ir A3 2

(PV'Cn'Ss'vTBbfELM't'R'A'I)

Gradient instabilities in high-pressure plasmas A.B. Mikhailovskii and V.S. Tsypin, Plasma Physics ^ (1971) 785.

?DSHSY-§2yH£S: The thermal energy of the plasma.

Main^features: A hiph-beta plasma with spatial gradients is treated by means of a refined system of transport equations. This leads to some new types of flute modes, partly being excited by the gyro-relaxation effect.

Stabilization: At certain values of the parameters involved in the dispersion relations. A3 3

(PV'Cn'Ss'vTBb'f'E'LMR'A'I)

Hydrodynamic electrostatic instabilities in a collision dominated plasma A.B. Mikhailovskii, Sov. Phys. JETP 2JL (1967) 831.

"thermal energy of the plasma.

S-fSå^USSiElectrostatic modes in a collision-dominated fully ionized, magnetized plasma with temperature and density gradients are considered in the hydrodynamic two-fluid approximation. A number of the most "dangerous" modes with respect to shear stabilization are investigated, depending upon the frequency range and the inclusion of various collisional effects such as viscosity, thermal conductivity, and heat exchange by ion-electron collisions

i: According to the specific dispersion relations obtained for the various modes; shear stabiliza±ion plays a relatively small role here.

.,* A34

(PV'Cn'Ss'vTBb'f'E'LM't'R'A1!)

Drift-dissipative electrostatic instabilities S.S. Moiseev and R.Z. Sagdeev, Sov. Phys. JETP 17^ (1963) 515; F.F. Chen, Phys. Fluids £ (1965) 912; N.A. Krall, Advances in Plasma Physics (Ed. by A. Simon and W.B. Thompson) Interscience Publ., New York 1968, Vol. 1 p. 153.

transverse drift motion.

Main_features: The drift-dissipative modes resemble the universal modes, but their growth is connected with collisions rather than with resonant particles. The instability mechanism is due to the fact that the potential and density fluctuations are put out of phase by collisions. This creates, in its turn, a guiding centre motion due to the perturbed electric field which is in phase with the density fluctuations and produces a growing disturbance.

increasing the temperature and decreasing the plasma resistivity.

T A ^ {( •* A3 5

J (PV'Cn'Ss'vTBb'f•E'L'M't'R'A'I}

Finitie orbit instability O.P. Förutse, Sov. Phys. JETP 2_5 Q967) 1021,

Energy source: The thermal energy of the plasma in form of drift motions.

M§in.fmatures: ;.n "a straight magnetic field, classical diffusion by collisions leads to guiding centre displacements of the order of a Larmor radius. In toroidal systems as those of the Tokamak or bumpy torus type, however, "banana" diffusion of trapped particles enhances the influence of collisions by producing guiding-centre displacements being much larger than those by classical diffusion. This leads to a kind of drixt-dissipative instability which occurs at much smaller collision frequencies than the usual instabilities of this type. The present mode belongs to the class of dissipative trapped-particle instabilities in a magnetic field.

Stabilization: The joint effects of a large shear, as well as minimum~average~B properties, reduce the growth rate appreciably. A36

(PV'Cn'Ss'v^Bb'f »E'LMR'A'I)

Resistive rirpling instability H.P. Furth, J. Killeen and M.N. Rosenbluth, Phys. Fluids 6 (1963) U59.

electric current in the plasma*

Main_features: In a limited region of a magnetized plasma there exist localized perturbations for which the effect of the vxB term in Ohmfs law is small, i.e. where k-13 is nearly zero and k is the corresponding wave number. Here the effects even of a small resistivity become significant, as well as the corresponding diffusion of plasma across the field. Thus, the electric current which supports the plasma can become channeled locally by the circulatory motion which produces ridges of lower resistivity in such a region. The rippling mode is therefore an electrostatic current-convective instability. This is somewhat similar to the conditions producing the screw instability in a fully ionized plasma, but the present mode develops on a smaller scale.

losses due to heat conduction. The growth rate is reduced by increasing the plasma temperature. A37

(PV'Cn'Ss'v'TBb'fE'LMR'A'I)

Screw instability in fully ionized plasma B.B. Kadomtsev, Sov. Phys. Tech. Phys. £ (1962) 882.

Energy source: An imposed axial electric current.

Main features: An electric current is flowing along a homogeneous magnetic field in a fully ionized plasma having temperature and resistivity gradients perpendicular to the field lines. Due to finite resistivity, matter can slip across the magnetic field. By a perturbation, matter of lower resistivity can then be brought into a region initially having higher resistivity. This corresponds to a change in the perturbed electric field which, in its turn, leads to an accelerated velocity perturbation of the plasma. The development of the instability can also be understood as an effect of the resistivity gradient which leads to an equivalent extra electric current by which the magnetic field lines no longer lag behind the plasma motion, as in a damped case, but instead lead it and convert the damping into a growth. This type of instability is somewhat similar to the resistive rippling mode, in the sense that it is associated with current channeling. Contrary to the rippling mode, however, it is of relatively large scale and involves large parts of the plasma body.

§£l§£2 By losses due to heat conduction which tend to eliminate the resistivity perturbations. A38

(PV'Cn'Ss'v'TB'E'LM't'R'A'I)

Run-away instability H. Dreicer, Phys. Rev. 11_5 (1959) 138-, G. Nicolis and Ph. Sels, Phys. Fluids 10 (1967) 4m.

imPosed electric field.

Main_features: The cross section of Coulomb collisions decreases with increasing mutual velocity of colliding charged particles. As a consequence, a group of plasma electrons will "run away" at increasing velocity when the applied electric field is chosen above a certain critical value.

keeping the electric field below critical run-away level. A39

(PV'Cn'S'l'TB'E'LMR'A'I)

Acoustic instabilties V.N. Tsytovich, Sov. Phys. Tech. Phys. 1£ (1965) 605; V.G. Makhankov and V.N. Tsytovich, Plasma Physics 1£ (1970) 741.

Energy source: A transverse imposed beam of electromagnetic waves.

Main_feature_s: An intense high-frequency beam of electromagnetic waves is sent through an unmagnetized plasma, making the latter weakly turbulent. The turbulent state of the plasma leads, in its turn, to "anomalous" dispersion properties in the low-frequency region. As a result,unstable plasma and sound waves are excited. This corresponds to "anomalous" dissipation of the beam energy, such as to increase the dissipation and heating beyond the usual Ohmic value in certain cases. These results are, among other things, of interest in connection with the interaction between laser beams and matter

£ili52i22 Within certain ranges of the wave number of the low-frequency oscillations. The .growth rate also decreases with decreasing energy of the beam. AU O

J (PV'CSsTBb'f 'ELMR'A'I)

Thermal convection (Rayleifth-Taylor) instabilities S. Chandrasekhar, Hydrodynamics and Hydromagnetic Stability, Oxford, Clarendon Press 1961, Ch. IV-, C.P. Yu, Phys. Fluids 11 (1968) 756.

Energy source: The buoyancy force due to an imposed temperature gradient in a fluid layer heated from below.

Main_features: A viscous electrically conducting liquid or "incompressible" fluid is heated from below, in presence of an immersed magnetic field which may be vertical or horizontal. Various types of boundary conditions are also of interest in this connection, yielding a number of critical conditions for the onset of instabilities in the form of convection cells.

Bv reducing the temperature gradient, increasing the magnetic field, or choosing the rest of the parameters involved within suitable r^. (PV'CSBb'L'A'I)

Trapped particle dissipative instabilities in a magnetized plasma B.B. Kadomtsev and O.P. Pqgutse, Nuclear Fusion 11 (1971)67,

EnergY.sources: Similar to those driving the instabilities in a simple magnetic field geometry without rotational transform.

Main_features^ These modes are related to the collisionless trapped-particle instabilities in a closed field-line toroidal geometry, with the exception that collisions are taken into account. Due to neoclassical diffusion effects, the collisions have a more pronounced influence on the growth of the instabilities than in the case e.g. of an ordinary mirror-type magnetic bottle.

: Bv effects similar to those in an equivalent ordinary magnetic bottle.without rotational transform. A42

(PV'C'SsTBbf 'ELMR'A'I)

Ballooning instability R.M. Kulsrud, Plasma Physics and Controlled Nuclear Fusion Research, IAEA, Vienna, Vol. I (1966) 127; R.M. Kulsrud, Plasma Instabilities, Proc, of the Int. School of Physics, Course XXXIX, Varenna (Ed. by P.A. Sturrock) July 1966, Academic Press, New York and London; H.P. Furth, J. Killeen, M.N. Rosenbluth and B. Coppi, Plasma Physics and Controlled Nuclear Fusion Research, IAEA, Vienna, Vol. I (1966) 103.

Gravity, or equivalent forces due to the magnetic field curvature.

Main_features^ A plasma is supported against gravity by a horizontal magnetic field. Perturbations of a given wave-length in the magnetic field direction then become unstable when a "bending"of the magnetic field lines produces a restoring force being too weak to oppose an interchange of the fluid elements under the action of gravity. j : i i By increasing the magnetic field strength. (PV^'SsTBbf »ELMR'A'I)

Buckling instability

K.O. Friecirichs, Rev. Mod. Physics 32. (I960) 889#

Energy source: The magnetic energy of a pinched buckled plasma column.

Main features: A fully ionized pinched plasma column with a helical magnetic field has not only an equilibrium in the form of a straight cylinder. In addition, there sometimes exist buckled equilibrium states. The stability of these states can be analysed by means of the energy principle. If is found that these states become unstable in a way similar to the sausage and kink modes of an unbuckled column.

Stabilization: The buckled states are unstable. AU 4

(PV'C'SsTBbf'ELMR'A'I)

Sausage instability M.M. Rosenbluth, Los Alamos, Report LA-2030 (1956); M. Sato, Inst. of Plasma Physics, Nagoya University, Nagoya, IPPJ-7, March 1963, p. 57.

Sie.r.6JLs.2lir.Se.: magnetic energy.

Main_features: In a cylindrical plasma column the radial pressure gradient is balanced by the jxI3 force produced by an axial electric current. Disturbances in the form of locally pinched regions enhance the corresponding local jxB. force and magnetic pressure, thus leading to an enhanced growth of this pinching process. Instability occurs only at high beta values.

Stabilization: By introducing a strong axial magnetic field which reduces the radial compressibility of the column, provided that the resistivity is small enough. This makes the magnetic energy of the perturbed state larger than that of the unperturbed one. A4 5

(PV'C'SsTBbf »ELMR'A'I)

Surface (Suydam) instability B.R. Suydam, Proc. Second United Nations Int. Conf. on the Peaceful Uses of At. Energy, United Natins , Geneva 1958, Vol 31, p. 157, M.N. Rosenhluth, Proc. Second United Nations Int. Conf. on the Peaceful Uses of At. Energy, United Nations, Geneva 1958, Vol. ^1» P- 85; D.J. Rose and M. Clark, Plasmas and Controlled Fusion, Mil Press and Wiley, New York and London 1961, p. 341.

Energy source: The plasma pressure gradient.

Main__features: In the surface region of a pinched column the axial and azimuthal magnetic fields become mixed and the resulting field ES has a helical shape with high shear. There are localized perturbations of wave number k for which k#^ becomes small, leading to weak coupling between plasma and field within a limited region. This leads to a type of dissipation-free ballooning mode, localized to a small region around the surface where k#jB=O. The corresponding instability may occur at the surface layer, or in the interiors of a pinch in cases where field mixing occurs.

: The growth rate is reduced at strong magnetic fields and small dimensions of the mixed-field region. A46

(PV'C'SsTBbf»EL»MR»A1I)

Flip instability C. Bartoli and T.S. Green, Nuclear Fusion 3^ (1963)

magnetic energy.

D-£S^t\i^t a 'tne'ta pinch with a reversed field produced by plasma currents, the latter give rise to a magnetic moment directed oppositely to the externally imposed magnetic field. A small tilt of the plasma body then leads to a torque which tends to reinforce the tilt, thus making the whole body flip around an axis perpendicular to the external magnetic field.

Qv making the axial extension of the plasma body large. AU 7

(PV'C'SsTBbf'EL'MR'A'I)

Haas-Wegson instability F.A. Haas and J.A. Wesson, Phys. Fluids ^ (1966) 2472 and 10 (1967) 22U5.

equivalent gravity force due to magnetic field curvature.

Main features: The present instability develops in a theta pinch with a sharp plasma boundary containing surface currents. The radius of magnetic field curvature is a function of the position along the pinch, and has both positive and negative values. Perturbations which leave the magnetic field unchanged are stable for all m in the case 3=1 where the plasma is completely separated from the magnetic field. Electromagentic perturbations, however, are unstable for sufficiently large m, and arise through a decrease in the magnetic field energy resulting from a displacement of the plasma surface in a region where the magnetic field decreases away from the plasma. When $<1, all perturbations m>0 become unstable. This instability is a type of collective macroscopic high-beta ballooning mode.

5ili5ä£i22 Conducting walls have a stabilizing effect oh modes with m>l. A48

(PVlCfSsT3blfEfLMRfA'I)

Flute (electrostatic interchange or Rayleigh-Taylor) instabilities M.D. Kruskal and M. Schwarzschild, Proc. Roy. Soc. A 223 (1954) 348; E. Teller, Project Sherwood (Ed. by A.S. Bishop), Addison-Wesley Publ. Com., Reading, Mass. (1954) p. 85; C.L. Longmire and M.N. Rosenbluth, Phys. Rev. 103, (1956) 507; S. Chandrasekhar, Hydrodynamic and Hydromagnetic Sta* bility, Oxford, Clarendon Press, 1961, Ch. IV.

Energy source: Gravity, or the plasma thermal energy and its associated centrifugal force due to the curvature of the magnetic field lines, or the centrifugal force due to a macroscopic rotation of the plasma body.

Main features: The difference in guiding centre drifts of ions and electrons in a magnetic field produces charge separation and an electric field which in its turn, gives rise to a plasma motion across the magnetic field, thus enhancing an initial plasma disturbance. The associated flute-type perturbations leave the magentic flux unchanged and do not produce any induced electric currents These modes are often denoted as "interchange" instabilities. There are, however, also electromagnetic modes which lead to an interchange of plasma elements, thereby lowering the potential energy of the system.

The thermal convection (Rayleigh-Taylor) instability is related to this type of modes, in which case thermal expansion in a layer heated from below gives rise to a driving gravitational force.

Stabilization: By "hard"methods due to which the available energy is removed by choosing flminimum-B" and equivalent geometries. 3y "soft" methods due to which the charge separation mechanism is affected by proper choice of the . density and magnetic field gradients, by finite-Larmor- radius effects, and by magnetic line-tying. Growth rate is reduced by viscosity due to ion-ion collisions. AU 9

(PV'C'SsTBb'fE'L'MR'A'I)

Flute instability; collective mode M.N. Rosenbluth and C.L. Longmire, Ann. Phys. 1^ (1957) 120; B.B. Kadomtsev, Plasma Physics and the Problem of Controlled Thermonuclear Fusion, Akad. Nauk, USSR 3 (1958) 285.

: The equivalent integrated gravity force due to magnetic field curvature.

Main_features: Due to magnetic line-tying by electric currents along the magnetic field lines, "good" regions of locally stabilizing curvature become coupled to "bad" regions of locally destabilizing curvature. The net force driving the instability arises from the integrated effect of the "good" and "bad" regions.

Stabilization: By the use of minimum-average-B configurations with fully developed magnetic line-tying. A50

(PV'C'SsTBb'f'ELMR'A'I)

Gravitation instability, collisionless macroscopic mode B. Coppi, Phys. Lett. 11 (1964) 226 and 1£ (1964) 213; H.P. Furth, Advanced Plasma Theory, Proc. of the Int. School of Physics, Course XXV, Varenna, Academic Press, New York 1964; R.J. Hosking, Phys. Rev. Lett. 1£ (1965) 344.

Energy source: Gravity, or its equivalent due to the magnetic field curvature.

^fe^ajture^s^: This instability is similar to resistive gravitational mode, but the effects of resisitivity in Ohm's law are replaced by these due to electron inertia, Hall, or pressure effects.

ti-LteätlS.1!.1 The growth rate is reduced when the higher order terms in Ohm's law become small. A51

(PV'C'SsTBb'f»ELMR'A'I)

Tearing instability; collisionless macroscopic mode B.Coppi, Phys. Fluids £ (1965) 2273.

magnetic energy.

ft^: This instability is similar to the resistive tearing mode, but the effects of resistivity in Ohm's law are replaced by those due to electron inertia, Hall, or pressure effects.

Stabilization: The growth rate is reduced when higher order terms in Ohm's law become small. A52

Hall instability R.J. Hosking, Phys. Rev. Letters 1£ (1965)

gravitation field or its equivalent.

Main_features^ A fully ionized collisionless plasma is supported against gravity by a magnetic field. For ballooning-type electromagentic disturbances, instabilities can then arise due to the Hall effect. This occurs even if there is stability in the frozen-in field approximation in absence of the Hall term. This mode is a special case of the larger class of collisionless gravitaional modes produced by deviations from the zero-order form of Ohm's law.

: Within certain ranges of values of w. and of the strength of the gravitation field. A53

(Py'C'SsTBb'f'ELMR'A'I)

Kink and Krushal-Shafranov instabilities H. Alfvén, Tel lus 2^ (1950) 74; S. Lundquist, Phys. Rev. 8_3 (1951) 307; M.N. Rosenbluth, Los Alamos, Report LA-2030 (1956); M.D. Kruskal, J.L. Johnson, M.B. Gottlieb» and L.M. Goldman, Phys. Fluids 1 (1958)421; D.J. Rose and M. Clark, Plasmas and Controlled Fusion, MIT Press and Wiley, New York and London 1961, p. 43 5.

magnetic energy.

T**e AXH f°rce balances the radial pressure gradient in a cylindrical confined plasma column. Kink- type perturbations which bend the column increase the local 2_xB force and magnetic pressure at the inside of the kinks, and decrease it at the outside. This leads to an enhanced growth of the perturbations. The instability also occurs at the Krushal-Shafranov limit where the rotational transform in toroidal geometry becomes any multiple of 2ir, and the perturbed cross section of the plasma column is able to match up after one turn-around the magnetic field. This applies not only to the m=l kink mode, but also to instabilities at higher m-values and even at low beta-values.

^ili5äi2 introducing a strong axial magnetic field which tends to "straighten11 the column, and makes the magentic energy of the perturbed state larger than" that of the unpa±urbed one. A5U

(PV'C'SsTBb'f »ELM'tRpA'D

Gravitation instability; collisionless microscopic mode E. Frieman, K. Weimer, and P. Rutherford, Plasma Physics and Controlled Nuclear Fusion Research, IAEA, Vienna 1966, pp. 600-605; B. Lehnert, Plasma Physics £ (1967) 301. l£e.r.SY._§o.\ir.2e.i Gravity, or its equivalent due to the magnetic field curvature.

2_£.§^!i.§§ This mode is similar to the collisionless macroscopic gravitational mode, with the exception of refinements due to kinetic theory, such as resonant particle effects and Landau damping. To the author's knowledge, no final result has yet been worked out in detail for this mode.

The growth rate is expected to become reduced by Landau damping, finite Larmor radius effects and other higher-order effects. A55

(PV'C'SsTB'E'LMR'A'I)

Radiation instability S.G. Alikhanov, Sov. Phys. Doklady 12_ (1967) 601

Energy source: The heat content of the plasma.

Main features: A lack of radiative equilibrium is an additional source of instability. Thus, in an optically transparent plasma where a perturbation arises in the form of a local increase in density, the local radiation losses increase as the square of the density. The high- density regions will then be cooled, causing an inflow of the surrounding plasma, and a further increase in local density.

Stabilization: By heat conduction which tends to equalize the temperature. A56

(PV'C'SsT'BbfELM'tRpA'I)

Loss-cone instability; finite beta electromagnetic mode L.C. Himmell, Phys. Fluids 14. (1971) 1419

Energy source: The deviations from isotropy due to a loss- cone distribution in velocity space.

Main features: A high-beta spatially homogeneous plasma is considered which is confined in a magnetic field and has a loss-cone distribution, being peaked in velocity space at a transverse velocity Wj_>0. Electromagnetic waves are then found to become unstable at frequencies close to the ion cyclotron frequency and its harmonics. The waves have a wave vector perpendicular to the magnetic field, somewhat in analogy with a kind of highly resonant flute mode.

^he growth rate tends to zero with the beta value. A57

(PV'C'SsT^bf »ELM't'R'A'D

Alfvén wave (fire-hose) instability E.N. Parker, Phys. Rev. 109 (1958) 1874; M.N. Rosenbluth and K. Wilson, Proc. of the Second Int. Conf. on the Peaceful Uses of At. En., Geneva, United Nations, New York, vol. 33L (1958) p.89; R.Z. Sagdeev and V.D. Shafrarov, Sov. Phys. JETP, 1£ (1961) 130.

: An anisotropy Tw - Tx of the plasma temperature.

Main_features: When the parallel temperature exceeds the transverse sufficiently, self-excited Alfvén waves will arise in a plasma immersed in a homogeneous magnetic field This is due to the magnetic field curvature produced by a perturbation, leading in its turn to a corresponding centrifugal force on the particles moving along the field lines. The situation is analogous to that of a wriggling fire-hose.

£!iEä-imaking the magnetic field strong compared to the plasma pressure anisotropy. A58

(PV'C'SsT'Bbf »ELM't'R'A'I)

Mirror instability M.N. Rosenbluth, Los Alamos, Report LA-2030 (1965)

Energy source: An anisotropy - T» of the plasma temperature ,

Main^features: Magnetic mirror action tends to concentrate the charged particles of a plasma in the weak field regions of an inhomogeneous magnetic field. In addition, an increasing transverse plasma pressure in these regions pushes the field lines apart and increases the mirror ratio, thus enhancing the mirror actbn. At sufficiently large temperature anisotropies and strong plasma pressure gradients this leads to growing disturbances. These can arise in the localized form of a periodic pattern in space.

making the magnetic field strong compared to the plasma pressure anisotropy. A59

(PV'C'SsT'Bbf »ELM't'R'A'D

Self-gravitational instability in anisotropic magnetized plasmas S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford Univ. Press, London 1961; J.N. Tandon and S.P. Talwar Nuclear Fusion 3^ (1963) 75; P.K. Bhatia, Phys. Fluids 110 (1967) 1652, and Nuovo Cimento, Vol. LVI B, N.I (1968) 23; A.K. Sen and C.K. Chou, Canadian Journ. of Physics M_£ (1968) 2553.

self-gravitation energy.

Main^features: As already shown by Jeans, an infinite homogeneous medium tends to condense into limited regions of high density, due to the self-gravity of these regions. In absenca of other forces than self-gravity and gas pressure this occurs, provided that the expanding force of the latter overcomes the condensing force of the former In presence of a magnetic field and the rotation of an anisotropic plasma, the criterion for the onset of thi's instability is modified. This occurs under certain conditions where the magnetic field and the rotation have a stabilizing effect on the condensation process. In addition, the Hall effect has a destabilizing influence.

high plasma pressures and under certain conditions by the additional effects of a magnetic field and a rotation. A60

(PV'C'SsT'Bt^f »ELM'tRpA'I)

Temperature-anisotropy transverse wave instability R. W. Landau and S. Cuperman, J. Plasma Physics U_ (1970) 13

temperature anisotropy T,j -

Main^features: Electromagnetic waves are considered propagating across a magnetic field in a plasma with anisotropic temperature. This case, which is different from the fire-hose instability for which the waves propagate along the field direction, also leads to unstable growth under certain conditions. The present instability has a higher threshold of electron pressure to occur than the fire-hose mode, but leads on the other hand to larger growth rates.

By decreasing T or choosing other plasma parameters in a stable range. A61

(PV'C'SsT'Bb'f 'ELM'tRpA'I)

Whistler instability; pressure-driven mode R. N. Sudan, Phys. Fluids £ (1963) 57; P. D. Noerdlinger, Phys. Fluids 9^ (1966) 222H; J. Jacquinot and C. Leloup, Phys. Fluids 14 (1971) 24U0.

s : anisotropv of the plasma tempera- Energy 2H££- T..,, - ture.

Main^features^ The present mode is somewhat related to the Alfven wave instability, but is associated with the high- frequency range of a whistler wave. Since the transverse phase velocity of such a wave can be much less than the speed of light, there exist particles which can travel with the wave and produce a growing disturbance.

Stabilization: By removing the anisotropy. Further, relativistic effects are strongly stabilizing. A62

(PV'C'SsT'Bb'fE'LrrtRpA'I)

Harris Cion cyclotron resonance) instability E.G. Harris, Phys. Rev. Letters 2^ (1959) 34; G. K. Sooper and E.G. Harris, Phys. Fluids J3 (1965) 984; A. V. Timofeev, Sov. Phys. JETP 12 (1961) 281; L. S. Hall, W. Heckrotte, and T. Kammash, Phys. Rev. A13 9 (1965) 1117.

Energy source: The thermal energy of the Larmor motion.

Main__features: In the case of the basic Harris mode, a homogeneous plasma is considered having zero particle velocities in the direction parallel to an immersed magnetic field, and velocities of the transverse ion (Larmor) mo-

tion being sharply peaked at a certain value wp. As a result, a strongly anisotropic velocity, distribution arises, having a "hump" from which energy can be fed into a growing electrostatic wave at multiples of the ion Larmor frequency. This mechanism works also in the more general case of finite ion and electron temperatures, provided that the temperature anisotropy Tj/T» is large enough.

: By reducing the anisotropy. A63

(PV'C'SsTBb'fE'LM'tRpA'I)

Negative-energy instability in inhomogeneous mirror geometry H. L. Berk, L. D. Pearlstein, J. D. Callen, C.W. Horton, and M. N. Rosenbluth, Phys. Rev. Letters 2_2 (1969) 876.

Energy source: The energy due to deviations from a Maxwellian distribution, being released and carried away by a positive- energy wave.

_: Negative-energy waves are excited by removal of energy from the system being considered. In the present mode such a removal takes place through the variation of plasma density and magnetic field strength along a magnetic field of mirror geometry, with a loss-cone distribution. At the centre of the configuration there then develops a negative energy wave which transforms at the mirror ends into an outgoing positive-energy electron-plasma oscillation. Since the outgoing wave carries off energy from the interior, it serves as a dissipation mechanism, driving the negative-energy wave unstable. The present mode is of the absolute type. It is related to the loss cone instabilities.

i22: By Landau damping. A64

(PV'C'SsT'B^E'LM'tRpA'I) j

Loss cone (maser) instabilities; electrostatic modes M. N. Rosenbluth and R.F. Post, Phys. Fluids £ (1965) 547.

deviations from a Maxwellian velocity distribution.

Main features: Due to the loss cone of a mirror machine, a lack of particles and a "hump" is created in the velocity distribution. This provides an energy source of a growing wave, somewhat similar to the two-stream and trapped particle instabilities. At low plasma densities the loss cone instabilities are further characterized by frequencies being multiples of the ion Larmor frequency, and also become related to the Harris mode of cyclotrcn instability. The present modes are partly of the convective type in the form of a wave propagating almost perpendicularly to the magnetic field, partly of an absolute flute-like type. !

5ili5éJ2Bv Landau damping and by making the length of configuration short in the magnetic field direction, in connection with magnetic line-tying at the mirror ends. In the case of the absolute modes, the plasma is unstable for a large range of wave numbers, and is therefore less readily stabilized by shear and similar effects. A65

(PV'C'SsT'B'ELM't'R'A'I)

Transverse wave (Weibel) instability E.S. Weibel, Phys. Rev. Letters _2 (1959) 83; N. Rostoker in Plasma Physics in Theory and Application (Ed. by W.B. Kunkél) Me Graw-Hill Book Comp., New York (1966) p. 138

anisotropy of the electron velocity distribution.

D£§££a plasma with a sufficiently anisotropic velocity distribution, self-excited transverse electromagnetic waves can arise which involve only the electrons, This can be described as the spontaneous formation of pinches with currents parallel and antiparallel to the axis of maximum thermal velocity. There are no trapping instabilities for transverse waves, because there are no particles going with the phase velocity of these waves which exceeds the velocity c of light.

Stabilization: By imposing a magnetic field perpendicular to the low-temperature direction. A66

(PV'C'Ss'vTBbfE'LM'tRA'I)

Negative energy wave instabilities of shock wave C.N. Lashmore-Davies, Phys. Fluids 1U_ (1971) 1481.

Energy source: The drifts produced by the large gradients of the plasma density, magnetic field, and electric potential at the shock front.

Main features: A collisionless shock wave propagates perpendicularly to a magnetic field. Partir.-.s drifts arise from the magnetic field gradient, the ._ectric field, and the plasma density gradient of i ne shock. As a result, instability is produced ei »er when a negative energy Bernstein wave comes i*-« frequency resonance with an ion acoustic wave, c when resonant ions absorb energy from the negative euergy wave. These instabilities may provide the necessary mechanism for a collisionless shock transition.

Stabilization: Growth in case of resonance with an ion acoustic wave can only take place when T.<

(PVfCfSsfvTBbf'ELM'tRpA'I)

Ion drift wave instability J.P. Freidberg and J. A. Wesson, Phys. Fluids 13^ (1970) 1009»

: A gravitation field or its equivalent due to the centrifugal force or the magnetic field curvature.

-.^I!S§ A high-beta plasma is supported against gravity by a magnetic field. Ballooning-type perturbations with k*B»tO> where k is the wave number and 13 the magnetic field, are then found to be unstable on account of an ion resonance with an ion drift wave. Thus, the flute disturbances with k*IJ=O which can be stabilized by finite Larmor radius effects are under certain circumstances not the most "dangerous" ones; the present mode leads to instability in the range 0<]c'B

Stabilization: Electron Landau damping has a stabilizing effect, but the instability still persists at sufficiently high beta-values. (PV'C'Ss'vTBbf »ELM'tRpA'I)

Tearing and sheet pinch instability; collisionless microscopic mode G. Laval, R. Pellat> and M. Vuillemin, Plasma Physics and Controlled Nuclear Fusion Research, IAEA, Vienna, Vol. II (1966) p. 259; F. C. Hoh, Phys. Fluids 9_ (1966) 277; K. Schindler and M. Soop, Phys. Fluids 11_ (1968) 1192; M. Dobrowolny, Nuovo Cimento, LV B, N.2 (1968) 427.

Energy source: The kinetic energy of particle motion near the neutral plane.

Main features: The magnetic field has a neutral plane in a sheet pinch configuration. A group of electrons drifting close to this plane are involved in a resonance mechanism of the inverse Landau type. Energy is transferred to these electrons. As a result,the parallel and antiparallel line structure breaks up into pinches in the neighbourhood of the neutral plane. This occurs within the framework of Vlasov theory, even though the resistivity is zero, i.e. somewhat like the case of the collisionless macroscopic tearing mode. The instability is absolute and it develops at a critical wavelength of the order of a few times the sheeth thickness. The existence of a neutral plane in the equilibrium magnetic field is a necessary condition for the instability.

§1:§i?i i i 5§l:i2Q: The instability vanishes when the sheet thickness becomes large. A69

(PV'C'Ss'vTBb'fE'LMRp'A'I)

Diocotron (slipping-stream) instability 0. Buneman, C.V.D. Report Mag. 21 (194H) and J. Electronics 2 (1957) 1; W. Khauer, Journ. Appl. Phys. 21 (1966) 602; 0. Buneman, R. H. Levy and L.M. Linson, Journ. Appl. Phys. 3]_ (1966) 3203.

Energy^source: The slip velocity. ! i \ Main_features: This instability occurs in unneutralized 1 charge sheets of finite width in a plasma in presence of a magnetic field. The sheet charges "slip" parallel to the sheet surface, due to the ExB> drift of the electric field IS created by the spare charges. Since the intensity of the space-charge field increases across the sheet, the ExB drift and the corresponding slip become sheared. The instability results from the interaction between two waves which propagate along each of the sheet surfaces. The mutual wave interaction produces a single exponentially growing mode whan the two surface waves are in resonance.

Stabilization: Growth rate decreases with sheet widths A 70

(PV'C'Ss'vTBb'fLr AfI>

Drift instabilities; flute-type collisionless modes N.A. Krall, Advames in Plasma Physics (ed. by A.Simon and W.B. Thompson), Interscience Publ., New York 1968, Vol. 1, p. 153.

Energy sources^ The drift motions due to the inhomogeneities in plasma pressure and magnetic field, or due to an equivalent gravitation force.

J2i—£.§£H..:: For this class of instabilities the drift ' motions couple to growing flute-like electrostatic or \ 1 electromagnetic disturbances in the plasma. In particular, j the macroscopic flute instability has here its refined : microscopic counterpart in a mode which also involves jj finite Larmor radius and other higher order effects. The drift of this instability is due to the antiparallel guiding centre motions of ions and electrons in a gra- j vitation field. There are both high- and low-frequency ! modes belonging to this class. \

Stabilization: By Landau damping, finite Larmor radius effects and various changes in the parameters which \ reduce the driving forces. A71

(PV'C'Ss'vTBb'f'ELMR'A'I)

Kelvin-Helmholtz instability; macroscopic electromagnetic mode A.K. Sen, Phys. Fluids £ (1963) 1154 and 7^ (196U) 1293; A.K. Sen and C.K. Chou, Canadian Journ. Phys. Uj[ (1968) 2557.

source: The gradient of the unperturbed velocity field.

Main_features: This instability is similar to the electrostatic Kelvin-Helmholtz modes, with the exception that perturbations of the magnetic field and the Hall effect are taken into account here. There is a destabilizing influence by the Hall effect, which vanishes for propagation perpendicular to the magnetic field.

Stabilization^ Strong magnetic fields reduce the growth rate. A72

(PV»CfSslvTBblf»ELM'tRpA'I)

Universal Alfven wave instability Y.A. Tserkovnikov, Sov. Phys. JETP S (1957) 53; N.A. Krall, Advances in Plasma Physics (Ed. by A. Simon and W.B. Thompson) Interscience Publ., New York, Vol. 1 (1968) 187.

Energy source: The transverse drift motion produced by a plasma pressure gradient.

Main featurs: This instability resembles the universal low- frequency drift mode, apart from the fact that there is here a resonance between a drift wave and an Alfvén wave instead of a sound wave. The present mode also requires a temperature gradient.tc be present. The perturbed electric field produces a guiding-centre drift which brings in a net number of particles from high-density regions, such as to counteract the Landau damping and make the wave grow. Further, since the Alfvén velocity is much larger than the drift-wave velocity, the Alfvén wave front has to propagate nearly at right angles to the magnetic field to provide a resonance with the drift wave.

Landay damping. A73

(PV'C'Ss'vTBb'flElLMRpA'I)

Beam-centrifugal instability A.B. Mikhailovski and V.S. Tsypin, JETP Letters 2 (1966) 158; A.M.Rozhkov, K.N. Stephanov, V.A. Suprunenko, V.I. Farenik, and V.V. Vlasov, Plasma Physics 1£ (1970) 519.

: The differential azimuthal motion of ions and-;electrons in a rotating plasma.

Main^features: In a rotating "zero temperature" plasma pro- j duced by crossed static electric and magnetic fields, \ .E and 13, the centrifugal force produces a differential I azimuthal drift motion between ions and electrons, in j 2 addition to the velocity ExB/B . This generates electrostatic oscillations by means of a resonance between the drifx motion and the cyclotron motion of the ions. The maximum growth rate occurs when the gyro frequency

\ i A7U

(PVlC'SslvTBbtflEtLMRlAtI)

Kelvin-Helmholtz instability; macroscopic electrostatic mode N. D'Angelo, Phys. Fluids £ (1965) 1748.

: The gradient of the unperturbed velocity field

M§H}_features: Ä plasma is streaming along a homogeneous magnetic field at a velocity being sheared in a direction perpendicular to the field lines. Electrostatic perturbations are then found to become unstable on account of the transverse velocity gradient, somewhat like the instability caused by wind over water. The instability is also somewhat related to the two-stream mode.

Stabilization^ By the thermal spread at high plasma temperatures . A75

(PV'C'Ss'vTBb'f»E'LM'tRpA»!)

Drift-beam instability M.V. Nezlin, Nuclear Fusion, Special Suppl. 1969, p. 257. ii£££S¥-§2l££S§§Kinetic energy of an imposed monaenergetic beam and of drifts due to the plasma inhomogsneity.

Main_features: The present instability arises from a combination of beam and drift motions in an inhomogeneous plasma, thus being more close to the physical conditions of some experiments than each of the pure beam and universal instabilities alone. The instability differs from the universal mode in the sense that it arises at a certain threshold of beam density which depends on the plasma parameters and the magnetic field.

: By imposed changes of the beam parameters. A76

(FV'C'Ss'vTBb'f »E'LM'tRpA'I)

Drift cyclotron resonance instability A.B. Mikhailovskij and A.V. Timofeev, Sov. Phys. JETP 12 (1963) 626; N.A. Krall, Advances in Plasma Physiea (Ed. . by A. Simon and W.B. Thompson) Interscience Publ., New York, Vol. 1 (1968) p. 172.

i The transverse drift motion produced by a plasma density gradient or a magnetic field gradient.

Main_features: A plasma with inhomogeneous density and homogeneous temperature is confined in a uniform or inhomogeneous magnetic field. High-frequency drift waves are considered, in the range of the ion Larmor frequency. Density perturbations yield with the Larmor motion a corresponding perturbation in the flow of momentum which "blows" stronger on the "back" side than on the "front" side of a wave. This effect becomes pronounced when the drift wave is in resonance with the Larmor motion.

Stabilization: At small density gradients and Larmor radii. A77

(PV'C'Ss'vTBb'f'E'LM'tRpA'I)

Kelvin-Helmholtz instability; microscopic electrostatic mode C.G. Smith and S. von Goeler, Phys. Fluids 11^ (1968) 2665.

*"ne gradient of the unperturbed velocity field.

Main_features: This mode is similar to the macroscopic electrostatic Kelvin-Helmholtz mode, with the exception that resonant particles, finite Larmor radius effects, and Landau damping are involved. The main result of Landau damping is a shift of the short-wavelength onset to smaller wave numbers and a decrease in the maximum growth rate.

5§2: 3v tJle thermal spread, finite Larmor radius effects, and Landau damping. A78

(PV'C'Ss'vTBb'f'E'LM'tRpA'D

Temperature drift instability M.N. Rosenbluth, Plasma Physics Division, Am. Phys. Soc. 8th Annual Meeting, Boston, Mass (Nov. 1966), Invited Paper 8L-10; M. Porkolab, Nuclear Fusion £ (1968) 29.

5n.e.r.&Y._§2Hr.9S: The drift motion due to a temperatuer gradient perpendicular to the magnetic field.

Main_features: The ion temperature gradient in an inhomogeneous magnetized plasma produces a drift wave which can become coupled to an ion acoustic mode of short wave length. This instability is believed to be one of the most dangerous to plasma confinement, since it cannot be stabilized by shear or by ion Landau damping. However, in most cases it is expected to be localized near the very edge of the plasma, where the characteristic length U-sT/IVT! of the temperature is shorter than the corresponding length L =n/|^n| of the plasma density.

When Lm

(PV'C'Ss'vTBb'f»E'LM'tRpA'I) Universal low-frequency drift instability L.I. Rudakov and R.Z. Sagdesv, Sov. Phys. Tech. Phys. !L0i (1960) 452; N.A. Krall and M.N. Rosenbluth, Phys. Fluids £(1963) 25»*; N.A. Krall, Advances in Plasma Physics (Ed. by A. Simon and W.B. Thompson) Interscience Publ., Wiley, New York, Vol. 1. (1968) p. 153; F.C. Hoh, Phys. Fluids 8! (1965) 968.

Energy source: The transverse drift motion produced by the plasma pressure gradient perpendicular to the magnetic field. Main features: A plasma with density and temperature gradients in the x dirsction is confined in a magnetic field B directed along z, An electrostatic wave is considered, the phase velocity w/k of which satisfies the condition w.</k . How- ever, in presence of antiparallel density and temperature gradients in the x direction, a perturbed electric field -E.y, and a drift ~Ey x--Bo in the x direction will send in more ions from regions of high density to "push" the wave, and less ions to "lean" on the wave, thus counteracting Landau damping and making it grow. The instability can also be considered as the result of a resonance between a sound wave and a drift wave produced by the gradients in the plasma.

Landau damping, A80

(PV'C'Ss'vTBb'f'E'LM'A'I)

Impurity ion drift instabilities B. Coppi, H.P. Furth, M.N. Rosenbluth and R.Z. Sagdeev, Phys. Rev. Letters 37 (1966) 377.

Pressure gradient and drift motion of impurity ions.

{?§££_££§££££§: Impurities are assumed to arise near the boundary region of a magnetized plasma due to wall inter- I action. If the impurity density distribution is peaked near the wall, an instability arises which leads to their enhanced transport into the central parts of the plasma. Thus, if the density of impurity ions is not too small, an impurity-ion sound wave is growing due to a wave-particle resonance. There is also a non-resonant mode which arises when the impurity density gradient is of the order of that of the hydrogen plasma.

Stabilization: Shear is rather ineffective for stabilization,

? • A81

(PV'C'Ss'vTBb'f'LM^'I)

Drift instabilities; ballooning-type collisionless low- frequency modes N. A. Krall, Advances in Plasma Physics (Ed. by A. Simon and W.B. Thompson), Interscience Publ. , New York 1968. Vol. p. 153.

sources: The drift motions due to the inhomogeneity in plasma pressure and magnetic field.

atures: ^or "this class of instabilities the drift motions couple to growing ballooning-type electrostatic or electromagnetic disturbances in the plasma. The disturbances have characteristic frequencies being much smaller than the ion Larmor frequency.

jb±l±zat±on: By Landau damping, finite Larmor radius effects, and various cnanges in the parameters which reduce the driving forces. A82

(PV'C'SsWTB'E'LM't'RpAI)

Ion beam instability; nonlinear mode T.M. O'Neil, Phys. Fluids 10 (1967) 1027.

Energy source: The energy of an imposed ion beam.

Main features: An ion beam is assumed to travel through an unmagnetized plasma. For waves propagating parallel to Hie beam, the parameters can be chosen such as to make these waves stable, i.e. electron Landau damping domi- nates over ion induced Landau growth. However, if a spectrum of large-amplitude wave disturbances is excited in the region where 3f-/9w>0, quasi-linear theory yields unstable solutions. This is the case when the total resonant kinetic energy available for growth is much larger than that for damping. Of course, initial wave amplitudes being large enough to introduce mode coupling and electron trapping, correspond to a situation being excluded here.

Landau damping. A83

(PV'C'Ss'vTB'E'L'MR'A1]:)

Electromechanical co-and counterstreaminq instability F.D. Ketterer and J.R. Melcher, Phys. Fluids 11 (1968) 2179.

Energy source: The electrostatic energy.

Main features: In the unperturbed state there are two parallel highly conducting i\Luid streams having a constant potential difference relative to each other, as well as to a pair of parallel electrodes positioned on either side of the streams. Perturbations in the form of deflections of the streams are accompanied by an unbalance in the Coulomb forces and lead to instability. Absolut0 or convective instabilities may arise, depending on prevailing conditions.

: The boundaries have a stabilizing influence in certain cases; i.e. systems of small finite length become unstable at stronger electric fields than those of large lengths.

?'* A84

(PV'C^sWT'Bb'fE'LM'tRpA'I)

Drift-velocity space instability; electrostatic collisionless mode A.B. Mikhailovskij, Nuclear Fusion 5» (1965) 125.

Energy source: The drift motions due to a plasma inhomogeneity, as well as a non-Maxwellian ion velocity distribution.

Main_features: The resonant coupling between drift waves and the cyclotron motion is considered for an inhomogeneous plasma. This mode is similar to the drift-cyclotron resonance mode, apart from the fact that the unperturbed state consists of an anisotropic ion distribution, like that of the loss-cone instability.

^ili55^i2D shear ana< by having cold plasma of finite density between the hot plasma and the conducting ends of an open magnetic bottle. A85

(PV'C'Ss'vT'Bb'fE'LM't'R'A'I)

Negative mass instability C.E. Nielsen, A.M. Sessler, and K.R. Symon, Int. Conf. on High Energy Accelerators and Instrumentation, CERN, Geneva (1950) p. 2U0; R.A- Dory, Plasma Phys., J. Nucl. En. Pt. C, £ (1964) 511: J.F. Clarke and G.C, Kelley, Phys. Rev. Lett. 21 (1968) 10U1.

EnergY_source: The energy of a unidi.rectional circulating ion beam.

Main^features: An ion beam is circulating around the axis of an axisymmetr5.c inhomogeneous magnetic field configuration, and is electrically neutralized by an electron background. A space charge perturbation results in acce- leration, a radial displacement of the orbit, and in azimuthal bunching of part of the particles. Thus, particles being accelerated will correspond to a perturbed electric field component in the backward azimuthal direction, as if these particles would have a negative mass. This is the case when the magnetic fi.eld strength decreases in the radial outward direction. As a result, the bunching and the amplitude of the disturbance increases. It can finally be shown that the negative mass and two-stream instabilities are effects being mathe- matically identical in the limit of applicability of the Vlasov equation.

Stabilization:By changing the radial magnetic field gradient or by removing the space charges along the field lines. A86

(PV'C'Ss'vT'Bb'f'RLM'tR'A'I)

Kelvin-Helmholtz instability in anisotropic plasmas; electromagnetic mode S.P. Talwar, Phys. Fludis £ (1965) 1295.

Energy_source: The gradient of the unperturbed velocity field.

iD-££^£H£f:§ is closely related to the macroscopic electromagnetic Kelvin-Helmholtz instability of isotropic plasmas. A compressible, unbounded plasma having a tangential discountinuity of velocity is considered in the Chew-Goldberger-Low approximation. This leads to overstability in some cases, not having a counterpart in the isotropic case.

£ili5§£i2£ ^ stable state is reached within certain ranges of the relative speed, magnetic fielc strength, and anisotropy. A87

(PV'C'Ss'vBf »LM'tRpA'I)

Hybrid instabilities D.E.T.F. Ashby and A. Paton, Plasma Physics 9^ (1967) 359-, F.A. Haas, Third European Conf. on Controlled Fusion and Plasma Physics, Utrecht 1369, Wolters-Noordhoff Publishing, Gröningen, p. 50.

1 The i°n beam velocity at the plasma boundary due to an ion diamagnetic current.

Main_features^ A collision-free low-beta plasma sheath is f considered having a thickness of the order of one ion J Larmor radius. The sheath consists essentially of a J homogeneous ion beam moving along the sheath at a I constant velocity, and of electrons of zero temperature. j The ion beam is perpendicular to the magnetic field. It 5 represents the macroscopic flow of matter due to the pressure gradient and Larmor motion at the boundary of a plasma body confined in the magnetic field. Due to resonant coupling between the ion beam and electrostatic oscillations at the hybrid frequency, a growing wave develops. This mode is similar to the loss cone instability, but here the driving force is due to the anisotropy in the ion velocity produced by the ion diamagnetic current, rather that to a loss-cone effect.

It is further possible for the beam to couple to non-electrostatic oscillations in the plasma, and to extend the theory to a high-beta case. I

Under low-beta conditions an instability always occurs, but under high-beta conditions stability is achieved in certain cases. Ad 8

(P^'C'Ss'vB'E'LM't'R'A'I)

Plasma diode instabilities M. Dobrawolny, F. Engelmann, and A. Sestero, Z. Naturforsch. 24a(1969) 1235.

energy of "free" particles. i Main_features^ A fully ionized collisionless plasma is j situated between tv/o plane, parallel electrodes of finite | Spacing. "Free" particles can be emitted from and be absorbed by the electrodes, thereby staying only for a \ finite time within the system. In addition, there are j "trapped" particles moving back and fort?, between the electrodes indefinitely. The stability properties of this system differ significantly from those of an infinite plasma, because the latter does not contain "free" particles. Thus, a coupling arises between the "free" particles and the longitudinal plasma oscillations. On the other hand, resonant and trapped particle effects do not alter the situation as compared to that of an infinite plasma.

2iii5^i2D growth time of an instability is short compared to the transit times of the bulk of the "free" particles, the coupling becomes weak and the dynamics approaches that of an infinitely extended system without electrodes. A89

(PV'C'Ss'v'TBb'f'ELMI^A'I)

Cross-stream instability in a magnetized plasma D.M. Spero and A.K. Sen, Phys. Fluids 11^ (1968) 1524.

Energy^source: A monoenergetic electron beam directed along a magnetic field.

Main^features: This mode is closely related to the Weibel and cross-stream instabilities in an unmagnetized plasma, with the exception that the growing electromagnetic wave is now affected by an imposed magnetostatic field B. The latter has an inhibitory effect on the particle motions transverse to B . —o ^ii55^i22 "thermal spread of an increased temperature. A90

(PV'C'Ss'v'TBb'f'E'LMR'A'I)

Rippling instability; collisionless macroscopic mode H.P. Furth, Advanced Plasma Theory, Proc, of the Int. School of Physics, Course XXV, Varenna, Academic Press, New York 1964; B. Coppi, Phys. Fluids £ (1965) 2273; T.E. Stringer, Plasma Physics and Controlled Nuclear Fusion Research, IAEA, Vienna, Vol I (1966) 571; R.J. Hosking, Fhys. Rev. Lett. 15 (1965) 344.

: The electric current in the plasma.

^features: This instability is similar to resistive ! rippling mode, but the effects of resistivity in Ohm's law are replaced by those due to electron inertia, j Hall current, or the pressure. 1

li growth rate is reduced when higher order terms in Ohm's law become small. A91

(PV'C'Ss'v'TBb'f •ElLMltRpAlI)

Universal drift instability, Current-driven mode B. B. Kadomtsev, Plasma Physics, J. Nucl. En, Part C, !5 (1963) 31.

Energy source: The longitudinal drift motion produced by an electric current along the magnetic field.

^i2-l^§^Un2: The mechanism of. this instability is similar to that of the universal low-frequency drift mode, with the exception that the destabilizing action of the tern- perature gradient is now replaced by the destabilizing effect due to an imposed electric current. The latter produces a "hump" in the velocity distribution which now counteracts the stabilizing effect of Landau damping.

Stabilization: By Landau damping. A92

(PV'C'SsWTEb'f 'E'LM't^pA'I) Ion wave instability in a magnetized plasma; current-driven mode I.B. Bernstein and R.M. Kulsrud, Phys. Fluids 3^ (I960) 937.

: An electron current along the magnetic field.

Main_features: In a plasma with zero ion temperatur, ion plasma oscillations can be excited by an electron current along the magnetic field lines, similar to the case of the two-stream instability. The phase velocity of the excited waves is at an angle to the magnetic field, but independent electrostatic oscillations are set up on each flux tube.

ili5§£i.2i A large thermal spread "smears out" and damps the disturbances. A93

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Cross-stream instability in an unmagnetized plasma H. Momota, Progress in Theor. Phys., ^i (1966) 380.

^he mo"ti°n of a monoenergetic electron beam.

2.££ä^éES§ A charged particle stream is sent through an unmagnetized plasma having isotropdc temperature. A bunching of the beam structure produces an induced magnetic field which, in its turn gives rise to a Lorentz force and to further bunching. As a result, a transverse growing electromagnetic wave arises. The instability is absolute and resembles the Weibel mode of an anisotropic plasma.

the thermal spread of an increased temperature . A94

(PV'C'Ss'v'TB'E'LM't'RpA'I) Two-stream and beam-plasma instabilities; basic modes J.R. Pierce, Proc. IRE 37^ (1949)980; A.V. Haeff, Proc. IRE 3J7 (1949)4; J.R. Pierce and W.B. Hebenstreit, Bell System Tech. J. 2£ (1949)33; D. Bohm and E. Gross, Phys. Rev. 7^ (1950)992; 0. Buneman, Phys. Rev. Letters 1 (1958)8; J. Briggs, Advances in Plasma Physics (Ed. by A. Simon and W.B. Thompson) Interscience Publ. , New York, Vol 4_ (1971)43; N. Rostoker, Plasma Physics in Theory and Application (Ed. by W.B. Kunkel), Me Graw-Hill Book Com., New York 1966, Ch. 5.

?n.e.r.SY.-s.2Hr.£e.: ^he kinetic energy of monoenergetic beams, ..futures: Two nionoenergetic parallel beams of charged particles move at different velocities through each other. A space-charge disturbance and its associated electric field will then affect the particle motion in such a way that bunching occurs which in its turn reinforces the space charge and the field, thus leading to a growing disturbance. For this mode all particles are in resonance with the space charge wave, and there is no particular group of particles trapped in an electrostatic potential well. The growth rate is therefore independent of the plasma properties.

Under more complex conditions , there may arise instabilities which are in principle of the same type; i.e. where the beam energy is converted into growing perturbations in the form of oscillating fields and/or various types of plasma wave phenomena. The details of the excited oscillations depend on the specific plasma conditions-

i§i A thermal spread comparable to the mutual beam velocity destroys the coherence and suppressed the instability. A95

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Current chopping instability A.A. Ware, Am. Phys. Soc. Meeting, Boston (1966) paper 7L-9-, J.E. Faulkner, A.A. Ware, W.L. Woodie, and C. Dames, Eight Int. Conf. on Phenomena in Ionized Gases, Vienna (Ib67) page 142; H. Alfven and P. Carlquist, Solar Physics 1 (1967) 220-, D.T. Tuma and A.A. Ware, Phys. Fluids 11^ (1968) 1206.

Energy source: An imposed electric current.

Main^features: A constant electric current is passed through a plasma column. A decrease in plasma, density, or a con- striction of the cross section, is assumed to take place within a limited region of the column. To keep the current constant, an extra voltage is required to accelerate the electrons within this region, which then becomes bounded by two double space-charge sheaths. The sheath at the cathode end causes electrons to enter from this side as a beam having a mean energy equal to the sheath potential plus the mean thermal energy. Further, the hump in electric potential thus formed causes the ions and electrons to be accelerated in a way to lower the density even more in the hump; i.e. the initial disturbance is enhanced. This instability is somewhat related to the Pierce and two-stream instabilities.

a thermal spread and by particle collisions. A96

(PV'C'Ss'v'TB'E'L'MRpA'I)

Finite length instabilities M. Cotsaftis, Rep. Euratom-CEA, F.C. 392, U_ (1966); J. Olivain, Journal de Physique 3C[ (1969) 187

mutual velocity of counterstreaming electron beams.

Main_features£ Two counterstreaming electron beams of finite length are considered. Electrostatic growing oscillations can then occur in a configuration of finite length, under conditions which would lead to stability in a corresponding case of infinite length. The underlying mechanism in the finite length case is due to the transit times of particles moving back and forth through the system.

r* Cv increasing the length, and by changing the density and velocity appropriately. A97

(PV'C'Ss'v'T'Bb'fE'L'M'tRpA'I)

Ion resonance instability in a non-neutral plasma R H. Levy, J.D. Daugherty and 0. Buneman, Phys Fluids 12_ (1969) 2616.

The electrostatic field due to an electron cloud.

Main_features: An electron cloud of finite size and axisymmetric shape is contained in an axial magnetic field. Ions of high energy and Larmor radii exceeding the transverse dimensions of the configuration are trapped in the strong electrostatic potential well produced by the electron cloud. The oscillatory motion of the ions which traverse the entire well then becomes coupled to the azimuthal ExI3 drift of the electrons, thus producing growing disturbances.

Dv choosing the magnetic field strength between two limits given by the electron d uisitj. ' large ion-electron mass ratio is advantageous to stability A98

(PV'C'Ss'T'E^M'tRpA'I)

Trapping and "bump" instabilities; basic modes N. Rostoker, Plasma Physics in Theory and Application (Ed. by W.B. Kunkel), McGraw-Hill Book Comp.. New York 1966, Ch. 5; N.A. Krall and M.N. Rosenbluth, Phys. Fluids £ (1962) 1435; G. 3ekefi, J.L. Hirschfield, and S.C. Drown, Phys. Fluids 4_ (1961) 173.

s.2\i£9e.: A deviation from the Maxwellian velocity distribution, often in the form of a "bump".

Main^features: These modes result from resonances between the phase velocity of the various possible plasma oscillations and the drift velocities of the plasma particles. Thus, inverse Landau damping arises when there is a"bump" in the velocity distribution, as defined by 9f(w)/3w>0 with f being the distribution function. Then, par tides which go slightly faster than the phase velocity of a wave in the bump region will outnumber those which go slower, and this produces a growing wave under certain conditions. In particular, "negative temperatures", as well as radiation in the microwave range on account of instabilities, can arise when the corresponding absorption cross section R(w) satisfies the condition 3(Rw4)/3w<0.

various damping mechanisms such as by dissipation. A99

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Collective electrostatic instabilities in a two-dimensional field B. Coppi, Advances in Plasma Physics (Ed. by A. Simon and W.B. Thompson) Interscience Publ. , New York 1971, Vol. »+ p. 173; B. Coppi, Physics Letters 28A (1969) 518.

: The pressure gradient in the plasma and its associated drifts and magnetic field curvature effects.

Main_features: This class of electrostatic modes occurs in a low-beta plasma confined in a two-dimensional configuration being the straight correspondence to multipole de- vices. Here the macroscopic field geometry of the system

is taken into account?leading to integral equations where the coordinate along the magnetic field lines is involved. There are periodic particle orbits of both the transit and trapped particle type. The equivalent magnetic moment M of the gyrating particles is conserved. There are several types of modes depending upon whether the action integrals J.i and J e associated with the motion of ions and electrons along the magnetic field lines are conserved or not. With the "bounce" frequencies o>, • and UK for ions and electrons moving along the magnetic field, there are "hydromagnetic" modes having a frequency u satisfying

o), .«(!), <

kinetic modes with a><v •<

(PV'C'ST'Bb'LM'tRpA'I)

Cyclotron instabilities in a plasma with anisotropic temperature H. Sen, Phys. Rev. 8£ (1952) 816-, K.E. Zayed, Plasma Physics 10 (1968) 673.

££. anisotropy of the thermal velocity distribution.

-ut A fully ionized plasma is confined in a homogeneous magnetic field and has perpendicular ion and electron temperatures exceeding the corresponding longitudinal ones. The cyclotron motion can then couple to a number of wave modes of both longitudinal and transverse character, such as plasma oscillations of the Langmuir type, Bernstein modes, and electromagnetic waves.

: Depends on the degree of anisotropy, on the unperturbed state of motion, and on the particular parameters involved. A101

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Trapped-particle collisionless instabilities in a magnetized plasma B.B. Kadomtsev and O.P. Pogutse, Nuclear Fusion 11(1971) 67.

Main_features: In closed field-line toroidal geometry there are two groups of plasma particles, i.e. transit and trapped particles. The latter are trapped between the magnetic mirrors formed by the rotational transform or by magnetic field inhomogenities. Tn certain cases trapping also occurs between the equipotential surfaces produced by an electric field component along the magnetic field direction. The transit and trapped particles react quite differently to low-frequency perturbations with small phase velocities. Thus, the trapped particles are contained in a tube of force lines between two magnetic mirrors, and are isolated from other regions of the plasma. They are therefore comparable to the trapped particles in an ordinary mirror-type magnetic bottle, and give rise to equivalent instabilities due to unfavour- able magnetic curvature and other destabilizing effects. The present modes are collective in the sense that they result from an average of the plasma and field properties taken along the field lines between twc turning points.

Stabilization: By effects similar to those in an equivalent ordinary magnetic bottle* A102

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