11. A. Ap4HMOBHY

~TO KAm.LJ.blV1 ttlVl3V1K

.LJ.OJlmEH 3HATb

o nnA3ME

L.A. ARTSIMOVICHAPHYSICIST'S

011 PLASMA

Translated from the Russianby Oleg Glebov

MIR PUBLISHERS/MOSCOW

First published 1978

Revised from the 1976 Russian edition

Ha an2AllUC~O.M, Sl3bl1'ie

© AToMH3AaT, 1976

© English translation, Mir Publishers, 1978

CONTENTS

Preface 7

1. Definition of Plasma 9

2. Motion of Electrons andIons in Plasma in the Absence

of External Fields 17

3. Plasma Behaviour in Electric Fields 28

4. Plasma Behaviour in Magnetic Field 37

,- 5. The Effect of the MagneticField on Characteristics of the Plasma 48

6. Results Derived fromMagnetohydrodynamic Equations 58

5

7. Development of Experimental Target 65

8. Conditions for Existence of a Plasma Ring 72

9. The Theory of Stability 85

10. Plasma Behaviour in Trapswith Magnetic Mirrors 93

11. Stability of Plasma Configurations 101

12. Shear Stabilization 110

13. Other Plasma Instabilities 112

Bibliography 124

PREFACE

This small book written by the late Lev Artsimo­vich, a Full Member of the USSR Academy of Scien­ces, is the lecture course he delivered for the physi­cists interested in plasma physics. The book presentsthe fundamental information on the high-temper­ature pl asma physics. These, at present largely well­established results, comprise, in effect, the sum ofknowledge indispensable for any physicist witha wide enough sphere of interests.

However, it should be clearly understood thatthe structure of the high-tern perature physics hasjust been outlined and is still being developed.A major contribution to this field of modern physicshas been made by the Department of Plasma Physics,headed by L. Artsimovich, which is a part of theI. Kurchatov Institute of Atomic Energy. L. Ar­tsimovich headed the research work with the toka­mak-type installations, adiabatic traps, rapid pin-

7

ches and other machines designed for harnessingthe high-temperature fusion reaction. He was direc­tly involved in actual work in a number of theseresearch projects and in development of the correctconceptual system and understanding of the plasmaphenomena. This work, of course, could not fail tobe reflected in the book; when discussing the expe­riments with tokamak-type installations and theadiabatic traps, L. Artsimovich described in moredetail the most recent results available at thattime. By now, some material in the book hasbecome somewhat dated; for instance, plasmasgenerated in Tokamaks and adiabatic traps havenow more impressive parameters and we have nowa markedly better understanding of the processesin them. However, since the general conceptualsystem has not undergone any major changes wehave deemed it unwise to alter the original author'spresentation of ideas and, therefore, no significantchanges have been made in the original text.

Hence, the material presented in the book com­prises, in the opinion of L. Artsimovich, the fun­damental results obtained through many years ofexperimental and theoretical research in the high­temperature plasma physics.

Prof. B. B. Kadomtsev

1.

The term plasma is applied to the ionized gas all(or a considerable part) of the atoms in which havelost one or several of their electrons and convertedinto positi ve ions. This definition of plasma as a spe­cific state of matter is only a tentative one, it willbe elaborated upon below. Generally, plasma can beassumed to be a mixture of three components ­free electrons, positive ions and neutral atoms (ormolecules).

Plasma is the most widespread state of matter inthe universe. The Sun and the stars can be regar­ded as enormous lumps of hot plasma. The Earth'satmosphere is surrounded by a plasma envelopeknown as the ionosphere. The circumterrestrialspace outside the ionosphere contains peculiar plas­ma formations - the so-called radiation belts. Inthe terrestrial environment we encounter plasma inlaboratories and technology in various gas dischar-

9

ges since any gas discharge (lightning, spark, arc,etc.) invariably generates plasma.

The research in plasma physics was typicallystimulated by potential practical applications ofthe results. At first, the scientists were interested ina plasma as a peculiar conductor of electric currentand as a light source. Nowadays, we should considerthe physical properties of plasma from another view­point and new aspects of plasma will be revealedto us. First, the plasma is the natural state of mat­ter heated up to a very high temperature; and,secondly, the plasma is a dynamic system subjectedto the action of electromagnetic forces. The newapproaches in the studies of the plasma behaviourstem directly from the large-scale technologicalproblems involving physics. The major problemsare the controlled nuclear fusion and the magneto­hydrodynamic conversion of heat energy intoelectric energy. In the near future plasma physicsmay play a significant role in accelerator techno­logy.

The studies of plasma phenomena are of not onlypractical interest. 'The plasma is a material mediumconsisting of a group of particles which interact be­tween themselves via the simplest mechanism, name­ly, the Coulomb electrostatic forces. A physicisthas to understand the mechanisms of various pro­cesses which occur in a plasma proceeding fromthe known microstructure of plasma. The basictheoretical concepts here are extremely clear. Herewe remain invariably within the domain of theclassical physics since the quantum effects do not

10

play any noticeable part in the typical plasma.Nevertheless, the theoretical analysis of the plasmaeffects is far from being completed and many gapsstill remain in our understanding of the plasma.

The major experimental efforts at present areaimed at developing the techniques for generatingplasmas with high parameters, namely, hightemperature and high density. Our subject of stu­dy - the high-temperature plasma - is producedby ourselves and we try to provide optimum condi­tions for the plasma existence when the plasma isin the quasineutral stable state.

Before continuing further, let us elaborate uponthe definition of a plasma. The. electric forces whichbind the opposite charges in a plasma provide forits quasineutrality, that is, an approximate equali­ty of ..the concentration of electrons and ions. Anyseparation of charges when electrons shift withrespect to ions must give rise to electric fields whichtend to compensate the perturbation. These fieldsare the higher, the higher is the particle concentra­tion and for a high-density plasma they are veryhigh, indeed.

To estimate a field produced by a perturbation ofplasma neutrality, let us assume that charges havebeen completely separated in a certain volume sothat it contains only one type of charges. The elect­ric field in this volume satisfies the Poisson equation

div E = 4np

where p is the charge density.If the linear dimensions of the volume are of the

11

order of x and the concentration of the charged par­ticles in the plasma is n, we obtain

div E ~ Elx »: 4nneand, hence,

E ~ 4:n;nexThe variation of the plasma potential in the volumeof charge separation is V ~ Ex ~ 4nnex2 •

Let us consider the following example. Assumethat the fully ionized plasma is produced fromhydrogen which is, initially, at the normal tem­perature and under thepressure of Lrnm Hg. A cubiccentimetre of such a plasma contains 7 .1016 ionsand electrons each. In these circumstances E is about1011 VIcrn. Therefore, if quasineutrality is sharplydisturbed in a volume with a diameter of about 1 mm ,the resulting electric field is more than 1010 V/cmand the potential difference across this volume isof the order of 109 V. Such a separation of chargesis, clearly, unfeasible. Even in a plasma of a muchlower density a sharp disturbance of quasineutralityin such volumes will be immediately suppressed bythe developing electric fields. The electric fieldwill push out the charges of one sign from the volumewith separated charges and draw the opposite char­ges into it. However, in a small enough volume 0

plasma, quasineutrality may be violated if theelectric field generated by an excess of particles ofone sign is too weak to have a significant effect onthe motion of particles. For a given concentrationand temperature of the plasma, it is described bya characteristic linear parameter ~: when x ~ ~,

12

then separation of charges within a volume with thelinear size x does not produce a significant effect 011

the motion of particles; when x ~ 6, then concen­trations of the opposite charges in this volume arealmost equal.

The characteristic length 6 can be estimated in thefollowing way. When the charges in the volume ofthe linear size 6 are completely separated, thepotential energy of a charged particle is of the orderof the energy kT of the thermal (heat) motion ofparticles; here T is the plasma temperature in abso­lute degrees *. Thus, we obtain the relationship

U == eV ~ 4nne 262 ~ kT

Hence we find

fJ ~ [kTj(t}J{ne 2) ]1/2 (1)

The parameter 6 is derived also from the analysis ofelectric field screening in a plasma. Assume thata "test" point charge q has been introduced intoa plasma. The potential generated by this chargeat a small enough dis lance r from it is q/r. However,polarization of plasma due to the electric field ofthe charge alters the form of the potential functionat large distances from the charge.

* The thermal energy of plasma particles canalso be expressed in electron-volts usingthe relationship kT = e8/300 eV = 1.6.10-128 eVwhich yields T == 11 600e eV. The parameter Bis referred to as the plasma temperatureexpressed in electron-volts. The temperaturedefined in this way is identical to the parametercharacterizing the thermal energy.

13

In a state of statisticalequilibriuni, the spatialdistribution of electrons and ions in the vicinity ofthe test charge is given by the Boltzmann formula(n r'Vexp (-UlkT), where U is the potential ener­gy. U has opposite signs for ions and electrons).This can readily be seen to result in screening ofthe electric field in this region. The concentrationof particles with the charges opposite to q is higherin the vicinity of the test charge q, that is, for com­paratively high values of U/kT; this should resultin a sharp drop in the electric field. Calculationsusing the Poisson equation and the Boltzmann dis­tribution law show that at large distances from thecharge q the potential decreases exponentially andthe electric field is high only in the sphere with theradius of the order of B around the charge q. Debyewas the first to introduce the characteristic length 0in his study of strong electrolytes. Later, this conceptwas applied to plasma physics. The parameter 0 isreferred to as the Debye radius or the Debye length.Inserting the numerical values of the constants intoEq. (1) for li, we obtain

6~7(Tjn)1/2 (2)

Here T is the plasma temperature which is assumed,so far, to be the same for the electron and ion com­ponents of the plasma. While the Debye radiusdescribes the spatial scale of the decompensation re­gions, the lifetime of these ~egions is found by di­viding 6 by the velocity of faster particles (electrons):

r = OjUe :::::;; (kT /4rr.ne2) 1/2(me/kT )1/2 ==; (n~p/4nne2)1/2

14

The parameter 1/"& has the dimensions of frequencyand is identical to the frequency of the electrosta­tic plasma oscillations which occur in plasma whenthe electron groups are displaced from the equilibri­um distribution. This frequency

(4 2/ )tJ2i(00 -::= It ne me (3)

is known as the plasma or Langmuir frequency.The higher the density of plasma the smaller the

sizes of decompensation regions and their lifetimes.In a high-density cold plasma quasineutrality canbe violated only within a fairly small region. Ina low-density hot plasma the Debye length can beconsiderably larger than the size of the plasma. Ifthis is the case, the motions of ions and electronsare independent and no automatic equalization ofconcentrations of the opposite charges occurs.

We can now redefine the term plasma in the fol­lowing way making use of the concept of the Debyeradius - the plasma is an aggregation of freelymoving particles with opposite charges, that is, anionized gas is called the plasma if the Debye lengthis small compared to the size of the volume occupiedby the gas. This definition was put forward byLangmuir who was the pioneer of the plasma science.

The two plasma parameters we have introduced ­concentration and temperature - should be com­mented upon.

1. Generally, the electron and ion concentrationsneed not be identical since a plasma can contain notonly singly charged but also multiply charged ions.

15

If the concentration of ions with a unit charge isdenoted by n1 , the concentration of ions with thecharge of 2 by n 2 , and so on, the electron concentra­tion n.; is given by n1 + 2n2+ .... However, belowwe shall mainly discuss the plasmas in which theelectron and ion concentrations are identical. Inparticular, this is the case for the hydrogen plasma.Normally, it is not too complicated to take intoaccount the effects of the multiply charged ions onthe basic processes in the plasma.·

2. The concept of the plasma temperature T isvalid only if electrons and ions have the sam e kine­tic energies. Generally, the plasma must be charac­terized by at least two temperatures - the electrontemperature T e and the ion temperature T i- Theplasmas which are generated in a laboratory environ­ment or in technology have typically the tempera­ture T e which is usually much higher than T i' Thedifference between T e and T', is due to an enormousdifference between the masses of electrons and ions.The external electric power sources that generatea pl asm a (in various gas discharges) transfer theenergy to electrons since it is the electrons that arethe current carriers. Owing to collisions with rapidlymoving electrons, the ions increase their thermalenergy. In such collisions the fraction of the kineticenergy of electrons which can be transferred to theion is no more than 4.me/mb where me and m., arethe masses of an electron and an ion. The meanfraction of the electron energy transferred to the ionin a collision is even lower. Since me is much lessthan nu, the electron has to undergo very many

16

(thousands) collisions to transfer...completely to theions the excess energy it has. Since the thermal ener­gy exchange between electrons and ions proceedsconcurrently with the energy transfer from the elec­tric power sources to electrons and at the same timethe energy leaks from the plasma via various heat­transfer mechanisms, typical result is a sharp tem­perature difference between electrons and ions inan electric discharge in gases. This energy differencedecreases, typically, with increasing plasma concen­tration since the frequency of collisions betweenelectrons and ions in a given plasma volume increa­ses with the squared concentration.

Under some special conditions T t can becomemuch higher than T e in a strongly ionized plasma.For instance, such conditions exist in short-durationhigh-power pulse discharges which result in devel­opment and cumulation of shock waves in theplasma.

2.

MOTION OF ELECTRONS AND IONSIN PLASMA IN THE ABSENCE

OF EXTERNAL FIELDS

The character of this motion is determined by theinteraction between the particles. In the highlyionized plasma the principal type of interactionbetween the particles is the classical Rutherfordscattering in the Coulomb field. We shall dealwith three basic types of elementary scattering pro-

2-0323 17

cesses, namely, scattering of electrons by ions,electrons by electrons and ions by ions. Otherelementary processes either occur with photon emis­sion and their probability is relatively low or theyinvolve also the neutral particles and their signi­ficance diminishes with increasing degree of ioniza­tion. The processes of the first kind are illustrated byemission of bremsstrahlung electrons in the electron­ion collisions, the processes of the second kind areillustrated by the ionization phenomena and atomexcitation by electron impact and by the chargeexchange processes occurring between ions and atoms.If we deal with a non-hydrogen plasma we have,generally, to take into account the interaction be­tween electrons and ions in various energy states.The intensity of radiation emitted by the excitedions can prove to be very high and can make a signi­ficant contribution to the energy balance of theplasma processes. We shall limit ourselves to ana­lyzing particle interactions in a fully ionizedplasma.

Let us assume that a charged "test" particle istravelling in the plasma (such a particle may be anyplasma electron or ion whose path we shall trace).When travelling, this particle will be scattered inthe Coulomb field of plasma electrons and ions whichit encounters along its path. If we consider a lightparticle travelling among heavy ones (an electronamong ions), then the scattering centres are assumedto be stationary. In this case the probability ofscattering into a gi ven angle is determined by theclassical Rutherford formula.

18

Every scattering of a test particle travelling inthe vicinity of the scattering centre results in theparticle's trajectory being rotated by an angle 8,that is, its velocity along its initial direction ofmotion decreases from v to vcos 8. In most cases scat­tering occurs at large distances and typically resultsin a very small change in the trajectory (this is typi­cal of the Rutherford scattering in the electricfield of point charges!). Therefore, here we cannotvisualize, as is customarily done in the kinetic theo­ry of gases, the trajectory of a particle as beinga broken line composed of straight segments or"paths" which connect the points of "collisions".Instead, we have a gently twisting line whose direc­tion varies owing to the effect of the numerous butvery weak impetuses due to "collisions" with otherparticles. In effect, these impetuses generate a con­tinuous influenceof the plasma "microfield" on a tra­velling particle. This microfield is a superpositionof the electric fields of individual particles.

I t is natural here to introduce the concept of thefree path 'A of a particle defining it as the distance atwhich the particle changes its initial direction ofmotion. This definition yields the following equa­tion:

dv = - odxl): (4)

Here dv is the mean variation of the velocity com­ponent along the initial direction of motion whentravelling the distance dx. This definition can beused to express A in terms of the integral over theangul ar distribution of scattered partie] es. Jf the

1U 2*

velocity vector rotates by the angle 8 after colli­sion, then the velocity component along the initialdirection of motion decreases by v (1-cos 8). Whena particle travels the distance dx in the volumewith n scattering centres per 1 em", the particleis scattered nf (8) dQ dx times into the angle close to 0within the solid angle dQ.· The function f (8) isknown as the effective differential cross-section ofscattering and is determined by the nature of inter­action forces. Hence, the mean value of dv isgiven by

dv = - vndx ~ f (8) (1 - cos 8) dQ (5)

From Eqs. (4) and (5) we obtain

1/f.. ~ n ) /(0) (i-cos 0) dQ (6)

If the scattering centres are assumed to be stationarypoint charges, then the function f (8) is given bythe Rutherford formula

f (8) = 1/4 (QlQ2/mlv2)2(1/sin4 8/2) (7)

where ql is the charge of the test particle, m, is itsm ass and q2 is the charge of the scattering centre.

Substitution of Eq. (7) into Eq. (6) and integra­tion over angles from a minimum angle Groin to n)1ield

A=:-; 1/4nn (m1v2/ Q1Q2)2 (1/ln (2/011l i n ) ) (8)

1 he value of 8m tn may be estimated as follows. Theelectric field of the scattering centre will be assu-

20

med to be a Coulomb one only at distances smallerthan the Debye radius fJ. At larger distances thepotential decreases expon ant.ially. Therefore, wehave to ignore the collisions in which the particlepasses: the scattering- centre at distances larger than B.Now.Inote that the distances of approach of theorder of B correspond to extremely small scatteringangles in the plasma. Under such conditions thetrajectory of the test particle is almost a straightline and the closest approach distance is practical­ly identical to the impact parameter. For small­angle scattering in the Coulomb field the impactparameter b is rol at od to the angl e 8 by the well­known equation

(9)

Substi tu ti on of b ---=: <S into Eq. (9) yields 8 m i n •

Since this calculat.ion yields a very hig-h log quantityin Eq. (8) for interactions of particles in the plasma(in allthe cases of interest it varies from 104 to 108) ,

a rough approximation of the estimated minimumscattering angle practically does not affect the com­putation of 'A.

The assumptions used for deriving the free path 'Aare valid when the test particle is an electron inter­acting with plasma ions. Denote by Api the meanfree path of electrons colliding with plasma ions.The path !vei is derived by averaging Eq. (8) overthe energy spectrum of electrons. When all theplasma ions-are singly charged an -1 the energy distri­bution-of electrons is g-iven by the Maxwell formula,we obtain Ihe following expression for the mean

21

free path:

T 2, 4 r: 105 I' 1A·- Jo --e t -- • n L

c(10)

Here L; is the so-called Coulomb logarithm whichis derived by substituting into In 2/S rn1n the mini­mum angle found from Eq.' (9) for b === 8, mv2 === 3kTe ann q2 == - ql === e. The value of L;varies from 10 to 20 for very wide variation rangesof nand T e- Since fairly rough estimates of theparam et ers of particle collisions can be used inplasma physics we shall assume below that L; :=:

=:= 1!1. Several other parameters describing the col­lisions between electrons and ions can also be intro­duced in addition to Api' The effpctive cross-sectiona ei for such collisions is given by the relationshipAei === 1/naei ' and the mean time period betweentwo successive collisions is Lei =--= AeillJe, where V eis the mean thermal velocity of electrons. The colli­sion frequency 'V e i is the reciprocal of 'rei' The aboveparameters may be calculated from 1h~ followingexpressi ons (for L; == 1f»:

aci ~ 3· 10-5/T~; 'rei ~ 3·1 0-2T;/2/n; v.. = 20n(!/T~/2

(11)

Tho above parameters can be readily modified fordescribing the collisions with multiply charged ions.For these processes the effective cross-section (T e i

increases as the squared ion charge and the otherparameters vary in a similar way,

The collisions between electrons and ions arc themost significant particle interactions in ~ the plasma;

22

in particular, they determine the mechanisms ofsuch processes as electric conduction and diffusion.

To make a more comprehensive description of theCoulomb interaction between the plasma particles,we have to introduce also the parameters whichwill characterize the statistical effect of the colli­sions between identical particles (electron-elec­tron and ion-ion collisions). Here the calculationsare more complicated since the analysis of theelementary collisions should take into account themotion of scattering centres. However, this canaffect only the numerical factor in the mean freepath expression, while the temperature dependenceshould remain the same. For instance, the expres­sion for App' (the mean free path for the electron­electron collisions) must be the same as the expres­sion for 'A e i differing only by a numerical factorfairly close to unity. The expression for Au (themean free path for the ion-ion collisions) is derivedfrom the formula for 'A ee by substituting T i for T e.

The parameters Tee and 'rei are close to each other.The ratio LUI1:ee is lrmi1me If T~/T~. When theelectron and ion temperatures are equal, the ion­ion collisions occur much rarer than the electron­electron or electron-ion collisions.

Let us summarize. The above analysis' has attem­pted to describe" the interactions of charged parti­cles in the plasma in terms of the elementary kinetictheory of gases; we have replaced the smooth l1~adu­

ally weaving trajectories of electrons and ions byarbitrary broken lines and reduced the statisticaleffect of numerous weak collisions to one abstract

23

strong impact. The usefulness of such, not too rigor­ous approach, lies in the fact that the expressionsfor the mean free path, the mean time between twosuccessive collisions, etc. help to visualize the basicphysical processes in the plasma. There existsquite a rigorous technique for analyzing the Coulombinteractions between the plasma particles usingthe mathematics of the kinetic equations theory.

Let us now discuss the exchange of thermal energybetween electrons and ions in the plasma. Let us,at first, consider the simplest case. Assume thata fast electron with the momentum p = meVetravels in the vicinity of a stationary ion and isscattered into the angle 8. In the process the ionreceives the momentum 8p ==2p sin 8/2. As a re­sult, the ion starts moving with the kinetic energy

(12)

To find the kinetic energy transferred by a fastelectron to stationary ions per unit time, multiplyL\W by nvef (8)dQ and integrate over angles. Theresult is

(13)

(it is assumed here that the ions are singly charged).The expression for the transferred energy can berewri tten as~

Lc"(41tne4./miVe) =(2mr/mir(4nne4!m~v:) (n~('v:/2) ~

(14)

24

Here "ei is the number of collisions between theelectrons wi th the kinetic energy We and stationaryions per unit time. The mean relative fraction ofenergy lost in one collision is 2m e/mi as should beexpected from the model of the elastic collisionbetween two spheres. The mean energy transferredfrom the plasma electron to the ions in 1 s is foundfrom Eq. (13) by integrating over the Maxwellvelocity distribution *:

- dWeldt == 1.2 .10-17nIAl( Te (15)

where A is the atomic mass of the ionized gas. Thisequation is valid only for T e ~ T i ; if T e is com­parable to Tb Eq. (15) must be replaced by thefollowing equation:

- dWe/dt ~ (1.2 .10- 17neIA) (Te- T i )/T: /2 (16)

To estimate the efficiency of heat exchange betweenions and electrons in the plasma, let us discussa specific case. Let the electron temperature in thehydrogen plasma be maintained at 106 K and theelectron concentration be n = 1013• Under theseconditions heating of the plasma from 0 K to T', == 1/2 T e takes 1.2 ms. Note that these calculationsare meaningful only if in the time period underconsideration the ions retain all the energy they haveobtained. Since an ion with an energy correspondingto a few hundreds of thousand degrees travels about

... The Coulomh logarithm Lc is taken to be 15.

25

100 m in 1 IDS, the heating of ions under such condi­tions necessitates efficient thermal insulation of theplasma.

A major characteristic of matter is the equationof state, that is, the relationship between the pres­sure, density and temperature. The equation ofstate for the plasma with isotropic distribution ofvelocities of charged particles has the same form asthe equation of state for a two-component idealgas:

p = nek (T e + T i ) (17)

where p is the plasma pressure which is the sum ofelectron and ion pressures. The equation of state isknown to have such a simple form only if the mutualpotential energy of particles is negligibly small com­pared to their thermal energy. In the plasma, this ispractically always the case. Indeed, the potentialenergy of a charged particle in the fluctuatingplasma microfield should be of the order of e2/a,

where a is the mean distance from a given particleto the next one (a "'" 1/Y2ne) . It. can be readilyseen that the ratio

is much smaller than unity except when we considera plasma with the particle concentration of 1018 andhigher and the temperature not lower than 104:~K.

The energy distribution of the gas particles" isusually assumed to have the Maxwellian c.~ form.This assumption cannot be"automatically applied

26

to the plasma. The Maxwellian energy spectrum isestablished owing to collisions between the gasparticles.

The Maxwell distribution in a wide energy rangecan be established for a given group of particleswith an arbitrary initial energy distribution, that is,the Maxwellian "tail" for the particles with energiesmuch higher than kT can grow only during a certaintime period in which the particles must undergo,in the average, several collisions with each otherand these collisions must be between the identicalparticles. For instance, the energy distribution ofelectrons will become practically Maxwellian aftera time period which is about 10 times the meantime 'tee between two successive electron-electroncollisions. For ions the Maxwellian distribution isestablished after about ten ion-ion cullisionsThus, when T i is about T e "maxwellization" forions proceeds much slower than for electrons. How­ever, for each plasma component, maxwellizationdue to collisions between the identical particlesproceeds faster than the development of thermalequilibrium between both components. Therefore,both ions and electrons in the plasma have theMaxwell distributions but with two different tem­peratures. At the same time, if the charged plasmaparticles have a short lifetime, the energy distri­bution of ions in the range of energies much higherthan kT can differ sharply from the Maxwelliandistribution (the number of such fast ions will bemuch less than it should be according to the Maxwellformula).

27

(18)

3.

PLASMA BEHAVIOURIN ELECTRIC FIEI;DS

An electric field or a pressure gradient in a plasmagives rise to directed particle fluxes (flows)- anelectric current in the first case and a diffusion fluxin the second case. When electric current is flowingin plasma, the ions can be assumed stationary.The current is carried by electrons. In the simplestcase of constant current there should be establishedan equilibrium between the force of the electricfield acting on electrons and the decelerating forcedue to collisions between electrons and ions. Thelatter force is given by the mean directed momentumlost by electrons in collisions with ions per unittime. An electron has Vei collisions in 1 s and ineach collision it loses the momentum m.u, where uis the directed electron velocity. Hence, the dece­lerating force is meuVei and the equilibrium is es­tablished when

eE -== muv.;

The current density in plasma is

j == neu (19)Hence we obtain

j == ne2E/me'Ve i == ne2TeiE/me (20)

Eq. (20) is the Ohm law for a plasma. The parameter

Op; ==-= ne~ei/n~e (21)

28

is the plasma conductivity. Insertion of the expres­sion for 'tei and the numerical values of constantsyields for L; == 15

aE ~ 107T:/ 2 CGSE units (22)

This formula is valid for a fully ionized plasmawith singly charged ions (the hydrogen plasma).Note that conductivity grows rapidly with increas­ing T e. At temperatures about 108 K, the conduc­tivity of the hydrogen plasma must be higher thanthe conductivity of copper at room temperaturemore than by an order of magnitude. If plasmacontains multiply charged ions its conductivity isconsiderably lower. In the general case when theplasma contains ions with the charges Zl' Z2' Z3' .••

- •• , Zn with the concentrations aI' a 2 , a 3 , • • ., an,Eq. (22) for the conductivity is transformed into

(23)

Let us now discuss briefly the application of theOhm's law to a plasma-This law is valid if in plas­ma there is an equilibrium between the forces of theelect.ric field acting on electrons and the deceleratingforces. However, can such an equilibrium be estab­lished under any conditions?

The decelerating force action on an electron acce­lerated by the field is the lower the higher is theelectron '8 velocity. Take an electron from the far

2B

tail of the Maxwell distribution (HIe~ kTe). 'fhedirected velocity component acquired by the elect­ron between two "collisions" with ions is proportio­nal to r ei and, therefore, increases as 1/1• Therefore,if the therm al velocity v of our electron is highenough, its directed velocity u can become as highas v or even higher. Under these conditions it is nolonger possible to use a simplified model in whichthe electron accelerates to a relativity low directedvelocity along its free path and then completelyloses it in an instantaneous strong "impact". Actu­ally, acceleration and deceleration of the electronare concurrent. While the electron is acquiring thedirected velocity, the Rutherford scattering on ionsis gradually changing the direction of its motion.The electric field tends to straighten the electron'strajectory while the interaction with ions is bend­ing it. If the increase in the directed velocitycomponent is not compensated by scattering, theequilibrium of forces cannot be established and theelectron should accelerate continuously so that itsenergy continuously increases. With increasingelectron energy, the decelerating force decreases,hence, an electron involved in continuous accelera­tion by the field will be accelerated as long as itremains in the field. The field accelerates continu­ously those plasma electrons which have acquiredalong the free path A an additional velocity u whichis higher than their ini t.ial velocity v. This condi­tion for continuous acceleration may be written asfollows:

(24)

30

Since Tei is proportional to lfJ/n, Eq. (24) showsthat continuous acceleration occurs only whenEWe/n is higher than a certain boundary value. Ascan easily be seen, for the hydrogen plasma thisbound ary value is of the order of 3 ·10 -12, if E ismeasured in V/cm and We in eVe In the plasma ex­periments the condition (24) is typically satisfiedonly for electrons with the energy much higher thankTe • The share of such electrons in the electrr.n com­ponent is very low. In this case, the current com­prising the overwhelming majority of plasma elect­rons satisfies the Ohm law. However, there willexist in the plasma the current of the acceleratedelectrons for which the Ohm law is not valid. Forlarge values of E/n, the condition (24) will besatisfied also for the electrons with the mediumthermal velocity. In this case, continuous accelera­tion can involve the major part of the plasma elec­trons and the Ohm law will be sharply violated.As calculations show, the continuous accelerationprocess develops markedly when the ratio of themean value of u for the plasma as a whole to themean thermal velocity of electrons is greater than 0.1.The ratio u/v increases with v2 so that when u/vfor the electrons with the mean thermal velocityis 0.1, for the electrons with the energy of 10 kT ethe velocity u is close to u. Such electrons are closeto the threshold of continuous acceleration. It may beassumed that continuous acceleration of electronsis observed in ring-type electric discharges whena plasma is generated inside a toroidal chamber andaccelerated by a cyclic electric field. In these

31

experiments under special conditions a certain(relatively small) grollp of plasma electrons hasbeen observed to be accelerated to very high ener­gies at comparatively low voltages across the plas­ma ring.

A more detailed analysis of the accelerated elect­ron flows indicates that these flows can generateand amplify a variety of oscillations and wavesin the plasma and transfer their energy to them.This gives rise to a new mechanism of braking theaccelerated praticles which puts a stop to accelera­tion after the plasma electrons have acquired acertain additional energy of a directed motion.This automatic braking mechanism makes it impos­sible for all the plasma electrons to be continuouslyaccelerated. However, Eq. (22) cannot be used inthis case for determining the plasma conductivitysince braking of electrons through interaction withthe waves must increase the resistance. The measu­red plasma conductivities in the ring systems agreewith the above considerations. For high plasma den­sity and relatively low electric field the experimen­tal values of 0E coincide to the measurement accu­racy with the values found from Eq. (22). Theresistances are found to be abnormally high in low­density plasma at high temperatures.

The peculiarities of the plasma are especiallynoticeable when we consider its behaviour underthe effect of high-frequency electric fields. In thiscase the mechanical inertia gains particular signi­ficance. Let us consider the simplest case. Let theplasma be under the effect of the electric field

32

E == Eo exp (iwt) (complex notation simplifies theanalysis). If the field frequency is high enough thatduring one period of field variation the probabilityof collision between an electron and ions is smallenough, we can neglect the braking force in thefirst approximation in our analysis of the motionof electrons. Hence the equation of motion can bewritten as

m.» == - eEoexp (iffit) (25)

where x is the coordinate along the field. Integrationof Eq. (25) yields the velocity

u :-== - X =: eEoexp (iwt)/(intew) (2(i)Hence we obtain

. 1 ne»j=neu=-;--E (27)

z nl.eW

Eq. (27) can be rewritten as

E = imewj/(ne2

) (28)

'rhus, the phase of a high-frequency field is shiftedby 900 ahead with respect to the plasma current. Thismeans that plasma in a high-frequency field posses­ses its own "non-magnetic" inductance which is dueto the inertia of electrons. For low concentrationsthe non-magnetic inductance of a plasma conductorcan be higher than its normal ("magnetic") inductan­ce. Let us analyze the conditions required for that.For a cylindrical uniform plasma conductor withthe radius a, the non-magnetic inductance per unit

3-032:1

length is given by (in the CG-SE units)

1 IE I 1 IE I rUe meW 7 = na~w J == Jl,a~ne~ = Ne:::

where N is the number of electrons per 1 cm of plas­ma length. The normal inductance (self-inductance)for a conductor 1 em long is of the order of unityin the CGSIVI units. In the CGSE units the induc­tance is of the order of 1Ic~. Hence, the ratio be­tween the normal inductance and the non-magneticinductance for plasma is

Ne2/rnec

2 ~ 3·10-13N

This ratio has a very simple physical meaning: it isthe full number of electrons contained in a plasmaconductor segment with the length of the classicalelectron radius (re = e2Irn

ec2

) .

Eq. (27) relating the current density to the high­frequency field can be extended to take into accountthe braking of electrons due to collisions. Thus, thefollowing relationship is obtained between E and j:

E == j (p + iwLe) (29)

Here p = 1/0E and L; === nte/(ne2) is the non-magne­tic inductance.

Some comments should be made here on the die­lectric properties of plasma. These properties areal 0 related to the motion of electrons under theeffect of external electric field. In the above simplecase of the external electric field E === Eo exp (iwt),.. .the acceleration x and the velocity x of an electronare given by Eqs. (25) and (26). Further integration

34

over time yields the displacement

x == eEoexp (iwt)/(mew2) (30)

Eq. (30) shows that the electron displacement x hasa phase shift of 180 0 with respect to the force -eEacting upon the electron. This effect is the reverseof that encountered in normal solid dielectrics.When the phase shift between the displacement of .the charge and the force acting upon it is 180°, thepolarization of the material is opposite to the fieldand, hence, the dielectric constant is less than unity,The dielectric constant can be expressed in termsof the electric moment p of the uni t volume in a well­known way:

e = 1 + 4np/E

We have in the plasma

p = - nex

(31)

(32)

where x is the displacement of electrons (the displace­ment of ions can be neglected since it is very small).Eqs. (30)-(32) yield

e == 1-W~/W2 (33)

Here 000 is the plasma frequency given by Eq. (3).When ro is less than 00o, the dielectric constant isnegative. A consequence of this is that the electro­magnetic waves with the frequencies lower than W ocannot penetrate the plasma and are totally reflectedfrom its surface.

The diffusion processes in a plasma are, to a cer­tain extent, similar to those in normal gas. A diffu-

35 3*

sion coefficient can be determined for each plasmacomponent; similar to the kinetic gas theory thediffusion coefficient is AVtI3, where A is the meanfree path and Vt is the mean thermal velocity ofthe species.

However, the plasma quasineutrality conditionnecessitates that the electron flux be equal to theion flux across any surface in the plasma. Sinceelectrons have a much higher diffusion coefficientthan ions (the ratio is the same as the ratio be­tween the velocities for equal free paths), an electronconcentration gradient at first will give rise to anelectron flux from a higher-concentration region toa lower-concentration region greater than the ionflux. This will result in polarization of the plasmaand in development of an electric field that willbrake the electrons and accelerate the ions. Simplecalculations show that the electron flux will begreatly decreased and the ion flux will be somewhatincreased. This effect is known as the ambipolardiffusion. I t is frequently found in the experimentswith relatively weakly ionized cold plasma withdiffusion of electrons and ions through neutralgas.

The heat conductivity of plasma is determined byenergy transfer in collisions between particles withdifferent thermal energies in a region with tempera­ture gradient. The decisive part in heat transfershould be played by the collisions between theelectrons since their collision frequency is high.The electron heat conductivity coefficient in a fullyionized plasma with singly charged ions can be cal-

3()

cul ated from the following formula:

Bh ~ 1.2 .10-6T5/2 (34)

where fJh is expressed in erg/ern- s- deg. Eq. (34) isvalid when the electron mean free path is small ascompared with the size of the region where the tem­perature gradient is established. Note that the rela­tionship between the heat conductivity coefficient f:,hand the electric conductivity coefficient (IE is givenby the well-known Wiedemann-Franz law. The heatconductivity of a fully ionized plasma increases veryrapidly with temperature. Even when T is about105 K the heat conductivity of the hydrogen plasmais higher than the heat conductivity of silver atroom temperature.

4.

PLASMA BEHAVIOURIN MAGNETIC FIELD

A physicist making his first acqnaintance with thebasic plasma concepts can readily feel some disap­pointment at the:beginning since everything seems tobe too clear and almost self-evident. The work inplasma physics seems to be lacking in excitement,it looks like doing exercises with already knownsolutions. However, this attitude cannot developwhen one considers the behaviour of a plasma ina magnetic field.

Due to the effect of the magnetic field the plasmabecomes non-isotropic and radically changes all its

37

properties. A magnetic field can be used to establishclosed plasm a configurations contained in a limitedvolume and, so to say, suspended in a vacuum. Thisis something that is quite untypical of gas. Undersuch conditions the plasma properties are closer tothose of liquids. At the same time such a plasmapossesses the properties which distinguish it fromall other states of matter. Ultimately, all thecharacteristics of plasma processes depend on the mo­t.ion of particles; therefore, before we attempt themacroscopic description of magnetic properties ofplasm a, we have to consider in brief the effect of themagnetic field on the motion of electrons andions.

A charged particle in a uniform magnetic field isgenerally known to travel along a helical line. Theprojection of the trajectory on the plane perpendi­cular to the magnetic induction vector B is a circlewith the radius p == mu 1 c/(qB) , where v1- is thetransverse component of the particle's velocity.This is the so-called Larmor circle. The motion alongthis circle is the rotation with theLarmor frequen­cy roB = qB/(mc). The particle: also travels alongthe lines of force with the constant velocity VII.

Generally, the magnetic field is not uniform.However, in non-uniform fields typically consideredin plasma physics, induction B has almost a con­stant value and sense at the distances of the order of1he Larmor radius of particles, that is, microscopicvariations of the magnetic field are very small. Letus find how such weak non-uniformity of the fieldaffects the motion of particles.

38

At first, assume that the field varies along thefield line. The trajectory of the particle travellingalong this line noticeably changes its shape withina segment in which the magnetic induction B in­creases or decreases significantly. When the particletravels towards the increasing field, the trajectorybecomes more steep and it can be compared toa spring being compressed. When a particle travelstowards a decreasing field, its trajectory becomesle~s steep.

This effect can be readily explained. A chargedparticle which rotates along the Larmor circlegives rise to a circular current so that it is equiva­lent to an elementary diamag-netic with the magneticmom ent u == W 1-/ R, where W 1 is the kinetic energyof the transverse motion. Indeed, the Ampere theoryyields

i 2 __ qw 2_ q qBm 2vi c2 _ mv~ _ W.LII =-= - rrp - Pr c --2"C --~ mcq-B? - 2B -B (35)

The magnetic field whose intensity ~varies along thelines of force acts upon a diamagnetic with the force

F == - 'tldB/d1 (36)

(here differentiation is done in the direction of thefield). The action of this force results in variationof the longitudinal velocity vII according to therelationship:

dlJIl IV.L dBmdt= -BdT (37:

39

Multiplication of both sides of Eq. (37) by vn yields

dlV 11 W.1 dB dl W 1. dBrsr : - B dldt= -73 Cit (38)

When a particle travels in a magnetic field, W.1 ++ Wlf is constant. Therefore, Eq. (38) can be trans­formed into

dW.1 W 1. dBdt=sdt (39)

Hence, we find

dW.1IW == dBIB; W.L/B === const (40)

Thus, the ratio W-l IB is constant when a chargedparticle travels in the magnetic field whose intensityvariation along the field lines is not too sharp.

The constant W.1/B is usually referred to as theadiabatic invariant of motion. This emphasizes thefact that the particle travels in a slowly changingmagnetic field. The kinetic energy of the transversemotion is W 1- === W o sin" ct, where W o is the fullenergy of the particle and a is the angle betweenthe velocity direction and the field line. Since W ois constant and W.1/B is the adiabatic invariant,the ratio sin" alB is also an adiabatic invariant.This shows that the slope increases with increasing Barid in the region with a stronger magnetic field thehelical trajectory becomes more steep as discussedabove.

Assume that at a certain trajectory point a === a.,oand B === B o• Using these initial conditions, wecan find the value of a at any trajectory point from

40

the relationship sin" al B == sin" ao/E o, that is,

sin a ~ sin aoVBIBa (41)

When a particle travelling towards the increasingfield comes to the point where B == B o/sin2 ao,then the angle ex becomes 900 and hence the longitu­dinal velocity VII vanishes. This means that at thispoint the direction of the longitudinal motion isreversed. The particle is reflected from the regionof high field towards t.he region of a lower field.

Thus, high-field regions under certain conditionscan act as some magnetic mirrors for charged parti­cles. For instance, if the field increases in oppositedirections from a certain intermediate region, acharged particle can' be blocked between two magneticmirrors; it will oscillate along the field lines withina restricted space region. The particles with a largeenough angle a (sin ex > -VBmln/Bmax) will beconfined in this region.

Let us now consider the motion of particles ina non-uniform field whose intensity varies perpendi­cular to the field lines. Let us at first consider thesimplest case when the velocity is perpendicular tothe magnetic field. Fig. 1 illustrates the particle'strajectory under such circumstances. The magneticfield is perpendicular to the plane of Fig. 1.

The field increases along the axis x. Here the par­ticle's trajectory projection in the xy plane is nota circle, since the Larmor radius at the right is lessthan the radius at the left. Clearly, the trajectoryis not closed after a complete cycle of rotation. Each

41

Fig. 1.

Drift of a charged particle in a non-uniformtransverse magnetic field.

rotation cycle of the particle comprises a loop andwith each loop the particle spirals a distance ~y

along the axis y, that is, perpendicular to the mag­netic field gradient. The particles of opposite chargesspiral in opposite directions along the axis y.After several rotation cycles, it can readily be seenthat the particle's trajectory is a path of loopsstretching parallel to the axis y. Such a motion isknown as the magnetic drift. The drift velocity ofthe particle is small compared to its Larmor velo­city (it is assumed here that variation of the fieldintensity at the distances of the order of the Larmorradius is small).

42

Let us note the following feature of the driftmotion. In the course of this motion the particledoes not travel towards stronger or weaker fields. Onthe contrary, the particle travels along a narrowpath which stretches within a constant-field re­gion. Hence, the parameter W.lIB is an adiabaticinvariant in the case of the magnetic drift.

The magnetic drift velocity Vd is l1y/T, where T isthe Larmor rotation period. The ratio between Vd

and the rotation velocity, clearly, must be of theorder of p/l, where p is the Larmor radius and 1is the characteristic dimension of non-uniformityof the magnetic field:

1/1 "-/ (1IB) (dB/dx) (42)

Hence we findp dB mv~c dB

v« "-/ V1- B a; "'" qB2 ([i" (43)

Calculations which we omit here confirm Eq. (42)and yield the following expression for Vd:

1 mvle dBv« = 2" qf2 """!IX (44)

Eq. (44) gives the velocity of the motion in theplane perpendicular to the induction B. However,the drift motion in the non-uniform magnetic fieldcan also be related to the longitudinal velocity vnof the particle. The mechanism of such a drift isillustrated by Fig. 2. The thick lines in Fig. 2represent the field lines of the non-uniform magneticfield which are generally curved. The Lorentz force

43

Fig. 2.

Drift of a charged particle in a non-uniformmagnetic field with the velocity parallelto the field.

is zero at the point M1 where the particle's velocityis parallel to B. However, the particle will continueinertial motion and slip off the field line; thiswill give rise to a small velocity component perpen­dicular to the field at the point M 2. The transversevelocity will automatically give rise to the Lorentzforce. This force gives rise to a drift velocity whichis perpendicular to the plane of Fig. 2. Considera­tion of special cases (for instance, the motion ofparticles in a magnetic field generated by a recti-

44

linear current) readily shows that the particle'sdrift due to the longitudinal velocity VII has thesame direction asthe drift due to the transverse velo­city v.1.

The following expression can be derived for thedrift velocity in the general case, when vII =i= 0and V.l =1= 0:

1 2 2Vd==-R (vlI+V.l/2) (4r))

WB

Here, W n is the Larmor frequency and R is the radi­us of curvature of the field line. The velocity vectorhas the sense of the vector product [B X R]. Eq. (45)is valid when the following two conditions are met:Vd ~ Vvi + VI" that is, the drift velocity is muchsmaller than the velocity of motion of the particle;and the current density in the region where theparticle travels is zero (or low enough). When theseconditions are satisfied, the relationship between thegradient of the field intensity and the radius ofcurvature of the field line is simple enough to makepossible derivation of Eq. (45). If the secondcondition is not satisfied, the expression for thedrift velocity has a more complicated form. Notethat both components of the particle's velocity re­main constant in the course of the drift motion. Sincethe particle always drifts perpendicular to the gra­dient of B, the value of B along the drift path re­mains constant. Therefore the drift motion does notviolate the invariance of the parameter. H"l./B.

From the above it follows that the motion ofa charged particle in the non-uniform magnetic

45

field can generally be expressed as a superpositionof three types of motion:

(i) rotation along the Larmor circle with the velo­city v.l;

(ii) motion of the centre of the Larmor circlealong the field line with the velocity vII;

(iii) drift motion of the centre of the Larmor circlein the direction perpendicular to B and grad B.

The instantaneous positions of the centre of theLarmor circle comprise the axial line of the trajec­tory which, at the same time, can be considered tobe an averaged path of the particle. The form of thisline is a fuudamental geometric characteristic ofthe motion of a particle in the magnetic field.

The nature itself provides us with an excellentillustration of the motion of charged particles innon-uniform magnetic fields, namely, the so-calledearth radiation belts. The radiation belts werefound by the first earth satellites and space rockets;they consist of high-energy electrons and ionstrapped in the earth's magnetosphere. The motion ofelectrons and ions of the radiation belts in the mag­netic field of the earth is rather complicated. If weignore the Larmor rotation and consider only theaveraged trajectories, the resulting paths are shownby the thick line in Fig. 3. Charged particles in themagnetic field of the earth oscillate along the fieldlines and are reflected from higher-field regions nearthe magnetic poles. A slower drift due to the cur­vature of the field lines is superimposed on thisoscillatory motion. The drift motion is in the azi­muthal direction. The particles of opposite charges

46

Fig. 3.

Motion of charged particles in the magneticfield of the earth. The thick lineis the averaged trajectory.

travel in opposite directions going around the globefrom east to west and from west to east.

To end this brief summary of motion of particlesin the magnetic fields, we should consider the situ­ation when, along with the magnetic force, thereis a force of some other origin acting on the travel­ling particle. Of the highest interest is the casewhen such a non-magnetic force is perpendicular to B.We shall omit here very simple calculations which

47

show that a force F perpendicular' 10 B gives rise tothe drift with the velocity

v« == q~t [(F X B)] (4G)

When the force is due to an electric field E, wehave F === qE and

v« == ;~ (E X B] (47)

Here the drift velocity does not depend on the chargeand sign of the particle. The drift motion of thistype also does not violate the adiabatic invari anceof W1./B (the particle travels perpendicular to theforce F and therefore its Incan kinetic energy duringone Larmer cycle remains constant).

5.

TI-IE EF~'ECT OF rur MAGNE1'ICFIELD ON CHARACTERISTICS

OF THE PLASMA

Let us now analyze the effect of the magnetic fieldon the properties of plasma. Let us assume thata moderate-density plasma is in a strong magneticfield. Between two Coulomb collisions, each chargedparticle of the plasma travels in a helical trajectoryalong the field. If the field is uniform, the axial lineof the trajectory is a field line. The particles cantravel perpendicular to the field only owing to theCoulomb collisions. Each collision makes a particle

48

t.ravel a distance of the order of the Larmor radius.If the collisions are rare (low-density high-tempera­ture plasma) the particles are, so to say, tied to thefield lines. Such a plasma is known as a "magnetized"plasma. The degree of "magnetization" is given bythe ratio 'Alp, where 'A is the mean free path and pis the mean Larmor radius. WIlen Alp:}> 1 (rarecollisions and high field), the particle can travelan appreciable distance perpendicular to the fieldonly if it travels a considerable distance along thefield. When 'Alp ~ 1 (high-density plasma, weakfield) the motion of particles is practically isotro­pic; this means that the magnetic field has a weakeffect on the behaviour of the plasma.

The degree of magnetization can also be expressedin the following form:

A qB'Alp ~ - - = ffiBLv me

where ro B is the Larmor frequency and 't is the meantime between two successive collisions. The electronand ion components of the plasma have differentmagnetizations. Under normal conditions the valueof (i) BT for the plasma electrons is much higherthan for the plasma ions (when T i ~ T es the ratiobetween these parameters is of the order of V mi/me).Hence, magnetization of electrons is higher thanmagnetization of ions. I t can happen, for instance,that electrons are magnetized and therefore travelfreely along the field lines, while the magnetic fieldby itself does not have a noticeable effect on themotion of ions. Under such circumstances,

4-0323 49

electrons are tied to the magnetic. field, while ionsare confined to the same space region by the electricfield of the electron plasma component.

But we are primarily interested in high-tempera­ture fully ionized plasma; in such a plasma the ty­pical situation is when both electron and ion com­ponents are magnetized. The main parameters ofexperimental hot plasmas vary in the followingranges: concentration - from 10 10 to 1014 , electronand ion temperatures - from 106 to 107 degrees, andmagnetic field intensity - tens of kilogausses. Forthese ranges of n, 1" and B the magnetization para­meter co n't for the electrons varies from 105 to 1011

and for the hydrogen ions, from 103 to 109•

Since a strong magnetic field restricts the motionof plasma particles in the plane perpendicular tot.he vector H, such a field can be used as a screenpreventing the contact between the plasma and thewalls of the vessel containing the plasma (Fig. 4).A cylindrical column of fully ionized plasma isinside a chamber with a strong magnetic field. Thespace between the plasma surface and the chamberwalls contains only vacuum and the magneticfield lines. 1'110 electrons and ions of the plasmacannot penetrate this space. Thus, a strong magneticfield provides for an effective thermal insulation ofplasma. But strictly speaking, this is not an equili­brium state. Owing to Coulomb collisions, thepl asm a wi ll , sooner or later, spread allover thechamber's volume up to its walls. The lifetime ofan insulated plasma column is determined by thediffusion rate of the plasma particles in the magnetic

50

B B 8 B

Fig. 4.

A plasma column in a magnetic field.

field (in the plane perpendicular to B). The diffusiontheory shows that this lifetim e is of the order ofa2/aD 1.' where a is the radius of the plasma columnand D.l is the transverse diffusion coefficient. A veryrough estimate of D.L can result from the followingsimple calculations. During the period T of onefree path, the particle in the average makes oneCoulomb collision which results in the particle'sdisplacement in the plane perpendicular to B toa distance of the order of the Larmor radius p, To

51 4*

find the resulting displ acom eu t after numerouscollisions, we have to add up the squares of theindividual displacements according to statisticalrelationships. Thus, the displacement of a particleperpendicular to B during the tirn e t is

(48)

;\t the same time, the mean displacement duringthis diffusion process is of the orrler of Y D .It.:I fence, we obtain

(49)

The mean time between two successive collisions isproportional to T3/2 and therefore we find

(50)

For high values of /3 and T, the coefficient Ir ; musthe very small. But the motion of particles along theIicld lines is he same as when B == O. Thus, theplasma in a high magnetic field has a sharplymarked anisotropy with respect to diffusion.

The above qualitative discussion fails to accountfor one important feature of the diffusion mechanism.Higorous theoretical analysis shows that the trans­verse diffusion of plasma particles is due only tocollisions between different particle species, thatis, to collisions between ions and electrons. Thecollisions between the same species cannot give riseto a macroscopic variation of the concentrationprofile.

52

Rigorous analysis yields the following expressionfor the diffusion coefficient:

(51)

Here o; i.s t.he mean thermal velocity of the plasmaelectrons, roBe is the electron Larmor frequency,and Do is the diffusion Coefficient in the absence ofthe field.

Clearly, the heat conductivity of plasma in thedirection perpendicular to B also must sharplydecrease with increasing field intensity. In contrastto diffusion which is determined by the collisionsbotween ions and electrons, the heat transfer acrossthe field lines in plasma occurs primarily in theion-ion collisions (if T', is not too small comparedto Te). This is oxplained by the fact that the inten­sity of heat transfer is determined by the size ofthe region in which there meet the trajectories ofparticles with different thermal energies under theconditions of tho temperature gradient.

The thermal conductivity coefficient in the direc­tion perpendicular to B is proportional to the squ (l­

red size of this region and this size is of the order ofthe Larmer radius. Therefore, the ion component isprimarily responsible for heat transfer.

The coefficient of ion thermal conductivity in thedirection perpendicular to the field lines decreasesby a factor of the order of (eoBi'tii)2 compared tothe coefficient in the absence of the magnetic field.The coefficient x J- of the transverse t.herm HI conduc­t ivit y for the hydrogen plasm a is given by the Iol-

53

lowing expression:

Xi == 2 .10- 16n2/(B2l!T i )

The rate of equalization of temperature along theradius in the cylindrical plasma column proves tobe much higher than the rate of equalization ofconcentration (approximately, by a factor of1/mi/me). Hence, a temperature gradient in thedirection perpendicular to B in the plasma columnmust vanish long before the plasma spreads allover the space owing to transverse diffusion.

Low diffusion rates make it possible, in principle,to establish in the magnetic field plasma configura­tions which are confined to a finite volum e of spacesurrounded by vacuum on all sides and have longlifetimes (of the order of seconds or even tens ofseconds).

Let us attempt a macroscopic approach to analyzethe behaviour of such confined plasma configurations.The first problem that arises here is as follows. Theplasma has a gas-kinetic pressure p = nk (T e +-t- T i); therefore, a plasma configuration witha certain spatial distribution of pressure can existfor a long time only if in any plasma element theforce due to the gradient of gas-kinetic pressure isbalanced by forces of other nature. What are theseforces? For instance, if we have a single plasmacolumn with a constant pressure p, what is theforce that compensates the pressure at the boundaryof the column?

In the light of the above discussion of the motionof charged particles in magnetic fields, the answers

54

to these questions seem almost self-evident. A con­fined plasm a configuration must be contained byelectrodynamic forces which arise due to the dia­magnetic behaviour of the plasma in the magneticfield. The plasma is diamagnetic since the Larmorcurrents of the rotating charged particles generatein each plasma element a magnetic moment directedopposite to the external field. This decreases themagnetic field in plasma and gives rise to a ponrie­romotive force which compensates the pressuredifference between the interior of the plasma andits boundary.

The ponderomotive force action on unit volum eof the plasma can be expressed in terms of thedensity j of the diamagnetic current in the pl asm aand the magnetic induction B. This force is fj X Bl/c.The equilibrium condition has the following form:

grad p == [j X Bl/c (52)

Note that the current density j can be found as thesum of the elementary currents due to rotation ofelectrons and ions in the magnetic field B.

Tho mechanism of plasma confinement can beillustrated by the following simple example. Fig. 5shows a diagram of the Larmor currents in a cylin­drical plasma column inside which t he pressure pis constant. Under these circumstances, the Larrn orcurrents inside the plasma column cancel out eachot her so that j == O.

But at the boundary of the column there is a netcircular current. The force result ing from interac­tion of this current with the magnetic field com pen-

55

Fig. 5.

Larmor currents in the plasma.

sates the pressure difference at the boundary of theplasma column.

Eq, (52) derived for equilibrium conditions can beextended to analyze the dynamics of plasma configu­rations. The second Newton's law for the motionof a unit plasma volume can be written as

pdv ldt === [j X B]/c - grad p (53)

where p is the plasma density and v is the velocityof the plasma volume. The right-hand member of theequation is the force acting on unit plasma volume.

56

Eq. (53) lacks all characteristics of the plasma asan ionized gas. This equation is generally known asthe fundamental equation of magnetic hydrodyna­mics. I t can be used for analyzing the processeswhich occur in any conducting medium in a strongmagnetic field when the medium is capable ofchanging its form under the effect of external for­ces, for instance, for analyzing the phenomena ina high-conductivity fluid. In plasma physicsEqs. (52) and (53) give an approximate model ofthe physical process which is rather rough. How­ever, this model yields veryrgood results in the ana­lysis of statics and dynamics of plasma configura­tions if we ignore some finer features of the processesand some aspects of the dissipation phenomena inthe plasma.

The next approximation is given by a two-fluidmodel of plasma which accounts for the electronand ion components of the plasma. The followingequations describe the dynamics of plasma proces­ses in the two-fluid model:

Pe d~e = -ne (E++(ve X B]) -grad Per: Fei (54)

Pi d~i =ne (E++(Vi xBl)-gradPi+Fei (55)

where Pe and Pi are the densities of the electron andion components, V e and v i are the averaged veloci­ties of electrons and ions, E is the electric field in­tensi ty, and F ei is the friction force acting on theelectrons in unit volume due to their collisionswith ions. This force is given by the momentum

57

transferred per unit time from electrons to ions:

(5G)

The physical meaning of Eqs. (54) and (55) isquite clear as the right-hand members of theseequations are the net forces acting on the givenparticle species in unit volume of the plasma.

The two-fluid model of plasma can be used toanalyze such processes as electric current in a mag­netized plasma and the development of a varietyof non-stationary plasma phenomena. It should beemphasized that both Eq. (53) and Eqs. (54) and (55)are valid only when the plasma pressure is isotro­pic so that p is a scalar quantity. This condition issatisfied only for such plasma formations in whichplasma can freely spread along the lines of force ofthe magnetic field.

6.

RESlJJ~TS DERIVED FROMMAGNETOHYDRODYNAMIC

EQUATIONS

Let us now consider some results which can bederived from general magnetohydrodynamic equa­tions. A simple case to analyze is the equilibriumof a plasma column in the longitudinal magneticfield (see Fig. 4). The field lines here are straightand parallel to each other so that the fieldintensity has only one component, which we denote

58

as Bz- The intensity Hz is a function of the coordi­nates x and y which are perpendicular to the field.The equilibrium condition tor the plasma column is

gradp~~(jXB]=-41 [rotBxB] (57)c n

Hence we obtain

8p _ 1 B en, 1 a B 2 • }ax - - 4n z"7fX= - 8n a; z, (58)!L __1_ B aB z 1_!..- H2f)y - 4n - Z By - 8n dY Z

so thatP + B2/8n = const (59)

Eq. (59) shows that the magnetic pressure B2/8noutside the plasma is higher than inside it by p.The maximum pressure at which the plasma can beconfined by the field of the given intensity B o isfound from the condition

Pmax == B~/81t (60)In this case the plasma must completely push outthe field from the volume occupied by the plasma.

If the field lines are curved, the sum of gas-kineticand magnetic pressures is not generally conserved.However, Eq. (60) regarded as the boundary con­dition remains valid. If the pressure at the curvedsurface of the plasma configuration drops to zero,then the following condition must be satisfied:

p +BfIBn== B~/8n

Here B, and Bo are the field intensities at both sidesof the boundary.

59

A high longitudinal current'passing through a plas­ma can produce a quasi-stationary cylindricalplasma column. In this case the plasma pressure iscontained by the magnetic field whose circular linessurround the plasma column. Let us limit ourselveshere to the simplest case when current flows onlyalong the thin surface layer [of the plasma (strongskin effect). In this case the field intensity inside theplasma is zero and the pressure in it is constant.This pressure can be found from Eq. (60), where B ois the intensity of the field generated by the currentat the outer surface of the plasma. If the current is I(in the CGSE units) and the plasma radius is a,Eq. (60) yields

1 (21) 2 [2P ~ 8n W == 2na~ (61)

Further, let us assume that the ion and electrontemperatures in the plasma column are the same (T).We have here

p == 2nkT (62)

Eqs. (61) and (62) yield the following relationshipbetween the temperature and the current:

T = ]2/4c2N kT (63)

Here N == rtna2 is the number of particles of thesame species per 1 em of the length of the plasmacolumn. Eq. (63) has been derived with a specialassumption that the current flows in a thin surfaceLayer. But a more careful analysis shows that thisformula is valid for any current. distribnt.ion overthe cross-section of 1he pl asm a column.

60

'rho formation of a plasma column stabilized bythe magnetic field of the current flowing in it isknown as the linear pinch effect. A current passingthrough a straight discharge tube filled with gasshould, at first, give rise to a plasma (owing toionization of the gas) which will be compressed byelectrodynamic forces so that a plasma column canbe Iorm ed. In these circumstances the current doesthree things at once - it generates a plasma, heatsit by the Joule heat and compensates its pressure inthe compressed plasma column by the pressure ofits own magnetic field. At the early stages of high­temperature plasma physics, this technique forgenerating high-temperature plasmas seemed to behighly promising owing to its apparent simplicity.It was thought that a sufficiently high pulsed currentpassing through a gas at low pressure can produce ina discharge tube over a negligibly short period oftimea plasma column with a tremendous temperature.

Indeed, assume that a current of 106 A passes ina discharge tube with a 10 em diameter filled withhydrogen at the initial pressure of 0.01 mm Hg(the initial concentration of atoms no === 7 .1015 em -3).

If all the gas is ionized and a plasma column isformed whose pressure is compensated by electro­dynamic forces, the plasma temperature, accordingto Eq, (63), must be

1 l~ 1 ]2T~.-:-: 400 Nk = 400 na 2nok- ~ 3·107

J(

In practice, however, the highest temperature obta­ined with this technique was only about 106 degrees

Gf

(and that was only for a few microseconds). Experi­ments have shown that a high-current short-dura­tion pulsed discharge does not generate a quasi­stationary state described by the equilibrium condi­tion (57). Now it can be seen that we could haveexpected that if we had a keener insight.

At the initial stage of discharge, the current flowsonly in a comparatively thin layer of the ionizedgas near the walls of the discharge tube owing tothe skin effect. Under these circumstances, theelectrodynamic forces are not compensated by thegas-kinetic pressure of the plasma, since the cylin­drical layer with the current surrounds the practic­ally unperturbed weakly ionized gas. Then theplasma layer will be accelerated by electrodynamicforces towards the axis of the tube. This stage ofthe process can be described very roughly byEq. (53) without the term grad p. The acceleratedmotion of the plasma towards the axis of the tubecan be regarded as the development of a cylindricalcompression shock wave. As the plasma layer travelstowards the axis, its mass steadily increases andnew gas layers before the plasma front become in­vol veel in the compression process. At the same timethe gas becomes ionized. At the last stage of com­pression, the plasma accelerated by the magneticpressure reaches the axis of the tube. At this stage,a considerable part of the kinetic energy of plasmaconverts into heat. Now it seems to be the timefor the next stage of the process at which the bal­ance can finally be established between the electro-

62

H

Fig. 6.Deformation of a plasma columncarrying curren t.

dynamic forces and the internal pressure of thehot plasma. However, here comes into play anoth­er physical factor, namely, the instabilit.y of theplasma pinch. Under typical experimental condi­tions deformations of the "bottleneck" and "snake"types develop in the plasm a pinch in a few micro­seconds (see Fig. 6). These deformations disrupt theregular geometrical structure of the plasma so that

ti3

Fig. 7.Compression of plasma bya rapidly increasing magnetic field.

it starts to interact strongly with the walls of thedischarge tube and rapidly cools down.

This case illustrates for us for the first time a ma­jor problem of modern plasma physics, namely, thestability of plasma configurations. The plasma isvery unstable in its behaviour; it tries in all possi­ble ways to get rid of the magnetic confinementslipping through the lines of the surrounding magne­tic field. At the end of the book we shall discuss thegeneral factors which determine stability or insta­bility of plasma configurations.

A rapidly increasing longitudinal magnetic fieldcompressing the plasma provides for more effectiveheating of it than in the case of a linear pinch(see Fig. 7). This process also starts with the initial

G4

non-stationary compression stage at which a shockwave travelling to the axis is generated. But whenthe plasma cylinder radius ceases to oscillate,a balance of forces is established for some time andadiabatic heating of plasma occurs with furtherincrease of magnetic pressure.

A magnetic field which increases up to about100 kOe in a few microseconds makes it possible toproduce a short-lifetime plasma with a temperatureabout 107 degrees and a concentration over 1016 cm -3.

The longer the discharge tube and the coil whichgenerates the field the longer the lifetime of thehigh-temperature plasma. But as the plasma canflow out of the open ends along the field lines, thelifetime of high-temperature plasma is still short­for the above values of nand T it cannot be morethan a few tens of microseconds in a system about1 m long.

7.DEVELOPMENT OF EXPERIMENrfAL

TARGET

The fundamental problem in the experimentalphysics of high-temperature plasma is to generatea medium to be studied. Serious experimental studiesof the main plasma phenomena can be feasibleonly when there have been developed the techni­ques for generating high-temperature plasma whosequasistationary state can exist for relatively longperiods. In recent years, some advances have been

5-0323 65

made in development of such techniques. Thereare two main fields of research here. The first oneis concerned with magnetic systems which makeit possible to generate closed plasma configurationsof a toroidal shape. Such machines are often calledthe toroidal magnetic traps. The second one isconcerned with the systems in which the high-tem­perature plasma is confined in open mirror-typemagnetic traps. In such traps the magnetic fieldincreases along the Held lines in both directionsaway from the plasma region. As discussed abovethe higher-field regions act as mirrors which reflections and electrons of the plasma.

Let us discuss at first closed plasma configurations.In such a configuration the plasma spreads freelyalong the field lines. This means that the field linesmust lie within the closed surfaces which are insidethe chamber in the region where plasma is generated.A gas-kinetic pressure p of the plasma is a scalar.When the forces acting on the plasma are balanced,the pressure must be constant along the field linesince as shown by Eq. (52) the pressure gradient isperpendicular to B. Superficially, there would seemto be a very simple technique for magnetic con­finement of the circular plasma pinch. This techniqueconsists in winding a regular coil on the surface ofa round toroidal chamber which generates a fieldwith circular field lines (Fig. 8). This is the fieldthat will confine the plasma which will be generatedinside the chamber in a certain way. ,

However, this technique is essentially defectiveas can readily be shown by a simple theoretical argu-

oG

z

Fig. 8Plasma ring in the fieldof a toroid al coil.

men l . The magnetic field in this case is not uni­Iorrn , since Is ~ 1/R so that ions and electrons ofplasma will drift perpendicular to the field linesto the chamber walls. The balance equation (52)also shows that plasma cannot be confined in a sim..pIe toroidal field. To supporl this assertion, let usdiscuss a special case.

Let the plasma completely push out the magneticfield from the region i t occupies. Then the followingcondition must be satisfied everywhere at the boun­dary of the plasma ring] - p = B2/SJt . But thefield is different at the points J.111 and M 2 (see Fig. 8)

U7 5*

Fig. 9.

A figure 8 configuration.

so that balance is impossible. I t can be seen from thebalance Eq. (52) that this is so in the general casewhen P =1= B2/8.Jt. The above simple techniquefor confinement of a circular plasma pinch can bemodified in several ways. One modification is tomake a figure 8 out of a toroidal chamber with themagnetic coil on it as shown in Fig. 9. Then theplasma pinch also has the figure-S shape and thedrift of particles has opposite directions at theopposite curved parts of the pinch, so that thedisplacement due to drift during one cycle will becancelled out. Thus, the laws of motion of theplasma particles in the toroidal "figure 8" do notpreclude equilibrium of the plasma. The resultsof the macroscopic analysis of equilibrium usingEq. (52) should be expected to show that the systemsof this type can be utilized as magnetic traps forclosed plasma rings.

The toroidal figure 8 was, in fact, the first stepin the development of a wide variety of magnetic

68

Fig. 10.

Superposition of the fields Be and B,-p.

syst.ems for generating closed plasma configurations.These systems based on an elegant concept olSpitzer are known as stellarators. In stellaratorsthe equilibrium of high-temperature plasma ismaintained by using external magnetic fields.

The stellarator studies have been for many yearsan important field of the high-temperature plasmaphysics. However, we cannot discuss here this veryinteresting work because this would lead us awayfrom plasma physics towards the analysis of geo­metry of complex magnetic fields.

Equilibrium toroidal configuration of anothertype can be generated using superposition of twofields (Fig. 10). One of the fields is produced bya toroidal coil (1Io) and the other field BfP is produ-

69

ced by longitudinal circular current flowing in theplasma itself. The field in this system has simplehelical lines which wind about the axis of theplasma pinch where the magnetic field of the cur­rent vanishes. The drift displacements of particlesare compensated owing to the curvature of thefield lines. To understand the mechanism of thiscompensation without calculations, we shall illustrateit by a special case when the following two con­ditions are satisfied: (1) the magnetic field generatedby the current in the plasma is low compared withthe toroid 31 field generated by the external coils;and (2) a plasma particle has a sufficiently highlongitudinal velocity and therefore can freely travelalong the field line without being reflected fromthe higher-field regions. Under these circumstancesthe motion of the centre of the Larmor circle is thesuperposition of two motions -the motion alongthe field line (always in the same direction) andthe drift due to non-uniformity of the toroidalmagnetic field. Fig.it illustrates a simple modelfor superposition of these two motions in whichthe continuous drift along the axis y (the principalaxis of the toroidal system) is expressed by twojump-like displacements. We see that the motionof the particle alorg the winding field line resultsin closing of the projection of its trajectory on thecross section of the plasma pinch so that the netdrift displacement is zero.

Analysis of the motion of particles in the super­imposed fields of the above type shows that driftdisplacements are componsated not only for the

70

y

o

Fig. 11.

Model of the drift motion of part iclesin a helical field.

particles with high longitudinal velocities whichfreely travel along the helical field lines, (theso-called "through-going" particles) but also forthe so-called "confined" particles which oscillatealong the field lines owing to the reflection from thehigher-field regions.

It should be stressed here that the compensationof drift displacement does not preclude any effectof the drift motion on the shape of the trajectory.Even the schematic model of Fig. 11 shows thatthe drift motion results in some deformation ofthe trajectory. This deformation is even morestrongly marked for the trajectories of the confinedparticles. We shall discuss som e of the resul tingeffects below when mentioning the problems ofthe rates of diffusion and heat truusfer.

71

8.

CONDITIONS FOIlEXISTENCE OF A PLASMA RING

Let us now discuss in more detail the conditionsfor the existence of a plasma ring in a longitudinalext.ernal magnetic field and the magnetic field gene­rated by the current flowing in the ring. We illustratethe general problem by analyzing in detailthis special case because, first, this is the sim plestexample of a magnetic confinement and, second,the longest times of high-temperature plasma con­finement have been obtained in the experimentswith the machines utilizing this concept.

Before we analyze the forces acting on the plasmaring, let us look at some of the characteristics ofthe magnetic field geometry in the toroidal trapunder consideration. As mentioned above, thehelical field lines wind about the circular axis ofthe plasma ring. Let the plane S be the cross sec­tion of the trap. The field line through the point M 1(Fig. 12) in this plane after passing around thetoroidal magnetic system will intersect the plane Sat the point M2 , next time it will intersect it at thepoint M 3 , etc. The set of points Aft, M 2 , M 3 , •••

is>, generally, infinite. But for some field lines thisset can be finite so that they are closed by them­selves. Such lines are known as "degenerate" lines.The axis of the ring at which the field is zero issuch a degenerate line; it is shown by point 0 inFig. 12.

72

M 2 .... ---- M,,- ....... ,1»: ... - ,

I \I • J\.. 0 I F... /

M3 " ,/,-.... ..... -.-...-.."",-

MA

Fig. 12.Rotation of a field line(rotational transformation).

The behaviour of the field line which winds re­peatedly about the toroid is determined by thepositions of the "descriptive" points M 1 , M 2 , M3 , ....

Assume that the point AIn after n cycles is closeto the initial point 1111• Then after the next cyclethe angle of rotation about 0 will be more than360°. Where will be the next point? It is naturalto suggest that the point M n+l will be between thepoints M 1 and AI2 , the point Mn+2 between thedescriptive points M 2 and M 3 and so on. Thus, afternumerous cycles, the points in the cross sectionof the trap will make up a smooth closed curve(curve F in Fig. 12). Thus, we can say here that thereis a toroidal magnetic surface which, so to say, con­sists of (or is tightly woven of) one infini ;e fieldline.

Analysis of the equations for a magnetic fieldsupports the assumption about magnetic surfaces

73

Fig. 13.

Magnetic surfaces in a helical field.

in the s ystem with the external toroi dal field B 6 andthe field BrfJ of the plasma current. As shown bythe analytical results, for regular distribution ofthe current density over the cross section of theplasma ring the ;ufficient general condition forthe existence of magnetic surfaces is the axialsymmetry of the field. When the axial symmetryexists, the equations for the magnetic field linescan be integrated yielding the analytical expres­sion for magnetic surfaces. The magnetic surfacescomprise the set of embedded toroids (see Fig. 13).Any toroid out of this continuum is comprised of onefield line. However, this set of magnetic surfacesincludes a countable set of "degenerate" surfacesgenerated by closed field lines. In this case thesurface is woven of a continuous set of field linesshifted with respect to each other.

74

Spitzer has suggested that magnetic surfacesexist also for such toroidal fields in which thehelical field lines winding about. the circular axisare generated by the ex.ternal current sourcesusing special helical windings or by deforming thewhole magnetic system into, for instance, a figure 8.

However, theoretical analysis shows that forsuch asymmetrical fields the concept of embeddedmagnetic surfaces can be used only to some appro­ximation. The structure of the magnetic field is,generally, rather complicated. The individual mag­netic surfaces are separated by a system of toroidaltubes ("filaments") between which the behaviour ofthe field lines is quite chaotic. Clearly, the existenceof magnetic surfaces is very important since itdetermines the prospects for utilization of m agne­tic systems of various types for confinement of hotplasma. Since the plasma freely spreads along thefield lines, its pressure p must be the same at diffe­rent points of the same magnetic surface. Thus,magnetic surfaces are farnilies of plasma isobars.If the point.s M I , M 2 , M 3 , • •• fill up a certainwide region in the cross section of the plasma ringinstead of com.prising a smooth curve, then theplasma within this region can be at equilibriumonly if grad p == O. The fact that the pressuregradient vanishes at the cross section of the plasmaring is, clearly, equivalent to the development ofabnormally high diffusion in the direction perpen­dicular to B.

Let us now return to the analysis of the equilib­rium conditions for a circular plasma ring proceed-

75

I

I~ R

Fig. 14.

Cross section of a pl asma ring.

ing from the equations of magnetic hydrodynamics.In the first approximation the cross section of theplasma ring can be assumed to be the circle of ra­dius a. The plasma pressure and the current den­sity vanish outside the circle of radius a. The toroi­dal surface of the radius a is the boundary surfaceof the plasma ring which has no contact with thewalls of the vacuum chamber in which it has been

7G

generated (Fig. 14). This surface is also one of themagnetic surfaces of the toroidal trap. Anotherfundamental geometric parameter of the system isthe major radius R of the plasma ring. Below weshall always assume that a is much less than R as isinvariably the case in experimental environments.Note that when the ratio a/R is small, the charac­teristics of the plasma ring are close to those of aninfinite plasma cylinder. Since the problem hastwo fundamental geometric parameters a and Rfor two degress of freedom of the plasma ring, wehave to find two equilibrium conditions -for theminor radius and for the major radius. For thefirst one we can neglect the effect of the toroidalstructure for the minor radius. Thus, we can usethe equilibrium equation for the plasma cylinder.Since the plasma in this case is under the effect oftwo fields, the equilibrium equations can be synthe­sized from Eqs. (61) and (62) which were derivedfor either the longitudinal field or the field of thecurrent. We can derive such equilibrium equationsfor the plasma cylinder assuming that both thefields and the plasma pressure depend only on r,that is, on the distance between the given pointand the axis of the plasma cylinder.

It can be easily shown that with the above con­dition, equilibrium necessitates the identity be­tween the sum of the plasma pressure p and thepressure B~ (r)/8:rt of the longitudinal magneticfield in the plasma averaged over the cross sectionof the plasma column and the sum of the pressuresof the longitudinal field and the field of the current

77

at the boundary of the plasma column:

p +B~ (r)/8n === B~ (a)/8n + B~ (a)/8n (64)

Eq. (64) can also be written in the following form:

2na2p== J2/C2 -t- [B~ (a) - B~ (r) ]/4 (ti5)

I t can be demonstrated by detailed theoreticalanalysis that Eq. (65) is also valid when we takeinto consideration in the first approximation thetoroidal structure of the system. The resultingequilibrium condition differs only by the correctedparameter Be (a), which, under rigorous treatment,is the longitudinal field at the surface of the plasmafor cp = n/2 (see Fig. 14).

To derive the equilibrium condition for themajor radius, we have to analyze the character andmagnitude of forces which can result in variationof R. Let us restrict ourselves here to a roughqualitative analysis of these forces. There arethree various factors gi ving rise to such forces.

(1) Electrodynamic Stretching of the Ring Cur­rent. According to the fundamental laws of electro­dynamics, the ponderomotive force acting on aconductor carrying current always has such a direc­tion that it tends to increase its inductance. In ourspecial case the inductance increases with themajor radius R.

Thus, we encounter here tensile stresses. This isalmost self-evident, though. Since the pondcro­motive force is due to the interaction between thecurrent and its magnetic field it is proportional to12 for given geometric parameters.

78

(2) Stretching of the Toroidal Ring under theAction of tlte Internal Pressure. 1'0 find the tensileforce due to the above factor, we have to calculatethe work performed by the plasma pressure forinfinitesimal variation of R and to divide theresult by 6R. The total tensile force proves to beproportional to a2pe

(3) Radial Ponderomotive Force due to the Diffe­rence Betuieen the I ntensities of the LongitudinalMagnetic Field Inside and Outside the Plasma. WhenB~ (a) is higher than B~ (a), this force is directedtowards increasing R.

The evaluation of the net effect of the above forcestaking into account the equilibrium conditions forthe minor radius shows that the net effect is toincrease the radius R, that is, the pinch carryingcurrent must stretch. This means the lack of equi­librium. But one step more and we reach our goal.The tensile force can be compensated by applyingthe magnetic field B..L perpendicular to the equato­rial plane of the plasma ring (see Fig. 14). If we haveappropriately chosen the direction of B1..' it willact with the force 2nRIBl-/c on the ring carryingthe current I (referred to the total length of thering). This force can compensate the net effectof the tensile forces discussed above.

In the experimental machines in which toroidalplasma rings are generated, the tensile forces arecompensated automatically, since the vacuum cham­ber has a thick conducting (metal) body. A dis­placem ent of the centre of the cross section of theplasma ring with respect to the centre of the cross

79

section of the toroidal conduct ing chamber bodygives rise to eddy currents in the body. Eddy currentsgenerate the transverse magnetic field which com­pensates the tensile forces. It can readily be shownthat the compensating compressive force is pro­portional to [2 and the displacement o. Hence,equilibrium is reached at a certain value of o.

V. Shafranov has analyzed this equilibrium pro­blem theoretically and found the equilibrium dis­placement 6*:

O=~{ln~+(1- a~) r 8np +Li - 1j } (6H)2R a P'" B~ (a) 2

Here b is the radius of the cross section of the con­ducting body, Bq> (u) is the magnetic field generatedby the current at the boundary of the plasma ring,and L, is the internal inductance of the plasmaring per unit length. Eq. (66) determines the posi­tion of the boundary magnetic surface of the plasmaring. Since the system is toroidal, the cross sectionsof the magnetic surfaces inside the plasma ringmust not be represented by concentric circles (as isdone for the plasma cylinder). Plasma carryinglongitudinal current which is inside any of theinternal magnetic surfaces has the same equilib­rium conditions with respect to the outer plasmalayers as the conditions discussed above for theplasma ring surrounded by the conducting chamber

• A rigorous definition of 6 is the distancebetween the geometric centre of the cross sectionof the plasma ring and the geometricalcentre of the cross section of the body.

80

Fig. 15.

Displacement of magnetic surfaces in a torus.

body. Therefore, we expected the centres of thecircles representing the cross sections of the magne­tic surfaces to be displaced outwards, towards theincreasing R the larger, the smaller their radiuses(see Fig. 15). The largest displacement with respectto the centre of the cross section of the conductingbody corresponds to the magnetic axis of thesystem where the current field B(fJ vanishes.These are the main results of theoretical analysisof equilibrium of the plasma rings carrying CUf­

rent.Let us note an interesting fact. The longitudinal

magnetic field does not have any effect on Eq. (66)which determines the equilibrium position of theplasma ring in the chamber. It is as if this fieldis non-existent, I t seems that logically the nextstep is to get rid of the longitudinal magnet ic field

6-0323 81

in the ring system altogether. This is quite admis­sible in the framework of the equilibrium theory.But once we become concerned with the stabilityof the plasm a ring, the longitudinal field becomesagain indispensable.

Whon there is no longitudinal magnetic field, thehending and bottleneck-type deformations can de­velop without limitations so that in a very shorttime (of the order of alu, where Vi is the thermalvelocity of ions) the regular shape of the plasmaring will spread out and the plasma will rapidlycool down owing to intensive intoraction with thechamber walls. A. strong longitudinal magneticfield produces a sort of a rigid frame of the fieldlines which stabilizes the plasma ring. Thus, eventhough the longitudinal field is indifferent withrespect to the equilibrium conditions, it is instru­mental in making the plasma ring stable. Thispoint, clearly, has to be discussed in more detailas is done below.

Plasma is a good conductor; hence, when it tra­vels rapidly in a magnetic field and crosses itslines, induced currents appear which change thedistribution of the intensity of the magnetic field.The general effect of this change is as if the plasmain its motion carries with it the field lines whichbehave as if they were "frozen" into the plasma or"glued" to it. To get a better understanding of the"freezing" of the field lines, let us consider thefollowing special case of plasma motion.

Let a magnetic field parallel to the axis z beapplied to a volume of plasma. The field varies as

82

a given function in the plane xy perpendicularto the axis z. The plasma density is constant alongthe field lines but can vary with the coordinates xand y. Assume that the plasma travels perpendicu­lar to the field lines. Let us consider a small plasmacolumn parallel to B whose cross section in theplane xy is dS. During the motion of plasma, thisplasma column will be compressed or stretched sothat its cross section will be changed. Since wedeal here with rapid processes when diffusioncan be neglected, the total number of particlesin the column must remain constant. Hence, weobtain

~dS = const (67)

On the other hand, the magnetic flux throughthe cross section dS must remain constant, too, ifthe motion is rapid. If the plasma conductivityis high enough, even a slight variation of the mag­netic flux, if it occurs rapidly, gives rise to a highinduced current. The magnetic field of this CUf­

rent compensates variation of the magnetic flux.Conservation of the flux means that

BdS = const (68)

Comparison of Eqs. (67) and (68) shows that underthese circumstance the ratio Bin remains constantwith the motion of plasma. If the plasma is com­pressed, the magnetic field in it increases and ifplasma is expanded, the field decreases. Redistri­bution of the plasma density due to rapid deforma­tions results in appropriate variation of spatialdistribution of the magnetic field. The "freezing"

83 6*

of the field lines is just the effect of this relation­ship between deformations of plasma and varia­tions of the field which follows from conservationof the magnetic flux in a travelling volume ele­ment of the plasma.

It should be noted, however, that the ratio Binis not always constant; this is so only in the specificcase discussed above. In different geometric condi­tions the relationship between the magnetic fieldand the density in the rapidly travelling plasmacan differ from the above one. For instance, in anaxially symmetric plasma jet travelling under theeffect of electrodynamic forces due to current in theplasma, the parameter that remains constant foreach plasma element during the motion is Blnr ,where r is the distance between this element andthe axis of the system.

There is another noteworthy point. The "freezing"of field lines is not an effect which is unique toa plasma. A similar effect can be found in anygood conductor travelling in a magnetic field withhigh enough speed.

The stabilizing effect of the longitudinal field onthe plasma ring is due to the "freezing" of field linesinto the plasma. If a part of the ring gets more nar­row, the longitudinal field in this part increasesowing to conservation of its flux; hence, the magne­tic pressure in the plasma in this part also increases.

This force prevents the development of deforma­tion. Bending deformations result in elongation ofthe "frozen" lines of the longitudinal field and hencein increase of the Maxwellian stresses. This effect

84

can also prevent the development of deformations.The effect of the longitudinal magnetic field on

some types of deformations of the plasma ring isa special case of those effects which are analyzedby the general theory of stability of plasma con­figurations. Later we shall discuss some of thegeneral aspects of the theory but until that weshall sometimes note some results obtained by ap­plying the theory of stability to the specific problemswe are interested in.

9.

THE THEORY OF STABIJ--JITY

This theory can be applied, for instance, toselecting a particular technique for confinement andthermal insulation of the plasma ring using thecombination of the longi tudinal magnetic fieldand the field of the plasma current. As"predictedby the theory, the system has the highest stabilitywith respect to the most dangerous large-scaledeformations when the longitudinal magnetic fieldBa is much higher than the current field Bcp. Thestability condition can be written in the followingform:

(Be/Rep) (r/R) > 1 (69)

Here r is the distance from a given point inside thepinch to the magnetic axis at which the field Bcpvanishes. The stability condition (69) has beenderived independently [hy M. D. Kruskal andV. D. Shafranov. Condition (69) must be satisfied

85

throughout the plasma ring. If we take into accountthe character of distribution of the current densityover the cross section of the plasma ring (this distri­bution is always bell-shaped), we find that inequa­lity (69) is satisfied at any point of the pl asma if itis satisfied with a margin at the plasma boundary,that is, for r = a. Hence, the Kruskal-Shafranovcondition can be written in the following form:

(Be/B~) (aiR) > q (70)

The quantity q is known as the margin of stability.It seems that stability is good enough when q ismore than 3 unless the cnrrent distribution is veryunusual.

For a fairly long time, the experimenters in theUSSR have been studying toroidal systems, whereplasma is confined by the magnetic field of thecurrent circulating in the plasma and the plasmaring is stabilized by a very high longitudinalfield. Machines of this type are known as "Toka­maks". Fig. 16 shows schematically a tokamak-typeinstallation. The toroidal discharge chamber encir­cles the core of the transformer which inducescurrent in the plasma ring. The chamber has aninner shell made of thin-sheet stainless steel andan outer shell of thick-sheet copper. There is anelectric insulation between two shells and thespace between them is evacu ated. Before each expe­riment, long-time heating is used to degas theinner shell, the so-called "liner", inside which theplasma is generated byt.heinduced electric discharge.The outer copper shell stabilizes the plasma

86

Fig. 16.

General view of a Tokamak.

by eddy currents generated with displacement ofthe plasma ring. The longitudinal field B a is gene­rated by the magnetic coils installed on the cham­ber. In a medium-scale Tokamak (T-4), the radiusa of the cross section of the plasma ring can be ashigh as 15-18 em for R = 100 em (the maximumvalue of a is determined by the size of the diaphragminside the liner).

The experiments with the T-4 Tokamak werecarried out with the highest longitudinal field ofabout 37 kOe and the current density varied from100 to 200 A/cm2

• The duration of the currentpulse could be varied from a few milliseconds to50-60 ms, The mean plasma density in these expe-

87

riments was varied from n; ~ 1.1013 to 4.1013 cm "by changing the initial gas pressure in the chamber.The experiments with other Tokamaks were carriedout in a wider density range (from 2.1012 to1 .1014 em -3). All the principal experiments weredone with hydrogen and deuterium.

The plasma in Tokamaks is heated naturally dueto the Joule heating of the current flowing in it.Under these circumstances, the current densityin the plasma is the principal factor determiningthe heating of plasma. The Kruskal-Shafranov sta­bility condition shows that the permissible currentdensity increases as the longitudinal field. Hence,the higher Be the higher the plasma temperatureth at can be obtained.

The experimental results from Tokamaks fullysupport the validity of the Kruskal-Shafranovcondition. The results show that if the stabilitymargin q is sufficiently large (over 3 or 4) all thesigns of the large-scale instabilities disappear andthe plasma can be heated up to a high temperature.The current experiments with the T-4 machinewith a current pulse of 25-60 ms and the longitudi­nal field of 37 kOe produce the hydrogen plasmawith the following optimum parameters: n; ~~ 4.1013 cm :", T'; ~ 1.107 degrees, T, ~ 5.106

degrees. Analysis of the energy balance of the plas­ma yields energy liberated in the plasma ring owingto the Joule heating and the thermal energy fluxfrom the plasma ring. The magnitude of this fluxis a gauge of the efficiency of thermal insul ation ofplasma which is related to the magnetic field. The

R8

efficiency of thermal insulation can be characte­rized by the parameter known as the mean timeof energy retention in the plasma and is definedin the following way

'tE == WIQ (71)

Here W is the total energy stored in the plasmaring, Q is the energy leaving the plasma per unittime (lost power) and 'tE is the mean time of energyretention. The time 'tE in Tokamaks can be aslong as 15 IDS under the optimum conditions (highlongitudinal field, high plasma current, and highenough plasma density).

The mean lifetimes of fast electrons and ions inthe plasma can also be determined in experiments.In the T-4 machine, these lifetimes are more than20-30 ms under the optimum conditions. An iontravels about 10 km in the chamber during thisperiod and undergoes about a hundred Coulombcollisions. A comparison of the confinement timefor particles in Tokamaks and the time predictedfrom the diffusion theory using the expression forthe diffusion coefficient in the magnetic field Begiven above shows that the experimental confine­ment time is by a few orders of magnitude smallerthan the predicted time. A few years ago this factwas assumed to indicate the existence of "anoma­lous" diffusion due to development of a "slow"instability of plasma. This idea was supportedby the measured heat losses from the plasma ring.These losses were by a few orders of magnitudehigher than those predicted by the theory of plasma

89

heat conductivity in a magnetic field using in theexpression for the heat conductivity coefficientthe value of B equal to the longitudinal field ina Tokamak.

It would seem that the anomalous heat conduc­tivity can be attributed to the development of"slow" inst.ahilities. We cannot be quite sure,however, that high loss rates of plasma energyand particles in Tokamaks are due to any insta­bilities. This is because rigorous classical theoryof diffusion and heat conductivity shows that thediffusion and heat conductivity coefficients inTokamaks should be higher by about three ordersof magnitude than it was assumed earlier. Intui­tively, it can readily be seen from":;'qualitativeconsiderations that the formulas for 'coefficientsof diffusion and heat conductivity for the Tokamakmagnetic systems should be modified. Particlesin Tokamaks travel in a composite magnetic fieldand the longi tudinal field Be alone cannot holdthem. Hence, the current field BCfJ must play themajor part in the transport phenomena and, sinceB(p ~ Be, the transport rates should be expectedto be higher than in the uniform field B 8-

The correct quantitative theory of transportphenomena in toroidal systems was developed intwo steps. At first, B. B. Kadomtzev and V. D. Shaf­ranov took into consideration the effect of toroidaldrift on the trajectories of particles and foundthat the coefficients of diffusion and heat condu­ctivity must be increased by a factor of about 2q2 ,

where q = fBz/B qJ (a)l (aIR) if the plasma density

90

A,

Fig. 17.

Trajectory of a confined particle.

is high enough and there is no physical meaning inclassifying particles as through-going and confined.Thus, the diffusion and heat conductivity coeffi­cients under normal conditions of the experimentmust be higher by a factor of 20-30.

The next theoretical advance was done by R. Z.Sagdeev and A. A. Galeev. They noticed that atoroidal drift had an especially strong effect onthe confined particles which oscillated along thelines of force being reflected from the high-fieldregions. Fig. 17 shows such a trajectory and itsprojection on the plane in which the system's axislies and which rotates by the angle ewith the particle.The particle also oscillates along the radial directionso that the trajectories of particles in the magnetic

91

field are mixed and the rates of diffusion and heattransfer increase greatly. These rates are higherthan the earlier predictions (B. B. Kadomtzevand V. D. Shafranov) by a factor of (R/a)3/2, thatis, about 30 times under the experimental condi­tions. This effect occurs when collisions in theplasma are rare enough; therefore, we can clas­sify the particles into confined and through-goingones. In Tokamak experiments, collisions aretypically rare, hence, the coefficients of diffusionand heat conductivity must be higher by threeorders of magnitude than assumed earlier.

It is yet early to assert that the experimentalresults are in a very good agreement with the modi­fied classical theory. The current experimental andtheoretical accuracies of the values of the diffu­sion and heat conductivity coefficients are suchthat if they differ by a factor of two or three, thisdoes not necessarily mean contradiction in theresults.

The available results suggest that at least undersome experimental conditions when a plasma hasa relatively high ion temperature and a not-tao-lowdensity, the measured rates of diffusion and heattransfer are close to those predicted by the theoryof R. Z. Sagdeev and A. A. Galeev.

However, we still cannot rule out the possibilitythat the slowly developing so-called "drift" insta­bilities also contribute to the loss of particles andheat energy from a high-temperature plasma. Buteven now it is clear that the plasma temperatureand the time of energy retention increase conside-

92

rably with further increase in the magnetic fieldand the cross sect.ion radius of the plasma ring inTokamaks.

In experiments with toroidal systems, the pri­mary aim remains to be the same-to increasethe plasma parameters (temperature, pressure andlifetimes of particles) further. Only when stablehigh-temperature plasma configurations have beengenerated, we shall be able to study in earnestthe fine physical effects typical of plasma, in par­ticular, those sophisticated tunes that could bereadily played with unique resonator.

10.

PLASMA BEHAVIOURIN '"fRAPS WITH MAGNETIC MIRRORS

Plasma is confined in such systems owing toreflection of particles from high-field regions. In themagnetic mirror traps the distribution of velocitiesof plasma particles is non-isotropic.

Let the maximum intensity of the magneticfield along one of the field lines is B m ax • This valuecan be assumed to be the same at both ends of thetrap since, if the values were different at the ends,the confinement of particles would be determinedonly by the reflection of particles from the mirrorwith a lower field.

Let us consider a small segment of the field lineat which the fleld intensity is B. For a long lifetime

93

Fig. 18.

Loss cones in a trap with magnetic mirrors.

of the plasma in the trap, only those charged par­ticles, for which the angle Ow between the velocityand the field line is larger than arc sin a == 11B/Bm ax , can be within this segment. Allthe particles with smaller angles a will leave thetrap within a very short period (of the order ofl/vh where I is the trap's length and Vi is the thermalvelocity of ions). In particular, near the magneticmirrors, where B is close to B max only such particlescan be found whose longitudinal velocit y is rela­tively small and the angle a is close to 90°. Fig. 18presents the sphere of all possible directions of theparticle's velocity. The velocity vectors of theparticles which freely leave the trap lie within

94

two cones in this sphere. These cones are calledloss cones. In such a trap the plasma pressure, clear­ly, is not a scalar quantity. We have to distinguishbetween the longi tudinal pressure

-2- 2)PII =-= n (mivlli + mevlIl?

and the transverse pressure2 "2

Fs: =-= n (miV.l i +mev.l e)/2

1£ the velocity vector of a particle gets into a losscone owing to a Coulomb collision, this particlewill immediately leave the plasma. If the angleof the loss cone is large enough, practically oneCoulomb collision is sufficient to expel a confinedparticle from the trap with magnetic mirrors. There­fore, the mean lifetime of particles in such trapsmust be fairly close by the order of magnitude tothe mean time Lii between two successive ion-ioncollisions. The ion-ion collisions play here a deci­sive role, since electrons, the collisions betweenwhich are more frequent, cannot leave the trapindependently of ions. At the moment of plasmageneration in the trap, a relatively small numberof electrons will leave it so that the plasma willhave a positive charge and the flows of particlesof both signs will be balanced.

The above results show that even under optimumconditions when the loss of particles is due only toCoulomb collisions the open-ended traps withmagnetic mirrors have a much lower confinementtimes than the toroidal traps in which the mea-

95

sured confinement times are higher than t'ii by twoorders of magnitude. But an important experimen­tal advantage of the open-ended traps is the possi­bility of using a variety of techniques for generat­ing a high-temperature plasma. In open-ended traps,a plasma can be generated by injection of fastparticle flows, capture of plasma jets, high-frequencyheating of cold plasma, production of fast ions incold plasma jet by specially generating instabili­ties, etc. These traps also make it possible to varyin wide ranges the energy spectra of electrons andions and to generate a plasma with high-energyions and low-energy electrons or, on the contrary,with very fast electrons and relatively cold ions.

At an early stage of the experimental open-endedsystem studies of plasma, the only systems usedwere those with two magnetic mirrors at the ends.The magnetic field in such traps increases alongthe field lines in both directions from the centralregion where the plasma should be. At the sametime, the field decreases radially. Numerous expe­riments have shown that this simplest open-endedtrap is not suitable for long confinement of plasmawith a hot ion component. For the plasma densityover 107-108 ern -3, the lifetime of plasma in a sim­ple two-mirror trap is not more than a few tens ofmicroseconds. The plasma flows out from thetrap owing to the development of the so-calledflute instabilities. This instability is due to thefact that the plasma is a diamagnetic and thereforeit can easily travel towards a weaker field by pro­ducing "tongues" which seep between the field

96

Fig. 19~

Diagram of a magnetic systemwith a combined field.

lines (between the "tongues" there appear fluteswithout plasma).

To prevent the development of flute instability,in the simplest magnetic mirror traps such tech­niques have to be applied which by themselvesresult in considerable deterioration of thermalinsulation of plasma which leads to its cooling(low vacuum, intense low-voltage arc in the spacefilled with plasma, etc.).

In 1961 M. Ioffe was the first to experiment withan open-ended trap with the magnetic field increas­ing in all directions from the plasma region. Fig. 19illustrates the magnetic system of such a trap. Themagnetic field is produced by coaxial coils and

7-0323 97

Fig. 20.Field of linear conductors.

six linear conductors carrying current which areplaced symmetrically with respect to the centralaxis of coils. Currents flowing in the neighbouringconductors have _-,opposite directions. This systemwithout linear conductors would be a simple mir­ror trap with a radially decreasing field. The linearconductors generate the field whose geometry isshown in Fig. 20. This field varies as r2,where ris the distance to the symmetry axis. In combinedfields of the coils and linear conductors, the result­ing field intensity increases in all directions fromthe central region of the trap where a plasma isgenerated. This is known as the minimum-B con­cept.

98

The experiments with minimum-B machines haveshown that they make possible stable confinementof plasma with a density up to about 1011 em-3

and the ion energy of the order of 1 keY (T i ~

-..,. 1.107 degrees). The lifetimes of plasma particlesin such machines amount to tens of milliseconds.Thus, a technique has been found for suppressing oneof the most dangerous rapidly developing insta­bilities typical of hot plasma. Therefore, all modernopen-ended hot-plasma systems are designed pro­ceeding from the minimum-B concept.

It is yet unclear whether higher temperaturesand plasma densities in the minimum-B machinescan give rise to other types of plasma instability.Theoretical predictions give some food for thesethoughts. Analysis of plasma behaviour in open­ended machines with magnetic mirrors shows thatthe peculiar distribution of particle velocites insuch systems gives rise to various mechanismswhich generate so-called kinetic instabilities. Thekinetic instabilities are due to amplification of theplasma oscillations by directed flows of ions orelectrons. These instabilities can readily developif the velocity distribution function has a maximumat a nonzero velocity. The existence of such a maxi­mum is equivalent to the existence of a directed flowof part icles.

If the plasma particles have the Maxwell velocitydistribution function (the dotted line in Fig. 21),the oscillations are not amplified. But in an open­ended machine, the velocity distribution functionfor confined particles must have a distinct maximum

99 7*

o

Fig. 21.

Velocity distributions of particles:Maxwell distribution-dashed line;distribution in the presenceof a loss cone-solid line.

for the transverse velocity component owing tothe existence of the loss cone (see the solid linein Fig. 21). With this shape of velocity distribu­tion, there can occur amplification of ion-electronoscillations resulting in a sharp increase of theplasma loss from the trap.

The first to analyze theoretically this type ofinstability were Rosenbluth and Post; they haveshown that this instability develops when thelength of the plasma clot trapped in the machineis more than a few hundreds Larmor radiuses.This instability can, apparently, be stabilizedin a short magnetic system.

100

The theory predicts another type of instabilitywhich is due to non-uniform distribution of plasmadensity in the direction perpendicular to the field(this non-uniformity is always present merely becauseof the finite cross section of the plasma configu­ration). The drift motion of ions in non-uniformplasma can give rise to amplification of ion oscilla­tions of a frequency close to the Larmor frequency(J) tu- The oscillations are amplified if the Larmorradius of ions amounts to about a [B2/(4nnmic2)]2/3,where a is the radius of the plasma.

The experimental plasmas in open-ended Ina­chines have the parameters close to values at whichthe conditions for the development of the aboveinstabilities are satisfied. However, up till nowsuch instabilities have not been observed. There­fore, we can hope that we can attain higher tem­peratures and densities for stable plasma confinedin the minimum-B open-ended machines.

11.

srABII~.ITYOF PIJASMACONF'IGURATIONS

Even a very brief discussion of plasma physicscannot ignore the central problem of this science,namely, the stability of plasma configurations.The advances in stability studies determine theprospects for increasing the temperature and life­time of hot plasma.

101

OUf general discussion of plasma stability willlack mathematical rigour. With this in mind, letus turn to our subject. Let us assume that a con­fined plasma configuration is suspended in vacuumby a magnetic field and is at magnetohyctrodyna­mic equilibrium. In such a plasma, the temperatureand density remain constant and Eq. (52) is satis­fied at each point. Will the equilibrium state bepreserved for a long time or the random fluctuationsin the plasma will generate increasing instabilitiesand the plasma will rapidly spread out over theavailable volume? This is the generally formulatedstability problem. A significant factor in thisproblem is that different instability mechanismsproduce perturbations with widely different ratesof increase in the plasma.

The development of theory should, to a certainextent, take this fact into account. Thus, theorigin of instabilities and the techniques for theirstabilization should be considered, starting fromthe instabilities which give rise to rapid macrosco­pic motion of plasma and are, hence, the most dan­gerous. When these, the most significant instabili­ties have been analyzed, we can then analyze lessdestructive instabilities and so on. I t can appearthat if we take into consideration all the types ofinstability of plasma confined in a magnetic field,we shall find that an absolutely stable plasma con­figuration is unfeasible. But absolute stability is,maybe, not essential. In practice, we must learnhow to inhibit the development of various insta­bilities, so that non-stabilized weak and slow-deve-

102

loping instabilities would not seriously hinderaccumulation and long-time retention of thermalenergy in high-temperature plasma.

The most dangerous instabilities are those whichgive rise to motion of macroscopic plasma elementswith the velocities of the order of thermal ion velo­city. For such a rapid motion, the plasma in themagnetic field can be fully described by the modelof an ideally conducting liquid. Therefore, rapidlydeveloping large-scale instabilities are known asthe magnetohydrodynamic instabilities. They de­velop owing to the diamagnetic properties of plas­ma. The plasma tends to shift towards a decreasingfield; therefore, if the plasma is in a region wherethe field decreases in the outward direction fromthe plasma boundary, this boundary can prove tobe unstable. As discussed above, in open-endedmachrnes, we can generate the minimum-B mag­netic field which provides for the magnetohydrody­namic stability of the plasma in the machine.

For the closed traps, however, it has been estab­lished that it is impossible to generate a magneticfield increasing outwards from the plasma boun­dary at each surface point of a toroidal plasmaconfiguration. The grad I B I component normalto the plasma surface has different signs at diffe­rent points of the boundary. For instance, in Toka­rnaks the value of B decreases outwards at theoutside of the plasma ring and increases at the insi­de of the ring. Thus, a question arises whetherthe plasma can flow towards a decreasing field inthe form of individual "tongues". To answer this

103

Fig. 22.

"Tongue" on the plasma surface.

question, we have to differentiate between a high­pressure plasma for which p r-- B2/8n and a low­pressure plasma for which p <{: B2/8n. On thesurface of the high-pressure plasma, there candevelop local perturbations in the form of "ton­gues" (Fig. 22). Since the field is "frozen" into theplasma, the development of the tongue results inan increase of magnetic energy and in curving offield lines. An appropriate work will be done bythe expanding high-pressure plasma at the expenseof its thermal energy. If the tongue extends towardsa decreasing field, it will extend further so thatthe plasma boundary will be unstable. In this caseinstability has a local nature since it is determinedby the local field geometry. In toroidal traps therealways exist parts of the plasma ring where thefield geometry encourages t.he development oftongues. Hence, the high-pressure plasma is un­stable in such systems.

If p ~ B2/8'Jt as is the case in modern toroidalmachines (for instance, Tokamaks and Stellarators),

104

Fig. 23.

Development of flutes on the plasma surface.

plasma perturbations cannot distort noticeablythe shape of the field lines. The plasma just lacksan extra thermal energy to do this. Hence, smalllocal perturbations of th/~ tongue-type are stabili­zed automatically and the only, perturbations insidethe plasma or at its boundary that can occur are therearrangements of the whole field tubes and theformation of flutes (Fig. 23).

The plasma which filled the field tube formedby a thin bunch of field lines tends to expand andwill, therefore, migrate towards the side wherethe tube volume will inCrease. This volume is

\ f>Sdl, where 6S is the area of the" tube cross sectionJand dl is the elementary length of the field line.Since the magnetic flux 6<D is constant along thelength of the tube, we have

Bv= JBSBdl/B = JB<Ddlf/J = M> Jaun (72)

The flux 8(1) remains constant for all movements of

the tube. Hence, the tube: volume varies as Jdl/B.

105

Since a plasma, as any other gas, tends to increaseits volume in motion of a tube with the plasma,

the parameter u = - ~ dUB plays a part similar

to that of the potential energy. Movements ofindividual plasma elements with the exchange ofthe field tubes are known as rearrangement or con­vective deformations. When they reach the plasmaboundary with an external field, the plasma surfaceacquires a fluted structure oriented along the fieldlines. These are the "flute" instabilities' briefly men-tioned above. --

Making use of the concept of "potential energy"u (this is a purely arbitrary term), we can formu­late for-the low-pressure plasma thestability criterionwith respect to flute instabilities. The plasma boun­dary is stable if u increases along the field tube withthe plasma towards its intersection with the plasmaboundary; that is, if the following inequality issatisfied:

l) ~ dUB < 0 (73)

The variation of the integral is taken along theperpendicular to the plasma surface between twoinfinitesimally close field lines. If the inequalityis not satisfied, the boundary is unstable.[1The stability condition (73) means physicallythat the necessary and sufficient condition of stabi­lity is the increase of the averaged field along thefield line outwards from the plasma boundary.This is a modified minimum-B concept. Note that

106

according to Eq. (72) \ dUB = dvlfJrJJ. Therefore,

the quantity ~ dUB ca~ also be named the specific

volume of the field tube.To be able to use the above stability criterion,

we have, at first, to eliminate the uncertainty inthe expression for u, where no limits of integrationalong the field line are given. There is no uncer­tainty when the field lines are closed at the plasma

surface. In this special case, the integral ~ dUB,clearly, must be taken along the length of thefield line. We can introduce here the followingimproved definition of the potential energy for themagnetic surface with closed field lines:

u = -fiN ~ dUB (74)

Here N is the number of turns around the magneticsystem after which the field line is closed uponitself. This can be naturally extended to the caseof closed field lines:

u = lim lIN JdUB (75)

Here N and the length of integration tend to infi­nity. The parameter u can be shown to be a defi­nite characteristic of the magnetic surface. There­fore, the expression for u can be rewritten to includeonly the parameters of the given magnetic surfaceso that any uncertainty in the length of integrationis eliminated.

107

The volume V corresponding to a closed magneticsurface can be treated as a function of the flux Q)eof the longitudinal magnetic field passing throughits cross section: 11 = V(Qle). Using this notationwe can write the potential energy as u == -V' (<1>e),so that the stability condition can be written inthe following form:

V" ((1)8) < 0 (76)

The form (76) of the stability condit.ion is validnot only for the systems in which the field linesare closed, but also for the general case when thefield lines on the magnetic surface are infinite.

For each pl asma configuration we can plot a"magnetic well", that is, the curve u (p), where pis the distance between a given surface and themagnetic axis (at a sm all distance from the axiswe have <De ---- p2). The relative depth of the mag­netic well is

Su] 1 U o I == [u (Po) - u (0)]/ I u (0) I)

where u (Po) and u (0) are the values of u at theplasma surface and at the magnetic axis. Thehigher the parameter ~u/ 1 U o I the better the sta­bility of plasma with respect to the convectiveinstabilities. If t his parameter is negative, that is,the system has a magnetic "peak" instead of thewell, the plasma is unstable. Analysis of t.he fun­ction u (p) (or the equi.valent function V' (cDa»shows that the existence or lack of the magneticwell is determined by the arrangement of magneticsurfaces with respect to the magnetic axis.

108

(0) (b)

Fig. 24.

Displacement of magnetic surfaces inwards (a)and outwards (b)from the magnetic axis.

Fig. 24a and b illustrates two different arrange­ments of magnetic surfaces in toroidal machines:at the left diagram the cross sections of magneticsurfaces are shifted inwards from the magnetic axis,that is, towards the geometrical centre of thesystem as a whole and the displacement is thelarger the larger is p; at the second diagram thesituation is reversed and the magnetic surfaces aredisplaced outwards from the magnetic axis. In thefirst case u increases with p: hence, there is a mag­netic well and the plasma is stable. In the secondcase there is no magnetic well and the plasma isunstable. These results follow from the rigoroustheoretical analysis bu t qu al i tati vel y they are

109

almost self-evident. If the magnetic surface isdisplaced inwards, it gets into a stronger longitudi­nal field. Therefore, the flux increases faster thanthe volume and V" (<De) must be negative, whichmeans that the plasma is stable. When the magne­tic surfaces are displaced outwards, the effectmust be reversed.

Let us illustrate the above discussion by theexamples of Tokamaks and Stellarators. As mentio­ned above, in Tokamaks the magnetic surfacesare displaced inwards with respect to the magneticaxis. Therefore, the plasma ring in Tokamaks mustbe stable with respect to convective magnetohydro­dynamic instabilities and, in particular, to fluteinstabilities at the surface. However, in the existingTokamaks the relative depth of the magnetic wellis very small (of the order of a few pel cent).

Analysis of the geometry of magnetic surfacesin stellarators shows that magnetic surfaces inthem are displaced outwards. Hence, we haveV" (<De) > 0 and the plasma ring can be unstable.

12.

SHEAR STABILIZATION

The magnetohydrodynamic instabilities can besuppressed not only by means of a magnetic wellbut also by the so-called shear, that is, crossingof the field lines. I f there is the shear in a systemwhere the field line rotates around the magneticaxis, this means that the rotation angle of the field

110

lines is a function of r, that is, it varies from themagnetic axis to the periphery of the plasma ring.The rotation angle depends on the geometry of themagnetic system in the presence of a plasma ring.For instance, in Tokamaks it can readily be seen thatfor aiR ~ 1 the angle of rotation of the field linearound the magnetic axis is (per unit length)

It = Bcp/rBa

The shear exists when J.L is a function of r, that is,if df1ldr is non-zero. To compare shears in diffe­rent plasma systems we have to make use of thedimensionless parameters. In the publications onthe stability theory the usual shear parameter is

8s =-= r 2dfl ldr (77)

In the existing experimental machines the shearas is not more than a few per cent.

I t is widely believed that the shear is some uni­versal technique for suppressing a very extensiverange of plasma instabilities. Let us attempt illu­strating the shear stabilization by discussing themagnetohydrodynamic instabilities.

Assume that we have a plasma with a low pressure8np/B2 <t 1. If the shear 88 is close to zero andthere is no magnetic well, then. the convectiveinstabilities can freely travel along the radius byrearrangement of the field tubes with the plasmaand come out on the surface of the plasma ringin the form of flute instabilities. With an increas­ing shear such instabilities will be stabilized, since

111

a radial displacement of plasma in this case willresult in a strong distortion of the field. When8s =1= 0, then a deformation which had at a certainsurface the form of a protrusion coinciding withthe field line must curve at the neighbouring magne­tic surface since the field lines have rotated. Theresult must be a plasma tongue of a complicatedshape. I ts edges are as though fixed along the inter­nal field line and the radially farthest ridge is paral­lel to the external field line. The extra magneticenergy required for this distortion of the field's shapecan be obtained only at the expense of the work ofplasma pressure (remember that the field lines arefrozen into the plasma). Hence, with a finite shear,perturbations in the plasma can be stabilized ifthe ratio 8np/B2 is small enough. I t can happen,however, that with a low shear a hot plasma canbe stabilized only at a very low density.

13.

OTI-IER PLASl\1A INSTABILITIES

The above discussion dealt with the rapidlydeveloping large-scale instabilities for which plas­ma is an ideal conductor. The real plasma has afinite conductivity. Therefore, for slowly develop­ing instabilities, the field lines cannot be assumedto be frozen into the plasma and are not carriedwith it. The "thawing out" of the field lines must

112

facilitate propagation of perturbations in the plas­Ina and can make the effectiveness of the above sta­bilization Inechanisms insufficient. This extendsthe range of possible plasma instabilities. This canbe illustrated by the so-called current-convective,or "corkscrew" instability which develops in theplasma when there is a current parallel to the mag­netic field. This type of instability has the follow­ing origin. A temperature fluctuation in a plasmawith a longitudinal current changes the electricfield in the plasma owing l·o the temperature de­pendence of conductivity. Under certain condi­tions, this can give rise to a drift motion perpendi­cular to B which invol ves the entire plasma layers.If the plasma has a temperature gradient, the driftscan, in their turn, amplify the initial thermalfluctuations owing to the replacement of colderlayers by hotter ones.

The instabilities due to finite conductivity ofthe plasma are usually known as dissipative insta­bilities. The perturbations with these instabilitiesare large in scale but have relatively low rise incre­ments. The dissipative instabilities can decreasethe efficiency of plasma heaLing owing to increas­ing heat losses. But with increasing plasma tempe­rature, their significance decreases (since the pl asmaconductivity increases with temperature).

The effect of dissipati ve instabilities on energyretention in a hot plasma seems to be practicallyconfined to the electron temperature range below106 degrees. When we have passed the initial heat­ing stage and generated a plasma with a higher

8-uJ23 113

electron temperature, the dissipation processespractically cease to affect the plasma stability.

If we have managed to produce a plasma confi­guration in which there is no development of mag­netohydrodynamic instabilities including the dis­sipative ones, it still does not follow that diffusionand heat conductivity in this plasma are due onlyto pair collisions of charged particles. There isanother type of instabilities which can make theconfined plasma "chronically sick" and result inits. premature "demise" following spreading outor cooling. These are the so-called drift instabili­ties.

Before discussing the mechanism of drift insta­bilities, I would like to mention a problem of a moregeneral nature. In a plasma there can develop andpropagate a variety of oscillations and wavesin a very wide frequency range. For instance,when electrons are displaced with respect to ions,this gives rise to high-frequency electron oscillationswith an angular frequency Wo == V4nne2jm

e _Theseoscillations can propagate in the plasma as longi­tudinal electrostatic waves. There can also be genera­ted waves of a much lower frequency due to the longi­tudinal oscillations of ions -the so-called ionsound. Along the field lines of the magnetic field,in the plasma there can propagate transverse ele­ctromagnetic waves (the so-called Alfven waves)whose velocity is B/V 4nnmi and is much lowerthan the light velocity when the plasma densityis high.

114

ot---y-----+--~---+---+......

x-~tk

Fig. 25.

Interaction between a particle and a wave.

We shall not go here into the details of waveproblems but mention only one general question,namely, the interaction between waves and particleflows in the plasma. The character of this interac­tion will determine amplification or attenuation ofwaves of a given type. Assume that a longitudinalwave parallel to the axis x has been generated inthe plasma. The wave process here is the periodicoscillations of the electric potential travellingwith the velocity v j == elk, where co is the cyclicfrequency and k is the wave vector (k == 2n/A).

Let us analyze the interaction between the plasmaparticles and this running periodic potential bar­rier. Electrons or ions whose velocity componentalong the axis x greatly differs from Vj will have smalloscillations of velocity in the region of variablepotential with the conservation of the mean energy.But if a particle has the velocity component V x closeenough to Vj, the particle and the wave will exchangeenergy. To understand this energy exchange, let usmake use of the reference system which travels withthe wave. Fig. 25 shows variation of the potential

'} 115 8*

energy of the particle in this system. Let the ampli­tude of oscillations of the potential energy be Urn­

The component of the kinetic energy of the particlealong the axis x in the travelling reference systemis m (vx - CJ)/k)2/2. If. this energy is less than Urn'

the particle will be reflected by the potential bar­rier. We have to distinguish between two caseshere. In the first case we have

oo/k+ V2umlm > vx > oo/k (78)

and in the second case we have

(J)/k>vx>oo/k- V2u nJnt (79)

In the first case the particle 1 catches up withthe wave and in the second case the wave catchesup with the particle 2. Following the collision withthe potential barrier, the velocity component V x ofthe first particle will reverse its sign and will bew/k - Vx • In the laboratory reference system itwill be 200/1£ - V x • Eq. (78) shows that this velocityis less than Vx• Thus, the particle's velocity hasdecreased following the collision, so that a partof its kinetic energy has been transferred to thewave. On the contrary, the collision of the secondparticle with the potential barrier will increaseits velocity owing to the energy exchange withthe wave. This simple analysis shows that theamplitude of the longitudinal electric wave in theplasma will increase or decrease depending on themajority of the plasma particles catching up withthe wave or falling behind it. In other words, ampli­fication or attenuation of the wave will be determi-

116

( 0) (b)

Fig. 26.

Velocity distribution of particles:a - Maxwell distribution;b - distribution in the presence of a beam.

ned by the shape of the velocity distribution ofparticles for the velocity component parallel tothe wave propagation direction.

Fig. 2G presents two different velocity distribu­tions for a given direction (the first distributioncorresponds to the Maxwell velocity distribution).Here we have dt/dvx < 0 in the whole velocityrange. Hence, for any phase velocity of the waveand any amplitude of potential oscillations, therewill be more particles that fall behind than thosethat catch up with the wave. Therefore, the wavewill be attenuated. This mechanism is known asthe Landau damping. The second velocity distri­bution function has the maximum at V x == vo. Tothe left of this maximum, we have dl/dvx> O.Hence,the waves with the phase velocities in this range

117

will be amplified owing to the energy transferfrom the particles ..

Note that if the velocity distribution for thevelocity component Vx has a maximum at Vx =1= 0,this can be interpreted as a result of a directedflow of particles in the plasma. (This means thatthe distribution in Fig. 26b can be represented asa superposition of a steadily decreasing functionand an individual peak.)

It is clear that the amplification of waves inthe plasma is due just to the energy transfer fromthe particles of the directed flow. The energy trans­fer from a directed flow of electrons or ions to wavesin the plasma decelerates the flow and increases theperiodic electric field in the plasma.. Various speci­fic effects due to this physical mechanism are knownas the beam instabilities. They are encounteredin various phenomena occurring in plasma. Howe­ver, the development of such an instability doesnot mean by itself that the thermal insulation ofplasma has been disrupted and the diffusion ratehas increased. The drifts contributing to the lossof particles and energy from the plasma can deve­lop only if the electric field of the wave due to thebeam instability has a component perpendicular to B.

This is just the case for the beam inst abilitiesdiscussed below. These instabilities are, in a sense,universal, since they are due to the fact that theplasma has non-uniform density and temperaturein the direction perpendicular to the magnetic field ..Any confined plasma configuration" is not uniformin this respect for the simple reason that near the

118

(83)

(82)

plasma boundary the concentration should sharplydecrease and the temperature should also drop.

Let us consider the simplest plasma configura­tion -a long cylindrical column of a low-pressureplasma in the longitudinal magnetic field. We shallassume the magnetic field to be uniform. The equi­librium equation yields

j J- == clB I grad p I (80)

where j 1- is the density of diamagnetic current inthe plasma. In the case of cylindrical symmetry,Eq. (80) can be written as

i , = clB It I (81)

Here the current will be perpendicular to the pres­sure gradient and the vector B, that is, it will flowin the azimuthal direction. Since the diamagneticcurrent is due to the Larmor motion of electronsand ions, we can write

j 1. ~ne (u, - u.)

where u, and u, are the averaged drift velocities ofelectrons and ions. The drift velocities have alsothe azimuthal direction. We have here

u· - .s.. I~ I· u - .s: I dPe I~ ---- neB dr ' e - n,~B dr

" If electrons and ions have close temperatures, sothat we can Introducafhe general plasma. tempera­ture T~ we can write-" -

ckT 11 'dp I' , . ckT_u··~u ~---- =--

z e eB p dr eaB

119

Here a is the characteristic linear size of pressurenon-uniformity in the plasma:

a-1 = _.!~p dr

If the concentration drops from the axis to theboundary of the plasma column more sharply thanfloes the temperature, we have

-1 1 dna ~ --ndr

Iff.he temperat.ure drop is more sharp, we obtain

-t 1 dTa ~ -1'([;

Analysis results suggest that when in the plasmathere develop periodic perturbations of the wavecharacter which propagate at a small angle to theazimuth, the phase velocity of this wave wlkcp isckTleaB and such waves are known as the driftwaves. Since the drift azimuthal velocities ofparticles are close to the phase velocity of wavespropagating in the same direction, the particlesshould be expected to interact strongly with eachother so that conditions can be established forrapid amplification of oscillations. Increase in theamplitude of oscillations results in developmentof strong azimuthal electric fields, therefore, whenthe plasma has no uniform density and temperature,there must develop radial flows of heat and particleswhich shorten the lifetime of a confined plasmaconfiguration. That is the effect of drift instabili­ties.

120

Detailed theoretical analysis shows that the ampli­fication of drift waves due to non-uniform plasmadensity is effectively inhibited by a high enoughshear. The drift instability due to the temperaturegradient is highly dangerous. Instabilities of thistype are known as drift temperature instabilities.Such instabilities develop when d (InT)/d (In n) > 1and considerably increase the heat losses from theplasma. Though an additional heat loss due tothe drift temperature instability in closed toroidalsystems can be sharply reduced by shear, it stillcan be higher than the heat losses due to the clas­sical mechanism of pair collisions. However, thoughthe availab le theory can predict the conditionsfor development of the drift temperature insta­bility, the anomalous heat conductivity coefficientcan still be calculated only to an order of magnitudeaccuracy. Very roughly, the anomalous heat con­rluctivity coefficient in Tokamaks can be estimatedto be 10-20 times as high as the coefficients predictedby the classical t.heory taking into account the roleplayed by the confined particles. The experimen­tally measured plasma heat losses lie somewherebetween the classical results and the calculatedanomalous heat losses. Therefore, the drift instabi­lity theory cannot be said to be well substantiatedby experiments. This inconsistency of the theoryand experiment reflects the current state of researchinto plasma instabilities. The theoretical classi­fication of various instability types in the plasm ahas been in the main completed. The mechanismsof instabilities have been analyzed and the condi-

121

tions have been found under which small perturba­tions for a given instability decrease or increase.However, the linear-approximation analysis ofsuch processes is only the first stage of the theory.

The next, much more difficult stage is to determinethe coefficients of diffusion and heat conductivityin plasma with developing instability. Such a plas­ma is typically referred to as the turbulent plasma.There has been made only a preliminary analysis ofdiffusion and heat conductivity in the turbulentplasma yielding primarily qualitative results whichare difficult to compare with experimental results.The only. processes that are comparatively simpleto analyze are transport phenomena under theeffect of magnetohydrodynamic instabili ties. Inthis case large regions of plasma must move withthe velocities of the order of the thermal velocity ofions. If the magnetohydrodynamic instabilitiesare eliminated, all the processes in the plasma aresmoothened down. Only drift-type instabilitiesremain, they give rise to plasma turbulence andincrease the rates of the transport processes. Tomake, at least, rough estimates of the rates of diffu­sion and heat transfer daring assumptions haveto be' made, for instance, the hypothesis' of the"relay" transfer of perturbations in the plasma fromone magnetic surface with closed field lines toanother magnetic surface of the same type. Howe­ver, at the very best; such an approach .yields onlyqualit.ative results, .

On the other hand, we should note that. we' stillcannot make accurate enough measurements of ~..he

122

diffusion and heat conductivity coefficients inexperiments. To find these parameters we mustknow the distributions of density and temperatureover the cross section of the plasma and to comparethe measurements with the predicted values, wehave to know in detail the geometry of the mag­netic field in the plasma. The available experimen­tal techniques are largely unsuitable for that. The­refore, we have to be content with determiningsuch general characteristics as the mean lifetimesof particles in the plasma and the mean time ofenergy retention in the plasma. These characteristicsmust be compared with the predicted values proceed­ing from fairly arbitrary assumptions about distribu­tion of density and temperature in the plasma. Sothe reader can see that the physics of plasma is stilla rather crude science and it is only the variety ofsophisticated equations derived by theoreticiansthat to a cert.ain extent can mask its immaturity.

123

BIBLI OG·ItAPI-IY

1. Artsimovich, L. A. "Closed plasma configurations", lectu­res delivered in France, 1967.

2. Artsimovich, L. A., Mirnov, S. V., Strelkov, V. S.Atomnaua energiija, 17, 170, 1964.

3. Afrosimov, V. V., Petrov ]\1. P. Zhurn, eksper, i teor. liz.,37, 1995, 1967.

4. Galeev, A. A., Sagdeev, R. Z. Zh.urn, ek sper, i teor,fiz., 53, 348, 1967.

5. K ad OIn tzev, B. B., Pogu tze, O. P. Zhurn. eksper . i teor,fiz., 53, 2025, 1967.

6. Shairanov, V. D. A tomruuja energuja, 13, 521, 1962.7. Shcheglov, D. A. Zhurn . ek sper, t teor., fiz: pisma, 6,

949, 1967.

124

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