P/2214 USSR

Dynamics of a Rarefied Plasma in a Magnetic Field

By R. S. Sagdeyev, B. B. Kadomtsev, L. I. Rudakov and A. A. Vedyonov

The nature of the motion and properties of a hightemperature plasma in a magnetic field is of particularinterest for the problem of producing controlledthermonuclear reactions. The most general theore-tical approach to such problems lies in the descriptionof the plasma by the Boltzmann and Maxwell equa-tions that connect the self-consistent electric andmagnetic fields with the ion and electron distributionfunctions.

If the mean free path of the particles is appreciablylarger than the characteristic dimensions of thesystem and if the time between collisions is greaterthan the appropriate characteristic time, then thecollision term can be neglected in the kinetic equationor considered to be a small correction.

In the absence of collisions the plasma particlesinteract through long-range electromagnetic forcesand the particle trajectories can be described by anequation of motion with self-consistent electric andmagnetic fields. This is expressed mathematicallyby the fact that in the absence of collisions the solu-tion to the kinetic equation

df/dt + v.V/+ {e/M){E+c-1vxB)-df/dv = 0 (1)

can be represented by an arbitrary function of thefirst integrals of the characteristic equations for themotion of the individual particles

di = dx/v - dv/(e/M){E + r ^ x B ) .

The exact equations for the motion of a plasma inan electromagnetic field can only be solved in certainsimple cases especially because the fields are influencedby the collective motion of all the particles. For acertain class of problems it is possible to work out aprocedure for decreasing the number of variables andthus simplify the characteristic equations.

The physical system can be characterized by "fast"and "slow" variables. If we are investigating changesin the system over a long period of time, we caneliminate the fast variables and describe the state ofthe system in terms of the slow variables. The method

of eliminating the fast variables from the kineticequation was developed by Beliaev.*

We shall eliminate the "fast" variables for the parti-cular case when the electric and magnetic fieldschange slowly in time and space; i.e., the conditionsRB/L^I and l/coT<Cl are satisfied, where RB isthe Larmor radius, œ is the Larmor particle frequencyand L, T are the characteristic size and characteristictime of interest. The net particle motion consists ofthe "fast" Larmor rotation and the "slow" drift of theguiding centers.

The procedure for excluding the Larmor phase;i.e., the "fast" variable, from the equation of motion bymeans of a transformation to new " drift " variables(v±,vlh R, a) was worked out by Bogoliubov andZubarev. г Omitting the algebra, we shall presentthe final result, limiting ourselves to the zero andfirst order approximations in RB/L and 1/OJT:

dvjdt = {vJ2œ)dco/dt

dvjdt = — e0 . [ - (e/M)E

+ (v±2j2œ)Vco +

dR/dt - eovu + (l/w)

(2)

+ dw/dt + vude0/df] (3)

- (e/M)Edvr/dt + vnde0/dt], (4)

±where

da/dt E= - w, eo~ B/B, w = cE X B/B2,

- ^ - = ^ - + v,,(€o.V) + (w.V).at ct

(5)

Since the characteristic equations are known, wecan easily write the kinetic equation for the distribu-tion function f(vu, v±> R, o¿, t) in terms of the driftvariables :

df/dt + (dR/dt).Vf + {dvjdt)df/dvu

+ {dvJdt)dfjdvL + {da/dt)df/da = 0. (6)

By expanding the resulting kinetic equation inpowers of CUT1, or more precisely, in powers of RB/Land 1/coT; we see that /0 in / = /0 -f- f± +..., doesnot depend on oc and one obtains an equation whichcontains only the "slow" variables:

Original language: Russian.

* Ed. note: S. T. Beliaev, Kinetic Equation for a Rarefied Gasin Strong Fields] AN SSSR, Plasma Physics and Problems ofControlled Thermonuclear Reactions, 3, 50, Moscow (1958).

151

152 SESSION A-5 P/2214 R. S. SAGDEYEV et al.

dfldt + Vnfo-Vf) + (w.V/)+ e 0 . [{e/M)E — (v^l2w)Va> — dw/dt]df/dvn

+ (vJ2<o){da)ldt)8fl8vJL = 0. (7)

If rare collisions are taken into account, a smallterm on the right-hand side of the kinetic equationappears.

The electromagnetic fields and the distributionfunctions are connected through Maxwell's equations.It is easy to show that to first order in cor1 the velocityof the centre of gravity of an elementary volume ofplasma, пЧ = J/(v, r, t)vdv, does not coincide withthe mean drift velocity, пч =and is given by

nV = J ( — Vx(V/2w) e0

With this expression, Maxwell's equations can bewritten

V.E = 4rre J (/</ — f^)vLdvLdv^dat (8)

V.B = 0 (9)

V x E - - (l/c)8B/8t,

V x B = (4тте/с) J [ - VX

+ dR/df] (/oi - foe

(10)

. (11)

It may be seen from this equation that the plasmapossesses diamagnetic properties.

The set of Eqs. (6)—(11) completely describes theplasma when collisions are neglected. However, itshould be stressed that this system of equationsfollow from the first non-vanishing approximation tothe exact system of plasma equations when thisexact system is expanded in terms of a small para-meter, ay-1. This is the so-called drift approximation.

EQUATIONS FOR THE MACROSCOPIC MOTION

OF THE PLASMA

Strictly speaking, conventional hydrodynamic con-siderations are valid when the plasma has a highdensity, low temperature and a corresponding collisionmean free path which is much less than the character-istic dimensions of the plasma volume. However, Chew,Goldberger and Low 2 and also Watson and Brueckner 3

have shown that it is possible to obtain a closedsystem of equations for the first few moments of thedistribution function. These equations are formallyanalogous to the usual system of magnetohydrodyna-mic equations with a non-isotopic pressure tensorwhen collisions are neglected. These authors ob-tained the moment equations by expanding the kineticequation in powers of со"1 = eH/Mc or in powers ofthe mass to charge ratio, M/e.

The similarity in the results obtained for thesetwo apparently opposite cases is explained by thefact that the magnetic field deflects the particles andsymmetrizes their velocity distribution in the planeperpendicular to the magnetic field. In this sense,the effect of the magnetic field is analogous to theeffect of collision.

The gas-dynamic equations for the ion and electrongases can also be deduced from the kinetic equationin the drift approximation,5 as will be shown below.

The moments of the distribution function are definedby the following integrals

n = Bjfd*v

nu = В I V\\fdsv

Ji - MBJ (vL*/2B)fd*v

p = MB Г (vn — uYfcPv

Q = MB Г (vn — u)*fd?v,

where d4 = dfidv^doc and /x == vL

2/2B. Note thatBd3v is the true volume element in velocity space.

If we assume that there is no heat flux Q along themagnetic field then we can multiply the equation forthe drift motion (4) and the kinetic equation (5)by various powers of vn and v± and integrate over thevelocities. This procedure leads to a closed system ofequations which describe the motion of the ioniccomponent of the plasma

8n\8t + V • n(eou + u) = 0 (12)

Mdu/dt = — {oj/n){e0 . Vifi/co))

+ €0 . [eE - (JL/n)VB - Mdw/df] (13)

d{fiB*¡n*)ldt = 0 (14)

d(jl¡n)ldt = 0 (15)

dR/dt = ue0 + (l/w)€o • [ — ( # ) E + udejdt

+ dwfdt + (p/nM)(eo.V)eo + (/iVB)/»]. (16)

The equations for the electron component differfrom these equations only in sign of the charge andin the value of the mass. It should also be notedthat the equations of motion (13) and (16) can bewritten in terms of the an isotropic pressure tensor,

//IB 0 0 \pii=[ 0 iiB 0 1 .

V о о p)The relations which govern the pressure variation

are given by Eqs. (14) and (15), which correspond tothe adiabatic law with у = 3 and 2 for the longitu-dinal and transverse motions, respectively. Thetransverse adiabatic invariant is the ion (or electron)magnetic moment. Maxwell's equations (8)—(11)can also be rewritten by introducing the moments ofthe distribution function

V.E — 4ле{щ — ne) (Sa)

V.B = 0 (9a)

V x E = - {l/c)8B/8t (10a)

V x B = (4ne¡c) { ~ с V X (ju + jüe)€0/e

+ fiiV* — neve}. (lia)

In order to describe the motion along the lines offorce of the magnetic field by means of these equa-tions, a special condition must be satisfied; namely,there must be no heat flux along the magnetic field

DYNAMICS OF A RAREFIED PLASMA 153

lines of force. If this condition is violated then theequation d(pB2/nB)/dt = 0 is not valid, and the wholesystem of equations of motion (12)—(16) are inap-plicable. In these cases a rigorous treatment hasto be carried out on the basis of the more complicatedsystem of Eqs. (7)—(11).

HYDRODYNAMICS OF A LOW PRESSURE PLASMA

The problem of determining the equilibrium stateand stability of a magnetically confined plasma isperhaps one of the most interesting and importantproblems in plasma physics. If the magnetic fieldis very strong this problem can be attacked by usingthe magnetohydrodynamic equations with a non-isotropic pressure tensor. However, if the plasma isconfined for a time greater than the collision timethen the longitudinal and transverse pressures will beequalized. The pressure can then be regarded asisotropic in calculating the equilibrium state of theplasma. The anisotropy of the pressure should betaken into consideration only while investigatingthe stability for increments of time shorter than thecollision time. However, it can be shown 4 that theanisotropy of the perturbation pressure for an isotropicequilibrium pressure can only improve the conditionsfor stability. Therefore, when considering the sta-bility of an isotropic plasma, one can use the ordinaryhydrodynamic equations: the stability conditionsobtained in this way are somewhat over-stringent.

We shall now consider the hydrodynamic descriptionof a plasma when its pressure is appreciably lowerthan the magnetic pressure, i.e., ft = 8irp/B2 <C 1.In this case the magnetohydrodynamic equations canbe simplified. Indeed, in such cases, the magneto-hydrodynamic wave velocity с А = (В2/4ттр)1//2 ismuch greater than the sound velocity cs = (8p/p)1/2.Therefore if we do not specifically consider the hydro-dynamic waves we may take с A to be infinite. Thismeans that any perturbation of the plasma leadingto a transverse displacement of the line of force ispropagated instantaneously along the line of force andall the plasma along the line starts to move at once.The equation for this motion may be obtained byexpanding the hydrodynamic equations in a powerseries in /3. Let us assume that p = pQ + ер± + ••-,P = Po + ePl + ..., v = v0 + 6Vi + - , В = e-K {BQ

+ eB-L + ...), substitute these expansions into themagnetohydrodynamic equations, compare coefficientsof the same powers of e and then put e = 1. Thefollowing system of equations is obtained by thisprocedure :

[VxB0]xB0 =

= Vx[voxBo]

= (1/477) ([VxBJxBo +

dpjdt + V-/>oVo = 0

(17)

(18)

(19)

(20)

= 0. (21)

By retaining only the zero-order terms we eliminatethe other equations and can drop the subscript zero.If we do not consider force-free fields we obtain fromEq. (17)

V x B 0 - 0

V - B 0 = 0; i.e., В = I1/¿V^,i

where the Ji are the external currents and the ф{are the corresponding scalar potentials which, gener-ally speaking, are not single valued.

For simplicity we shall suppose that the currentsdo not change with time and that the conductors areimmobile. Then d"B/dt = 0 and from Eq. (18) weobtain

v = eovl{ + eoxV(f>/B,

where vu is arbitrary an function of r, t and ф is afunction of r and t which satisfies the condition€0 • V ф = 0. Inserting this into Eq. (19) we mustconsider Eq. (19) to be an equation defining Bv

Furthermore, as is well known, the condition for theexistence of a solution must be fulfilled : the left-handside of Eq. (19) must be orthogonal to the solutionsof the homogeneous equation conjugate to the equationLBi = [ V X B x ] x B 0 = 0. This condition leads to

pdvu I dt + Pe0. (v . V)v + e 0 . Vp = 0 (22)

J V - [(р/В2)Ч(дф/Щ ~ (p/B) €0x (v . V)v

- (l/B)eoxVp]dl/B, (23)

where the integral is taken along any line of force.Together with Eqs. (20) and (21), these equationsdescribe the behaviour of a low pressure plasma inthe zeroth approximation. It should be noted thatthis approximation appears to be inadequate if themagnetic lines of force are not closed but cover thewhole surface. Indeed, in this case, ф must beconstant on the surface, whereas v retains only twodegrees of freedom.

If we put v = 0 in Eqs. Í22) and (23) then conditionsfor equilibrium are obtained which can be transformedinto the form

= 0; U=-jdl/B, (24)

i.e., the pressure p must be constant on the surfaceU = constant.

Equations (2) through (23) can be used for investi-gating stability. We shall put p = p0 + p', p = p0

+ p'} v = dt\ldtf Y) = Г1\+Г1^=7]\1ео + [еохЧф]/В,where p0, p0 are the equilibrium pressure anddensity and p', p', r\ are small quantities. Substitutingthis into Eqs. (20)-(23) and neglecting higher orderquantities, we obtain linearized equations whichdescribe small deviations from equilibrium. It ispossible to show that these equations can be obtainedfrom the variational principle S /L dt = 0, whereL = T — V, T = ^ Çpo(dri/dt)2dic is the kinetic energyand V is the potential energy of the plasma. If weintroduce curvilinear coordinates (£lt f 2, f3) so thatthe lines | 3 coincide with the (closed) lines of magneticforce, then V can be written as follows

154 SESSION A-5 P/2214 R.S. SAGDEYEV et al.

(25)

where g33 is a component of the metric tensor andg= d et gilc. For plasma stability it is necessary andsufficient that the potential energy V be positivedefinite; i.e., its minimum value be positive. Mini-mizing Eq. (25) with respect to rju we obtain

y .* mm — _ _ _

(26)

which shows that a necessary and sufficient conditionfor the stability of a low pressure plasma is t

Vpo^VU + {ypo/U)(VU)2<O. (27)

This condition is a hydrodynamic analogue to the sta-bility criterion obtained from the kinetic equation byTserkovnikov 5 for a plasma in a magnetic field witha cylindrical symmetry. It is also similar to thecondition for the convective stability of a nonuniformlyheated gas in a gravitational field, where the conditionthat the drop in the gas temperature with heightshould be within a certain limit is analogous to thecondition that the plasma pressure must not rise toofast with increasing U.

If the plasma has a sharp boundary the first termin Eq. (27) is much larger than the second term and,consequently, the condition for stability is expressed

by Vpo'XfU < 0. This condition was obtainedearlier by Rosenbluth and Longmire 6 from an energyprinciple.

INSTABILITY OF A PLASMA IN A MAGNETICFIELD WITH AN ANISOTROPIC ION

VELOCITY DISTRIBUTION

If the equilibrium pressure is nonisotropic anotherkind of instability appears in the plasma. In orderto investigate this type of instability in its pure formwe shall consider a homogeneous plasma that isstable when the pressure is isotropic. This case canagain be analyzed with the nonisotropic magneto-hydrodynamic equations, but a more exact analysiswould involve the use of the kinetic equations. Bothcalculations lead to qualitatively consistent criteriafor stability but the quantitative results are somewhatdifferent. The situation here is quite analogous tothe Langmuire plasma oscillation problem.

Let us first make the assumptions that we canneglect the particle collisions, that the wavelength ofthe perturbation is much larger than the mean ionLarmor radius and that the interval of time is muchless than the Larmor frequency. Under these condi-tions, the drift approximation (Eqs. (4)—(11)) canbe used.

The unperturbed distribution function is chosen tobe of the form

/o K2> V ) = - {{M/2){ví*¡Tí

On linearizing the initial equations for a small perturbation and performing a Fourier coordinatetransformation and a Laplace time transformation, it is easy to obtain the characteristic equation forP = P(ky,kz):

м- i ] -

Ц (28)

where the z axis is in the direction of the unperturbedfield Bo and j8±, /3,, are defined by

j3 ± ^

The perturbations increase exponentially with time if

PiWJTu) - 1] > (1 + j81 - A , ) № ) 2 + 1. (29)

In the limiting case Bo -> 0, the instability conditionis TL > Г,, and for ky = 0 the characteristicequation is

[pjhY = ( i y w 0 M ) ( i + p± - pM). (30)

This corresponds to the Alfvén magnetohydro-dynamic branch. When j8M > 1 + j8±, the plasma isunstable and, when Bo -> 0, this instability conditionreduces to T\\ > T±.

t For a magnetic field with axial symmetry this conditionhas also been obtained by a slightly different method.4

We shall now discuss the physical interpretationof the instabilities which arise from perturbations ofa plasma with a nonisotropic temperature. Considera density perturbation in an otherwise homogeneousplasma. At points where the density is high themagnetic field is decreased because of the diamagne-tism of the aniso tropic plasma. There also appearsa gradient in the magnetic field and a curvature ofthe magnetic field lines as a result of the perturbation.If the curvature is small (kz <C ky) and TL > Г„,then the main force acting on the perturbed plasmawill be -— jitV»B. This force tends to increase theamplitude of the perturbation. If TL < Гц, thenthe force — \x V*B is compensated by the longitudinalpressure and the amplitude of the density perturbationdoes not continue to grow.

If the curvature of the perturbed force lines islarge so that kz > ky (ky is negligible), then forTu> TL there are large uncompensated centrifugalforces — fiR/R2, where R is the radius of curvatureof the perturbed line of force. It is not difficult to

DYNAMICS OF A RAREFIED PLASMA 155

see that these forces tend to increase the amplitudeof the initial perturbation.

All the calculations described heretofore are basedon linearized equations and, naturally, they give noinformation about the limiting amplitude of theundamped oscillations of an anisotropic plasma.

We have carried out the analysis to second orderapproximation for a number of simple limiting casesand have shown that the appearance of the instabilitydescribed above leads to a transfer of kinetic energyfrom the transverse to the longitudinal motionwhen TL > Tn, and in opposite direction if TL < Tu.Therefore, it is reasonable to assume that the insta-bility tends to develop until the longitudinal andtransverse energies are equalized; i.e., until thetransverse and longitudinal temperatures are ap-proximately equal.

STABILITY OF A PINCHED CYLINDRICALPLASMA USING THE KINETIC EQUATION

In this section the problem of the stability of ahigh-temperature, pinched, cylindrical plasma istreated. We shall assume that all the conditions forthe validity of the drift approximation are satisfied.For simplicity we shall neglect the electron tempera-ture in comparison to the ion temperature. With thisassumption, we can use the starting equations of theprevious section.

We shall consider that in the equilibrium state, theplasma cylinder is uniform along the z axis and alongthe azimuth ф. Outside the cylinder r = a the non-zero components of the field are H<peW (r) = 21/re, HzeM= constant. Inside the cylinder, we have Вф^°) = 0and Дгг(0) = Bo = constant. The indices i, e relate tothe fields internal and external to the cylinder, respec-tively. The ion distribution function /0 is defined at thebeginning of the previous section. We neglect the electricfield EZ) assuming that it is eliminated by the motionof the electrons along the z axis. This is true whenthe condition d/dt <C wpe is satisfied, where cope is theLangmuir electron frequency.

On the surface of the plasma cylinder the followingcondition should be satisfied:

where

The motion of the boundary obeys the drift equationVdr = cE X B/B2. The perturbed quantities are taken tobe of the form F(r, ф, z, t) ~ F(r)expi(kz + тф + œt).One then obtains the following dispersion relationin the usual way:

1 -(tn + khe)

2

alm{a)

+ p± + (kn¡B0*)

Шш-^ЩК^к) + m

, (31)

where Im and Km are Bessel functions of animaginary argument. The parameter a is determinedby the relation

ft - A,) -

where

d*v '

(32)

эпаЫ=В0/Нфе(°)(а),

The form of this equation has much in commonwith the dispersion equation obtained in the magneto-hydrodynamic approximation by Shafranov.7

We shall now consider a number of particular cases.1. TL = Т„: In this case the instability criterion

coincides precisely with the criterion obtained byShafranov for у = 2. However, the dependence ofthe perturbed quantities on the wave vector willdiffer from the dependence obtained from the hydro-dynamic approximation.

2. 1 = 0, НфеЮ =0, TL^TU: The magneto-hydrodynamic treatment did not show any instabilityunder these conditions. In our treatment, however, itis possible to have an instability in a plasma cylinderconfined by a longitudinal magnetic field when thereis no axial current. The instability criterion can bewritten explicitly when the perturbations along theaxis are greater than the radius of the magneticallyconfined plasma. The plasma column will be unstableif

- Ги0. (33)

This condition can be fulfilled only if T± < Tu.

In the limiting case of short wavelength perturba-tions, an instability may occur with TL > T,,. Thecriterion for the existence of such an instability isgiven by

[ a ( w = 0)]2 < 0. (34)

3. The vibrational branch: Because of the thermalmotion there is a damping of magnetosonic wavesin the cylinder analogous to Landau damping.For small k, when the damping is small, the imaginarypart of со is

Im8ТТП0Т ( He

X- {M/2T)(œ/k)2

2(w + k)(2m-\-(35)

The instability associated with temperature an-isotropies could be observed in experiments on so-called adiabatic heating when only the transversetemperature decreases with decreasing magnetic fieldand the longitudinal temperature remains constant.This situation might arise at high temperatures wherecollisions can be neglected.

In experiments on the fast compression of a plasmacolumn the compression time can be less than themean time between ion collisions. This automaticallyleads to a temperature anisotropy that can make thecondition for stability more rigid.

156 SESSION A-5 P/2214 R. S. SAGDEYEV et ai.

NON-LINEAR ONE-DIMENSIONAL MOTIONOF A RAREFIED PLASMA

We shall now investigate the motion of a rarefiedplasma far from an equilibrium state We shallrestrict ourselves to the case when all the variableswill depend only on one space coordinate x per-pendicular to a magnetic field directed along the zaxis В = (0, 0, B(x)) The system of hydrodynamicequations (12)-(16) are valid subject to the conditionsthat the space and time gradients are small, i e , theLarmor radius and period are small compared to theappropriate linear dimensions and time Howevergenerally speaking, these conditions for the arbitrarymotion of a plasma will not be fulfilled after a certaintime

Indeed, the formal similarity between the systemof Eqs (12)-(16) and the ordinary gas-dynamicequations with у = 2 enables us m the case of ararefied plasma m a magnetic field to immediatelygeneralize Riemann's solution corresponding to theformation of a shock wave Steep velocity andtemperature gradients m an ordinary shock frontlead to a considerable increase of the entropy due toviscosity and heat conduction However, m a rare-fied plasma the formation of discontinuities oughtto take place m a different manner since collisions areof no importance

A hypothesis concerning the possible existence of ashock wave was advanced in an earlier paper 8 wherethe non-conservation of the ion magnetic moment in afield with large gradients was considered as a mecha-nism leading to non-adiabaticity

The rigorous treatment of this problem is verycomplicated since the motion even of a single chargedparticle m an arbitrary nonumform field cannot bedetermined without the aid of a computer

The simplest idealization of the processes m sidethe wavefront would be to consider the change m theLarmor energy across a magnetic discontinuity Itappears, however, that the non-adiabatic increase ofthe Larmor energy of the individual ions does notimply that the random energy is also non-adiabaticallyincreased since there is a grouping of the Larmorphases of the ions

This grouping of phases leads to a local oscillationof the ions In a self-consistent problem this mightalso give rise to oscillations Damping can in principletake place m the absence of collisions, e g , due to theLandau mechanism or on account of phase mixing ina nonumform magnetic field, and the damping lengthwill determine the width of the shock wave Withthe curvature of the velocity profile increasing, thecondition T ^ of1 is violated before the conditionL > RB since T~L [{В2/4ттР) + wa]-% Thereforethe wavelength A of the oscillation may be expectedto be of the order A ~ [(Б2/4тг/>) + u^œ^r1 For

a low pressure plasma this becomes 1tupi] = CA/CÜPI, where eoPi

2 = 4ттпе2/М However,it may be shown by using the exact equations for theion and electron motion that for vanishing plasmapressuie, the assumption of plasma quasi-neutrahtyand electron adiabaticity is sufficient to ensure theapplicability of Riemann's solution, instead of thecondition a>T ^> 1 (the condition L >> RB IS thenfulfilled identically since pL = 0)

In ordinary gas dynamics the only type of stationarymotion is a stationary shock wave In a rarefiedplasma there exists still another solution which is atravelling magnetosonic wave with a finite amplitude

If the thermal motion is neglected the equationconnecting the wave velocity with the amplitude of themagnetic field has the form и2 = (Вг + £0)

2/16тгр0

where the field amplitude Bx of the wave cannotexceed 3B0

The correction to the velocity due to thermalmotion is i

2 V - У - 2«o(V -У(36)

where

This equation shows that the transition to smallamplitude waves leads to absurd results since thedenominator vanishes This fact is probably asso-ciated with the existence of a damping m the linearapproximation analogous to Landau damping

ACKNOWLEDGEMENT

The authors wish to thank Prof M A Leontovichfor the valuable advice and continuous attention hegave to this work

REFERENCES1 N N Bogolyubov and D N Zubarev Ukramsky mate

matichesky zhurnal VII 5 (1955)2 G Chew M Goldberger and F Low Proc Roy Soc

A 236 112 (1956)3 К Watson and К Brueckner Phys Rev 702 12 (1956)4 I Bernstein E Fneman M Kruskal and R Kulsrud

Proc Roy Soc A 224 17 (1958)5 Y A Tserkovmkov Zhur Eksptl i Teoret Fiz 32 67

(1957)6 M Rosenbluth and С Longmire Ann Phys / 120

(1957)7 V D Shafranov Atomnaya Energiya No 5 38 (1956)8 R S Sagdeyev and A A Vedenov Mechanism, of Injec

tion in the Atmosphere of Stars Address at the Vlth AllUnion Conference on Cosmogony (June 1957)

* Ed note In another version of this paper the term3wo/4 is written 3uJ2