stability of plasma in static equilibrium

P/365 USA

Stability of Plasma in Static Equilibrium

By M. D. Krusiial and С R. Oberman *

Our purpose is t ^ derive from the Boltzmannequation in the small mje limit x, criteria useful in thediscussion of stability of plasmas in static equili-brium. At first we ignore collisions but later showtheir effects may be taken into account. Our approachyields a generalization of the usual energy princi-ples z> 4>5 for investigating the stability of hydromagne-tic systems to situations where the effect of heat flowalong magnetic lines is not negligible, and hence tosituations where the strictly hydrodynamic approachis inapplicable.

In the first two sections we characterize our generalmethod of approach and delineate the properties ofthe small mje limit which we use to determine theconstants of the motion and the condition for staticequilibrium. In the next two sections we calculatethe first and second variations of the energy andconclude with a statement of the general stabilitycriterion. In the final three sections we state severaltheorems which relate our stability criterion to thoseof ordinary hydromagnetic theory,5 we show howto take into account the effect of collisions, andbriefly discuss the remaining problem of incorporatingthe charge neutrality condition into the presentstability theory.

GENERAL METHOD

Our method consists of writing down the energyof the system to second order in the perturbationfields of / and B, where / is the distribution functionin x,v space and В is the magnetic field intensity.We eliminate the terms involving the second orderperturbation / by employing certain constraints,namely, that certain constants of the motion have theirequilibrium values. The constants of the motion weemploy are time-independent and are functionals of /and В which are regular at, and permit expansion about,their equilibrium values.

The resulting expression for the energy is a quadraticform in / and Ç jointly whose positive-definitenessprovides a sufficient condition for stability. (Moreabout Ç later, let it suffice for now to say % describes

the displacement of magnetic lines of force awayfrom their equilibrium positions.) We rid ourselves ofthe dependence on / by minimizing the energy withrespect to it, subject to the constraint that all con-stants of the forementioned type have their equili-brium values. We then have a sufficient conditionfor stability involving \ alone.

Generally, the constants of the motion of the typewe employ do not specify the motion completely sothat there exist many motions evolving from the sameequilibrium (at t = —сю). By restricting the con-stants of motion to their equilibrium values, the onlypossible motions other than the equilibrium behaviorare instabilities (the pure modes of which have anexponential time behavior and hence vanish att= —oo).

To illustrate the method we consider the simplest ofexamples, the equation of motion

x = Àx. (1)

* Project Matterhorn, Princeton University, Princeton,New Jersey.

This system has one time-independent constant,the energy

There exists another constant (the initial phase) ofmore complicated behavior and involving the timeexplicitly. We do not employ this latter constant.

The first order variation of the energy vanishessince in static equilibrium x(t) vanishes. The secondorder variation leads to the form

for the perturbation. Clearly if Я > 0 then theenergy is a positive-definite form and the system isstable, for there exist no motions away from equili-brium. The stable oscillatory motions must necessarilyincrease the energy from its equilibrium value andhence are disregarded. If Я < 0, however, the formis indefinite, (3) can be satisfied nontrivially, and thereexist (exponential) motions away from equilibrium.

This method was suggested by a technique used byW. Newcomb 2 to show stability in the much simplercase of a plasma with a Maxwellian equilibrium distri-bution.

137

138 SESSION A-5 P/365 M. D. KRUSKAL and С R. OBERMAN

SMALL m/e LIMIT

In the present investigation we obtain these criteriaby examining the second order variation of the energy

S = f dH\{B2 + E2)

/J/ dvded*x[mf№ + W + vB)](4)

from its equilibrium value. Here E and В are theelectromagnetic field intensities, / is the distributionfunction in x, v space of a particular species of chargedparticles, the summation is over all species, and

v = a + v± +B(x, t) = В (х, t)n(3

a=ExB/B!,

q = V»n,

v = v^\1B,

s = a2/2 + vB.

?n,

Ct)

(5)

(6)

(7)

(8)

(9)

(10)

The quantity (Б/д) dvde represents the volume elementin velocity space. We assume for simplicity that anyboundaries present are such as to present no complica-tions, e.g., rigid and perfectly conducting walls withВ entirely tangential. The properties of the smalltn/e limit we employ are:

(a) v is constant following a particle motion.(b) f is rotationally symmetric in velocity space

about a line parallel to В and passing through thepoint a.

(c) a is the common drift velocity of all particles.

This last fact, as is well known,1' 3> 4> 5 permits theintroduction into the formalism of a displacementvector Ç(x, t) which governs not only the developmentof the field quantities but also the transverse motionof the particles. We defer consideration of theadditional property of charge neutrality until later,but we remark at this time that in contradistinc-tion to the Chew-Goldberger-Lowx (CGL) theory,where one particle species is taken to have a muchsmaller mass than another in order to satisfy thecondition that E*n vanishes, we treat all particlespecies on an equal footing, regard E and В as parti-cipants in the mje expansion and find that E»n isindeed zero to lowest order in m¡et which is all thatis necessary for the evolution of the expansion.

The equilibrium distribution function we denoteby g(v, s, L) where L labels the line of force passingthrough a point in space. We take g to be mono-tonic in s with

ge<0 (11)

for reasons we shall see later. The equilibrium condi-tion is

0 = - V-P+ (VxB)xB= - V.(/>+l + £_nn)+

where P is the stress dyadic, I is the unit dyadic,

with p± and pn given by

andpL - mjj(B/q)dvd8 vBg. (15)

FIRST ORDER VARIATION OF ENERGY

The first order change in given by

g?= B.B ) - ш í JX [{g

dvdecPx

) + Bq(f+ efe)].(16)

Here we take for convenience the change followingthe displacement Ç rather than at a fixed point, andwe now take equilibrium quantities without desig-nation and denote perturbation quantities withcircumflexes. (We shall frequently perform an inte-gration by parts with respect to e, as we have inarriving at (16), by making use of the fact that \\qequals qe, which follows immediately from (7). Wedo this in order to avoid the appearance of non-integrable integrands like l/<73.) The perturbedmagnetic field intensity is given up to second order in

= В + [B.V Ç -+ i {В [(V-Ç)2 +

(17)

- 2(y.Ç)B-VÇ},

and the volume element at the displaced point isgiven by

Ç) = d4{\ + V-Ç(18)

A.

However, we find S vanishes trivially when we makeuse of the general constraint condition that all con-stants of the motion have their equilibrium values.Indeed, to first order, we have

| Cj C(B¡q)dvded*x G(/, v, L)|A = 0, (19)

where G(f,v,L) is an arbitrary function of the distri-bution function / (remember / = 0), v, and L whereagain L labels a line of force passing through a pointin space. That magnetic lines of force maintain theiridentity during a displacement 6 is a consequence ofthe fact that E«n is zero (to lowest order in m/e) ;that particles stick to magnetic lines of force is aconsequence of (c). We may write condition (19) as

0 - - J fjdvd8d4Gf(g{v, e, L), v, L)

X

VB2 - B.VB (12)- (B/qffl

(20)

PLASMA STABILITY IN STATIC EQUILIBRIUM 139

and now regard Gf as an arbitrary function of e, v, Lbecause, by (11), g is monotonie in e. Accordinglywe may strip (20) to the basic constraint condition

X ( - V.Ç + nn:VÇ)(<?2 - vB)ge - Bf/q],(21)

where the integration is over a thin tube of force T.(We may, in general, transform integrals over thintubes of force of flux dtp to integrals along lines of forceaccording to the prescription

JT<Px BA (x) - dy)fLdl A (x), (22)

for arbitrary A(x).

mfffdvded*x

Here Q is the mass density. If we write the constraintcondition (19) to second order,

0 =

G'fV

(G'f> + G'fjy

Bq*)ee + UG>f)ee(Bq*)

I)2 - V? :

(26)

and make the same choice (23)for Gf, we can eliminateЪ -s

/ in the same way / was eliminated in first order, andS then becomes a quadratic form in Ç and / jointly.(We have not explicitly introduced the next ordercorrection to the displacement \ since its contribution

to S vanishes in second order, as the contribution of

\ to S vanished in the first order.) We now have

ÔW (27)

with ÔW defined by

ÔW =

+and

+ (Bf) * }, (28)

(29)

We find after using (10), (17), (14), and (15) that (28)becomes

We now make the particular choice

Gf(g(v,e, L), v, L) = — me (23)

for Gf in (20), add the resulting expression to (16) in

order to eliminate /, and obtain

f + B.B)- mf f fdvded*xg [BqV-% + {Bq)"]. (24)

The right-hand side of (24) now vanishes identically asstated when use is made of (10), (17), (12), (14), and(15).

SECOND ORDER VARIATION OF ENERGY

The second order change in energy is given by

B.B

Щ3))* (Чее + 2ge)

(eL + 2fe)+Bq*(efee + 2fe)}.

+ (VxB).(ÇxQ)

(25)

- [ -

X nn:VÇ)2]. (30)

We are now prepared to state our stability criterion :

ÔW is a quadratic form in \ and / jointly, otherwisedepending only on equilibrium quantities. If thisform is positive definite (i. e. positive for all nontrivial

permissible \ and /), then our system is stable. Indeed,

the only \ and / for which ê can vanish (as it mustsince S is a constant of the motion) are the trivial onesand hence no instability can develop. We hope toshow this condition is necessary as well as sufficientfor stability.

Let us now minimize this expression with respect to

/ (find the worst / from the point of view of stability)subject to the general constraint condition (21).To do this we multiply (21 ) by the Lagrange multiplierÀ(e, v, L), integrate over v and s and then integrate(sum) over tubes of force to obtain

0 = J J fdvded*xl(v, e, L)[BqgeV-%

- vB) ~ Bf/q).(31)

We now add this expression to (30) and then vary

with respect to /, obtaining the Euler equation

- f/g6 + Я = 0. (32)


If we now use this to eliminate / i n (16), we find

X = Г dH(B¡q)[q^-% + ( - V.Ç + nn:VÇ)

X ( « f - d4{Blq). (33)

These give the minimizing / in terms of Ç, so equation(30) now represents a quadratic form in Ç alone, andotherwise involving only equilibrium quantities. Inthe hydromagnetic г> 5 fluid theory, where the pressuredevelops according to the adiabatic laws

d {p „B*lQ*)jdt = 0 d {pJqB)jdt = 0

the corresponding expression SWD is

n = i¡<Px{Q* + (VxB).ÇxQ + \p( Ç ) ( № ) C

(34)

- 2 :VÇ + 3 (4(nn:VÇ)2

+ n.V|.(Ç.Vn)]}.

We can now write (30) as

ÔW = ÔWD + I -

(35)

where

/ = _ i mffJ{B/q)dvde<Pxge{iï - v* В*

(37)

(This expression for aW can be shown to be inde-pendent of the component of Ç parallel to В as itshould be on physical grounds.)

COMPARISON THEOREMS

If our condition is necessary as well as sufficient forstability, it is easy to prove that stability under the

- h

= (15/4)Я\¡Bm

dsxdyBp (

~0Г=ГуЩг{

where Bm\n is the minimum value of В along a line offorce. For this isotropic case we now have

+ (VxB).ÇxQÔW = \

This result has been also obtained independently byM. Rosenbluth7 using another method. Since theintegrand in 1г is positive, we may take Schwarz'inequality in the opposite direction, perform theу integration, and obtain

present particle theory implies stability under theCGL. fluid theory. For by means of Schwarz' in-equality

2vB

(38)

If we now insert this inequality into (37) we find

(39)

When the right-hand side of (39) is expressed in termsof p- and p+, it becomes precisely the last integralon the right-hand side of (36). Hence,

ÔW (40)

We can obtain an important inequality in theopposite direction when the equilibrium distributionfunction is isotropic (gv = 0). In this case

g(et I) ,

/35) and we may write

(42)

X =

where

-уВ)~Щ1 - |yB)nn:VÇ

+ \yB V. %]/ f BdH{\ -yB)-*,

у - v\z. (43)

If we now take у and e as variables in velocity spacerather than v and s we find we may write I in terms of themoment p after an integration by parts in e andobtain

-УВ)-Щ\ -

¡TBd4(\

ÔW > ÔWH =

(44)

(46)

where

We may conclude that if for a y = 5/3 hydromagneticfluid we can show stability (ÔWH < 0) then we mayconclude the system will indeed be stable under ourmore refined particle picture.


COLLISIONS

In case collisions are not negligible, the situation issomewhat altered in that we lose most of the constantsof the motion. Those of the type in (19), for which Gis independent of / and v, remain. However the fact,that the equilibrium distribution function is nowlocally Maxwellian (as it must be for static equilibriumwith collisions) enables us to proceed with the argu-ment. We do not lose the property that particlesstick to magnetic lines of force, however, since thesize of the step away from a magnetic line of forceafter a collision goes to zero with m/e.

The Boltzmann Ж function

Ж = (jj d*x{B/q)dvdeO{L)f{e, L) In /

Жо= + Ж +Ж +.. (48)

has the well-known property

Ж < 0. (49)

We now assume that all regular, time-independent,phase functions have their equilibrium values att = — oo and in particular obtain

and

Now

and, therefore,

= - oo) =

§ = 0

Ж < 0.

(50)

(51)

(52)

But аЖ/dt is linear in / and hence a reversal in the

sign of / leads to аЖ\&1 > О. We conclude, therefore,

Ж = 0

and finally obtain

(53)

(54)

Now S is still a constant of the motion and we may

use the expression for Ж to eliminate / from (25),obtaining expression (28) for this Maxwellian case

with the additional positive term —Ж on the right-hand side. We could have used the theorem that8 S+ Ж is a minimum for the Maxwellian distributionwith modulus в, to arrive at this result. We minimize

this expression with respect to / now with the con-straints that the Ж function for each tube of force isconstant to first order (see (53)) and the number ofparticles in each tube is constant. That is, using(16) and (20) we may minimize subject to the con-straints

0 = J f fT dfixdvde{B¡q)e[q*V*Cge (55)

+ ( - V.Ç + nil VÇ) (q2 - vB)ge - /]

T dHdvde (B/?)fea

+ (-V-Ç + nniV

This leads to

(56)

- vBgt) - / j .

(57)

for the value of the integral involving /2. (Themeaning of <y • Ç> is the same as in (46).) It follows atonce that

(58)

i.e., stability is not destroyed by the occurrence ofcollisions.

CHARGE NEUTRALITY

We conclude our presentation with a brief discus-sion of the charge neutrality condition which is alsoa consequence of the small m\e limit. This conditionis

0 = Si

nl el

= — S e* Íf/Bqdvds. (59)

We must now minimize (25) with respect to / subjectto the present constraint as well as (16). This leadsto a coupled set of linear integral equations forthe multipliers with which the constraints are intro-duced. We have not solved these equations and deferfurther discussion to future work.

ACKNOWLEDGEMENT

We are indebted to M. Rosenbluth for a valuablediscussion.

REFERENCES

1. G. F. Chew, M. L. Goldberger and F. E. Low, The Boltz-mann Equation and the One-Fluid Hydromagnetic Equa-tions in the Absence of Particle Collisions. Proc. Roy.Soc. A, 236, 112 (1956).

2. I. B. Bernstein, Plasma Oscillations in a Magnetic Field,Phys. Rev. 109, 10 (1958).

3. S. Lundquist, Magneto-Hydro static Instability, Phys. Rev.83, 307 (1951).

4. S. Lundquist, Studies in Magneto-Hydrodynamics, Ark.Mat. Ast. Fys., 5, 297 (1952).

5. I. B. Bernstein, E. A. Frieman, M. D. Kruskal and R. M.Kulsrud, An Energy Principle for Hydromagnetic StabilityProblems, Proc. Roy. Soc. A, 244, 17 (1958).

6. W. Newcomb (to be published).7. M. Rosenbluth (private communication).8. R. C. Tolman, The Principles of Statistical Mechanics,

p. 550, Oxford University Press (1948).


Mr. Kruskal presented Paper P/365, above, at theConference and added the following remarks :

I should like to summarize the development of thetheory of plasma stability and, in particular, tocompare the work of Rosenbluth and Rostocker, asdescribed in Paper P/349 presented to this Conference,with the Kruskal-Obermann theory as expounded inPaper P/365.

The main problem in the production of controlledthermonuclear energy is the confinement of a com-pletely ionized plasma of hot nuclear fuel by means ofstrong magnetic fields. A major obstacle to suchconfinement, however, is the possible existence ofinstabilities which may quickly disrupt an otherwisesatisfactory equilibrium configuration. Accordingly,it is a major task of theory to predict the stability ofgiven configurations, as well as to devise ever bettermeans for so doing.

In the earliest theoretical approaches the plasmawas treated as a continuous hydrodynamic fluid descri-bed at each point of space-time by a few parameters, forwhich, together with the electromagnetic fields, onecould write a complete set of partial differentialequations determining the possible motions. Thistype of description is appropriate when there is amechanism which keeps neighboring particles closetogether, so that a set of neighboring particles canform a coherent element of fluid. In many appli-cations the frequent collisions between the particlesconstitute such a mechanism.

At fiist stability was treated by a number ofinvestigators on the basis of the hydromagneticequations, using the method of normal modes. Thisinvolves linearizing the equations for the perturbationsof the equilibrium state under investigation, andlooking for solutions of these linearized perturbationequations which are purely exponential in theirtime-dependence, with real or complex exponents.The equilibrium is unstable if growing exponentialsolutions exist.

Except for rather simple equilibrium states, however,this normal mode method is very difficult to carrythrough. Following the pioneering work of Lund-quist,3 several groups of authors therefore developedan energy principle which greatly enlarges the class ofstability problems which can be investigated practic-ably. This energy principle states that a staticequilibrium is stable if and only if a certain associatedhomogeneous quadratic form is positive definite. Thequadratic form represents physically the second-ordervariation in the potential energy of the system due toan arbitrary virtual displacement of the fluid.

The energy principle provides a very satisfactorygeneral stability theory, at least for static equilibria,when there is some mechanism (almost necessarilycollisions) that permits the plasma to be treatedhydrodynamically. In controlled thermonuclearenergy applications, however, the effects of collisionsare negligible for processes that take place as quicklyas the growth of most instabilities. There is, instead,a mechanism which is in a sense two-thirds effective

in keeping neighboring particles close together;namely, a strong magnetic field forces the chargedparticles to gyrate around a point which sticks toand moves with a line of force. As a result, the par-ticles cannot disperse in the two directions perpendi-cular to the magnetic lines of force, but only in adirection parallel to the field lines.

Several groups of authors, notably Chew, Goldberger,and Low,1 have shown how to treat the plasma in themathematically appropriate way when collisions arenot important. They employ the so-called collision-less Boltzmann equation with a term {q/fn)(E + v X B) in place of the usual collision term.The electric and magnetic fields satisfy the Maxwellequations and the current and charge density areexpressed as sums over the different species of integralsover velocity space, utilizing the distribution functionwhich satisfies the Boltzmann equation as a weightingfunction in the integrand.

One now wishes to obtain the limiting form ofthese equations as the radius of gyration of a particlein the magnetic field becomes very small compared tothe characteristic length of the system under considera-tion, and the period of gyration becomes very smallcompared to a characteristic time. This limitingprocess may be formalized in various ways, thesimplest of which is to treat the charge q as beingvery large. It is not trivial to carry this programthrough systematically. However, it has been shownthat the resultant reduced system of equations iseasier to handle than the original system. The reducedsystem, obtained in this fashion, may be used toinvestigate the stability of equilibria by the normalmode or equivalent methods. This is essentiallythe approach adopted by Chandrasekhar, Kaufman,and Watson. Unfortunately, compared to the hydro-magnetic equations, it is more difficult to carry throughthe normal mode method with the reduced systemwhen the equilibrium is not simple.

In the two papers which I wish to discuss here,(P/365 and P/349) an energy principle for determiningstability is derived based on the reduced system.The methods used differ considerably from each other.The conclusions reached in each paper are in somerespects more general than those in the other, butwhere they overlap they agree.

In the paper by Rosenbluth and Rostoker, theequilibrium state is assumed to have isotropic distri-bution functions to lowest order in the gyrationradius. The first part of the paper is devoted toobtaining the equations of a normal mode, and it isshown that the first-order perturbations from equili-brium of all quantities are proportional to the exponen-tial of cot. The quantity œ, which may be complex,is the characteristic growth or frequency parameter ofthe mode. In this analysis, all the first-order pertur-bations are expressed in terms of the usual first-ordervector \, which describes at each point the perpendi-cular displacement of the magnetic line of force.One obtains finally an integro-differential equationfor \ analogous to the differential equation for Ç ob-tainable from hydromagnetic theory. The presence


of the integrals, which are taken along equilibriumlines of force, is of course a consequence of the spatiallynon-local character of the treatment. The normalmode equations constitute an eigenvalue problemfor со.

The next step in the Rosenbluth-Rostoker methodis to construct the second-order change in energydue to the first-order perturbation. This can beexpressed as a quadratic functional W in Ç. For œ = 0,the integrand of the functional contains three termswhich are the same as in the hydromagnetic casebecause they represent the second-order change in theenergy of the magnetic field. The fourth term in theintegrand involves two integrals which, for eachpoint x and each value of the dummy variable ofintegration, are line integrals taken along part of theequilibrium line of force through x.

It turns out that the quadratic functional Wvanishes and is stationary for a vector field \ whichis not identically zero if and only if \ is a solution ofthe eigen-value problem described earlier and hasthe eigenvalue œ = 0. This suggests strongly thatif W is negative for some Ç, then there is an instability.A proof is given that if W is positive-definite then theequilibrium is stable.

In the other paper, we (Oberman and Kruskal)have made no effort to obtain conditions for instability,but have sought the weakest conditions we could forstability. To do this we look for constants of themotion of the system which do not depend explicitlyon the time and which are regular near the equilibriumconfiguration. Since any purely unstable motionhas been arbitrarily close to the equilibrium state farenough back in time, such constants of motion musthave the same values for an unstable motion as forequilibrium. That is, the first and second-orderperturbations of these constants of motion must

vanish. This leads to severe restrictions on thefirst-order perturbations of the physical quantities;when they are so severe that these perturbations mustvanish, stability is assured.

We do not assume that the equilibrium distributionfunctions are isotropic, but we do require of the equili-brium that the mass velocity vanish to lowest orderin the gyration radius for each species of ion. Weobtain finally a quadratic form in Ç, the positive-definiteness of which implies stability. In the case ofisotropic equilibrium distributions it reduces exactlyto the form W obtained by Rosenbluth and Rostoker.

Our method of using constants of the motion canbe applied even when the collision terms of theBoltzmann equation are retained. There are thenfar fewer suitable constants available, but this iscompensated for by the necessary restriction to thoseequilibria which are invariant during collisions tolowest order in the gyration radius. Such equilibriaare those having Maxwellian distributions with constanttemperature along lines of force. The final resultturns out to be the same energy principle as before.

Both papers also give essentially the same com-parison theorems. One of these is that W is boundedabove by the simpler result derived from the standardhydromagnetic theory with two distinct pressures, oneparallel and one perpendicular to the magnetic field,each governed by its own adiabatic equation of state.The other comparison theorem is that in the isotropiccase, W is bounded below by the result of standardhydromagnetic theory with one scalar pressure.Because of these comparison theorems, fortunately,we have demonstrated in these two papers that thestability results previously obtained from standardhydromagnetic theories still have considerable signi-ficance when viewed in terms of our more accuratecalculations.

stability of plasma in static equilibrium

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