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P/365 USA Stability of Plasma in Static Equilibrium By M. D. Krusiial and С R. Oberman * Our purpose is t ^ derive from the Boltzmann equation in the small mje limit x, criteria useful in the discussion of stability of plasmas in static equilibrium. At first we ignore collisions but later show their effects may be taken into account. Our approach yields a generalization of the usual energy principles z> 4>5 for investigating the stability of hydromagnetic systems to situations where the effect of heat flow along magnetic lines is not negligible, and hence to situations where the strictly hydrodynamic approach is inapplicable. In the first two sections we characterize our general method of approach and delineate the properties of the small mje limit which we use to determine the constants of the motion and the condition for static equilibrium. In the next two sections we calculate the first and second variations of the energy and conclude with a statement of the general stability criterion. In the final three sections we state several theorems which relate our stability criterion to those of ordinary hydromagnetic theory,5 we show how to take into account the effect of collisions, and briefly discuss the remaining problem of incorporating the charge neutrality condition into the present stability theory. the displacement of magnetic lines of force away from their equilibrium positions.) We rid ourselves of the dependence on / by minimizing the energy with respect to it, subject to the constraint that all constants of the forementioned type have their equilibrium values. We then have a sufficient condition for stability involving \ alone. GENERAL METHOD There exists another constant (the initial phase) of more complicated behavior and involving the time explicitly. We do not employ this latter constant. The first order variation of the energy vanishes since in static equilibrium x(t) vanishes. The second order variation leads to the form Generally, the constants of the motion of the type we employ do not specify the motion completely so that there exist many motions evolving from the same equilibrium (at t = —сю). By restricting the constants of motion to their equilibrium values, the only possible motions other than the equilibrium behavior are instabilities (the pure modes of which have an exponential time behavior and hence vanish at t= —oo). To illustrate the method we consider the simplest of examples, the equation of motion x = Àx. This system has one time-independent constant, the energy Our method consists of writing down the energy of the system to second order in the perturbation fields of / and B, where / is the distribution function in x,v space and В is the magnetic field intensity. We eliminate the terms involving the second order perturbation / by employing certain constraints, namely, that certain constants of the motion have their equilibrium values. The constants of the motion we employ 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 quadratic form in / and Ç jointly whose positive-definiteness provides a sufficient condition for stability. (More about Ç later, let it suffice for now to say % describes * Project Matterhorn, New Jersey. (1) Princeton University, Princeton, 137 for the perturbation. Clearly if Я > 0 then the energy is a positive-definite form and the system is stable, for there exist no motions away from equilibrium. The stable oscillatory motions must necessarily increase the energy from its equilibrium value and hence are disregarded. If Я < 0, however, the form is indefinite, (3) can be satisfied nontrivially, and there exist (exponential) motions away from equilibrium. This method was suggested by a technique used by W. Newcomb 2 to show stability in the much simpler case of a plasma with a Maxwellian equilibrium distribution. 138 SESSION A-5 P/365 M. D. KRUSKAL and С R. OBERMAN SMALL m/e LIMIT where P is the stress dyadic, I is the unit dyadic, In the present investigation we obtain these criteria by examining the second order variation of the energy 2 with p± and pn given by 2 S = f dH\{B + E ) /J/ dvded*x[mf№ + W + vB)] (4) from its equilibrium value. Here E and В are the electromagnetic field intensities, / is the distribution function in x, v space of a particular species of charged particles, the summation is over all species, and v = a + v± + ?n, B(x, t) = В (х, t)n(3Ct) (5) a=ExB/B!, (7) q = V»n, (8) v = v^\1B, (9) 2 s = a /2 + vB. (6) (10) The quantity (Б/д) dvde represents the volume element in velocity space. We assume for simplicity that any boundaries present are such as to present no complications, e.g., rigid and perfectly conducting walls with В entirely tangential. The properties of the small tn/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 the point a. (c) a is the common drift velocity of all particles. This last fact, as is well known,1' 3> 4> 5 permits the introduction into the formalism of a displacement vector Ç(x, t) which governs not only the development of the field quantities but also the transverse motion of the particles. We defer consideration of the additional property of charge neutrality until later, but we remark at this time that in contradistinction to the Chew-Goldberger-Lowx (CGL) theory, where one particle species is taken to have a much smaller mass than another in order to satisfy the condition that E*n vanishes, we treat all particle species on an equal footing, regard E and В as participants in the mje expansion and find that E»n is indeed zero to lowest order in m¡et which is all that is necessary for the evolution of the expansion. The equilibrium distribution function we denote by g(v, s, L) where L labels the line of force passing through a point in space. We take g to be monotonic in s with ge<0 (11) for reasons we shall see later. The equilibrium condition is 0 = - V-P+ (VxB)xB = - V.(/>+l + £_nn)+ VB 2 - B.VB and pL - mjj(B/q)dvd8 vBg. FIRST ORDER VARIATION OF ENERGY The first order change in g?= B.B given by )- шíJ dvdecPx ) + Bq(f+ efe)]. X [{g (16) Here we take for convenience the change following the displacement Ç rather than at a fixed point, and we now take equilibrium quantities without designation and denote perturbation quantities with circumflexes. (We shall frequently perform an integration by parts with respect to e, as we have in arriving at (16), by making use of the fact that \\q equals qe, which follows immediately from (7). We do this in order to avoid the appearance of nonintegrable integrands like l/<73.) The perturbed magnetic field intensity is given up to second order in (17) = В + [B.V Ç + i {В [(V-Ç)2 + - 2(y.Ç)B-VÇ}, and the volume element at the displaced point is given by Ç) = d4{\ + V-Ç (18) A. However, we find S vanishes trivially when we make use of the general constraint condition that all constants 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 distribution function / (remember / = 0), v, and L where again L labels a line of force passing through a point in space. That magnetic lines of force maintain their identity during a displacement 6 is a consequence of the fact that E«n is zero (to lowest order in m/e) ; that particles stick to magnetic lines of force is a consequence of (c). We may write condition (19) as 0 - - J fjdvd8d4Gf(g{v, e, L), v, L) X (12) (15) - (B/qffl (20) 139 PLASMA STABILITY IN STATIC EQUILIBRIUM and now regard Gf as an arbitrary function of e, v, L because, by (11), g is monotonie in e. Accordingly we may strip (20) to the basic constraint condition 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 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 thin tubes of force of flux dtp to integrals along lines of force according to the prescription JT<Px BA (x) - dy)fLdl A (x), + B.B) f - mf f fdvded*xg [BqV-% + {Bq)"]. (24) The right-hand side of (24) now vanishes identically as stated when use is made of (10), (17), (12), (14), and (15). (22) SECOND ORDER VARIATION OF ENERGY for arbitrary A(x). The second order change in energy is given by B.B mfffdvded*x Щ3))* (Чее + 2ge) (eL + 2fe)+Bq*(efee + 2fe)}. Here Q is the mass density. If we write the constraint condition (19) to second order, (G'f> + 0 = (25) + (VxB).(ÇxQ) G'fjy - [- Bq*)ee + UG>f)ee(Bq*) I) 2 - V? : G'fV X (26) and make the same choice (23)for Gf, we can eliminate -s Ъ / in the same way / was eliminated in first order, and S then becomes a quadratic form in Ç and / jointly. (We have not explicitly introduced the next order correction 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 = nn:VÇ) 2 ]. (30) We are now prepared to state our stability criterion : ÔW is a quadratic form in \ and / jointly, otherwise depending only on equilibrium quantities. If this form 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 must since S is a constant of the motion) are the trivial ones and hence no instability can develop. We hope to show this condition is necessary as well as sufficient for 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 multiplier À(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-% + + (Bf) * }, - vB) ~ Bf/q). (28) (31) and (29) We find after using (10), (17), (14), and (15) that (28) becomes We now add this expression to (30) and then vary with respect to /, obtaining the Euler equation - f/g6 + Я = 0. (32) 140 SESSION A-5 P/365 M. D. KRUSKAL and С R. OBERMAN present particle theory implies stability under the CGL. fluid theory. For by means of Schwarz' inequality If we now use this to eliminate / i n (16), we find X = Г dH(B¡q)[q^-% + ( - V.Ç + nn:VÇ) d4{Blq). X («f- (33) 2vB These give the minimizing / in terms of Ç, so equation (30) now represents a quadratic form in Ç alone, and otherwise involving only equilibrium quantities. In the hydromagnetic г> 5 fluid theory, where the pressure develops according to the adiabatic laws d {p „B*lQ*)jdt = 0 d {pJqB)jdt (38) If we now insert this inequality into (37) we find (34) = 0 (39) the corresponding expression SWD is When the right-hand side of (39) is expressed in terms of p- and p+, it becomes precisely the last integral on the right-hand side of (36). Hence, n = i¡<Px{Q* + (VxB).ÇxQ + \p (Ç) ( № ) C - 2 :VÇ + 3 ( 4(nn:VÇ)2 (40) ÔW (35) + n.V|.(Ç.Vn)]}. We can now write (30) as We can obtain an important inequality in the opposite direction when the equilibrium distribution function is isotropic (gv = 0). In this case g(et I), ÔW = ÔWD + I /35) and we may write where (42) / = _ i mffJ{B/q)dvde<Pxge{iï - v* В* -уВ)~Щ1 X = (37) (This expression for aW can be shown to be independent of the component of Ç parallel to В as it should be on physical grounds.) COMPARISON THEOREMS If our condition is necessary as well as sufficient for stability, it is easy to prove that stability under the - h = (15/4) Я \¡Bm dsxdy Bp ( ~0Г=ГуЩг{ - |yB)nn:VÇ + \yB V. %]/ f BdH{\ -yB)-*, where у - v\z. (43) If we now take у and e as variables in velocity space rather than v and s wefindwe may write I in terms of the moment p after an integration by parts in e and obtain -УВ)-Щ\ - ¡TBd4(\ (44) where Bm\n is the minimum value of В along a line of force. For this isotropic case we now have ÔW = \ + (VxB).ÇxQ This result has been also obtained independently by M. Rosenbluth7 using another method. Since the integrand in 1г is positive, we may take Schwarz' inequality in the opposite direction, perform the у integration, and obtain ÔW > ÔWH = (46) where We may conclude that if for a y = 5/3 hydromagnetic fluid we can show stability (ÔWH < 0) then we may conclude the system will indeed be stable under our more refined particle picture. PLASMA STABILITY IN STATIC EQUILIBRIUM COLLISIONS 141 and In case collisions are not negligible, the situation is somewhat altered in that we lose most of the constants of the motion. Those of the type in (19), for which G is independent of / and v, remain. However the fact, that the equilibrium distribution function is now locally Maxwellian (as it must be for static equilibrium with collisions) enables us to proceed with the argument. We do not lose the property that particles stick to magnetic lines of force, however, since the size of the step away from a magnetic line of force after a collision goes to zero with m/e. The Boltzmann Ж function T dHdvde (B/?)fea + (-V-Ç + nniV (56) - vBgt) - / j . This leads to (57) for the value of the integral involving / 2 . (The meaning of <y • Ç> is the same as in (46).) It follows at once that d*x{B/q)dvdeO{L)f{e, L) In / Ж = (jj Жо= + Ж +Ж +.. (58) (48) has the well-known property Ж < 0. CHARGE NEUTRALITY (49) We now assume that all regular, time-independent, phase functions have their equilibrium values at t = — oo and in particular obtain = - oo) = i.e., stability is not destroyed by the occurrence of collisions. We conclude our presentation with a brief discussion of the charge neutrality condition which is also a consequence of the small m\e limit. This condition is (50) 0 = S nl el Now i § = 0 (51) Ж < 0. (52) = — S e* Íf/Bqdvds. and, therefore, But аЖ/dt is linear in / and hence a reversal in the sign of / leads to аЖ\&1 > О. We conclude, therefore, Ж = 0 We must now minimize (25) with respect to / subject to the present constraint as well as (16). This leads to a coupled set of linear integral equations for the multipliers with which the constraints are introduced. We have not solved these equations and defer further discussion to future work. (53) and finally obtain ACKNOWLEDGEMENT (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 righthand side. We could have used the theorem that 8 S + Ж is a minimum for the Maxwellian distribution with modulus в, to arrive at this result. We minimize this expression with respect to / now with the constraints that the Ж function for each tube of force is constant to first order (see (53)) and the number of particles in each tube is constant. That is, using (16) and (20) we may minimize subject to the constraints 0 = J f fT dfixdvde{B¡q)e[q*V*Cge (59) (55) + ( - V.Ç + nil VÇ) (q2 - vB)ge - /] We are indebted to M. Rosenbluth for a valuable discussion. REFERENCES 1. G. F. Chew, M. L. Goldberger and F. E. Low, The Boltzmann Equation and the One-Fluid Hydromagnetic Equations 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 Stability Problems, 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). 142 SESSION A-5 P/365 M. D. KRUSKAL and С R. OBERMAN Mr. Kruskal presented Paper P/365, above, at the Conference and added the following remarks : I should like to summarize the development of the theory of plasma stability and, in particular, to compare the work of Rosenbluth and Rostocker, as described in Paper P/349 presented to this Conference, with the Kruskal-Obermann theory as expounded in Paper P/365. The main problem in the production of controlled thermonuclear energy is the confinement of a completely ionized plasma of hot nuclear fuel by means of strong magnetic fields. A major obstacle to such confinement, however, is the possible existence of instabilities which may quickly disrupt an otherwise satisfactory equilibrium configuration. Accordingly, it is a major task of theory to predict the stability of given configurations, as well as to devise ever better means for so doing. In the earliest theoretical approaches the plasma was treated as a continuous hydrodynamic fluid described at each point of space-time by a few parameters, for which, together with the electromagnetic fields, one could write a complete set of partial differential equations determining the possible motions. This type of description is appropriate when there is a mechanism which keeps neighboring particles close together, so that a set of neighboring particles can form a coherent element of fluid. In many applications the frequent collisions between the particles constitute such a mechanism. At fiist stability was treated by a number of investigators on the basis of the hydromagnetic equations, using the method of normal modes. This involves linearizing the equations for the perturbations of the equilibrium state under investigation, and looking for solutions of these linearized perturbation equations which are purely exponential in their time-dependence, with real or complex exponents. The equilibrium is unstable if growing exponential solutions exist. Except for rather simple equilibrium states, however, this normal mode method is very difficult to carry through. Following the pioneering work of Lundquist,3 several groups of authors therefore developed an energy principle which greatly enlarges the class of stability problems which can be investigated practicably. This energy principle states that a static equilibrium is stable if and only if a certain associated homogeneous quadratic form is positive definite. The quadratic form represents physically the second-order variation in the potential energy of the system due to an arbitrary virtual displacement of the fluid. The energy principle provides a very satisfactory general stability theory, at least for static equilibria, when there is some mechanism (almost necessarily collisions) that permits the plasma to be treated hydrodynamically. In controlled thermonuclear energy applications, however, the effects of collisions are negligible for processes that take place as quickly as 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 charged particles to gyrate around a point which sticks to and moves with a line of force. As a result, the particles cannot disperse in the two directions perpendicular to the magnetic lines of force, but only in a direction parallel to the field lines. Several groups of authors, notably Chew, Goldberger, and Low,1 have shown how to treat the plasma in the mathematically appropriate way when collisions are not important. They employ the so-called collisionless 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 Maxwell equations and the current and charge density are expressed as sums over the different species of integrals over velocity space, utilizing the distribution function which satisfies the Boltzmann equation as a weighting function in the integrand. One now wishes to obtain the limiting form of these equations as the radius of gyration of a particle in the magnetic field becomes very small compared to the characteristic length of the system under consideration, and the period of gyration becomes very small compared to a characteristic time. This limiting process may be formalized in various ways, the simplest of which is to treat the charge q as being very large. It is not trivial to carry this program through systematically. However, it has been shown that the resultant reduced system of equations is easier to handle than the original system. The reduced system, obtained in this fashion, may be used to investigate the stability of equilibria by the normal mode or equivalent methods. This is essentially the approach adopted by Chandrasekhar, Kaufman, and Watson. Unfortunately, compared to the hydromagnetic equations, it is more difficult to carry through the normal mode method with the reduced system when the equilibrium is not simple. In the two papers which I wish to discuss here, (P/365 and P/349) an energy principle for determining stability is derived based on the reduced system. The methods used differ considerably from each other. The conclusions reached in each paper are in some respects more general than those in the other, but where they overlap they agree. In the paper by Rosenbluth and Rostoker, the equilibrium state is assumed to have isotropic distribution functions to lowest order in the gyration radius. The first part of the paper is devoted to obtaining the equations of a normal mode, and it is shown that the first-order perturbations from equilibrium of all quantities are proportional to the exponential of cot. The quantity œ, which may be complex, is the characteristic growth or frequency parameter of the mode. In this analysis, all the first-order perturbations are expressed in terms of the usual first-order vector \, which describes at each point the perpendicular displacement of the magnetic line of force. One obtains finally an integro-differential equation for \ analogous to the differential equation for Ç obtainable from hydromagnetic theory. The presence PLASMA STABILITY IN STATIC EQUILIBRIUM of the integrals, which are taken along equilibrium lines of force, is of course a consequence of the spatially non-local character of the treatment. The normal mode equations constitute an eigenvalue problem for со. The next step in the Rosenbluth-Rostoker method is to construct the second-order change in energy due to the first-order perturbation. This can be expressed as a quadratic functional W in Ç. For œ = 0, the integrand of the functional contains three terms which are the same as in the hydromagnetic case because they represent the second-order change in the energy of the magnetic field. The fourth term in the integrand involves two integrals which, for each point x and each value of the dummy variable of integration, are line integrals taken along part of the equilibrium line of force through x. It turns out that the quadratic functional W vanishes and is stationary for a vector field \ which is not identically zero if and only if \ is a solution of the eigen-value problem described earlier and has the eigenvalue œ = 0. This suggests strongly that if W is negative for some Ç, then there is an instability. A proof is given that if W is positive-definite then the equilibrium 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 for stability. To do this we look for constants of the motion of the system which do not depend explicitly on the time and which are regular near the equilibrium configuration. Since any purely unstable motion has been arbitrarily close to the equilibrium state far enough back in time, such constants of motion must have the same values for an unstable motion as for equilibrium. That is, the first and second-order perturbations of these constants of motion must 143 vanish. This leads to severe restrictions on the first-order perturbations of the physical quantities; when they are so severe that these perturbations must vanish, stability is assured. We do not assume that the equilibrium distribution functions are isotropic, but we do require of the equilibrium that the mass velocity vanish to lowest order in the gyration radius for each species of ion. We obtain finally a quadratic form in Ç, the positivedefiniteness of which implies stability. In the case of isotropic equilibrium distributions it reduces exactly to the form W obtained by Rosenbluth and Rostoker. Our method of using constants of the motion can be applied even when the collision terms of the Boltzmann equation are retained. There are then far fewer suitable constants available, but this is compensated for by the necessary restriction to those equilibria which are invariant during collisions to lowest order in the gyration radius. Such equilibria are those having Maxwellian distributions with constant temperature along lines of force. The final result turns out to be the same energy principle as before. Both papers also give essentially the same comparison theorems. One of these is that W is bounded above by the simpler result derived from the standard hydromagnetic theory with two distinct pressures, one parallel and one perpendicular to the magnetic field, each governed by its own adiabatic equation of state. The other comparison theorem is that in the isotropic case, W is bounded below by the result of standard hydromagnetic theory with one scalar pressure. Because of these comparison theorems, fortunately, we have demonstrated in these two papers that the stability results previously obtained from standard hydromagnetic theories still have considerable significance when viewed in terms of our more accurate calculations.