Low Beta Rigid Mode Stability Criterion for an Arbitrary Larmor Radius Plasma
H. L. Berk and H. V. Wong Institute for Fusion Studies, The University o f Texas at Austin, Austin, Texas 78712
Z. Naturforsch. 42 a, 1208-1224 (1987); received May 25, 1987
Dedicated to Professor Dieter Pfirsch on his 60th Birthday
The low beta flute axisymmetric dispersion relation for rigid displacement perturbation of plasma equilibria with arbitrary Larmor radius particles and field line radius of curvature large compared to the plasma radius is derived. The equilibrium particle orbits are characterized by two constants o f motion, energy and angular momentum, and a third adiabatic invariant derived from the rapid radial motion. The Vlasov equation is integrated, assuming that the mode frequency, axial “bounce” frequency, and particle drift frequency are small compared to the cyclotron frequency, and it is demonstrated that the plasma response to a rigid perturbation has a universal character independent of Larmor radius. As a result the interchange instability is the same as that predicted from conventional M H D theory. However, a new prediction, more optimistic than earlier work, is found for the low density threshold of systems like Migma, which are disc-shaped, that is, the axial extent Az is less than the radial extent r0. The stability criterion for negative field line curvature x
1 x\al^ 8 dh______r ro ß/ 1 + 2 Z —--- j- Az co£jj where
I. Introduction radius, for the response of an equilibrium con figuration to long wavelength rigid mode perturba Under a variety of conditions the linear response tions. The assumptions made in the calculation are of hot particles with arbitrary Larmor radius may that there is azimuthal symmetry in the equilib differ appreciably from the predictions of MHD rium, that the magnetic field curvature radius is theory [1-7]. There is a variety of concepts, such as large compared to the plasma radius and beta is astron [8, 9], Elmo Bumpy Torus [10], field reversedlow. Because of the last assumption, we shall neglect theta pinch [11], tokamaks containing hot particles the equilibrium induced magnetic fields and con [12], and Migma [13, 14] that hope to exploit suchsider only electrostatic perturbations. The self- an effect to achieve favorable containment. In this consistent description with equilibrium magnetic work we present a theory for arbitrary Larmor fields and electromagnetic perturbation will be reported in a later work, although preliminary results of the fully self-consistent calculation will be discussed in the summary. Reprint requests to Prof. Dr. H. L. Berk, Institute for Fusion Studies, The University of Texas, Austin, Texas The primary result of our calculation is that the 78712, USA. response to a rigid perturbation has a universal
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Dieses Werk wurde im Jahr 2013 vom Verlag Zeitschrift für Naturforschung This work has been digitalized and published in 2013 by Verlag Zeitschrift in Zusammenarbeit mit der Max-Planck-Gesellschaft zur Förderung der für Naturforschung in cooperation with the Max Planck Society for the Wissenschaften e.V. digitalisiert und unter folgender Lizenz veröffentlicht: Advancement of Science under a Creative Commons Attribution Creative Commons Namensnennung 4.0 Lizenz. 4.0 International License. H. L. Berk and H. V. Wong • An Arbitrary Larmor Radius Plasma 1209 character [1, 15] independent of Larmor radius. As a discuss the consequences of the interchange instabil result the same interchange instability as predicted ity to the Migma concept. from conventional MHD theory is predicted. How In the summary we note that a rigid mode pertur ever, if the predicted M HD growth rate is less than bation can rigorously be chosen at sufficiently high the precessional frequency of the particles, stabiliza density. However, near the threshold density, the tion arises. Such stabilization can occur at low rigid mode eigenfunction cannot be rigorously densities,if the plasma frequency is appreciably less justified, and a modification of our threshold con than the ion cyclotron frequency [1] or if the ratio of ditions is to be expected. The method of obtaining hot particle density to cold particle density is the modification will be discussed elsewhere. sufficiently small [4], These stabilization mechan isms have been noted in small Larmor radius II. Equilibrium theories [1, 4], but we believe that this calculation is the first to rigorously demonstrate that the stability In cylindrically symmetric plasma equilibria criteria is independent of Larmor radius. where the current density J 0 is azimuthal, the The calculation is applied to the Migma con equilibrium vector potential A0 may be represented figuration, where the experimentally achieved in terms of the flux variable y/(r, z): equilibrium has an axial width, Lp, considerably less than the radial extent, Rp, which is of order two A0= if/VO Larmor radii. Because of the shape, the density where r, 0, z are the usual cylindrical coordinates. threshold for instability is shown to be larger by a The equilibrium magnetic field B is factor, Rp/Lp, than the case where Lp ^> R p, which is usually studied. The predicted particle number B = V(// x V#. ( 1) threshold is above the number of particles trapped The lines of force lie in surfaces of i// = constant and in the Migma experiment [14] and is therefore com 0 = constant. patible with experimental results, where the inter It is convenient to introduce a function /(r,z) so change mode has not been observed. that t//, 6, / form an othogonal set of curvilinear We expect the universality of the response of coordinates, where rigid modes to apply even in the finite beta fully electromagnetic case. Hence, if we apply known B = (2) results from small Larmor radius theory, we can so that / is determined by indicate the stability criteria for arbitrary beta. This 0/ rB2 n discussion will be given in the summary section. We V x B = - ; ee= 4 Joeee ■ also need to examine the validity of a rigid mode as oi// /. c an eigenfunction (see below). Note that The structure of the paper is as follows. In Sect. II 1 B Vi// = rB, |V0|=—, | Vzj =— . we discuss the equilibrium. In Sect. Ill we analyze r k the dispersion relation that is obtained for a rigid The equilibrium Hamiltonian for the particle displacement. The algebra needed to obtain the motion describing the particle dynamics in these formal response makes use of Hamiltonian pertur coordinates is bation theory. Use of a Hamiltonian formalism in the perturbation theory allows for a systematic analysis of complex configurations. Since we are ( 3 ) dealing with large orbits, and will need to find 2 M 2 Mr 2 M k ‘ adiabatic invariants in a non-conventional ordering, a Hamiltonian formalism is particularly useful [16]. where p v, pe, px are the particle momenta conjugate However, the details are somewhat involved, and as to (//, 0, /, and M the particle mass. The equations of a consequence we have included the algebraic motion are then as usual: reduction of the problem separately in Section IV. q j= dH /d p j, pi = — dH /d qj , In Sect. Ill the analysis of the electric field coupling for a disc-shaped plasma is presented. In Sect. V we qi=(if/,9,X), P i= (PV,P0 ,PX)- 1210 H. L. Berk and H. V. Wong • An Arbitrary Larmor Radius Plasma
For example In this analysis we also treat x/r independent of position, which is a justified approximation if Bx (z)
P¥ 6 . P e ~ ~ c y/ dr is quadratic in z2 and the mirror ratio sampled by p if/ r B + 2M dy/ M r: dij/ plasma particles is close to unity.
P e - e-y III. Electrostatic m = 1 Flute Perturbation e d B 2 + - P\ c M r 2 2M dy/~? III. 1 General Description of Method of Solution We are interested in low beta mirror equilibrium of Dispersion Relation where the vacuum magnetic field may be approxi In order to describe the electrostatic dispersion mated by a near axis expansion of the vacuum field relation or the perturbed system, we examine the flux function yj\ structure of the variational quadratic form to the “2 response of a rigid displacement. This procedure uz = — Bo + B\ (z) — —3 —j B\(z) + ... can be justified as follows: Y 2 8 dr" (1) The electrostatic dispersion relation is valid Thus for the threshold of instability as it occurs at such low beta that magnetic perturbations, which B- — Bq + B\ — -—-—; B\ ... = B ------r B primarily produce large positive energy perturba 4 dz 4 dz' tions, must be negligible in order to have instability. r d - (2) A symmetric quadratic form in the perturbed Br~~ 2 dz Bx' amplitude r d 2 - Vy/ linear eigenmode equation [17-19]. The substitution x = b ■ Vb = ------r B] (5) of a first order accurate representation of the eigen 2 B dz2 1 V y/ function gives a second order accurate expression We assume in our subsequent analysis that the for the eigenvalue. field line curvature and mirror ratio are small: (3) If the system has zero curvature, a rigid displacement with ip a /-exp(/9) would be an exact eigenfunction for a zero frequency mode. Further, Bo dz2 2 Bl dz for a disc shaped plasma, dip/d/ would dominate and that / = 1, which follows at very low beta. the electric response in the electric field energy The constants of motion are H and p e. In unless (p is nearly independent of y. Thus, to addition, we assume an adiabatic invariant arising examine the response of a low frequency mode for a from the rapid y/ motion compared to the / motion. system with weak curvature, we choose From standard Hamiltonian theory the adiabatic 1/2 c p o B invariant, Pv, is given by (p exp (i9)r/r0 Dfl P¥= § d The quadratic form is obtained by considering change instability. The last term is a more com the Poisson equation plicated kinetic term that would be small if co P cox. It is of the same structure that would be obtained V2cp = - 4 n Y J \d3pfj qj , in a zero Larmor radius kinetic theory and it is of j importance at sufficiently low density near the where f is the perturbed distribution function of threshold for interchange instabilities or if the species j of charge qj due to the test function we energetic particles are only a small fraction of the have chosen. Multiplying by cp+, and integrating plasma particles. Explicit Larmor radius terms do over all space then gives the quadratic variational not appear in this theory because of the use of the form rigid displacement perturbation discussed previously. D (cv) = J d 3/- V«p+ •Vfi - 4n ^ qj j d 3r d 3p cp+f= 0. In addition there is the electric field integration ‘ (7) in the first term of (8). We note that the electric field must be integrated over all space, and if the A critical part of the calculation is to calculate the plasma is a thin disc, the predominant contribution self-consistent f . However, as the derivation is to the integral comes from the region outside the somewhat involved, we present the final results plasma. To calculate this integral we observe here, and show the detailed derivation in Section IV. The final result is quoted immediately below. In J d3/- • V^+== j d3rV^-V^+ addition we note that ip must be known over all space. Hence, given D (co) 1 r , ^ ^ ^ r , M jN jC 2 r _ _ _ J drV#.V#+2£Jdr^pj with a given cp(r) = cp 0 — exp {id). Thus from Coulomb’s law, ;° p]Bl\ 2c1xq Defining 2 = rr' exp (/ £), then performing contour and then integrate deformations in the z-plane to the real axis, en j d r S/ip • V^+ circling branch points at z =0 and z = min(r, /■'), leads to the following integral when we choose over all r and 0, and — Az/2 ~P\Bl\2c2 x0 1 . . 4öf, Qh = 0 (15) -< F0J //0 + 1 + 2 Z r0 f2//cOpj Az 2m / co >o # 0 (w — cox) (12) with Z = 8/3 7t. Q t = <7, B0/Mj c. If there are no cold ions present, bh = 1 and stability 2 Z r0 Q 2 We also observe that if we are to consider a more requires (assuming - standard case of an extended plasma, where Az P r0, the factor Z is easily calculated. We use co;2i Az _ 2M hc2 \ d3r Nh(r,z) H. L. Berk and H. V. Wong • An Arbitrary Larmor Radius Plasma 1213 where we used that r0 = 2v± /,/ß/, in an ideal Migma = e - Q e , distribution. We note that the use of Z = 8/3 7i, dG dH0 0 r . . 9 rather than the Z for the cylindrical limit Qx = ~ r r = x + J d y/p¥ + (Z — Az/r0) gives a higher density threshold for ZP* dP, dH $ dP, instability. Using the Z of the cylindrical limit = x~Qx- would produce previously derived dispersion (20) relations as found in [1] and [15]. The new Hamiltonian H (Ph Q,) is obtained by IV. Derivation of Equations using the transformation to express H in terms of P„ Qi- We now derive the perturbed response of the dis H(Pi, Qd = H = H0(Pw, Pe, Px, Qx) + ..., (21) tribution /, to the rigid displacement perturbation. To derive the response it is convenient to define where higher order terms involving derivatives with new canonical variables that make use of the respect to Qx are neglected. adiabatic invariant Pv as one of the canonical The equation of motion may be approximated by momenta. We therefore consider the transformation dHo 1 = pv, y/,pe,9,px,x -> Px, Qv,?o, Qe,Px, Qx 0/V dy/ defined by the generating function B Pv x G (P¥, Pb, Px, yt, 0,x) = i P¥di//+ Peq+ Pxx, (17) dH0 dy/ where (18) r2 B2 py Pe~~¥ dH0 dy/ P*B< P2 b 2± jl H q(Pv, Pe,Px,x)~ * x dPx v r2 B2 py M 2m r2 2m P¥ = 0, Pv= § P A HO' p e, Px, V,x) dy/. (19) P e= 0 , Pv is an adiabatic invariant defined by integrating p v over one period in yj holding x constant. It is P = (22) explicitly evaluated in (A.7) of the Appendix. X 0 0 / Equation (19) determines the function H 0 = Ho{Pv, Pe, Px, x)- The equilibrium magnetic field where Qv = — Q ¥, etc. varies slowly in x- The transformation equations are The equilibrium distribution function F0 = F0(H q, P0, Pv) now satisfies the equation S G P¥ = — = p v/(H0,P e,P x,y/), [£o,tfo] = 0, (23) dG n where we use the standard Poisson bracket notation. Pe - ~ d ö - Pl” Now we consider the linearized equation for /, the perturbed distribution, with dG d H0 d V 0 V PV ~ -- = Py H------J dy/pv + — J d Now taking into account the variations of magnetic field, we find d \ c pv r (p0 i c I e \cp0 dr W\ = ~ — \------+ — \Pe — V — ) e ,col+'1 dt { co r0 co \ c ) r0 oy/ 2 P e - e~V c rtp0e -iu>t+i9 r2B 2 1 0ß pi 0 ■pl + — — + — B — B\ co r o M m r2 B 0 y/ M dy/ J i c r cp0 e,e 1(01 pv c^J \ I dr 1 . , ,, d I dr 1 -----™ ---- > )rB\------_ r3^2 (28) co r0 Mr I \ dy/ rB j dy/\dy/ r B jJ where H. L. Berk and H. V. Wong • An Arbitrary Larmor Radius Plasma 1215 The second, third, and fourth terms on the right-hand side of (28) are all of order curvature. This is a direct consequence of choosing nearly rigid perturbation of the form prescribed by (24). Since dr 1 1 1 1 B r Br Lp ^ H-- ~ x r --- dy/ rB rB : rB 2r Bz B: B. we will neglect the last term in (28). The method of integration can be continued, and if terms of higher order in the smallness parameters co/Q, cox/Q , xr, Az/r are neglected, where cox is the axial bounce frequency, it may also be verified that Pe~ ~ ¥ r con_____ dB ^id-icot ü ■ ,____ 1 w 2 = y r ~ P A P e - ~ m — dr 1 4 e co \ r0B2 dy/ 2 e co \ c J r0 B oy/ Pe~~¥) M e 2 I . px d c J n r 2 n or -I----- 1 — j Q) + B ---— - tv3, (29) e co \ M dx r2 B2 r0 Py/ r0 dy/ _ £ \3 / _ £ \2 1 c2 repo dB . J \Pe c ¥) V 0 c 7 / r2 B2 B , / e = ------— ------a *0 - ,w t {_ ------!— + j p 1------_|_ ------p i ------p 2 p ------y 4 e co r0 B dy/ M r* B v Mr2 M * MyvY c and that 2 e \ \Pe 11 d i e 3 cpo dB , + ... . W = ------—----eie-iwi ^ ( Pe~c ^+ r3 B2pl +irB pl^pe-~Vj + ^ J dt \ 4e2 co r0 B 2 dy/ (30) We may therefore combine (26)-(30) to obtain //, = — ^ , (0) + ^ , (1) = -ico ^ , (0) -I- [ ^H] (0), + i(I), (31) d t where 1216 H. L. Berk and H. V. Wong • An Arbitrary Larmor Radius Plasma , 2 -j P o- -V ... < i e , py d wnr c 1 dB rcp0 c p jl d \ r = gid-iiot' (33) This equation is valid for arbitrary Larmor radii. where Q ¥ is defined in Eq. (22) and We also note that as we discuss only flute-like difj B* = PX. perturbations,t K ♦ • forf whichU- u — d------r(Po(x) = 0, n we avoid Qxx = ^ v vPx(HQ,pv,P 1 v e,x )^rl -B~ r2 ^p¥^ M $X >'o a large contribution to the variational form. Let Let y/=y/Q + Si//, f=g- lF a,#f°>b (34) where Substituting this expression for f in (25). we obtain rL - r d~B\ - ic o g + [g,H] + [F0, ^ ] = 0, (35) V ° = j B 0, 6V = - — — where we used the Jacobi identity Expanding about y/ = y/0 and B = B0, we obtain [A, [B, C]] + [C, [A, B]] [C, +A]] [B, =0 rB p v= rB pov + rB bp0v + ... , and the equilibrium condition where [F0,//] = 0. This equation fo g is most conveniently solved by evaluating the Poisson brackets in the trans Pe~ ^ Vo B2 Px formed canonical variables Ph Qt. r2B2pL=2M Hn 2 M r 2 M The equilibrium Hamiltonian H is approximat ed by H = H 0(PV/, Pe,P x, Qx) + ... e d by/ rB Spot// — | 7 by/ — + 4 -j-r Px rBpoy. while 0/\ F0 = F0(H0,P v,P g), Qo may then be expressed as = g (Ho, Py, Pß, Qi», Qv) e~ •irP\ 0 dH, Qe ~ ~ dy/p¥ = Ü ¥ 9 = Q^(Oo + 6] + ...), (36) dPft (40) = e ~i w t+ iQ e + i Qe ^ (37) en = - d r r B p o ¥ , where ^ (1) expressed in terms of the transformed dP a variables may be approximated by e Öy/ d r --- rB p 0v [2MH0 - B2P2(H0, Pv. P„, * )] - )- dPa 2 co MB 0 0 r , 4 by/ P s -t — § d r~ jjrB p 0v dB r dFo0 . . dF(] mor radius. 9o is essentially the change in the = I l CO T T T " + I + 5 y /D g il dH n dP, dP¥ 0£„ azimuthal angle, neglecting the effects of curvature. H. L. Berk and H. V. Wong • An Arbitrary Larmor Radius Plasma 1217 in one “radial” bounce period. Thus, one can expect order is independent of /, and subsequently we shall that 90 = 0 or 2 n depending on whether the particle neglect the small / dependent terms, including the trajectory does or does not encircle the axis of last term in (43). symmetry (see Appendix A for a rigorous demon We can therefore average (43) first in the variable stration). The expression for 9\ is evaluated in Qy and then in Qx to obtain Appendix A (see (A. 10)). Let h = h + ..., dFn j ei80Q4, = ]t + . 0FO 9 ^ () i ei0oQ* + iQt> (4 2 ) (—i at + i cox) h = \i co + — Q ¥ dP¥ dH0 Q ¥ dPv Substituting in (39), we obtain dF0 = dF() -I- / (45) , . - • 0 h dh — i co h + i 9\ Q w h + Qy —--- 1- Q dPe, dP„ 1 ¥ * x dQ, '" 8 Qv where 0Fn ZQ* ^(1)e‘8oM — W CO Q ¥ dP¥ 0//o (46) dQx r d Q ¥ dF0 + i ^(1)e‘8oM + iöQd Qx Qv dPe ’ dP¥ dQ, Q, 0 1 dF0 ^ ( • ) e i0o(v) + iöQ0 ^ (4 3 ) = ~Q X q ¥ = 9 = dQx Q ¥ dPv ^Qx d Qx where J Qx (47), (48) d Qx Q¥ + Qe = 90 (t//) + ÖQe, Q¥ 9}. (49) Oo(y/) = - j dvßoy,, In summary, the perturbed distribution function /is dP6 f = g-[F0,^ ] , (50) 0 v 0Qd = - j dy/öpoy (44) dP 8 Fq m 1 dF0 9 = ^ ~ ^1( ° + ----- 1 ^1(1) + h e~i6aQ* +I U q - 1 CO t 8 H 0 Q ¥ 8 P¥ (51) jdy/öpoy J dy/py dP dH o The perturbed charge density q is + Q= X *S dh-f. (52) dH n Substitution in Poisson’s equation yields the self- We shall now neglect 0Qe/90(y/) in the evaluation consistent equation for the perturbed electrostatic of (43). potential cp: We consider the limit where V2(p=-AnY,e\^vf. (53) dh . dh_ As already discussed, instead of evaluating the Q \— i co h + i 6\ Q¥ h | "8 (2 , perturbed charge density directly, it is more con venient to construct a quadratic variational form that is, the radial bounce frequency is larger than since / will need to be known only to first order in the parallel bounce frequency, and the perturbation the curvature. frequency and curvature drift frequency are smaller We can construct a quadratic variational form if Qe 8 F0 we multiply (59) by the adjoint potential cp+ and than both. We note that ^ (1) to leading Q„, dP„t integrate over all space. 1218 H. L. Berk and H. V. Wong • An Arbitrary Larmor Radius Plasma With the boundary terms set equal to zero The adjoint equation forH\ analogous to (31) is because of the boundary conditions, we obtain — JdVV^.V^ = < / « !> H ' = i “ ^ (0,+ + [ ^ ,0U ] + *1(0,+ - (55) 4 n Thus (F 0[Hf, ^1(0)]> + 7 - 1 d 3/- Vcp • V^+ =(F 0 [Hf, ^ (0)]> +(F0 [tff\ °)+]>+ ( ( ^ + T T ^ I 4 7r \ \ 0 H 0 Q ¥ d P v ^ 0 +j_ 0 M +/wi._*wi Stf0 ß „ SPr j \ a/>„ dF,; (57) (CO - co*) Substituting for H\, <^(0) + and we obtain Me2 ,3 , \Pe—c V 1 dB + < ^ 0-- \r2B2p l + (58) e co \ r i-Q B2 dip 2 0 5 (F0[ ^ \ ^ +])=(F0c 2 -1 P0 --V C2 2 0£ - 2 ( F0^ ß - \ + I tf„ + ' ' 2M \~W ~d^ H. L. Berk and H. V. Wong • An Arbitrary Larmor Radius Plasma 1219 where we neglect the last two terms in (58) since xo is the curvature at rQ, rL is the Larmor radius, they are higher order in co/£20 1. and rG is the distance of the guiding center from the In evaluating the integrals to determine /, we axis of symmetry. neglect dQe in the exponent and axial inhomogene The curvature drift frequency cox is (see Appen ities in the magnetic field B. We then obtain (see dix A, (A.9)) Appendix A, (A. 11)) B lP 2 H + (65) MQ q ro \ 2M d Qx dy/ — ^ 0 )ei0o From (19), we deduce that r2 B2 p v / = dQx r dy/ Spy Qx r2B2pv dPx(Ho,Pe,Pv,x) dP„ dP» Spy 0 Py C Xp(p 0 BpP2x Ho + re (61) co rlB o 2 M 9Px_ rp d2B\ dPv dp* where xq = — (62) § dy/ 2 B0 dz2 0/\ and therefore T. 2 _r2 , 2 Pa 1 G - r L + (63) dPx - dPx x - 0 — ^ = 0. (66) dP¥ dP, J ± P2 ri = H n (64) 2M x Similarly, 3 Px 1 d p x + — r-1 =0. (67) n ° - m c ’ ^ ■ J e / V J e, 0//n dP„ Thus, the kinetic term in (57) may be integrated by parts to obtain 0FO , 1 dF0 \ , I dF0 x dF0 IF CO 1 ----- 1------— +' dH0 dP¥) \dPe dPvj J (co - cox) IF IF = - < co F, - { F t dHn \ co — co* P». P, dPa I CO — 0).„ Ho, Pi , 2 I //0 H— -—- (68 ) 2 I 0 2 M IF where the term involving may be neglected since it is of higher order in co/Q0- 0 / / o I CO — CO;. For the same reason, the third term in (5) is negligible. 1220 H. L. Berk and H. V. Wong • An Arbitrary Larmor Radius Plasma Equation (57) may therefore be approximated by 1 f ^ f , 2M N c2 PlB2o\ 2c2 A) V. Conclusions axis mid-plane value, and Z is approximately given by an interpolation formula Z = 8/3 n + A z/r0 We have rigorously derived the dispersion which goes over to the correct limits if either relation for the low beta flute interchange mode to a Az/i'o 1 or Az/r0< 1. Note that for a disc shaped perturbed rigid displacement for a plasma system plasma, the stability criterion is independent of Az. where the Larmor radius is of arbitrary size and the Results for the Migma experiments are reported curvature radius is large compared to the plasma in [14]. A disc shaped deuterion plasma was formed radius. The dispersion relation is given in (13). It is with a hot particle energy of 650 keV in a 3.17 T characterized by being independent of finite magnetic field. The plasma radius is r0 = 9.3 cm Larmor radius terms, and thus is valid for any low and the axial length is Az ~ 1.0 cm. It is noted that beta system where the curvature is weak. This dis cOyJQ = 0.04 from which it can be inferred, by using persion closely resembles those derived for the dis r0 ^ 2 a h, that x0 r0 ~ 0.3. Assuming <5*=1, Z = placement mode of small [15] and finite Larmor 8/3 n, we find that the stability condition is radius systems [10], except for how the electric field j^h = J d3/- Nh(r) < J 'a = 1.1 x 1012 particles. A maxi energy couples into the problem. We note that this mum of 3.2 x 1011 particles are reported to have energy term can significantly differ in disc shape been stored Hence the lack of experimental obser and cylindrically shaped plasmas. The coupling term, vation of interchange mode is consistent with denoted by the symbol Z, is given by Z = 8/3 n for a theory. Note that the theory for a long cylinder disc shape and Z = Az/r0 for a cylindrical shape if would predict instability if JV > 10" particles. Az /-Q, where r0 is the radius and Az the axial Thus, using the correct coupling to the electric field extent. energy is important in interpreting why the Migma The stability condition is given by (16). To experiment does not exhibit an interchange in interpret this relation easily, let us assume that the stability. cold and hot ions in the system have the same Above the particle threshold. stability mass M, and the pressure of the cold component is can be obtained by introducing cold ions. Alter negligible compared to the hot component. Stability natively feedback stabilization may be effective; at then requires lower density successful capacitive feedback tech l*o I al 8<5* niques have been developed [20], > (70) r0 rp Q The former method of stabilization, that of using 1 + 2Z Az co2x cold plasma, has successfully been used in hot electron experiments, and the stability criterion where ah = r_|_ /,/ß = hot ion Larmor radius, given by (70) when ZtOpj r0/2 Q 2 Az > 1 is essential ly the same as mechanism needed to explain the coi pi i 4 7i NMc n 2 B- stability of these experiments. At such densities, stability requires N = X I d3/' N)(r) / ti r\ Az = mean ion density of all species, _ Jh 1 I ! aj t>h = -- < ------(71) J, 8 /'o öh = j d V Nh(r)/Y j 1 d 3/‘ Ni(r) = hot ion density fraction, where , Vt is the total number of ions (cold and hot). x0 1 d2£(0) More specifically, in the Migma configuration of — = ~ -Tb To) — cf^’— 1 ^ ' re^err*n^ t0 on[14], where we take all the same parameters except H. L. Berk and H. V. Wong • An Arbitrary Larmor Radius Plasma 1221 the number of particles, stability would require Note that there is several orders of magnitude in the hot particle number where straightforward stabili Öl < 0.0 1 + (72) zation of the interchange could be difficult, as the plasma will tend to shield signals from active capacitive feedback, while passive magnetic feed The main penalty with this method of stabilization back with conducting walls is insufficient to provide is that for the same stored plasma energy the stabilization. It would appear that it is then neces- collisional slowing down time of the electrons is ary to develop magnetic feedback techniques to enhanced by a factor of 100 (due to the increased stabilize the interchange mode in this intermediate factor in electron density that must be introduced to particle number regime. charge neutralize the system), thereby increasing Finally, we note that our analysis has been the input power requirements needed to achieve a predicated on the assumption that a rigid mode given stored plasma energy. perturbation approximates the actual eigenfunction Feedback stabilization [20] should be effective in of the integral operator. However, one can show stabilizing the interchange mode, especially as the that for a system that is predominantly finite interchange instability is expected to only affect the Larmor radius stabilized, that the rigid perturbation displacement mode, while shorter wavelength is not valid near marginal stability. To see this modes should be stabilized by finite Larmor radius qualitatively, we note that a symbolic representation effect. However, it has been noted [21, 22] that there for the response to the interchange mode is of the are unstable density bands present even with strong form finite Larmor radius effects. Feedback by capacitive coupling should be effective if Q 2/co^ > 1, but the COr CO; o j x Q i 2 Z + ^ 4 + cp — 0, plasma may shield external electric potential signals Q j co (co — cox) (73) when ß 2/ojpj ^ 1. Further studies of this problem are needed. where &>* is an operator with a magnitude given by Conducting wall stabilization, through the mag co* ~ ahQi/rp, with the annihilation property that co COi netic interaction of the plasma with its image If ^ P 1, and i f ---r > 1, then —— current arising from the plasma perturbation, can Z Q f xo i p co stabilize [7] the interchange mode and other negative dominates Eq. (73) for co ~ (coxQ ,)y2, unless a per energy modes [8], such as the precessional mode. turbation proportional to re'9 were taken, where However, to be effective a substantial plasma beta upon the d> term is annihilated. This observation is required. Very roughly, stabilization from a justifies the rigid mode perturbation at high density. conducting wall situated just outside the plasma However, near marginal stability, where co = cox, radius requires the cbf/co term in Eq. (73) is comparable to the last term for a migma configuration. Hence, one cannot ß > i x0 r0 = ßCT justify choosing a rigid eigenmode perturbation. with i a numerical factor of <^(1), for which a Thus, further refinement of the theory is needed to careful calculation still needs to be performed. For find the condition for the onset of instability. definitiveness, let us consider ßcr=0.2. Then an However, it is clear that at a high enough density estimation of the number of hot particles, ■yfi, the unstable predictions of our theory are justified. needed to achieve wall stabilization from magnetic interaction is roughly Acknowledgements We would like to thank Prof. M. N. Rosenbluth ß cr B q /-q A z for useful comments. This paper is dedicated to — 8 E, D. Pfirsch, on the occasion of his sixtieth birthday. We note that he has been a longtime advocate of where Eh is the energy of the hot particles. For the the careful application of Hamiltonian formalism Migma parameters considered here we require for plasma theory, and we are pleased to provide a Az h > 3x 101 particles. concrete example where such a formalism has been essential in solving our problem. This work was 1222 H. L. Berk and H. V. Wong • An Arbitrary Larmor Radius Plasma supported by the U.S. Department of Energy con Let Z = el Appendix A. (A.4) Evaluation of Various Phase Space Integrals MQ rG rL dZ (Z 2 - 1)‘ $ — 4/ The equilibrium particle trajectories may be c z2[z+—\iz + rL fL r Gi approximated by the Hamiltonian H0(P^, Pe, Px, x) defined by Eq. (19). The “radial” momentum is If rL/rG < 1 (> 1), the poles at Z = 0 and Z = Pv{Hq, P0, Px, y/, x) defined by (18). rL/rG (rG/rL) contribute to the contour integral: If the field line curvature is neglected and the magnetic field is approximated by B = B (z)a and , A j M Q 7t r i , rL/rG < 1, y/ = j B(z) r2, the radial momentum is 3> dy/pov= ( . „ 2 / ^ , \ MQ 7i rG, rL / rG > 1 . I (Pe — eB r2/2c)2 J 1/2 rBfi0w=\2MH0- — --- 5 B Px Note that, if rL/rG < 1, this is the usual expression (A. 1) for the adiabatic invariant, but a different form from the conventional one arises if rL/rG > 1. The turning points (where pov = 0 in (A.l)) of the The change in the azimuthal angle 90 (Eq. (41)), trajectories occur at neglecting curvature, in one “radial” bounce period is ± _ , 2 , 2Pec\'n r = r- = W+— ± rL , Pg> 0, ’ o - (A.5) dP, 2\Pg\c 1/2 0 Pe> 0 r = r± = rL±\ ri Pe< 0, cB 2 7r Pp < 0 (A.6) where the Larmor radius rL is We have thereby explicitly confirmed the intuitive result of the text. rL = ~~ß (2MH q — P2 B2)U1. Now let us consider the change in the azimuthal eB x angle Q\ induced by the field line curvature in one Particles with Pe< 0 encircle the axis of symmetry “radial” bounce. From (41) we have while particles with Pe < 0 do not. The radial position of the particle guiding center rG is Ö2 r e 0] = ^ 2 J dr~ dy/rBpo¥ 1/2 o r e c 2 P g C rG = \rl + eB 0 0 ^v —§ dr ÖB rp0y, (AJ) In order to evaluate integrals over a closed particle trajectory, it is convenient to introduce the Larmor where phase angle cp given by ?i c /-2 dzB öy/ = , 0B = r2 = rG +r[ + 2rG rL cos tp. (A.2) 16 d r2 ’ 4 dz2 ‘ Thus, for example, the integral for the lowest order Since adiabatic invariant, (19), with use of (A. 1)-(A.3), § dr r5 B p 0v/ is given by = § d(p M Q rG rl sin2 6\ may be evaluated: absence of curvature: n e M Q d2BX ö .= rh rl (rG + d ) 16c dz2 dP2e i B p y / r B p y e 8 (po Xq C B 2P 2x The local curvature drift frequency wcurv is Ho + ■ co rl B 2M Q ^ P2B2 toCurv ' H0+— — I dcp U T 2 Tt M r0 Q 0 2M f — (,r g + >'l cos cp - i rL sin cp) 2 n >o d2B\ where x0 = is the local curvature at n>k>c I H + B2 P\ 2 B dz2 r = r0. 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