Some Stable Hydromagnetic Equilibria

Transcription

1 P/1875 USA Some Stable Hydromagnetic Equilibria By J. L. Johnson,* C. R. Oberman^ R. M. Kulsrud and E. A. Frieman The problem of hydromagnetic instability in connection with the controlled thermonuclear program was first raised in 1951 by Spitzer, who gave arguments that the pinch discharge should be unstable. A detailed analysis by Kruskal and Schwarzschild x showed that this configuration should, in fact, be unstable. This problem was raised again in 1954 by Teller 2 who gave strong intuitive reasons for suspecting all the devices to be unstable. An energy principle 3 for the treatment of hydromagnetic instabilities was developed and from it a criterion for the stability of general axisymmetric equilibria was obtained by the summer of It was seen by this criterion that the stellarator is unstable with respect to the " interchange " instability. This " interchange " can be visualized by considering an axisymmetric equilibrium in which the plasma and the magnetic field are imbedded in each other, and in which the magnetic lines of force lie entirely in r, z planes (in the usual cylindrical coordinate system). A displacement of the plasma, which interchanges lines of force in such a way that the magnetic field and its magnetic energy are unchanged, can be carried out by first specifying, on a cross section with z = constant, a mapping of the magnetic lines of force into themselves, and then extending this displacement throughout the volume by making each line of force continue to go into the same line as that assigned on this cross section. Since the magnetic field strength is given by the density of lines, the magnetic energy is clearly unchanged. However the material energy may increase or decrease. Consider a displacement in which two flux shells with equal amounts of flux dip are interchanged. If the volumes in these flux shells are not equal the plasma in one shell will expand and that in the other will compress. Thus the resulting change in the total compressional energy of the plasma will depend on the pressure profile and is given, 3 up to a positive constant factor, by (V"/V) [V"/V + fi'lyp]* primes denote differentiation with respect to \p, V(tp) is the volume in a flux tube with flux y) and y is the ratio of the specific heats of the gas. If the magnetic field lines are not straight * On loan from Westinghouse Electric Corporation. t Project Matterhorn, Princeton University, Princeton, N. J. and if the material pressure is sufficiently small, then V" is positive and non-zero while the factor in parenthesis goes to negative infinity at the boundary of the confined plasma so that the system is unstable. If the magnetic lines of force are perfectly straight as in an infinite cylinder and p' is negative, V" is negative and the system is stable with respect to this interchange. However, displacements can be found with respect to which the infinite cylinder is neutral. Since in the stellarator there will be small ripples in the magnetic lines of force due to the discreteness of the coils which produce the field, we expect the stellarator to be unstable. This argument (for instability) applies in general to more complicated situations such as stellarators, since nothing would prevent us from carrying out the " interchange " displacement unless the system is such that the magnetic lines of force return on themselves. In the latter case, our original assignment of the mapping of lines into lines would not match when we bring the lines around on to themselves. Nevertheless, one might ask whether even in re-entrant systems, interchanges can be constructed which lower the energy of the plasma and leave the magnetic field unchanged. The answer can be given in terms of the rotational transform angle i, á which can be defined on each magnetic surface as 2n times the average number of turns about the magnetic axis which a line of force makes as it is followed once around the system. (The magnetic axis is that line of force which closes on itself.) We consider first a system in which i is constant over the cross section; i.e., independent of theflux %p within each magnetic surface. If in some such system every magnetic line of force returns on itself after n times around the system (i = 2яш/п for every ip), an interchange can be selected so localized that it need not match until the wth time around, when it matches perfectly. If i is not a rational multiple of 2л; the matching can never be achieved exactly, but it can be so closely approximated by a rational multiple of 2n that the matching comes arbitrarily close. One can choose the displacement sufficiently close to an interchange to make the change in the magnetic energy negligible. Thus for any t, constant on all flux tubes, an effective interchange can be carried out with respect to which the system is unstable. 198

2 STABLE EQUILIBRIA 199 The situation is different if i changes with changing ip. In this case if we try to construct an interchange, the matching becomes worse and worse as we carry the mapping around the system. The first diagram in Fig. 1 indicates an attempt at assigning on a cross section, a mapping of the magnetic lines of force into each other indicated by the flow pattern. After carrying this mapping along the lines once around the system, the flow diagram becomes that of the second diagram. Here it is assumed that i is larger on the outer surfaces than on the inner ones, and since the lines are the same in the flow pattern, the outer lines move farther, shearing the flow pattern. After a second time around the system, the flow pattern (for the mapping) becomes that illustrated in the third diagram. It is clear that the possibility of matching becomes more and more hopeless as we carry the mapping around more and more times. Further, even by carrying out displacements differing from the interchange which manage to produce matching, the magnetic energy cannot be made negligible. Thus an i which depends on ip inhibits the unstable interchanges. Achievement of stabilization by use of " shear " magnetic fields, whose direction rotates out of the r,b plane with increasing separation r from the magnetic axis, was suggested by Spitzer 5 in A preliminary calculation, 6 in which the twisting of the magnetic field lines is produced by an axial current on the surface of the plasma, was encouraging. Stabilization by means of plasma currents is not applicable for a steady state machine (such as a stellarator) since an emf is necessary to drive the stabilization current. For this reason Spitzer 5 proposed that the stabilizing field be produced by pole pieces placed at right angles to the plasma. These pole pieces would be rotated in space in order to make the magnetic fielddepend on z as well as r and в, r, 0, and z are cylindrical coordinates with axes parallel to the main magnetic field. (A dependence on z is necessary in order to obtain an equilibrium situation.) These fields are to lowest order (in themselves) helically invariant and proportional to sin(0 hz) or cos(0 hz). It can be seen that stabilizing fields that are the same as these to lowest order can equally well be produced by wrapping two wires helically around the system, with currents flowing in opposite directions in the wires. It had been pointed out earlier (in February 1955) by Koenig 7 that a helical magnetic field of this type would produce a rotational transform, but it was not realized until later that this configuration would have a stabilizing effect because of the variation of transform Figure 1. Distortion of the flow pattern if di/c/ф > 0 angle with radial distance. We examined the nature of these equilibria and obtained necessary and sufficient conditions 8 for the system to be stable to all perturbations \ which are periodic over the helical period 2njh. In the calculation, the equilibria and the perturbations were expanded in the parameters P and a, /? characterizes the magnitude of the material pressure and ô characterizes the magnitude of the crossed fields. If /5 is of order ô the system is unstable. If f} ~ ô 2 a critical condition for stability is obtained, while if /? is proportional to a higher power of ô the system is stable. Spitzer 9 pointed out in 1956 that the rate of change of the pitch of the magnetic lines of force, with increasing distance from the axis, could be increased by considering more general helically invariant fields which vary as sin(z0 hz) orcos(/0 hz), since if their pitch is small they vary to lowest order as r 1-1. Clearly larger values of I produce steeper dependences of i on r for the same i, but the advantage for stability of this steeper dependence is off-set by the steeper dependence of the fields on r, so that if the fields are applied by wrapping wires helically around the system some distance from the surface of the plasma, the fields in the plasma will be correspondingly smaller. In consideration of these factors it seems reasonable to suppose that either the I = 3 or 1 = 4 fields (i > ' r l ~ 2 ) are most favorable for stability, although the 1 = 2 fields produce a large rotational transform which may have advantages in the equilibrium situation. Since the imposition of externally produced helically invariant fields makes the magnetic lines of force into small helices about their original position, an " interchange " perturbation will cause a larger decrease in the plasma energy with them present. There is, therefore, a critical value of /? associated with a given helical field above which the system is inherently unstable. It is found in the minimization of ÔW that if one expresses the perturbation Ç of the equilibrium in terms of its Fourier components in expi(md + nkz), k is 2n over the length once around the system, the worst \ takes on its maximum near the radius for which i = 2nn\m. Such a radius exists for finite n and m only if i is finite or, for infinitesimal i, only if m is infinite. Since we wish to treat the stability of a system with infinitely small helically invariant fields superimposed on a finite axial field and avoid infinitely large values of ж, it is necessary to consider the tube identified over an infinite length L in the lowest order (L ~ I/o 2 ), so that i can remain finite. We therefore obtained the stability criterion for systems in which there is a superposition of helically invariant fields imposed on a large axial field, with the Ç's Fourier analysed over the length of the infinitely long machine. 30 At the same time we determined the effect of these helical fields on the long-wavelength kink instability caused by the presence of a small axial current. We therefore were able to show 10 that by wrapping wires helically around the tube and passing current through them in opposite directions in alternate

3 200 SESSION A-5 P/1875 J. L. JOHNSON et ai. wires, we can produce fields which stabilize the interchange instability. The same fields can also raise the limiting axial current which can be present without setting up a long-wavelength kink instability. Equilibrium situations which would exist in a stabilized stellarator are discussed in the next section. In the following section, the integral ÔW, which arises in the treatment of stability by the energy principle, 3 is minimized with respect to all components of the perturbation Ç except the lowest order radial component, r. Finally, the " interchange " instability and the " kink " instability are treated separately and critical conditions for stability are obtained.î EQUILIBRIUM In the last section we suggested that certain classes of equilibria might be stable. In this section we calculate these equilibria and investigate those properties, such as the rotational transform angle, 4 which are related to stability. We use the usual hydromagnetic equations, 12 assuming that the plasma is infinitely conducting, so that Vp= jxb, (1) as well as the usual Maxwell equations, V.B = 0 (2) and VxB = i> (3) must be satisfied. In order to avoid the complications which are involved in the treatment of closed curved systems such as a torus, we imagine the system to be straightened out with the ends identified. In the stellarator, a large axial magnetic field is employed to confine the plasma. The system is made unstable with respect to " interchanges " by bulges which, for instance, may be due to the finite spacing between the coils which produce the main field. In order to obtain stability we superimpose on this field a helically invariant field by wrapping wires in a helical fashion around the tube and passing currents of alternate sign through them. Clearly an integral number of periods of thisfieldmust fit between the identified ends of the tube. Since it is difficult to solve (1), (2), and (3) for this system if plasma is present (V/> ф 0), we will assume that the amplitude of this helically invariant field is small compared to the main axial field. Further it will be shown that a small helical field can stabilize the system only if little plasma is present. We therefore expand all field quantities as power series in the small parameters ô and p which give a measure to the magnitude of the helical fields and the amount of plasma present, respectively. We define ô and p as follows: The zero-order situation consists of a perfectly conducting right circular cylinder of radius S in which * Rationalized Gaussian units with с = 1 are used throughout this paper. A more detailed treatment of this problem may be found in the Project Matterhorn report PM-S there is a uniform axial magnetic field 5 and in which no plasma is present. For the purpose of this calculation, the fields produced by the helically invariant wires are considered to be produced by deforming this cylindrical boundary with the condition that the normal component of the magnetic field must remain zero. In particular, we consider the simple deformation and define,z) = S -\- a s cos u, и = 16 hz, ô = a s /S. We will later generalize to the case many helically invariant deformations are superimposed. For simplicity, no two different values of I are associated with the same helical wavelength, 2n/h. Note that " bulges " in the magnetic field correspond to the choice I 0. We define the quantity /? as (4) (5) (6) p = що)/в<", (7) ^>(0) is the material pressure at the axis of the tube. (Note that we are using rationalized Gaussian units with с = 1 so that the usual factor An in the numerator of the right-hand side of (7) is omitted. Also, as we increase the value of the pressure at the axis, keeping д fixed, we cannot expect the pressure to scale linearly since the magnetic surfaces are changing. Thus p is not directly proportional to/3 and we expect фр 8, фрр 8,... to exist. The pressure must be zero at the surface of the plasma since it is contained by the magnetic field.) Since we do not take diffusion into account, the pressure distribution is arbitrary, subject only to the condition = 0, (8) which easily follows from (1). One other parameter must be specified to determine the system, the net axial current. For steady state operation it is necessary to require that 13 J fj z rdrde = 0, the integral being taken over any cross section contained within a magnetic surface ; that is, there shall be no net axial current in the volume contained by any magnetic suri ace. During the initial time of operation of the stellarator, before it has been heated to its steady-state condition, an axial current is set up in the plasma to heat it ohmically. 14 Since during ohmic heating all inertial terms are small, we may calculate the quantities by the equilibrium equations (1), (2), and (3). Further, since these change slowly in time, we may in treating stability regard the situation as an equilibrium one. We introduce the parameter r\ to represent the magnitude of this ohmic heating current. Due to the nature of our zero-order situation, the \ can be made independent of the pressure.

4 STABLE EQUILIBRIA 201 We assume in this work that two distinct regions are present. All material pressure and currents in the equilibrium situation are confined inside an inner region bounded by the surface r = R + Q* + Q** + QP* + QV* -f... (10) In the outer region bounded by this surface and the one given by (4), only vacuum fields exist. The boundary conditions which must be imposed are B.n = 0 (11) on both the outer surface and the interface (10), and ; p + = о (12) across the interface. Note that (11) determines the interface (10). Here n is the normal to the surface and [ ] indicates the jump in the direction of n. In the discussions of stability which follow we shall consider two physically different situations which have the same equilibrium quantities. In the first we assume that there are enough particles present in the exterior region so that it is infinitely conducting and therefore can support a current when it is perturbed. However, no currents are assumed in the external region in the equilibrium. Since not enough particles are present to represent a significant material pressure, we denote the exterior region a " pressureless plasma". In the second situation, the external region is assumed to be a vacuum and the magnetic field is determined by the usual boundary conditions. 3 For equilibria to exist, p and j must be constant on magnetic surfaces. However, in this expansion technique magnetic surfaces first appear in the equations in the ô 2 order a rotational transform is present, and hence the first-order quantities pp and j 7? must be influenced by second-order considerations. Mathematically, the higher-order equations are incompatible unless the first order quantities satisfy certain relationships. In the expansion of the equilibria we find that in calculating the first-order quantities, pp and jv are arbitrary functions of r and в. However, if pp and j 9? are chosen as functions of r alone, the higher-order quantities can be calculated satisfactorily except in the case in which two helically invariant fields with different values of I have the same period, in which case pp and p must depend on в (in a well-determined way) 15 in order that the second-order quantities satisfy the periodicity conditions. This removal of a degeneracy is similar to that in the expansion of the case a helically invariant field is applied to a toroidal system. 16 For simplicity, we exclude in this paper the case two helically invariant fields with different values of / have the same period. We use a prime to denote differentiation with respect to the argument and use the usual cylindrical coordinate system. Analytic continuation of the fields to the perturbed surface is assumed. The fields are thus to order zero: B = e z B (a constant), = 0, p = 0; (0 < r < S), (13) and to order /? : B = 0, }P = 0, pfi = 0; (R<r<S) (14) / (r/r) is arbitrary, such that / (0) = 0, / (1) = 1, \f[r/r) < 1. The quantity /3 is given by Eq. (7). To order rj, R Г 7]B [ i r\ R \ g R [^ \RR {0<r<R) rjb R ' \a'l J _tf = e z l], * = 0; (R<r <S) (15) g (r/r) is arbitrary, such that g(0) =0, g (I) = 1. The quantity rj = Bv(R)/B. To order ô, B»= %s Цх.) COS U s T^Y-COS u 9 ] ; I l\ X s) J (0 < Y < S) = 0; p' = 0. (16) The perturbed surface is 5 + 2s <?s d cos u, the ô s = cr 9 8 /S are all of the same order, ô u 8 = h s z, x s = h s r, X s ~h s S, Ij(x) is the modified Bessel function of the first kind, 17 I l (x) = Ii(x)/xlï(x), and the subscript 5 has been dropped from the Г s when they are themselves used as subscripts or superscripts. For the purpose of the following stability analysis only these lowest-order fields need be written down since all the properties of the higher-order fields which must be used can be obtained easily from these by (1), (2) and (3). Nevertheless, higher-order fields must be obtained to understand the equilibrium situation. These are given in Appendix I. We now proceed to find the rotational transform angle i in the blunt, but straightforward and pictorial, way by finding the average angle of rotation 13 of a magnetic line of force about the magnetic axis as one proceeds along the z axis. Since the length over which we specify the average angle of rotation is arbitrary we shall choose it to be the period of the helically invariant field 2n/h, and denote this average angle by ih. (The rotational transform angle i is usually defined over the identification length of the system.)

5 202 SESSION A-5 P/1875 J. L JOHNSON et al. We first compute the angle of rotation Ав for a length 5 such that a tube of force passing through a fixed constant-^ cross section of a magnetic surface shall have sampled once every point in this cross section when it is screw-projected along the z axis at the helical rate, so that (h/l)z - Ad = In. (17) The magnetic surfaces have cross sections perpendicular to the z axis which rotate about this axis at the constant rate, dd/dz = h/l, (18) as we proceed along this axis, as the lines of force rotate around the z axis at a much slower average rate dd/dz ~ <5 2. (19) Hence we must proceed in the z direction until a line of force first comes back to the same relative position in a cross section that it originally occupied. We are now assured that the average angle of rotation for all lines in a magnetic surface is the same as that of any one line. We thus have 13 i h = (Ад/z) (2n/h) + 2nn, (w = 0, ± 1, ± 2,...) (20) or, making use of (17) and adopting the convention n = 0, = 2n/l. (21) We can use (17) and (21) to compute. Since the fields are known as a power series in the expansion parameters, the equations of lines of force and thus will be given as power series in the expansion parameters. We find to the lowest significant order We note from (22) and (23) that V"(ip) is finite even for / = 0. If only one helically invariant field is present, an important and elegant formulation of the problem of obtaining equilibria, which is especially valuable when solutions for finite /? and ô are desired, can be given by utilizing the two-dimensional nature 18 of the problem. We introduce the scalar functions Y and /, such that We find from (25) B.VY = 0 f = 1В г Л- hrb e. that Ydldu = drb r, dy/dr = - d(ib e - hrb z ), (25) (26) (27) и is given by (5) and д is an arbitrary function. It follows from (2) that д is a function of Y only. For simplicity we choose д Ç ) = 1, any other choice merely constituting a relabeling of the constant Y surfaces. It follows from (1), (3) and (27) that / and the pressure ft are functions of Y only. The fields and currents are then given by В X /' hrf-l(3y/dr) 1 P + [hrf Z 2 + 2h 2 rlf 2 + [Z 2 +(Ar) 2 ] 2 J 2hl 2 f (28) (29) and + (P + x*)i'(x)*] (22) 2hlf I 2 + (hr) 2 [I 2 + (^) 2 ] 2 _ 1 д г д Id 2 = Tâ7Z 2 + (hr) 2 l)r + "~r 2 du 2 (31) [3-2(2 + Z 2 + % 2 )l l (x) + (3Z 2 + 2x 2 )f(x) 2 ]. (23) For convenience, Lh 88 and Rdi^88 /dr are tabulated for the limit л;< 1 < 1 in Table 1 for several values of Z. We now see from Table 1 that the rotational transform angle actually does depend on radius and therefore the helical wires do produce a varying rotational transform as surmised in the introduction. Higher orders in the power series representation of ih as well as relations between and other important equilibrium quantities 13 have been obtained. 11 In particular, if V(y)) is the volume enclosed by a surface of longitudinal flux tp, d krb ' (24) Here k = 2л/L, L is the length of the machine. and the prime denotes differentiation with respect to Y. Clearly Y and /, and therefore thefields,are determined by (30) if ^>(Y) and a condition on the axial current such as (9) are given. We can show u that Y is a flux related to ip, the flux the long way, and %, the flux the short way over one helical period around the system 13 by 2n MINIMIZATION OF ÔW (32) We determine whether or not our magnetostatic equilibria are stable by means of the energy principle, 3 which reduces the question of stability to the problem : Can the quadratic functional of Ç,

6 Table 1. 2ÔW = Limiting Values oft and di/dr for Small hr i h 88 (per length 2n h) Rdi h 88 jdr (per length 2n h) STABLE EQUILIBRIA 203, (33) (34) be made negative for any choice of Ç(r)? Here B, j, and p represent the equilibrium values, Ç is an arbitrary virtual displacement from the equilibrium, subject only to the condition Ç n = 0 (35) on a rigid boundary and ô W is the resulting second order (in Ç) change in the potential energy. If ôw can be made negative for some Ç, it can be made negative for a normal mode Ç, some potential energy can be converted into kinetic energy and the system is unstable. We shall first order our equilibrium parameters in such a way that we can carry through the stability analysis in terms of a single parameter X. We shall then expand áíf as a power series in X and carry through the minimization order by order. We shall find that the zero order ÔW is non-negative and can be made to vanish, the first order is then automatically zero, the second order is again non-negative and can be made to vanish, and the third order is again automatically zero. The fourth order ÔW contains the desired critical condition. We minimize it here over all components of \ except >. In the next section we carry through this final minimization for specific cases. We obtained equilibria in the last section as power series in several independent parameters. It is much easier, however, to solve the stability problem for a one-parameter equilibrium, and we express our equilibrium in terms of the single expansion parameter X by assuming the ordering f) ~ r\ ~ ô 2 < > X 2. The reason for this particular choice can be seen later. It is clear that different values of X characterize a family of equilibria. A Ç which makes ÔW negative for one value of X need not make it negative for another. Since we wish to determine the stability conditions for this entire class of equilibria, we must allow % to depend arbitrarily on X as well as r. We therefore expand Ç as a power series in X. In this treatment of the stability of the stellarator, we neglect any effect of the curvature of the tube. However, to keep partially the effect of the closed machine we demand that our equilibrium and our perturbation \ be periodic over a length L = 2n/k equal to the length of the machine. We must choose kr ~ X* (R is taken finite) to keep the rotational transform angle t, defined over the whole machine, finite as X goes to zero. It should be observed that any finite situation may be approached by a system expanded in X in any number of ways. For instance, if k ~ X 2, k will be finite when X becomes finite. Alternatively, the same finite situation could have been approached by keeping k fixed (independent of X), as X is increased from zero. The best expansion is the one that leads to the most rapidly converging scheme. Since we do not examine the situation beyond the lowest significant order, we cannot apply this criterion to our scheme. However, our choice of orders leads to an expansion which remains " uniformly " valid as we arbitrarily shift our choice of orders, while another choice does not. In particular, we have shown u that the results remain valid when hr is made transcendentally small compared to X. Further, our expansion readily yields itself to physical interpretation so in some sense it is the " best " expansion. We cannot expand our Ç's at a fixed point since they will have wave lengths of order L and the expansion would not be uniform over L. To get around this difficulty we first Fourier analyze Ç in z. Thus n,s with s and n finite integers. We have shown n that the assumption that Ç can have no other Fourier components cannot affect the question of stability since the non-vanishing of such components always makes ÔW positive. Now 2*(s, n) can be expanded uniformly in X. It should be noted that the operator д/dz acting on \ can change its order. We now expand ô W as a power series in X and examine the lowest order term which gives a decisive answer as to stability. The zero-order term, ôw, is positive definite, and can be made zero by restricting % (s,n), so that Ç ± (s,»)=o, (s^o) (37) V.Ç ± = 0. (38) Here we have introduced the notation R ± = e r Ar +е в А в ; A = e z A z. (39) We temporarily relax the restriction given by (35). It will be shown that if ÔW can be made negative, the 2* which does this can be chosen to satisfy this condition to all orders. The next order term, ôw A, vanishes trivially. By performing an integration by parts we show that dw xx is positive definite. It can be made zero by choosing C ll (s,w)=0, (s^o) (40) in the region the pressure is not zero, and ishb 0^ (s,n) = [Vx (Ç xb A )] ± (s,w), (41) B V.Ç_L A = [V x (Ç X В л )]. (42) That (41) and (42) are compatible can be seen by taking the divergence of the first and the derivative

7 204 SESSION A-5 P/1875 J. L JOHNSON et ai with respect to z of the second, and subtracting. We îiave shown u that at this point to the order we are interested, we can set Ç 2 (0, n) equal to zero every with no loss in generality. We show that àw xxx is zero by two straightforward integrations by parts. In the next order we can now write = f {(QAA)2 jfi ff0 Q/A J Г { AU + j V (^0 Q^AA (V.Ç)] AAAA } dr -^)}dt, (43) U is the volume in the unperturbed cylinder and AU" is the perturbed volume. This can be minimized with respect to f z * and Ç AA ± by setting and ish (s,n) = - AA = - [Vx(Ç xb (s,n) = [Vx (Ç x B AA + I х x B A )]! (44) (45) (46) As before, (45) and (46) can be shown to be compatible. We can now use (41) to express ÔW (henceforth we delete the superscripts) in terms of Ç x only. At this point it is convenient to Fourier analyze in terms of в. We write Ç ± (0' w ) = 2 % ± {m\n) е гт (47) m and note that ÔW breaks up into a sum over m and n with the m\n th term involving only % ± (m;n). We can use (38), so that except for those terms for which m 0, ôw m. fn is a function of r {m\n) alone. Further ôw 0 ;n is positive definite, and can be made zero by setting We therefore obtain f e o(o;«) = 0. (48) 2ÔW = 1àW m., n m>0 n (49) and 2ÔW m;n = nkrb dr Sm-n /Mm;» =»"m;»»- 0 (»i;w). ;n)*, (50) (51) X [1-27 z (saf) + (Z 2 + s*h*r*)i l {shr)*\ (52) a m -,n = m 2 + X{1-2[/ 2 + {shr)*]i l {shr) + I4 l {shr)*}. (53) The desired expression for ÔW in terms of an arbitrary r Q (m',n) is given by Eq. (50). Here, as usual, the prime denotes differentiation with respect to the argument and I l (x) is given in Eq. (16). The letter I should be written as l 8 to denote that for a given helical field period it is arbitrary. We have shown u that Ç's which satisfy our minimization condition also satisfy (35) if r AA (s Ф 0, n) is accordingly restricted. We are frequently interested in equilibria in which there is a sharp discontinuity at some surface in the plasma. We can treat the stability of such equilibria by including surface terms in Eq. (33), 3 or by regarding the surface of discontinuity as a volume region in which physical quantities vary very rapidly and then letting the thickness shrink to zero. It is necessary to make the jump in J^n across the discontinuity zero to prevent cavitation or interpénétration. We have considered n an equilibrium which has a discontinuity in the pressure, a surface longitudinal current and a discontinuity in the volume longitudinal current. We find that, denoting the radius at which the discontinuity occurs by R, Eq. (50) is replaced by the more general result T S rdrll V а/лт;п i {/u m;n ) r=r. X [1-2[Z 2 + {shr)*} P{shR) + l4 l {shr) 2 ] X {3-2[2 + Z 2 + [3Z 2 + 2{shR)*] P{shR)*} - (54)

8 STABLE EQUILIBRIA 20S /u m;n, v m;n and a m]n are again given by (51), (52) and (53). In (50) and (54) we can treat u m;n as the arbitrary function in ôw m. n instead of f (m,n) with the provisions : jn (S) is zero, i vanishes at all points v does and x' is continuous v vanishes. With this in mind, we can minimize ÔW in the region R < r < S over x such that Lt(R) is prescribed. If v does not vanish in this region, we get for this part of 2dW (55) If v does vanish in this region let ar be the smallest value of r at which it does. The contribution to 2ÔW from the integral between R and ar has the minimum value m Since ju r is continuous at ar, the contribution to ÔW from the region between ar and S is not zero. Since it is positive it can be made as small as desired, for instance by taking x=0 and letting к approach infinity. (57) (58) In the preceding work we have assumed that the external region R < r < S is filled with a pressureless plasma, capable of supporting a current. The energy principle 3 can be generalized to consider equilibria in which this region is a true vacuum. Instead of minimizing Q 2 in this region, we minimize (VxA) 2, subject to the conditions nxa = (i at the plasma interface and nxa = 0 (59) (60) at the external boundary. When we carry through the minimization we find that the contribution to ôw m;n from this region is again given by Eq. (55). The difference between the two cases is that the pressureless plasma can develop a sheet current at CLR, while the vacuum cannot. The vacuum is always unstable if the corresponding pressureless plasma case is, but the converse need not be valid. We can rewrite Eq. (54) in terms of the rotational transform angle, i, di/dr, and V", as _ krb ltn;n Ô ^ *я 71 v* m;n = 2лп/т rv* 1 d "I r {tn;n) kv* «** (62) (63) v (64) and A m;n is the contribution to 26W from the external region, given by Eq. (55) if the external region is a vacuum, or by (55) or (56) if it is a pressureless plasma depending on whether v* m. yn is zero in that region or not, respectively. Here the rotational transform angles, i 88 8 from the helically invariant field which depends on u s and fl from the axial current, and the term F" V ac are computed over the length of the machine (2л/k). The terms in jp and {ВЦ are similar to those which in the axially symmetric case 3 represent the energy released by the expansion of the gas. The terms (dju/dr) 2 and ( unt/r) 2 represent the energy increase due to the charge in the magnetic field, (ôb) 2 /2. The terms in fl represent the work done by the force term j X ÔB (computed at a fixed point). The field energy terms can be made somewhat more transparent as follows: The quantity r* represents the average rate at which the magnetic lines of force turn compared to the displacement Ç, and thus it represents an effective field across the displacement. For simplicity we consider as a model a straight tube of length L, with a large uniform B Zi with a B e independent of в and z, just sufficient to produce a rotational transform angle equal to v*, but with no axial current (so that one of Maxwell's equations is violated). We find that with a displacement f r independent of z but depending on в as exp into and g 0 such that V*Ç X = 0, the fields are changed by дв 0 =-д (v*rç r LB 0 )/dr. (65) (66) Squaring (65) and (66) and adding, we have the stabilizing terms of Eq. (61). The origin of the terms is made pictorial by considering ôb r as being due to compression of the lines of force by the displacement and 6B e as being due to shearing of the lines. The 2ÔW m. n = 8л2 + dis" 4л 2 В» R d(rhv) /л т;п 2 (61)

9 206 SESSION A-5 P/1875 J. L JOHNSON et al. actual situation and this model are analogous in the sense that the model is obtained by smoothing out the ripples in the lines of force (which the displacements in the actual situation automatically take care of) and untwisting the system so that the displacement is " untwisted ". One can see, by examining (61), (62), (63) and (64), the reason for the particular ordering of the parameters which we made. If c 88 or iv were not finite over the length of the machine, one or more of the terms in v* myn would have entered ÔW in a different order. Similarly if we had not chosen /? ~ ô 2, the terms in /? would have entered in a different order. In any case, any other choice of the ordering would have yielded a critical condition in which one of the parameters does not enter, or would have made the system stable or unstable with no critical condition. STABILITY CRITERIA In the last section we carried out the minimization of ÔW with respect to all components of \ except y (w;w) ехрг(тв + nkz). Since the final minimization is too difficult to do in general, we shall in this section discuss several special cases. We treat " interchange " instabilities and their stabilization in the first part of this section and kink instabilities and their stabilization in the second part. Interchange Instability We wish to determine the critical /3 for stability against interchanges when helically invariant fields are present. We assume p = 0, a condition clearly satisfied for steady state operation of the stellarator and assume only helically invariant fields with 1 = 0 or 3 and small hr. We consider in detail only the case the plasma fills the entire tube (R = 5), so that bt(r) is zero. In Appendix II we carry out the final minimization for this situation with an arbitrary pressure distribution which we then select to maximize the critical /5. One would expect any other distribution to quickly attain this optimum distribution if sufficient material is injected. We find that the critical /9 is X [1 - (Зф/4) ln(l + 4/Зф)] (67) for the optimum pressure distribution m = 1 - (30/4) ln[l + (4/Зф)] ' = r/r, di = (68) (69) (70) Qi 8 and Qf are respectively the amplitudes of the Fourier components of the distortions of the plasma surface with wavelengths such that pi helical and qi bulge fields can fit into the machine, and f(t) is related to the pressure by Eq. (14). If no bulges (1 = 0) are present the optimum pressure distribution is parabolic. If, in addition, only one helically invariant field is present the inherent stability is given (for 1 = 3) by fi e = И#) 2. (71) Since the maximum value of Q 8 /R in a system with I = 3 is about, on the basis of the present theory, p c would not exceed about 0.1. However, the theory is presumably not valid for so large a value of either Q 8 R or of p. We have shown previously u that if a pressureless plasma exists between the plasma boundary and the perfectly conducting walls of the system, the same results are obtained. If, however, this region is a vacuum, the system is unstable for interchanges in a very thin surface layer if /'(1) is not zero. We therefore expect that the pressure distribution will adjust itself so that /'(1) is zero and the critical fi will be somewhat lower than the /? c given by Eq. (68), which would obtain if the external region were a pressureless plasma. The problem can be carried through in the same way î for the case hr is finite, any combination of helically invariant fields (with any I's) are present (subject to the condition that no two fields with different values of I have the same wavelength), and the next axial current jv is zero. We find that the optimum pressure distribution is and f (Z s th t u {t)ldt)*tdt (73) t 88 s and V" are Jogiven by Eq. (22), Eq. (24), respectively, but here evaluated over the length of the machine. We note that as the machine is made longer by adding long straight sections which contribute nothing to V" for the entire machine, one might suppose that if dijdr per unit length is kept fixed, the stability would be unchanged. On the contrary, it is necessary to increase dc/dr in proportion to the length of the machine, V" always being the same, to preserve the same critical /?. This is also clear from our intuitive picture of the interchange instability, since the interchanges can unwrap themselves in the long straight section after being curled up by the dt/dr and can thus match themselves with less increase in the change in magnetic energy. With this caution in mind, we can produce the di/dr shear by wrapping helical wires only over part of the tube and still obtain stability. In minimizing ô W we found for /?>/? c that, as /?-/?c is made smaller, the minimizing Ç's become more nearly singular in the neighborhood of the radius at which i = 2nn/m and eventually change ap- * We have not yet completed the argument in connection with (93) that ju does not vanish between a and 1 in this general case. 11

10 preciably over a region small compared to an ion Larmor radius. The minimizing Ç thus represents a motion to which the theory no longer applies since it is based on equations which assume that the ion Larmor radius is the smallest length in the system under description. Thus we cannot assert that these systems (/? > j8 c ), which are only unstable to such sharp Ç's, are definitely unstable. It is therefore of interest to ask how large /3 can become before instability sets in, when we apply the added constraint that \ varies slowly over an ion Larmor radius. This question can be answered by minimizing dw over all Ír subject to the restriction that d r /dr < ( r )maxm, 1 is a length of the order of the Larmor radius. We find 11 that the critical /3 under this restriction is, under certain circumstances, as much as twice the critical (3 derived allowing unrestricted g r. The question of the validity of this constraint on \ must be settled by a more refined theory. Kink Instability The instabilities associated with a longitudinal current have been extensively studied. 19 It is found that if the axial field is much larger than the azimuthal field, the system is, for a uniform axial current distribution, unstable with respect to minimizing Ç's which vary as expi(mo + nkz), if n/m < iv/2n < n/(m I). 11 (74) Here if i 7! (the rotational transform over the identification length, evaluated at the plasma radius R and due to the axial current) is positive, n must be negative for instability. If the external region is a pressureless plasma the system is stable for m > 1. These " kink " instabilities are not so well understood physically as interchange instabilities. We shall first make plausible the existence of the kink instability for m = 1 by a simple force picture based on the fact that the lines of force are tied to the matter. We shall then consider the stabilizing effect that stabilizing fields can have on this instability. We consider a long cylinder of pressureless plasma of radius R and length L embedded in a large axial uniform magnetic field B o and in which a small axial current of uniform density / is flowing. As usual, the ends are to be identified. The current density / produces a field B e and an t, given in the plasma by B e = 2jr, i = B e /krb 0, k = 2TT/L. (75) Note that i is constant in the plasma. Let us subject this plasma to a displacement \ given by f, = cos {0 - kz), f, = sin(0 - fe), f* = (76) which moves each constant-^ cross section rigidly a constant distance f perpendicular to the axis, so that the tube of plasma is distorted into a helix whose pitch is L. Consider two cross sections oc and /3 a distance L/4 apart which are therefore displaced in STABLE EQUILIBRIA 207 perpendicular directions. If i = 2n, lines of force rotate through n/2 between these cross sections. That is, any line of force through a point S in a passes through a point T in 3 such that QT makes an angle л/2 with QS (see Fig. 2). In these circumstances, it is clear that any line of force passing through A in a and В in f} is displaced to a line of force passing through A' in a and B' in /3 О A' and QB' make a right angle with each other. But this means that the line of force through А, В is displaced into the position of a line of force which passed through A', B' in the un displaced equilibrium. Further, since the density of lines is unchanged because the displacement of each cross section is rigid, the field is unchanged by the displacement. Thus the situation characterized by с = 2л is neutral with respect to perturbation (76). Let us now consider the case in which (t 2n) is positive but small, subject the plasma to the same perturbation (76) and examine the same cross sections ос and /5, L/4 apart. Then any line of force of the equilibrium passing through a point S in a will pass through a point T in /? such that QT makes an angle L/4 > л/2 with OS (see Fig. 3). A line of force through A in oc and В in /3, О A and QB make an angle of /4 with each other, will then be displaced to a position passing through A' and B', О A' and QB' no longer make an angle of i/4 but make a slightly smaller angle. Thus the displaced line of force is rotating about the axis OQ at a slower rate (in z) than the line of force which passed through B' in the undisplaced equilibrium. Hence we see that B Q is weakened by an amount proportional to (L/4 n/2)g. If we consider other points in the cross section, we find that ÔB is constant in each cross section. Since the cylinder is long, ôj = V X ÔB is negligible and OF = j X SB + ôj X В ^ j X SB. We see, therefore, that OF is in the same direction as \ and tends to enhance the perturbation. Thus for i > 2n the system is unstable with respect to perturbation (76). We now consider the effect of a helical fieldon these instabilities. In the stellarator the axial current is applied only when /3 is small. We therefore set /? equal to zero in this discussion. We have shown n that the results are continuous as /3 goes to zero. For simplicity, we assume that the conducting walls are infinitely far away, the axial current is uniform in the plasma region and only one helically invariant field, with 1 = 3 and a small hr, is present Figure 2. Kink displacement if L = 2TT \B'

11 208 SESSION A-5 P/1875 J. L JOHNSON et a/. A r Figure 3. Kink displacement if i > 2n In Appendix III we carry through the calculation for the case the external region is a vacuum. The results are exhibited in Fig. 4. The unshaded region is stable for m = 1. The regions in the first and third quadrants in which M, > \iv\ are stable for all m. The other regions have not been completely investigated for higher m's. We have also carried through the calculation n for the case the external region is a pressureless plasma. The results are exhibited in Fig. 5. The unshaded region is stable for m = 1. Instabilities for higher values of w can occur only in the parts of the first and third quadrants for which \i bb \ < ^. It can be easily seen n that if the axial current is a sheet current at the surface of the plasma, the system is stable for all m if \fl\ < 4\L 88 \ and if both have the same sign. (2) the helically invariant fields affect the stability by introducing a shear (dt/dr) in i inside the plasma and by increasing or decreasing the shear outside the plasma according to the sign of di/dr. This effect is most prominent for the higher values of m\ finally, (3) the helically invariant fields may cause a surface to exist on which the effective i == t 88 + iv is 2лп/т. On this surface the electric field E, due to the perturbation, is parallel to В and along a line of force is always in the same direction. Therefore, this electric field leads to large currents on the surface so that the surface acts like a rigid perfectly conducting wall to the displacement. This effect can lead to increased stability. For m = 1 the first effect outweighs the latter two and the critical current is decreased. For higher m's the latter effects dominate and the stability is improved. SUMMARY We have thus been able to find and investigate the properties of equilibria which are hydromagnetically stable. These equilibria can be obtained, for example, by wrapping conductors helically around the stellarator tube. Systems with I = 3 or 4 are indicated to be optimum for stability purposes. In some cases an admixture of I = 2 fieldscan be advantageous for achieving equilibrium. ACKNOWLEDGEMENTS We wish to express our deepest gratitude to Dr. Lyman Spitzer Jr. for suggesting many of the problems treated in this paper and to Dr. Martin Kruskal for his constant advice and encouragement. We wish to thank all our colleagues in the Theoretical Division of Project Matterhorn for many helpful suggestions. Figure 4. Stabilization of the kink instability if external region is a vacuum. The unshaded region is stable for m = 1 We note that the helical fields have three effects on the kink instability which might lead to an understanding of these i 88 vs. 0 diagrams. The first effect (1) is to increase the effective i by t 88 (which may be negative) so that for m = 1 the kink is made more unstable if i 88 is positive (or less if i 88 is negative) ; Figure 5. Stabilization of the kink instability if external region is a pressureless plasma. The unshaded region is stable for

12 STABLE EQUILIBRIA 209 APPENDIX I HIGHER ORDER EQUILIBRIUM QUANTITIES For simplicity, we give the higher-order quantities for the case f(v R) = {r/r) 2 (a parabolic pressure distribution), g{r/r) = r/r (uniform axial current) and only one helically invariant field is present. To order jsjff: To order щ : 0; jw = 0; ^ =0. (Л < r < S). (77) B^ = e^ab ri-^yl; jw =e g 2^f r ; #" =0. (0 < r < R) B^ = 0; j^=0; ^ =0. (i?<r<s). (78) To order 0(5 : B» = e,-^^ [1-2 (P + X*) P(X)11^Шг sb 2«, 04 ~ ВЧ» (2X)[1-2 (P + X») I 1 (X)] ~щу cos 2u À2 Ш 1 (%) [l 2(P + X») /'(X)] [1 + 2 ^ 2 / ^ ) ^ 2 To order /fy : j** = 0; p** = 0. {0<r<S) (79) B^ = 0; j ^ = 0; p^ = 0. {0<r<S) (80) To order /?<5 : Щ + Cfi'I'i(x)\sinu ^ /z (%)j cos M - e z апцх) cos W; ^ /' (X) - ^ cos M ; 4W^cosu - (0<r<R + e + Q) L\ (A) x)] sin w + e e [С?Щх) + О^Щх)] cos и x - e 2 [CP 8 Ii{x) + DP 8 Ki(x)~\ cos w ; j ^ = 0; ^ = 0. {R < r < S), (81) 7; (Л) II (A) Ah'{hR)l DÍ* = - AR [СЩЩ and ^(л;) is the modified Bessel function of the second kind as defined by Watson. 17

13 210 SESSION A-5 P/1875 J. L JOHNSON et al. To order r]ô : Г B - e '{- sm и cos«; j " = e, r,ô 2 Bo 4- -^Йт sin «+ e gv d2b4xi l (X) 4~ ^TVV cos (0 < r < R) В" ' (ж) + К^^г' (ж)] sin м + е [С'*1 г (я) + D^7f ; (л:)] cos и (х) + Ô 7»*^ (л;)] cos и; ji* = 0; рч* = 0. (7? < г < S) 2 B.il, m fg>r ' ЩЩ ЩХ) J! APPENDIX II INTERCHANGE INSTABILITY If yv = 0, only helically invariant fields with 1 = 0 or 3 and small hr are present, and the plasma fills the entire tube (R = S), (50), (51), (52), and (53) can be written as and 2ÔW m;n = 87Г2 T(t) = (Lftiôi 2 ) (t 2 a 2 i a 2 = - п1\ъ (84) (85) (86) and the other quantities are defined by (70). Clearly /л(0) and /u(l) are zero, ju(t) must go to zero at a at least as fast as t a, and /л' (a) must be continuous. We shall determine the critical /? for each value of m and n for which a is in the range 0 < a < 1. We shall show later that other values of m and n do not make the situation worse so that the true critical /? is the least of these. The Euler equation which is obtained by varying (83) with respect to x is = 0. (87) (We need not carry a normalization condition explicitly in this discussion.) In the vicinity of the singularity at a, (87) behaves like / = 0. (88) The contributions to SW from the two regions t < а and t > a can be considered separately. For > a, the solutions of (88) are /л = (t - Q)\ exp [± (1 - A)i]n(t - a)] (89) A = Pf'(a)T(u)la<. (90) We now show that for values of m and n under consideration, the critical /3 is determined by setting A equal to 1. If A > 1, the solution л of (87) which vanishes at t = 1, is p={t- a)* cos { (4-1)* In (t - a) + y}, (91) for sufficiently near a. It therefore must possess at least one zero for t > a. For a particular value of A, A-y> \, we let t x be the largest zero below 1, and consider for this A x a function /x lf which we define to be identically zero for t < t ± and to be a solution of (87) for t > t ±. It follows from (83) and (87) that for this p lt ÔW = 0 if A = А г and (5TF < 0 if A > Л х. Therefore, we have shown that the critical /5 corresponds to a value of Л < 1. In cases A < 1, we first consider any x which is identically zero for all t less than some t x > a and vanishes at t = 1. We now show that for ^'s associated with this t lf an Л х > 1 exists such thai ÔW > 0 for all A < A 1} and therefore for A < 1 We consider the pressure distribution given by (10Г which we later shall find leads to the largest true critical p. For this distribution with m and A = /?//? set equal to 1, the general solution of (87)

14 STABLE EQUILIBRIA 211 i = [1 - (а/о 2 ] ПС + D Ы{Р - а 2 )], (92) cannot vanish between a and 1 since C/D must be chosen to make it vanish at t 1. Since dw is a continuous function of /?, the position t v at which the minimizing x must first vanish for a given ^ >1, must approach tt as ^ approaches 1. Since the solution of the Euler equation with no zeros between t x and 1 yields the minimum value of àw over all u's which vanish identically for а < t < t x and for A = A x this value is zero, all such ju's make dw > 0 for A А г. We therefore have shown that, for this pressure distribution, for any t x > a, an А г > 1 exists below which ÔW > 0 for any x which is identically zero for t < t ± and vanishes at t = 1. We must still show that for A < 1 no other / which vanishes at t = a and 1 (i.e., t x = a) can lead to instability. To do this, we assume that such a x exists which makes dw < 0, say dw = s. We first show that for any sufficiently small ô, the contribution to ÔW from the region between a and а + ô is positive, so that the integral from t = a 4- ô to t = 1 must be more negative than s. We shall then show that this integral from t = a -f- ô to t = 1 diners from a positive definite integral by an amount which can be made as small as we desire by taking ô sufficiently small. This contradiction will complete the proof of stability for A < 1. The contribution to ÔW from the region between a and а + ô is, from (83), x = t a, y* = [г*1^х, (y**)'\xdx, (93) (94) (95) and the prime indicates differentiation with respect to x. The first two terms are positive since A < 1. Since jy* (x) must go to zero as x goes to zero, we see that the last term is also positive for any y*. We now consider the value of SW which corresponds to a particular u, that is /2, which is zero for i < а + \ ô, increases linearly over the region between а + \ ô and а + ô, and from there to t = 1 is the same as the previous ju*, which was assumed to make ÔW < 0. Since A is less than the A x which determines the solution of (87) which first vanishes at t x = а + \ ô, this value of ÔW must be positive. It differs from the one for *, which had to be less than e, by the amount )]iliz}dx. (96) This integral is less than a fixed positive quantity times a, so that we can make it smaller than e by a suitable choice of ô. This completes the proof that no x exists which can make ÔW negative if A < 1 and, therefore, if only the region t > û is considered, the critical p is determined by setting A = 1 in (90). We can replace (89) by p=(a- 0* exp [ ± i (1 Л )* In (а - /)] (97) in the region t < a and repeat the entire argument to show that A = 1 again determines the critical /3. If a is zero or one, the argument can still be carried through. Before considering values of m and n for which a 2 is not in the range 0 < a 2 < 1, we shall determine the pressure distribution which maximizes the critical /3 with a 2 in this range. We shall then show that for this optimum pressure distribution, instabilities for which a 2 is not in this range lead to higher critical values of /?. If 0 < a 2 < 1, the critical /? is ^с = тт а [а 4 //Ча)Г(а)]. (98) We now determine the pressure distribution, i.e., f(t), so that for the worst volume of a, the critical fi is as large as possible. For any particular/ (t), (98) requires that Рс/'(а)<а 4 /Г(а). (99) We consider large enough values of m and n that we can treat a as a continuous variable. Integrating both sides of (99) with respect to a and using the boundary conditions (14) on /(/), we find that for any /, However, for = mina pûû 4 /r(a). (100) (loi) dt t*/t(t) =^ dt t*/r(t), (102) and this / is the optimum distribution. Since F(t) is given by (86) the optimum pressure and critical /3 are given by (67), (68) and (69). We must still show that instabilities for which a 2 is not in the range 0 < a 2 < 1 do not lead to lower critical values of /?. It can be seen at once, by inserting the pressure distribution (101) into (83), that ÔW > 0 if a 2 < 0 and /? < /? c. Similarly, if we set y = t)i, (103) it is immediately apparent that ÔW > 0 if a 2 > 0 and p <p c for this pressure distribution. APPENDIX III STABILIZATION OF THE KINK INSTABILITY If p = 0, if S = oo, if jv is constant in the plasma region, if the external region is a vacuum, and if only one helically invariant field with 1 = 3 and small hr is present, we find by minimizing (61) over the volume regions that

15 212 SESSION A-5 P/1875 J. L JOHNSON et al. 2ÔW M>n X m a 2m 1 (m;n) simplicity we shall not consider higher values of m. If fl < q < 5< * + ^ then a is not zero and q = 2яп/т, (104) (105) a a =(î-^)/ w, (0<a 2 <l) (106) and a is the value of r/i? / must vanish because <7 + ^( y ) + 5< *(f) does, and all quantities are evaluated at the plasma boundary, R. If a does not satisfy the inequality in (106), we set it equal to zero. We first consider cases 0 < ^ < t 88. If q < i 88, the last term in (104) is less than the first so that ÔW > 0. If t 88 <q< i 88 + i\ й is not zero and we can show that dw > 0 for all m. If q > i 88 + ^, both terms of (104) are positive. We have thus shown that the system is stable for all values of m if 0 < e < L 88. We next consider cases 0 < t 88 < fl. As before ÔW > 0 unless q is in the range t 88 < q < i 88 + iv. If i 88 <q < i 7!, then a = 0, and ÔW q X[? + ** + ***] (107) s negative for m = 1 for all ^'s in the range. For X [ -? + i M + (108) is again negative for w = 1 for these g's. We have therefore shown that if 0 < ** < ^, the system is unstable for m = 1 if and only if i 88 <C q <C t 88 + *Л Conditions for instability have not been obtained for higher m. We finally consider cases ^ < 0 < i 88. Again àw > 0 unless q is in the range i 88 + < q < i 88 (сч is negative). For these qs, a 2 > 1 and must therefore be set equal to zero in (104), so that (107) holds. For m = 1, ÔW is negative for all these q's. Again we do not here obtain stability conditions for higher m. If we replace t 88 by i 88, by ^, and q by #, (104) is unchanged. We have therefore shown that the system is stable for all m if \i 88 \ > i^ and i 88 and û have the same sign. Otherwise it is unstable for m = 1 for values of q which lie between i 88 and i 88 + ^. Higher w's have not been considered in these regions. It should be remembered that these results apply to the case the external region is a vacuum, incapable of supporting a current. We see from (105) that q is limited to the values 2nn m, n can be any integer and can thus construct Fig. 4. REFERENCES 1 M Kruskal and M Schwarzschild, Some Instabilities of a Completely Ionized Plasma, Proc Roy Soc A 233, 348 (1954) 2 E Teller, Talk at Princeton, N J (26-27 October 1954) 3 I В 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) 4 For a definition of a rotational transform see, for instance, L Spitzer, The Stellarator Program, P/2170, Vol 32, these Proceedings 5 L Spitzer, Talk at Los Alamos, New Mexico (June 1955) 6 R Kulsrud, E Frieman and J Johnson, Talk at Princeton, N J (17-20 October 1955) 7 H R Koemg, Confining Ionized Plasma with Helical Magnetic Fields, A E С Report No NYO-7310 (PM- S-20, 1956) 8 E A Frieman, Talk at Gatlmburg, Tenn (4-7 June 1956). 9 L Spitzer, informal communication 10 E A Frieman, Talk at Berkeley, Calif (20-23 February 1957) 11 J L Johnson, С R Oberman, R M Kulsrud and E A Frieman, On the Stability of Hydromagnetic Equilibria with Varying Rotational Transform, A E С Report No NYO-7904 (PM-S-34, 1958) 12 L Spitzer, Physics of Fully Ionized Gases, Interscience Publishers, Inc, New York (1956) 13 M D Kruskal and R M Kulsrud, Equilibrium of a Magnetically Confined Plasma in a Toroid, P/1876, this Volume, these Proceedings 14 J M Berger, I В Bernstein, E A Frieman and R M Kulsrud, On the Ionization and Ohmic Heating a Helium Plasma, P/363, Vol 32, these Proceedings 15 J M Greene and J L Johnson, unpublished 16 H R Koenig, informal communication 17 G N Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England (1952) 18 IB Bernstein, H R Koenig and E A Frieman, Talk at Gatlmburg, Tenn (4-7 June 1956) 19 M D Kruskal, J L Johnson, M B Gottlieb and L M Goldman, Hydromagnetic Instability in a Stellarator, P/364, Vol 32, these Proceedings, M D Kruskal and J L Tuck, The Instability of a Pinched Fluid with a Longitudinal Magnetic Field, Proc. Roy Soc, m press