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The influence of errors on plasma-confining magnetic fields

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1962 J. Nucl. Energy, Part C Plasma Phys. 4 253

(http://iopscience.iop.org/0368-3281/4/4/303)

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Plasma Physics (Journal of Nuclear Energy, Part C), 1962, Vol. 4, pp. 253 to 262. Pergamon Press Ltd. Printed in Northern Ireland

THE INFLUENCE OF ERRORS ON PLASMA-CONFINING MAGNETIC FIELDS*

D. W. KERST General Atomic Division of General Dynamics Corporation, San Diego, Calif., U.S.A.

(Received 2 March 1962)

Abstract-Endless lines of force confined in toroidal plasma containing geometries are subject to periodic perturbations by field errors which can produce integral, half-integral, third-integral, and quarter-integral, etc. resonant-like disturbances on the spiralling line of force. A formal analogy exists between the topology of a trace of a line of force in the co-ordinate space of a plasma-confining region and the trace of a phase point in the phase space for a particle orbit in an accelerator. The methods used for determining stability limits for particles in periodic accelerators are thus applicable to the problem of confining lines of force in stellarators and pinches. Examples are given and computer methods are described for typical stellarator configurations subject to a variety of field errors.

I N T R O D U C T I O N

FOR the construction of plasma-confining magnetic fields a sufficient knowledge of the necessary field tolerances is not available. However, something can be said about tolerances for a vacuum field or for a field modified by distributed currents exclusive of the question of how plasma magnetohydrodynamics may affect these tolerances. The case considered in detail is that of endless lines of force in a closed toroidal system like a stellarator or toroidal pinch. These systems are periodic in structure-in all cases with periodicity equal to the circumference. In stellarators there is an additional periodicity equal to the period of the helical winding’s field pattern. Thus, errors are periodically applied to a line of force, and this periodic aspect creates remarkable resonance-like effects, which can be much more serious than dis- tributed errors.

We can draw help for this tolerance question from the extensive experience with the similar tolerance problem for periodic structures used in particle accelerators. There is an analogy between the topology of magnetic surfaces (SPITZER, 1958) generated by stellarator magnetic field transforms in co-ordinate space and the topology of particle trajectories in the phase space of a periodic accelerator (KERST, 1959, 1960). Both are the results of periodic applications of transformations of initial conditions. The result is that, as in an accelerator, the stellarator transforms can be sensitive to integral, one-half integral, one-third integral, and one-fourth integral ‘resonances’ (and

253

possibly higher-order resonances) which are excited by certain types of periodic field errors. The lines of force may leave the aperture of the stellarator a t these resonances. In the L = 2 (quadrupole) case, the ‘stability’ limit has zero area for integral, half- integral, etc. resonances to which the transform is subject. Thus, for L = 2 at the ‘Kruskal limit,’ which is an integral resonance, the lines of force would leave the confining region due to the effect of field errors without regard to the magnetohydrodynamic stability. For the L = 3 (sixpole) case, the stability limit or critical surface is located at the area’s perimeter for which the transform is tuned to integral, half-integral, third-integral, o r fourth-integral rotations in one structure period; and the magnitude of the exciting error, respectively dipole, quadrupole, sixpole, or octupole errors, determines how far the lines of force migrate.

Analytical and computer methods for studying the tolerances and the topological properties of trans- forms can be taken from the accelerator literature. (COURANT and SNYDER, 1958; LASLETT and SYMON, 1956; HAGEDORN et al., 1956; STURROCK, 1958 and KERST et al., 1960). Some examples of the methods and of the magnitude of the effects of errors on the aperture or stability limit for a stellarator will be pointed out.

THE D I F F E R E N T I A L EQUATIONS First, the differential equations of lines of force of

a confining field and the differential equations of trajectories of phase points in an accelerator have some similarity. We apply a uniform Bz for the con- fining field on which is superimposed components B, and B, of a multipole field generated by poles or wires

* This work was carried out under a joint General Atomic- Texas Atomic Energy Research Foundation programme on controlled thermonuclear reactions.

254 D. W. KERST

slowly rotated about the 2 axis as the lines of force are followed in the 2 direction.

du=B_y dz B,

are the differential equations for a line of force. The period of the structure is so long that we neglect the slight variation of B,. The similarity to accelerator equations for co-ordinates and momentum was described in KERST (1959, 1960). SYMON (1962) has recently carried this similarity of magnetic field and phase space topology further. He uses the dependent variables x and

p =[Bz(x, y' , 4 dy' (2)

so that1the:flux;through a contour C is

0 = I p d x C

and this area in p and x variables is preserved as we follow flux lines in the z direction.

For a line of force

- * - - _ - _ - d i B, ap

dz B, - _ -

and

(3)

where H i s a Hamiltonian since

SYMON carries this development much further, employing some of the methods of dynamics to the Hamiltonian form of the magnetic field line problem. A simple example of a Hamiltonian form for a quadrupole magnetic field superimposed perpen- dicularly on a uniform field, B,, in the Z direction is:

BL = (Bll/rJY B, = (4l/rll).~

and p = yB, giving

H = (B,/r,,)[(p/B,)2 - s2].

This is analogous to a Hamiltonian for a particle of momentum, p , 011 a potential hill. The particle inoves

away from the unstable equilibrium point a t x = 0. Likewise, lines of force in the quadrupole field leave the region of the null point. If the quadrupole is twisted as the line of force progresses in the 2 direc- tion, then the signs in H alternate and the possibility of lines of force remaining stably near an equilibrium point exists-just as charged particles moving through alternately focusing and defocusing fields can be focused. This is the case examined in the subsequent discussion.

While the formal analogy to particle orbits may be suggestive, it is not pursued here. Rather the tech- niques used in the study of tolerances for orbits in periodic structures will be applied to some magnetic field examples.

Several cases of multipole fields will be examined, starting with the equations for multipole fields (SPITZER, 1958)

B, = Bo(;r-'sin (LO - z/A)

where A is 1/(27r) times the length of the period of the twisted structure t o repeat similar multipole field patterns (not the period for a particular magnetic member to come back to its original position except in the dipole case).

From (4) we obtain

B, = BO(kr-'sin ( [ L - 11 0 - ./A)

First, consider the quadrupole case, L = 2, where (5) becomes

B, = (Bo/ro)(y cos [ z /2 ] - x sin [?/I]) B, = (Bo/ro)(x COS [i/A] - y sin [i/R]). (6)

These can be combined with (1) to give the fourth- order differential equation

which gives a stable periodic solution x - cos R 3

with no exponential growth, provided

The influence of errors on plasma-confining magnetic fields 255

Geometrically we can see what is involved if a AB* is applied periodically as we follow the line of force (Fig. 1). This displacement, 6, can be roughly

At the condition of equality there is one value for s2

n, = 1/(2K). (9)

At this limit the line of force twists half-way around while the multipole pattern repeats once. Thus, the line can twist just as fast as a given pole. For (119;) > 2Bo/(roB,) we get two values of Q-one low frequency QL < Q, and another high frequency Q H > R,. Since QL determines the overall perio- dicity of a line of force, the highest value Q, can have is half the frequency of the field pattern.

This problem resembles the case of a Mathieu or Hill equation (as encountered in alternating gradient accelerators) in which the highest stable frequency is half the period structure frequency. There is thus a ‘stability limit’ on the magnitude of the rotational transform for a line of force.

The structure need not be continuously twisted to produce a transform; it may be composed of straight segments with the successive segments orientated differently. In this case, the transform can be de- scribed by the use of transfer matrices for the quadru- pole ( L = 2) case; or, for L > 2 , by computations using the differential equation or equivalent successive non-linear algebraic transformations if they can be found.

A simple case for which such non-linear algebraic transformations can be found is that of the multipole segments representing only a n infinitesimal portion of the 2 path-most of which has only B,.

The segmented L = 2 case is treated in the Appendix with the result that the limiting rotational transform period is twice as long as the field repetition period, the same as (9) shows for a twisted quadrupole.

E F F E C T S O F F I E L D PERTURBATIONS A confining field of a periodic (toroidal) structure

with a n amplitude independent transform angle I = 2nv per circuit around the structure will be sensitive to a transverse field applied at one point in the structure especially when Y is close to an integer. For v integral, the lines of force leave the structure because the perturbation is in phase with the trans- form period. I is constant for all distances from the centre for L = 2 or for a pinch discharge with. a uniform current density and a uniform B, or for any ‘shearless case’ (we neglect the effect of the curvature around a torus on the transforms).

Besides the resonant destruction of the confining field, a transverse field causes a displacement of the central or equilibrium line of force which closes on itself. This displacement can be calculated by methods described i n COURANT and SNYDER (1958).

UNPERTURBED LINE CLOSING UNPERTURBED LINE CLOSING

PATH OF CLOSED LINE-,/

FIG.l.--Path of the closed line of force in the cross sectionof a torus in which ABa is added every time theline completes one circuit around the torus. The case at the left is for v < 1 while the case at the right is v 1.

estimated by S = (ABEW)/(2B, sin - where Wis the

Z extent of a field error AB,. As I+ 2nn where n is an integer, we see that the equilibrium position does not exist.

If we plot the locus of a particular line of force at the same point in circumference of the torus, we get magnetic surfaces (Fig. 2).

OF FORCE

2 7

FIG. 2.-Locus of a line plotted every circuit around the torus. The circuits are numbered. The central fixed point

is the closed line of force.

If the field error AB, N X , then half-integral resonances are excited. Following a line we have the resultant resonance shown in Fig. 3. We have an exponential growth. Such an error can be caused by a quadrupole field error. If the tune is near nx = I , then a beating or throbbing (COURANT and SNYDER, 1958) of amplitude occurs near the unstable region for I.

256 D. W. KERST

I = n s FIG. 3.-A half-integral resonance, following the line of force around the torus, is excited by ABz which reverses

polarity on opposite sides of the origin.

While stellarator transforms are chosen to be well below 1 = 2n-, pinch discharge transforms may be much higher and may suffer resonance in certain plasma layers due to field perturbations; but for structures giving small transform angles one-third integral, one-fourth integral, etc. resonance may be excited when certain kinds of field error, depending on higher powers of X, are present.

This can be seen in the following way: if v is the number of complete field line revolutions per circuit, X- approximately describes the X co-ordinate of a line. If AB, ,- X”, then ABx-ei“”Q . If the structure error giving AB2 has a harmonic component e*ik+ describing its distribution around the torus, then AB, -ei(m”+k)+. If mv ,I k = &v, then we have a resonant application of AB, which causes a growth of the X co-ordinator of the line of force. Whether or not the line will leave the aperture depends upon changes of tune with amplitude and the magnitude of the field error.

For ABz N X 2 , we have m = 2 and thus a resonant possibility at v = 1/3 (one-third integral resonance). For AB, ,- X 3 , we may have trouble a t v = 1/4, etc.

ABx- X 2 would result from a n L = 3 or sixpole field error a t some azimuth around a torus; and ABr - X 3 would result from the presence of an octupole magnetic field error a t some point in the structure.

To demonstrate some of these effects, digital computer runs were made for a structure with L = 3.

( I ) and (5) give dX/dZ = (Bo/(B,r t )][2XY COS ( Z / 2 )

d Y/dZ = [B,,/(B,r,,’)][ZXY sin (Z//s) - (P - Y2) sin (Z/Z)]

-+ (P - Y 2 ) sin ( Z / 2 ) ] . (10)

Taking X and Y in units of r, and Z in units of 2 with B = (Bo2)/(roB,), we have x = X/ro, y = Y/ro, z = zp..

(10) becomes

dx dz -- - 8[2xy cos 2n-z - (x2 - yz) sin 2n-z]

_ - dy - b[2xy sin 2n-z + (x2 - y2) cos 2n-zJ. (11) dz

These differential equations were used for digital computer calculations.

Figure 4 shows the xo dependence of the number of sectors for a transformation I of 2n-. In general Z N The initial condition is yo = 0 and p = 1 in all examples subsequently given. For L = 3 an essential resonance occurs with the structure for 3 sectors for I = 2n-, and the lines of force leave the limit set for the computer (2 + y 2 = 25) at x, 2 2.0. Figure 5 shows this limiting magnetic surface for no field errors. The plot gives a line of force position every 10 sectors. This limiting x, = 2.0 is shown in Fig. 4 for 3 sectors per complete rotation of a line of force.

To test the sensitivity of the magnetic field topology to imperfections in the field, errors were imposed once every 10 sectors for an extent in the Z direction equivalent to one sector length. In some cases the x displacement of the line was applied and in other cases the differential equation of the line of force passing through the sector containing errors was actually integrated.

For sixpole or L = 3 transforms, Ivaries with x or r so that resonant tune is not maintained after x grows a little. However, a large enough error will drive the lines of force out. Figure 4 shows that the integral resonance, v = 1, immediately appeared (Ax = 0.05) and gave a maximum stable value of xo = 1.4. Larger errors, Ax = 0.1 and 0.2, brought the stable limit down further to xo = 1-1 and 0.6, respectively. This is the usual experience-that errors decrease stability limits. Figure 6 shows magnetic surfaces with Ax = 0.1, Fig. 7 with Ax = 0.2. As the limit is ap- proached, the erratic behaviour of the lines becomes evident. In Fig. 7 we see seven islands. These represent one bundle of flux closing on itself after seven transits around the torus, and there can be no pressure gradients along the contours around this bundle. This demonstrates the effect of errors in creating a topology which cannot support desired pressures. A shearless layer of lines occurs at the centre of the bundle (dl)(d.r,,) = 0. Figure 8 shows the effect of error magnitude on available area or

The influence of errors on plasma-confining magnetic fields 257

80 t

$ 70 0 T

N

t 60 z

50 d

> D:

U 40

e E E 30

20 0 D: : IO f

0

114 INTEGRAL RESONANCE

113 INTEGRAL RESONANCE

112 INTEGRAL RESONANCE

INTEGRAL RESONANCE

ESSENTIAL LIMIT I

0 1.0 2.0 X INITIAL DISTANCE FROM CENTRE

FIG. 4.-The number of sixpole (L = 3 and B = 1) periods for a complete rotation of a line of force is shown as a function of the initial position, xo, of the line (in all cases,y, = 0). The loci ofessential, integral, half-integral one-third integral, and one-fourth integral resonances are shown. In addition some ob-

served stability 1imits:are given for different types of field errors.

-2.2 -20 -1.8 -1.6 -14 -1.2 *W-F++

FIG. 5.-The limiting magnetic surface for no field errors. A line goes through 3 magnet periods before it circulates through 2 ~ . Successive points on the plot are

10 magnet periods or sectors apart.

258 D. W. KENT

0 5

-to -8 -6 -4 -2

-2 --

OB

018

-kO 1 06

FIG. 6.-Limiting magnetic surface for transverse; field error described by A x = 0.1. Field line positions are printed 10 magnet periods apart. Circled points are loci of an escaping line of force started outside the

stable region.

L

t -‘6 t - . e 1

FIG. 7.-Limiting magnetic surface for transverse field error Ax = 0.2. The stable limit is small and a bundle of flux closing on itself every seven turns around the IO-period structure is seen. A line through thecentre ofthese bundles is the

locus of a ‘shearless’ layer of lines of force (dI)/(dx(,) = 0.

The inff uence of errors on plasma-confining magnetic fields 259

B l \

0 0 .05 0.1 0.15 0.2

FIG. 8.-Shrinking of available aperture area as a function of the magnitude of the transverse field error.

Ax

aperture. To indicate the magnitude of these field errors, note that Ax = 0.1 is equivalent to a field error which is 0.1 of the multipole field strength at the distance Y = x = 1 and which is present on only one of the 10 sectors.

Quadrupole field errors were next applied after the passage of the tenth sector. The form of the error was Ax = Cx, Ay = -Cy. This is a real error field since it satisfies Maxwell‘s equations. However, it can also be considered a transformation x2 = x1 + Cx,, y2 = y1 - Cy, for which the Jacobian is 1 - C2. Since B, is constant, the density of points is to be constant, and we want the Jacobian unity. Tests were made with C = 0.05 and xo = 0.95. The vacuum field was unstable with the lines of force leaving the field after 34 passages through the field error. Since 1 - C2 = 1 - 0.0025 or a (0.0025) area error for one passage, the worst fractional area error after 34 passages would be 0.09 or a fractional radial excursion of -0.045. This excursion does not take the line of force out of the aperture. In fact, since the Jacobian is less than one, the area should shrink inwardly. Consequently, we conclude that the physical half- integral resonance effect a t x, = 0.95, and not the computational error, was responsible for the in- stability of the line of force.

This field error at x = 1 is 0.05 of inultipole field at x = r = 1, and the error is applied only on one sector length among the ten sectors. The test was not

extended to smaller errors to find the minimum error which would destroy containment at the half-integral Kruskal limit.

With A/? = 0.1 we excite one-third integral reso- nances and find no serious instability. Instead, in Fig. 9 we show three bundles of flux appearing, which means one bundle closing on itself after three transits around the torus. As mentioned earlier, this can be caused by a strengthening or weakening of a six-pole lens at some azimuth. This behaviour is similar to the three-bundle behaviour observed in the plasma of the stellarator (BERNSTEIN et al., 1959). Again in the centre of these bundles, (dZ)/(dx,) = 0 or a ‘shearless’ layer exists and no pressure difference can exist across the flux bundle. The existence of such bundles for a pressure-bearing plasma has been discussed by OHKAWA and KERST (1961).

For an octupole error, a Maxwellian 1 = 4 field was applied for one sector length and then the differential equation for the unperturbed L = 3 stellarator was followed for ten sectors, then the error field was again applied, etc. The line of force differ- ential equation in the sector containing a Maxwellian octupole error is,

dx/dz = &(x3 - 3 ~ ~ 4 dy/dz = .b3 - 3 ~ 2 ) .

Thirty-two Runge-Kutta steps were taken in the error sector. Figure 10 shows the results with

260 D. W. KERST

T “O

-1.0 - FIG. 9.-A 10 percent strengthening of theL = 3 field for 10 per cent of the torus circumference ( p -+ Bo -!- 0.1) causes a one-third integral resonance with stable lines of force at x,, larger than the resonance region. The resonance does not cause lines to leave the aperture as the one-half integral

resonance does.

-.9

FIG. lO.-Octupole errors (see text) cause the cross section of a flux bundle to appear four times at the one-fourth integral resonance. The error destroys confine-

ment outside the half-integral Kruskel limit.

The influence of errors on plasma-confining magnetic fields 261

x = 0.097. The one-fourth-integral resonance is ex- pected for xo = 0.7 and as expected the four passages of a flux bundle appear. The size of the error is not sufficient to cause instnbility for this xo, and thus a magnetic surface can be seen surrounding the separa- trix of the flux bundle. However, a t x0- 1.0, the half-integral resonance, the error is sufficient to destroy the confining field outside. With tl = 0.2 and xo = 0.7, the octupole error destroys the field containment outside xo.

A phenomenological estimate was made of the size of the flux bundles, assuming the transform angle of the structure is

For an error field I = 2%-v(r/r0>2'L -2) .

b, = bo(r/ro)(l-l) sin IO bo = bo(r/ro)(L-l) cos IO

we find the radial width of the bundle

2r0 A = 2/4.nv(L - 2)IB,r0/(b0W) - ( I - 2p

where yo is the radius at the flux bundle, B, is the main stellarator field, and W is the circumferential distance occupied by the 21 pole error. For the large A case, v = Ill,

2Y" A = d ( L - 2)4%-roB,/(boW) - ( I - 2)1*

The second term is usually much smaller than the first, We see that a quadrupole, L - 2, or a figure eight stellarator, has extremely large fieldline excursions or flux bundles. This formula predicted the width of the flux bundles within 10 per cent for the octupole error shown in Fig. 10.

Any particular stellarator design would require ad hoc testing for sensitivity to field errors. For example, a figure eight torus has the same transform angle, I , for all magnetic surfaces. The response to dipole, quadrupole, etc. errors would be different from the responses for an L = 3 structure, but the same critical v values would be present and would be much more destructive in the L = 2 or figure eight case.

If a current density is present in the z direction, it adds to or subtracts from I , thereby moving the operating point relative to the critical values of v. Non-uniform current densities would add (U)/(&) # 0, thereby bringing in the non-linear transformation effects seen for L > 2.

Ackriow/r~~r~ierits--lt is n pleasure to acknowledge the help of ERNEST COURANT i n the early studies with the digital computer for the application of techniques employed in particle accelerator

design. The handling of digital computer programming and experimentation was ably carried out by MR. PETER KAESTNER. I am also grateful for discussions on this topic with K. R. SYMON and M. ROSENBLUTH, who examined a pinch case, and especially with T. OHKAWA, who has studied these topological questions in the presence of pressure gradients (OHKAWA and KERST, 1961).

REFERENCES BERNSTEIN W., KRANZ A. Z . and JENNER F. (1959) Phys. Fluids

COLJRANT E. D. and SNYDER H. S. (1958) Ann. Phys. 3, 1. HAGEDORN R., HINE M. G. N . and SCHOCH A. (1956) Pro-

ceedings of the International Conference on High Energy Accelerators and Pion Physics Vol. I, p. 237, CERN.

2, 713.

KERST D. W. (1959) Bull. Amer.phys. Soc. 114, 352. KERST D. W. (1960) Bull. Amer.phys. Soc. 115, 352. KERST D. W. et al. (1960) Rev. sei. Instrum. 31, 1076. LASLETT L. J. and SYMON K. R. (1956) Proceedinps of the Inter-

notional Corlference on High Eiiergy Acceleritor; and Pion Physics Vol. I , p. 279, CERN.

O H K ~ W A T. and KERST D. W. (1961) Bull. Amer,phys. Soc. 116,

SPITZER L. S. (1958) Phys. Fluids 1,257. STURROCK P. A. (1958) Ann. Phys. 3, 113. SYMON K. R. (1962) To be published.

290.

A P P E N D I X The segmented case corresponding to (6) has

BO & = - Cy COS 2+ - x sin 2+) yo

B 5, = 2 sin 24 + x cos 2+)

r0

angular position of a segment. Then, with

dx dz a(y cos 2+ - x sin 2+)

2 dz = ct(x cos 24 + y sin 24).

- =

We choose four positions for the segments in one period or sector: = 0, +, = ~ / 4 , 4% = 71/2,+% = (371)/4.

x = xo cosh a(z - zo) i yo sinh a(z - zo)

y = yo cosh ct(z - z,,) + xo sinh ct(z - z,,)

The solutions for = 0 are

or in the matrix form,

1 ("")=MI(;). cosh a(z - zo) sinh ct(z - zo)

sinh afz - zo) cosh a(z - zo) yo

For +3 = 7112

i cosh ct(z - zo) -sinh u(z - 2")'

-sinh a(z - zo) cosh u(z - zo), M3 =(

and for = ( 3 ~ ) / 4

262 D. W. KERST

The determinant of all these matrices is unity, thus insuring the area preserving requirement for uniform B,.

The transformation for a complete sector is

v, here cosh2 a(z - zo) - eWz-zo) sinhz a(z - zo),

cosh a(z - zo) sinh a(z - za)[l - e- W - z o ) ] , M = (

cosh a(z - zo) sinh r ( z - zo)[l - e2x(z-z0)]

cosh* a(r - zo) - e-Wz--Zo) sinhz a(z - za) and the trace M = all + uZ2 is equal to twice cos p where p is the angle of rotation of a line of force in a complete sector.

cos p = 1 - 2 sinh4 a(z - zo). There is a range in which

-1 < c o s p < -1

showing stable oscillatory solutions for

0 < (B02'/4roEz) < 0.887

where 3 = 4(z - zo) is the total length of a sector composed of the four segments successively rotated 45". We cannot get stability with just two segments, M I M , = M or M,M, = M, since trace M = 2 so we cannot confine lines of force with just two segments per sector. With three segments per sector, we can have confinement of lines (stability) if 2 = 3(z - zo) and

0 < (B09)/(3r0Bs) < 1.32.

Notice that just as (9) shows for the twisted structure the trans- formed line of force can rotate as fast as a pole and no faster for stability, so also -1 = cos ,U shows that the limit for the seg- mented structure is for a line of force going half-way around for one field period which is just the same amount that a particular pole of a quadrupole rotates in one field period.

Gaps between segments where there is E , only do not alter the results since the transfer matrix is the unit matrix in such gaps.