J. Plasma Fusion Res. SERIES, Vol.1 (1998) 70-74

Spatial-A<is Stel la rators
SHAFRANOV Vitalii
Center "Kurchatov Institute" 123182, Moscow, Russia
ResearchRussianInstitute,Nuclear Fusion(Received:30Septemberl997/Accepted:22October1997)

Abstract

The short review of the theoretical results obtained in Kurchatov Institutein the field of helicalgiven. Some arguments are admagnetic confinement systems and which could be useful at present areon 3-D codes vMEC andbased
duced in favor of the Heliac-type quasisymmetrical systems. The result(P>- 3'/" in the four-periodTERPSICHORE is presented, demonstrating rather high ballooning stableparticles which are responslightly comrgated heliac-type system. The way to avoid the locally trappedof quasisymmetry is dissible for enhanced transport in stellarators, less restrictive than the condition:
surfaceson the base of "normal" systems having topology of lines B const' on the magnetic
cussedlike at axial, helical or poloidal symmetry, i.e. without local extremes of B on magnetic surfaces'

Keywords:
helical system, flux coordinates, plasma stability, quasisymmetry, normal systems

Here /[:

1. Introduction

current and magnetic flux and F(c), V(a), are poloidal
external cument and magnetic flux, a is a label of thetoroidal magnetic surface, e.g. the normalized radius

At present we observe a revival of the helicalplasma confinement systems which are directly connected with L. Spitzer invitation of the space-figureeight stellarator [1]. In Kurchatov Institute, the theoryof such systems starts in the middle of sixties in connection with construction of small figure-eight stellarator
consisting of four semi-tori. For calculation of theplasma shift, the Mercier quasicylindrical orthogonal
coordinate system was used [2]. The Mercier quasifluteinstability was studied in Ref. [3] on a base of curvilinear flux coordinates with both contravariant and covariant representations of the magnetic field B:

ZnBia(r) : const, or the value Z(a) inside the magreticsurface, or 1(a) : Y*- v(a) etc' Functions v and E

a: 0 and V ' B: 0. By definition, the rotational transform is t :are nec€ssary to satisfy conditions B 'Y:- rI''(a)la@)'@ .vql@ 'vg) l(a)lo(a):One of useful conclusion of those old researches
was discovering a possibility of the local sufficient stability criterion for the ideal MHD modes in the casewhen there is no net toroidal current [a]. This criterion
is obtained from simple consideration of the potential
energy of the ideal incompressible displacements and- 10, - v' {8, a {El'I(V0 x V6'Ya\'t, J(a), <D(a) are toroidal

I2nBi:{-v+ OEl0a, J* 0q100, F+ aElAll'In vector form: 2nB: Vt/x Vg + VO x V9,has a form [5]:ffi_s#A+zkl,'*0,(9-,)]2nB: M+ FVe- vYatYq,We,(thatcorrespondsto 2nA:andt kzlY al2 < 0.

'DV0+

B:VxA).

@1998 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author's e-mail: shafran@ qq'nfi' kiae'su70

IShafranov V. eral., Spatial-Axis Stellarators

Here A : (a')rt I v', (j . B> : (J, F- F, J\l V'. With_out the net current J(a) : 0, only the magnetic well
could provide fulfillment of this condition. Besides it iscos0), where r is directed along the axis principal nor_mal v. If in Boozer coordinates the line d" : const., 4 :const. has topology of the line I : const. on magnetic

desirable to have:surface, then the QS condition is:- small Pfirsch-Schltiter current (the first term),- small k2lYal2.
ff. f @) I 0, then the condition J,l > 0 is desirable.
A computational information about the all terms

0Bl0l,:0,
or in invariant form:(FB+ B x VW).VB:0.

of sufficient criterion could be useful for better under_
standing the way of confinement system optimization.

Here F(a) and V\a) are the current and magnetic flux

The less restrictive sufficient stability criterion,proposed by D. I_artz et al.

inside the contour d,[6], takes into account the: const., d: const. If the poloidal

strip 4: const. ofthe Boozer coordinates (a, go, (n) iscounted from the torus ,.hole", not rotating around thebending of the magretic field lines. By neglecting the all
explicitly positive terms in 6W except the first and themagnetic axis, thenlast ones, one obtains from functional dw(f) the next

0,: 0n* NCr,, B: B(a,4 + N6').

Lagrangian equation:

B .v(B .v I llv al2) + r(6 : 0.

It is the case of the quasihelical symmetry. It is natural
to consider the contours fu: const., (n: const, having
the same topology. In this case In: J,: /, (pr,: @":A finite solution corresponds to instability. At present,with existing codes it could be useful to analvze this@.

equation numerically.

From .B.V4: B.ye, + NB.Vgh and from co_variant representation of B it follows:

2. Ouasisymmetric Systems
Our study of the quasisymmetric (eS) toroidal
system was inspired by J. Ntihrenberg and R. Zille's

4:to*N,V":'!n- NO, 4:4 + N/.

[7]

discovering a possibility to satisfy condition of quasisymmetry, which was proposed by A. Boozer[g] as ameans to avoid the locally trapped particles due to con_servation of the canonical momentum of the drift par_The condition of the poloidal quasisymmetry is
0B 100,-ticle. In this case one will have deal with neoclassicaltransport like in axisymmetric tokamak. In our firstg, B: B(a, l,)in Boozer coordinates. or(tB+Bxv@).vB:0

paper concerning the QS systems[9], it was discovered
a possibility to have th" eS conditions at small numberof periods not only in systems of Helias-type but also ofin invariant form. At zero net current, ,f : 0, it is condition of the "isodynamic,' system, where the particles

the Heliac-type systems. Let recall the difference be_

drift along the magretic axis that leads to the classical(!) transport. The systems satisfying this condition at ktween these systems.

: 0 are known as orthogonal mirror systems. Unfortu_
nately, the term kx: kp cos? in expression for B could
not be excluded at & I 0 (in Boozer coordinates).

In Heliac-type systems the magnetic surface cross_section rotates mainly together with the magnetic axis
principal normal. Thus, being vertically elongated at themaximum axis curvature, it will be vertically elongatedalso at the middle of the system period. This allows toIn our consideration the form of the magnetic axis
is prescribed. Usually it is taken as a helical line r :conserye the favorable D-shape form for stability along
the all period. Because of this we have concentrated onro(s) on a reference torus. The radius-vector of the ar_bitrary point is described by r: ro(s) 1- peo,where theunit vector eo lies in the plane s: const. which is orthogonal to the axis, thus er.Vs : 0. I€t @ is the angle
between the principal normal v and eo, then the metricof the quasicylindrical coordinate system (p, @, E :the Heliac-type systems. In Helias-type systems themagnetic surface cross-section rotates twice slowlv thus
becomes horizontally elongated at the middle of perioa.

The types of quasisymmetry depend on directionof the lines B - const. on magrretic surface. They can2ns I L) is represented by the expression (see Ref. 10][)

go along the torus (*toroidal" quasisymmetry) or

dl2 : dpz t p2da2 * 2rcp2Rdadl *around the magnetic axis (..poloidal,' quasisymmetry).
Near the magnetic axis B:

B: B(a,0,),Bol(l - kx) = Bo(l * kpR'[(1 - Kp cos a)2 t rcz p2ldl2 .
71,

Shafranov V. et a/., Spatial-Axis Stellarators

Let the near-axis elliptical magnetic surface cross-0":@ + d(g)section rotates according to the relation
top ofthefromis
countedwhereQ6G),o) + nE:+the ellipse. Following C. Mercier [11] we denote its
semiaxis by a exp(-412) and a exp(q l2)' Thus theequation of the ellipse in parametric form isp cos ee: 4 exP(- q 12) cos 0!
p sin Q : a exP(412) sin 0f .

As a result we obtain for the linear in /<a term in expression for B * Bo(1 -f kx):

kx: kp cos(Q - d)
: ka[exP(- 1112) cos d cos 0f

It follows from this criterion that ballooning modes
slowly decreasing along the extended poloidal coordinate, are more stable than the Mercier modes, in systems with positive shear S: al lt> 0'
Such contradiction was discovered by Cooper'
Hirshman and Lee [16]. They found unstable modes
very localized in the extended poloidal representation'
thus not localized strongly in radial direction' Later, theballooning modes of such type were found to restrict
the plasma pressure in a number of modem stellarator
devices and future projects [17, 18], including the heliac-like quasi-helically symmetric stellarators, Ref' [ 19]'
To understand a reason of the localized ballooning
modes instability the role of magnetic surface geometry
was investigated in Ref.[20] for conventional stellarator
using the stellarator approximation. The 3D geometryof the magnetic surfaces and only 2D tokamak-like dipole secondary currents were taken into account' It was+ exP(q 12) sin d sin 0i ].
Returning to the poloidal variable Of; counted from theprincipal normal (thus QS condition to be B: B(o'shown that only 3D geometrical effects are responsible
for local ballooning modes instability' Up to now, there
is no any analytical description of such kind of ballooning modes. Perhaps, the analysis of the terms in thelocal sufficient criterion, and the D- I-nrtz et al' criterion, can help to make the problem more clear'0f, )), we get

kx: a(A, cos 0f, * Arsin ol),
whereAr: k(l){exP(-qlz)cos d'cos(ne + 1)
+ exP(tt 12) sin d ' sin(nf + i)],To overcome the low ballooning stable B-limit it is
to be refused from the condition of the toroidal quasisymmetry by adding bumpiness into system' Accordingly a topology of the line B : const' was changed toAr: kG){- exP(- q 12) cos d ' sin (n | * i )-r exP(rl 12\ sin d' cos(nf + f ))'poloidal one and rather high Mercier stable (4%) andballooning stable (3%) betas were obtained in such
Heliac-type quasi-mirror symmetrical system' For de-A1+ A1: k2[exP(-4)'cosz r)* exp(4)'sin2 dl.

tails see [19].

We introduced parameter ,i,(6) for the magnetic field
line to be straight.

3. NormalSystems

const. Using the condition Ar(0) :0, the predictionsof rather high Mercier stable (f) for Heliac-type QS

which the lines B : const. going along the all torus intoroidal or poloidal direction don't create the islands ona magnetic surface. In this case there is no local mag-:The QS conditions require: Ar : const', A,

Under "normal" we mean the toroidal systems in>8) weresystems with large number of periods (Ndone in Ref.[12] which agrees with the later 3-D com-

netic well, no locally trapped particles and corresponding enhanced transport. The formal condition of the"normality" could be formulated as a necessity of theputations, Ref.[13]. Meanwhile, the 3-D computationsof the ballooning modes have shown (fL to be smaller

straight lines B : const. in the flux coordinate withstraight field lines [21]. In this case one can choose thethan Mercier limit both in Helias{ike and Heliac-likeQS stellarators in contradiction with some analyticalnecessary condition of the ballooning modes stability
perturbation [15]):[14] (based on special probe: const coinciding
coordinate surfaces 0: const. or g:
condition of northewith surfaces B const' Thus
mality is as follows.
Toroidal direction:

I2 * ( p:4\ [u*,.r B? a *("P])'tp" olIaBlae:0, B: B(a,0)\B|lll"'in any straight field line flux coordinates' In vector+1s(?T)'2"\tnat ' oform:
72

Shafranov V. er a/.. SpatiaFAxis Stellarators{Q + aelaeB + Bx vr)}. vB: 0.

The condition of the normality A: B@,e*) is tulfilledat

Poloidal direction:0Bna{ y+ kfBo:o.

08100:O; B: B(a,l).
In vector form:

It follows from this condition that the magnetic axis

(J + aela|)B+ Bx vr)J. vB: 0.

curyature should be zero at the extremes of theBo: k :0 at ABo I AE: 0.

These conditions differ from corresponding
con_ditions of quasisymmetry by presence a free functiong, which obeys to the equation

The poloidal normality seems to open interesting
new possibilities for optimization of the helicalmag_netic systems.

B'YE-o,<82>v':
21r "- - -4", lE 'Transforming coordinates 0, I to the habitual Boozer

Acknowledgments
I woukd like to express my deep gratitude to M.
Mikhailov, M. yokoyama and A. Subbotin for help
inones [5],preparing this paper.
IU'

0:e-*,r'.
- -b <82>V,,t,,O'
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)73

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74