Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 (2004)
6 (2004)550–553
000–000
J. Plasma Fusion

Global Plasma Equilibrium in a Helical Systemwith Ideally Conducting Wall
PUSTOVITOV Vladimir D
Russian Research Centre “Kurchatov Institute”, Moscow, 123182, Russia(Received: 9 December 2003 / Accepted: 26 January 2004)

Abstract
The global plasma shift is calculated analytically for a helical system with an ideal wall. The derived expression
for the plasma shift, incorporating both the finite-β plasma expansion and the opposing reaction of the nearby idealwall, can be used for interpreting the observable high-β equilibrium effects in LHD and other helical devices.
Keywords:
stellarator, plasma equilibrium, finite-β effects, plasma shift, ideal wall
increasing β, as described by ∆β . Such a case can be realized
in LHD which is a superconducting device operating now
with high β [2,5,6].
Theory of current-carrying plasma equilibrium in a tokamak with an ideally conducting casing is described in [7,8].
These studies have been stimulated long ago by the earlyexperimental results showing the urgent need of providing
the plasma equilibrium along the major radius in a tokamak[9]. However, the same problem for stellarators has not yet
been analyzed. This is so because until recently the stellarators operated at rather low β, and the plasma equilibrium instellarators is provided by the original magnetic configuration which is not strongly distorted by the equilibrium plasma
currents at low β.
The problem of global plasma equilibrium in a stellarator was discussed in [10,11], see also the reviews [1,4]. The
present analysis is based on the models described there. Thenew element introduced here is the ideal wall at some distance from the plasma. In [1,4,10,11], the vertical field B⊥
was considered as a free parameter. Now we must find it
under the constraint of flux conservation due to the ideallyconducting wall.

1. Introduction
Analysis of free-boundary plasma equilibrium in a conventional stellarator shows [1] that the pressure-induced shift∆β of the plasma column must be fairly large at high β:∆βββ≤≤ 00 .
02β eqb2β eq(1)Here β 0ep = µ2bb/R, µb is the rotational transform at the plasmaedge, b is the averaged minor radius of the plasma, R is themajor radius, β is the volume-averaged ratio 2p/B20 with p
being the plasma pressure and B0 the toroidal magnetic field
at r = R, and β 0 = 2p(0)/B20 is the β value at the magnetic axis.
For Large Helical Device (LHD) with R = 3.9 m, b = 0.6m, µb = 1 [2] we have β 0eq = 0.15, and the lower bound in (1)is 0.1 for β = 3%, which gives 6 cm outward shift.
If such a large shift would appear in LHD, it certainly
could be observed, for example, when the Shafranov shift
was measured with soft X-ray CCD camera [3]. However,there is no mentioning of observations of large ‘global’ plasma shift in [3]. Therefore a question arises why the plasma
column shift in LHD is actually smaller than the above estimate.
The pressure-induced plasma shift can be suppressed by
the vertical field B⊥ as described by the formula (see [1,4]and references therein)∆ b = ∆β + ∆⊥ ,2. Brief introduction to the model and definitions
The model is based on the modified stellarator approximation allowing description of tokamaks and stellarators
within the unified approach [1,4]. We consider a ‘conventional stellarator’ with a circular planar axis and helicalfields, a device like LHD or CHS. Within the model, theplasma boundary is described as a circular torus perturbed by
the helical field. Large-aspect-ratio expansion is used in ana-(2)where ∆b is the observed shift of the plasma boundary or‘global’ plasma shift, the expressions for ∆β and ∆⊥ are given
below. The field B⊥ can be produced by the currents in thepoloidal windings, which can be controlled, and by the currents induced in the conducting structures around the plasma
when the plasma column tends to expand toroidally with

©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchCorresponding author’s e-mail: pustovit@nfi.kiae.ru
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Pustovitov Vladimir D., Global Plasma Equilibrium in a Helical System with Ideally Conducting Walllytical calculations here. The shift ∆b of the plasma boundaryis assumed to be small: ∆b /b << 1. For derivation of the basic
relations from the general equilibrium equations and moredetail see [1,10,11].
Magnetic surfaces ψ = const in stellarators (or tokamaks) are described by the functionψ = ψ v + ψ ext + ψ pl ,We describe the plasma boundary as l = b. If ψ is given as afunction of l and α,

ψ = A(l ) + B1(l)cosα ,the condition ψ = const at the plasma boundary is satisfied byB(b) = 0.
The plasma-generated part of ψ has been already calculated in such a form, eq. (4). We have to express two otherfunctions in (3) in variables l and α.
The poloidal flux of the external vertical field, eq. (8),can be rewritten as(3)where ψv is the poloidal flux of the helical magnetic field,
ψext is the poloidal flux due to the external axisymmetricfield, and ψpl is the plasma-produced poloidal flux. At thefirst step of our analysis we need the function ψ outside theplasma, in the vacuum region between the plasma and thewall.
The plasma-produced poloidal magnetic flux outside theplasma in a stellarator was calculated in [11]:ψ pl = 2π [ f0(l) + f1(l)cosα + …],

ψ ext = π B⊥ [(R + ∆ b )2 − 2(R + ∆ b )lcosα + l 2 cos2α ]. (11)Here we use the relations between the coordinates:r = R − ρ cosu = R + ∆ b − lcosα ,f1 (l) =⎞bbl ⎛ 8R
C−BJ ⎜ln− 1⎟ ,⎠⎝2ll

ρ ≈ l − ∆ bcosα ,~ ~^(5)ψ v ( ρ ) ≈ ψ v ( l ) − ψ v' (l )∆b cosα .

B(l) = 2 π f1(l) − 2 π RlB⊥ − ψ v' (l ) ∆ b ,(15)and the boundary condition B(b) = 0 gives

f1(b) − RbB⊥ + R ∆ bB* = 0.

(16)Here f1 is the function defined by (6), andB* ≡ −(7)ψ'v (b)2πR= µbbR

B0 .

(17)

In the model, the ideal wall is prescribed by ρ = ac,
which allows a shift of the plasma relative to the wall. To
apply the constraint (9) at ρ = ac, we must transform ψ to theformψ = D(ρ) + E(ρ) cosu ,(18)

so that the condition ψ = const at the wall is satisfied by E(ac)= 0.
Similar to (13), for ρ >> ∆b we can express l from (12) asl ≈ ρ + ∆b cosu.

(8)

(19)

After transformation of the functions ψpl and ψext to variables(ρ, u) we obtain

Next we must find the constraints from the boundary
conditions for the function ψ.

E(ρ) = 2 π [ f0' ( ρ) ∆ b + f1 ( ρ)] − 2 π R ρ B⊥,

3. Boundary conditions(20)

and, finally, from E(ac) = 0:

By definition, the function ψ must be constant at theplasma boundary and at the ideal wall. Thus, at these twoboundaries of the vacuum region we must satisfy the conditions

ψ = const.

(14)

Combining (4), (11) and (14) and comparing the resultwith (10), we obtain(6)

where r, ζ, z, are the cylindrical coordinates associated with^the main geometrical axis, f is the oscillating part of the integral of f over ζ, Bt is the toroidal magnetic field, and brackets〈…〉ζ denote the averaging over ζ. The most simple approximation for ψv is obtained in coordinates (ρ, u) related to thegeometrical axis: ψv = ψv ( ρ).
For our purposes, in the geometry assumed (circularaveraged cross-sections of the plasma and the conductingwall), it is sufficient to keep in ψext a term describing theexternal homogeneous vertical magnetic field:

ψext = ψ 0 (t) + π B⊥(r 2 − R 2).

(13)

which is valid for l >> ∆b. In this case, in linear approximationin ∆b /b,

BJ = J/(2π b) is the magnetic field of the current J, and C is aconstant which can be related to plasma parameters bymatching the solutions for the magnetic field at the plasmaboundary.
~The poloidal flux ψv of the helical magnetic field B
through the magnetic surface is calculated by the formula [1]ψ v = 2π r 2 < Bz Br / Bt >ζ ,ρ sinu = lsinα . (12)These relations give(4)where (l, α) are the polar coordinates with origin at the centerof the plasma cross-section,

f0 (l) = bRBJ ⎛⎜ln 8R − 2⎞⎟ ,⎠⎝l(10)(b/ac)[ f1 + f'0 (ac)∆ b ] − RbB⊥ = 0.

(21)

Relation (16) is a part of the equilibrium conditions for aplasma. This follows directly from ∇p = j × B and must be
satisfied in any case. Equation (21) is an additional constraint
resulting from the requirement that the wall is ideal.

(9)
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Pustovitov Vladimir D., Global Plasma Equilibrium in a Helical System with Ideally Conducting Wall

4. Equation for ∆btendency: the plasma shift ∆b is smaller for the wall closer tothe plasma.

Without the wall, we have eq. (16) which gives ∆b as afunction of B⊥ and f1 with B⊥ being a parameter. With idealwall, we have two equations: (16) and (21). Subtracting one
from another to eliminate B⊥, we obtain

f1 (b) −b

f1 (ac) =acbac

f'0 (ac) ∆ b − R ∆ b B*,

6. Current-free stellarator with circular shifted magnetic surfaces
If the surfaces ψ = const near the plasma boundary Γ arecircular (the relative shift is allowed), we obtain [11](22)H1 =which, by using (5) and (6), can be transformed into⎛ b2 ⎞⎛b2 ⎞b2aC ⎜ 2 − 1⎟ + BJ ln c = R ∆ b ⎜B* + 2 BJ⎟ .
⎝ ac⎠⎝
ac ⎠2b(23)bR ⎛∆b *b ⎞B −BJ⎟ .
⎜H 1 −2 ⎝b2R ⎠(24)( µh ∆)'⎪Γ = 2So, the equality (23) can be rewritten in a form convenient
for further use:

Bβ =Since no restriction has been imposed on BJ and B* inthe above derivation, expression (25) must be valid for any
ratio between B* and BJ. In other words, it can be applied
either to tokamaks, B* = 0, or to stellarators without current,
BJ = 0, which are two limiting cases. It describes, as well,
any intermediate configuration (stellarators with current).
For a tokamak, B* = 0, relation (25) is reduced to1 ⎞⎛a ⎡ ac ⎛b ⎞⎤∆b =+ ⎜ Λ + ⎟ ⎜1 − 2 ⎟⎥,⎢ln2R ⎣ b ⎝2 ⎠⎝ac ⎠⎦1 b p'(a) a 2CPS da,
B0 0 µ (a) b 2∫

H1 − B*∆ b / b = 2Bβ .

(30)

(31)

This allows us to rewrite (27) in a compact form

∆ b = ∆β (1 − b 2/ ac2 ),(32)∆β = Bβ / B*.

(33)

where2(26)The latter is the free-boundary pressure-incduced plasma
shift disussed in the Introduction.
Expression (32) explicitly describes the ideal wall effecton the pressure-induced shift of the current-free plasma column in a stellarator. The effect is stronger for the wall closer
to the plasma. However, the effect is noticeable even when
the gap between the plasma and the ideal wall is rather large.
For example, eq. (32) shows that for b/ac = 0.7 the wall effectresults in 50% reduction of the shift. This means that in theexample considered in the Introduction we would obtain 3
cm plasma shift with the ideal wall instead of 6 cm for thefree-boundary case.
Note that ∆b can be also written in the form (2) with ∆βgiven by (33) and with

where Λ ≡ − RH1/(bBJ). This is a well-known Shafranov’s
result [7] for a tokamak plasma.
The tokamak expression (26) cannot be used for currentfree plasma in stellarators (more precisely, for equilibrium
configurations with small BJ /B*) because Λ is not defined for
BJ = 0 when H1 is finite. Since B* and BJ enter eq. (25) in different ways, it is impossible therefore to draw a close analogy between tokamaks and stellarators in this case.
For a current-free plasma, eq. (25) gives

1⎛∆b *∆ ⎞⎛b2⎞
B = − ⎜ H1 − b B* ⎟ ⎜1 − 2 ⎟ .
b2⎝b ⎠⎝ac ⎠(29)and Cps is the coefficient describing the reduction of thePfirsch-Schlüter current when the plasma column is strongly
shifted relative to the geometrical center of a stellarator.
Usually Cps ≈ 1 in conventional stellarators. For the case considered, eq. (28) yields(25)

5. Two opposite limits: tokamak and current-free stellarator2c

R Bβ ,b B0where⎛a2 ⎞a2 ⎧a∆ b ⎜BJ + B* 2c ⎟ = c ⎨BJ ln c⎝b ⎠ 2R ⎩b⎡1b2 ⎞ ⎧R ⎛∆ b *⎞ ⎤ ⎛+ ⎢ BJ −B ⎟ ⎥ ⎜1 − 2 ⎟ ⎨,⎜H 1 −b ⎝b ⎠ ⎦ ⎝ ac ⎠ ⎩⎣2(28)which is valid for both the tokamaks and stellarators. If weassume that the model of circular shifted magnetic surfaces isapplicable for the description of the whole plasma column,
one can obtain from the equilibrium equations [1,4]The constant C, first appeared in (6), is given by [11]C=bb∆
BJ + B0 ( µ ∆' + µh' ∆ b ) + B* b ,RRb(27)

The quantity H1 can be found from the magnetic measurements outside the plasma [11]. The shift ∆b itself can bemeasured by the magnetic loops and probes [11, 12]. Thus,eq. (27) relating the measurable values allows experimentalverification.
Equations (25), (26) and (27) clearly show the effect of
the conducting wall on the plasma shift. In all cases, for ac =b they give a natural result ∆b = 0. This illustrates a general∆⊥ = bB⊥ / B*(34)(this follows from (16) for the current-free plasma).
Equations (32) – (34) imply that with ideal wall ∆ ⊥ = −∆β b2/a2c, so that in this caseB⊥ = Bβ b 2 / ac2.

(35)

The latter relation gives the value of the vertical field pro3552Pustovitov Vladimir D., Global Plasma Equilibrium in a Helical System with Ideally Conducting Wallduced by the currents induced in the ideal wall.

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7. Conclusion
The analysis shows that the ideal wall must strongly
affect the pressure-induced plasma shift in helical devices.
According to (32), the effect is strong even for b/ac = 1/2.
The plasma shift reduction due to the currents induced in thewall may be a reason why the Shafranov shift measured withsoft X-ray CCD camera [3] was small.
According to (32), the plasma shift may be completely
suppressed by the wall reaction in case only when the plasma-wall gap is negligible. In real devices, the gap is finite.
For LHD, a rough estimate can be b/ac = 0.6/0.9 [13]. In thiscase, the plasma shift must be twice smaller than it would bewithout the wall. For β ≥ 3% we obtain from (32) that theshift will be several centimeters in LHD. Such a shift is large
enough to be measured, which can be used for diagnostic
purposes. Therefore, the problem deserves more attention.

Acknowledgements
The work was supported in part by the Grant-in-Aid
from the Japanese Ministry of Education, Science, Sport andCulture. Useful discussions with Profs. K. Matsuoka, K.
Yamazaki and H. Iguchi are highly appreciated.

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