J. Plasma
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)500–503
000–000
J.

About the Possibility of Impurity Ion Accumulationin the Island Region in Helical Plasma
SHISHKIN Alexander
Institute of Plasma Physics, National Science Center “Kharkov Institute of Physics and Technology”, Academicheskaya str.1, Kharkov-108, 61108,
Ukraine and Kharkov “V.N.Karazin” National University, Svobody sqr. 4, Kharkov-77, 61077, Ukraine(Received: 9 December 2003 / Accepted: 23 February 2004)

Abstract
The possibility for ions to be accumulated in the island region of the toroidal plasma is shown here on the basis ofmagnetohydrodynamics (MHD) equations. There are analyzed the mass ion flow trajectories, which are the solution of
the equation dr × uα = 0, where uα is the velocity of the plasma species α. There are used analytical expressions for
two MHD variables: the velocity uα and the thermal flux qα of the same species as the functions of the plasma pressurepα, plasma temperature Tα and the electric field in plasma E. The island structure is described with the analyticalexpression for the function Ψ, which gives us the cross-sections of the magnetic surfaces in the vertical plane.
Keywords:
impurity ions, magnetic islands, accumulation of ions, MHD transport

1. Motivation of study
One of the methods, which can be used for the protection plasma from the penetration of impurity ions, is the producing the magnetic island structure with the use of the externally applied magnetic field perturbations. The magnetic
island diverters on heliotron Large Helical Device and stellarator Wendelstein −7 AS are examples of such possibility.
Can the islands effect on the plasma and impurity ion flows
and how can we observe these phenomena?
In order to answer these questions there are solved theequations for the mass flow trajectories, namely dr × uα = 0,
where uα is the velocity of the plasma species α. There areused analytical expressions for the velocity uα of the plasma
species α and the thermal flux qα of the same specie as thefunctions of the plasma pressure pα, plasma temperature Tα
and the electric filed field E in plasma. These plasma characteristics entered the MHD equations through the products (B× ∇pα), (E × B), (B × ∇Tα). The other new issue here is theintroduction of the magnetic surface function Ψ in order todescribe the magnetic island structure. Another very important question considered here is the possibility to change theisland geometry, particularly, to excite the second chain ofislands (adjacent resonance) in addition to the first chain ofislands in order to control the impurity ions more effectively.

− ∇pα + eα nα E + eα nα (uα × B ) − Fα 1 = 0(1)51nα∇Tα + eα nα ( qα × B) − Fα 2 = 02pα(2)−whereFα 1 =2 α β qβ∑l u − 5l p ,αβ11 β12βFα 2 =β2 α β qβ∑l u − 5l p .
αβ21 ββ22β(3)Here lijαβ are the matrix elements, which depend on the plasma parameters [1,2]. The friction forces are taken into
account in eqs. (1)–(3). From these equations the linear
analysis can be carried out. The variables uα and qα can beexpressed through the main magnetic field B and profiles ofthe density nα, the temperature Tα and the electric field E.
The effects of inertia and viscosity should be studied separately.

2.2 Analytical expressions for uα and qα
From the eqs. (1)–(3) the analytical expressions for variables uα and qα can be obtained and they have the followingform:uα =2. Main equations
2.1 Starting equations

1 ⎧5Bqα[(B × ∇pα) − eα nα (B × E )]⎨ nα (B × ∇Tα) +Duα ⎩ 2Aqα

For our analysis the starting equations are chosen in thefollowing form
©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: shishkin@vlink.kharkov.ua
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Shishkin A, About the Possibility of Impurity Ion Accumulation in the Island Region in Helical Plasma

Coefficients L6*β , L 7*β , L1*β ,…are finally expressed through lijαβ.
In (6) we remain three well-known physics mechanisms,
which effect on the plasma flow: combination of pressure
gradients (∝ ∇Pα and ∝ ∇Pβ), which makes the impurity ionspenetrate into the plasma core, the electric field E in plasma
and the temperature gradient, which can screen plasma fromthe impurity ions. The terms, which are proportional to B∇p,BE, B∇T, Buβ, Bqβ are important in the transport along themagnetic field lines but in the study below we take theimpact of vector products (B × ∇pα), (E × B), (B × ∇Tα) intoaccount. The contributions from the terms mentioned above
will be studied in future paper.

⎡+ eα nα B ⎢L1(B ∇pα) + L2eα nα (BE ) + L3 nα (B∇Tα)⎣+⎛qβ ⎞⎡∑ ⎜⎝L (Bu ) + L (B p )⎟⎠⎢⎣4ββ5βββ ≠α5+ L6( ∇pα − eα nα E ) + L7 nα∇Tα2+∑ ⎛⎜⎝(L l − L l )u + ( − 5 L l + 5 L l ) pαβ
6 112αβ7 212αβ6 12βqβαβ7 22ββ ≠α+ L8β (B × uβ ) + L9β (B ×qβ ⎞ ⎧)⎟ ⎨ ,pβ ⎠ ⎩(4)
2.4 Magnetic field model1 ⎧Auα 5qα=nα (B × ∇Tα )⎨(B × ∇pα) − eα nα (B × E ) −pα Dqα ⎩Buα 2

In our case the magnetic field is modeled in the following simplest way⎡+ eα nα B ⎢J1(B ∇pα) + J2eα nα (BE ) + J3 nα (B∇Tα)
⎣⎧
⎧1⎪ r
⎪
B0 = B 0 ⎨0, ι ( −r 2),⎨,Rr⎪
1− cosϑ ⎪⎩⎩R+⎛qβ ⎞⎡∑ ⎜⎝J (Bu ) + J (B p )⎟⎠⎢⎣4ββ5β

the magnetic surface has the form Ψ 0 =ββ ≠α5+ J6( ∇pα − eα nα E ) + J7 nα∇Tα2+αβ
7 212βαβ6 122β ≠α+ J8β (B × uβ ) + J9β (B ×2r, where −r =.
a(Fig. 1). The magnetic perturbation field is chosen in theform

B1m,n = B 0 {bm,n −r m − 1sin(mϑ − nϕ) ,∑ ⎛⎜⎝(J l − J l )u + ( − 5 J l + 5 J l ) pαβ
6 11r−2(7)αβ7 22qβbm,n −r m − 1cos(mϑ − nϕ), 0}.

βqβ ⎞ ⎧)⎟ ⎨ .
pβ ⎠ ⎩(8)(5)

Then the modified function Ψ in the case of one resonance
perturbation

These are the general expressions, which can be useful forthe linear analysis, particularly to compare the MHD velocityof the background ions ui and the MHD velocity of the impurity ions uI. In order to carry out such analysis it is necessary
to induce the expression for qα into the expressions for ui andu I. The matrix elements mentioned above are expressed
through the coefficients [1], which are obtained as momentsof the collision operators from the kinetic description [2].

R mr 2) − n]−r d−r + bm,n −r cos(mϑ − nϕ). (9)Ψ = [mι (−a∫The shape of the magnetic surface with one chain of island
described with (9) is shown in Fig. 2.

2.5 The description of the island structure
The function, which can describe the magnetic surfaces

2.3 Physics in the expression for uα
From the expressions (4) and (5), after the substitution
the expression for uβ and qβ, the following expression can beobtaineduα =51 ⎧⎨( ∇pα − eα nα E )L6α + nα∇Tα L 7α2Duα ⎩+ ( ∇pβ − eβ nβ E )L∗6β ++5nβ∇Tβ L∗7β25Bnα (B × ∇Tα) + qα [(B × ∇pα) − eα nα (B × E )]2Aqα⎧
+ N1β nβ (B × ∇Tβ ) + N2β (B × ∇pβ ) + N3β (B × E )⎨⎩(6)
Fig. 1 Flow trajectories in the magnetic configuration without islands.

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Shishkin A, About the Possibility of Impurity Ion Accumulation in the Island Region in Helical Plasmawith islands with the “wave” numbers: m,n and m',n', has thefollowing form:

Ψm,n,m,n'' =2.6 Flow trajectory equations
The mass flow trajectories can be obtained as the solution of the equations

1
22(r 2 − rm,n)(r 2 − rm,n' ') rdr
6rm,n

∫

dr × uα = 0.

+2
1 ⎛Δ rm,n⎞ 2⎛ r ⎞ mr 2 − rm,n' 'cos(mϑ − nϕ + δ m,n )⎟⎠ ⎜⎝ ⎟⎠⎜⎝
22 rm,n rm,nrm,n+2
1 ⎛Δ rm,n' ⎞' 2⎛ r ⎞ m'⎛rm,n' ' ⎞ 4 r 2 − rm,n⎟⎠ ⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠⎜⎝
2rm,n2 rm,n' ' rm,n'rm,n× cos(m'ϑ − n'ϕ + δ m,n'' )Plasma parameters are the functions of the magnetic surfacefunction, i.e. p = p(Ψ ), T = T(Ψ ), Φ = Φ (Ψ ), where Φ is theelectric field potential (E = −∇Φ), and Ψ is the magnetic surface function. Functions pα(Ψ ), Tα(Ψ ), Φ (Ψ ) enter in u∇Ψ anduB×∇Ψ, those give uα in the form:(10)(uα )r = u ∇ΨHere rm,n and rm',n' are the radii of the magnetic surfaces withthe rotational transform values ι(rm,n) = n/m and ι(rm',n') = m'/n'; Δrm,n and Δrm',n' are the half widths of the magnetic islands.
The half width is Δr−m,n =(11)∂Ψ∂r1 ∂Ψ ⎞⎛ 1 ∂Ψ
+ uB × ∇Ψ ⎜Bϑ− Bϕ⎟⎝ R ∂ϕr ∂ϑ ⎠R2bm,n (−r )ma m,n. The parameterdι −2m(r m,n)d (r−2)(uα )ϑ = u ∇Ψ

1 ∂Ψr ∂ϑ

1 ∂Ψ ⎞⎛ ∂Ψ+ uB × ∇Ψ ⎜Bϕ− Br⎟⎝ ∂rR ∂ϕ ⎠r m,n' 'is the ratio of the rational magnetic surface radii. Ther m,nshape of the cross-sections of the magnetic surfaces with twochains of islands are shown on Fig. 3.

(uα )ϕ = u ∇Ψ

1 ∂ΨR ∂ϕ∂Ψ ⎞⎛ 1 ∂Ψ
+ uB × ∇Ψ ⎜Br− Bϑ⎟⎝ r ∂ϕ∂r ⎠(12)

Here the parabolic functions of the equilibrium quantities p =p(Ψ ), T = T(Ψ ), Φ = Φ (Ψ ) are determined. In that case themagnitudes u∇Ψ and uB×∇Ψ are independent on Ψ to underline
the effect of the island structure. The more general cases andthe effect of the sign of the electric field on the transport ofions through the island region will be considered further.

3. Physics results
One can see the results of the solution of the eq. (11) on
the Figs. 1-3. The flow trajectories are shown for the configuration without islands (Fig. 1), in the configuration with one
m/n = 1/1 island (Fig. 2) and in the configuration with two
chains of magnetic islands: m/n = 1/1 and m/n = 3/2 (Fig. 3).
The principal thing is the following: the flow trajectories
change in the presence of islands. Flow trajectories concentrate in the region with the islands. It means that the islands
can be the transport barriers on the way of impurity ions into
the plasma core. However in the vicinity of X-points theflows are directed in the core of plasma (Fig. 3).

Fig. 2 Flow trajectories in the magnetic configuration with
m/n = 1/1 island.

4. Discussion
Some words about the restriction of the theory and calculation results. The equations for uα and qα are obtained in
the case of the axisymmetric main magnetic field B0 andmagnetic field perturbations B1m,n and parabolic profiles of thedensity nα, the temperature Tα and the electric field E. Theeffects of inertia and viscosity should be studied separately.
The considered here configuration is similar to the helical field configuration in the magnetic coordinates. The helical field can change the picture: the shape of the magnetic

Fig. 3 Flow trajectories in configuration with magnetic
island chains: m/n = 1/1 and m/n = 2/3.

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Shishkin A, About the Possibility of Impurity Ion Accumulation in the Island Region in Helical Plasma

cross-section from circular to elliptical, for example.
Nevertheless the principal features in the behavior of ions,when the islands are present, should be the same.
In the experiments on Large Helical Device it isobserved the accumulation of the test impurity in the region,
where the m/n = 1/1 island is produced [3]. The possible
physics explanation of this effect on the base of MHD
approach is proposed in this paper.

way of the impurity ions into the plasma core.
The change of the island geometry: the excitation of theadditional islands (adjacent resonance) can help to control theimpurity flows in the helical plasmas.

Acknowledgements
This work is partly carried out under the support fromthe Science and Technology Center in Ukraine, Project N
2313.

5. Conclusions
References

The theory of the accumulation of impurity ions in theisland region of the helical plasmas is given on the basis ofthe MHD approach. The mass flow trajectories as the solution of the equation dr × uα are obtained for the configurations with and without islands. One can see the strong effectof the islands on the mass flows.
Magnetic islands can be the transport barriers on the[1] A Shishkin, H. Wobig, R. Schneider and Y. Igitkhanov,14 th Intern. Stellarator Workshop, (Greifswald,Germany, September 22-29, 2003), Report P. Tu. 32[2] S.P. Hirshman, Phys. Fluids 20, 589 (1977)[3] Y. Nakamura et al., Nucl. Fusion 43, 219 (2003)

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