J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)299–302
000–000

Formation of the X-Point in Helical Plasmas
upon Injection of High Energy Particles
YAKOVLEV Mikhail, MOTOJIMA Osamu1, INAGAKI Shigeru1,
SHISHKIN Alexander2 and KOMORI Akio1
Department of Fusion Science, The Graduate University for Advanced Studies, Toki 509-5292 Japan
1
National Institute for Fusion Science, Toki 509-5292, Japan
2
Institute of Plasma Physics, National Science Center “Kharkov Institute of Physics and Technology”,
Academicheskaya str.1, Kharkov-108, 61108, Ukraine andKharkov “V.N. Karazin” National University, Svobody sqr. 4, Kharkov-77, 61077, Ukraine(Received: 9 December 2003 / Accepted: 23 March 2004)

Abstract
The injected high energy charged particles could drive the localized currents in plasma, and thereby they canmodify the magnetic field structure, particularly can form the X-point. The analytical description of X-point due tothe localized current is presented on the base of magnetic surfaces function with analysis. The formation of large∇n/n region in magnetic configuration with the X-point by localized current is predicted. The bifurcation of theelectric field is expected by change in ∇n/n. The helical magnetic configurations with X-points, which can be realized
on LHD (Large Helical Devices) are proposed.
Keywords:
X-point, magnetic island, magnetic configuration, plasma bifurcation

1. Introduction

()

1
dp
= F = q ⎡⎢ E + V × B ⎤⎥dtc⎦⎣

In the real experiment it is not so simple to achieve
improved plasma confinement (H-mode regime). The
bifurcation of the electric field and its effect on the plasma
confinement strongly depend on the target plasma state. It isnecessary to find the physics mechanisms, which can trigger
the bifurcation of the electric field. One of the candidates tocontrol the electric field in plasma is injection of the charged
particles proposed in [1,2]. To such candidates there can be
related the modification of the magnetic field and magnetic
surfaces structure, particularly the formation of X-point,
another type than island X-point, with the use of electron andion injection. The difference in the X-points produced withthe current in local island diverter coils and the X-point fromthe current in plasma is explained below.

(2.1)

dr= p = γ m0V , γ = (1 – V2/c2) is relativistic factor, rHere
dt
is the radius vector of the particle, V is vector velocity, p is
the relativistic momentum, q and m0 is the charge and massof the particles, E and B represents the magnetic and theelectric field respectively and c is the speed of the light.
Because the particles have high energy, the collisions are nottaken to account in this research.
2.2 Main magnetic field
The main magnetic field (B = ∇Φ) is modelled by usingof the magnetic field potential⎡RnΦ = B0 ⎢Rϕ −ε n, m (r / ah ) sin(nϑ − mϕ )m⎢⎣n⎤
+ ε1,0 r sin(ϑ )⎥ . (2.2)⎥⎦∑

2. Electron and ion paths and possibility toproduce the current in helical plasma
2.1 Equation of motion for charged particles
The motion of high-energy charged particles is studied
by using the Newton equation with Lorenz force taking into
account a relativistic factor that makes possibly to take into
account the Larmor radius also. This system of equations has
the following form

Here B0 is the magnetic field at the circular axis of torus, Rand ah are the major and minor radii of the helical winding
system, r, ϑ, ϕ are coordinates connected with the circularaxis of the torus, r is the radial variable, ϑ and ϕ are theangular variables along the minor and major circumference©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: yakovlev@LHD.nifs.ac.jp1299

Yakovlev M. et al., Formation of the X-Point in Helical Plasmas upon Injection of High Energy Particlesof the torus respectively; ϑ increased in the opposite direction
to the main normal of the circular axis of the torus. The
number of the magnetic field period along the torus is denotedby m and the helical winding polarity number denoted as l;εn,m are the coefficient of the harmonics of the magnetic field.
For results presented here the parameter of LHD [3] are taken:
B0 = 3T, R = 390 cm, a h = 97.5 cm, l = 2, m = 10. The
parameters εn,m and ε1,0 are taken in a way which is explainedin [3].
The perturbation magnetic field is described by scalarpotentialΦ p = B0 ah

ε n, m , p
mp

V|| /V and with decreasing this value the square of drift island
in meridian cross-section will increase. For injection in thevicinity of the X point of island and X-point of the helicalfield they become passing but the main part of trajectory
follows the drift surface and does not form the closed-looppath.

3. Formation of the x - point as themethod to effect on plasma parameter
gradient
X-point as the result of additional current in plasma canbe written in the following simplified way. Simple magneticfield description below gives us the picture of the effect ofcontrolled beam current on the magnetic configuration withthe rotational magnetic transform.
Magnetic surface with the current in plasma can bedescribed by the following expression.

(r / ah )m sin(m pϑ − n pϕ + δ m,n, p ) (2.3)
p

2.3 Simulation of injected particles motion
The simulation of the injected charged particle motion
has been performed with goal to find the path in theconfinement region in which particles could move closely to
one circular loop along circular axis of torus. As candidates
for such path production the next injection points arereviewed: the centre of island (O-point), X-point of the helicalfield and X-point of m/n = 1/1 island.
It is found that upon injection of electrons and +23He ionsin the centre of island for the most of values of V|| /V, whereV|| is the particle velocity parallel to magnetic field line at theposition of particle injection, the injected particles are“chaotically blocked” (after several oscillations the helicallytrapped electron periodically escape mirror filed and turn totoroidally blocked and vice versa) inside the island even if
V|| /V = 0.9. But they became passing when V|| /V = 0.98 andtheir trajectory is the closed-loop-like around the main axisof the torus Fig. 1. The current path for ions lies inside drift
island with very small square in the meridian cross-section.
The value of this square is connected directly with relation

2⎡⎛r⎞ ⎤r
Ψ = r ι(r ) dr − η ln ⎢1 − 2 cos(ϑ − ϕ − ϑ C ) + ⎜ ⎟ ⎥rC⎝ rC ⎠ ⎥⎢⎣⎦∫

2(3.1)
2J R

where η = cr B a . Here J is the value of the produced
C 0current, ι is rotational transform, a is the small radii of thetorus, rc and ϑC are the radial and angular co-ordinates of thecurrent position.
The equations for the singular points is the followingsin(ϑ − ϕ − ϑ C ) = 0

ι (r 2 ) + η

(3.2)

1=0
⎛
r⎞
r ⎜1 − ⎟⎝ rC ⎠(3.3)

The solutions of this equation can be presented approximately
with the following expressions

ϑ − ϕ − ϑ C = πk , k = 0, ±1, ±2,...

(3.4)( −1)η
η, r2 = rC + 22ι(r ) r = 0
ι(r ) r = r

(3.5)

r1 =C

The X-point is formed only if η < 0.
The radial variable r̄ as the function ϑ – ϑC on theseparatrix curve in the plane ϕ = const may be written in awayr = rS + ∆rS ,(3.6)

where rS is radial co-ordinates of the separatrix position and1− 2∆rS =⎛ r⎞rcos(ϑ − ϑ C ) + ⎜ ⎟rC⎝ rC ⎠21η ln2⎛ ι(0) + dι r 2 ⎞ r⎛ rS ⎞rSS212−−+cos()ϑϑ⎝⎠SC⎜r ⎟drrC⎝ C⎠(3.7)
Fig. 1 3D trajectory of the +23He ion with the energy W = 100
keV. This particle is injected in the centre (O-point) ofisland 1/1.

which can be treated as the dimension of the deviation of thecurves forming the separatrix from the curve r̄ = r̄S.
2300

Yakovlev M. et al., Formation of the X-Point in Helical Plasmas upon Injection of High Energy Particles

It is assumed that the density profile is connected withthe magnetic surface with formula⎛ ε (r ) ⎞C(1) ≈ ⎜ t ⎟⎝ ε h (r ) ⎠gk⎡⎛Ψ ⎞ ⎤n = n(0) ⎢ An − ⎜ ⎟ ⎥ ,⎝ Ψb ⎠ ⎥⎢⎣⎦g −1(n(r)[cm ] / 10 )−3⎛Ψ ⎞( −1)k ⎜ ⎟⎝ Ψb ⎠k −1∇ΨΨbharmonics. The collision frequency ν α , β =4 π Λ C e 4 Zα2 Z β2 nβ
3 mα Tα3 / 2,

1/ 25/ 2⎛ mi ⎞ ⎛ Ti ⎞ ⎤ ⎛ 3 1 * ⎞23 ⎡2⎢
X1 =1 + 2⎜ ⎟ ⎜ ⎟ ⎥ ⎜Λ n,Te −− q ⎟,31.5C(1) ⎢⎝ me ⎠ ⎝ Te ⎠ ⎥ ⎝ 2 2 ⎠⎣⎦(3.9)Γ e = Γi

13 2

where α,β = e,i and Zα,β is charge numbers of species.
The ambipolar equation (4.1) can have three roots of theelectric field(4.7)

We expect that the radial electric field Er can be effectedby the changes in the gradients of the density n(r) andtemperature Te(r) and Ti(r) after formation of X-point. Theambipolar electric field can be obtained from the equation

5/ 2 ⎤X2 =⎛m ⎞23 ⎡⎢1 + 2⎜ i ⎟Λ2n,Te −31.5C(1) ⎢⎝ me ⎠⎣X3 =1/ 25/ 2⎛ m ⎞ ⎛ T ⎞ ⎤ ⎛ 3 1 *⎞23 ⎡⎢1 + 2⎜ i ⎟ ⎜ i ⎟ ⎥ ⎜+ q ⎟Λ2n,Te −3
1.5C(1) ⎢⎝ me ⎠ ⎝ Te ⎠ ⎥ ⎝ 2 2 ⎠⎦⎣1/ 2⎛ Ti ⎞⎜ ⎟⎝ Te ⎠( )⎥ − q * , (4.8)⎥⎦(4.9)Here(4.1)

q* ≈where the expressions for the electron Γe and ion fluxes Γican be written in the form [4,5]

⎡⎛m ⎞
13
⎢1 − 2⎜ i ⎟+23 C(1)Λ n,Te ⎢⎝ me ⎠⎣

1/ 2⎛ Ti ⎞⎜ ⎟⎝ Te ⎠7/ 2

Λn,Ti ⎤⎥ , (4.10)
Λn,Te ⎥⎦

⎛3 A ⎞⎛3 A ⎞
Λ n,Te = λ n − ⎜ − 2 ⎟ λTe , Λ n,Ti = λ n − ⎜ − 2 ⎟ λTi .
⎝ 2 A1 ⎠⎝ 2 A1 ⎠3n ⎡ ⎛⎤
A1 λ n − λTe + X ⎞ + A2 λ Te ⎥ ,⎢⎝⎠2a pν ei ⎣⎦(4.11)(4.2)

2⎛ cT ⎞nΓi = ε t2 ε h3 / 2 ⎜ i ⎟⎝ eB0 r ⎠ a pν ii×(Ti (r )[keV])5

Here εt and εh are the coefficients of the main toroidal andhelical harmonics; εj are the coefficients of the satellite helical4. Control of the electric field with the Xpoint formation

22(4.6)

Calculations on the basis of formula (3.1) and (3.9) showsthat density gradient near X-point becomes steeper than
before it formation.

⎛ cT ⎞
Γe = ε t2 ε h3 / 2 ⎜ e ⎟⎝ eB0 r ⎠2

⎛ Te (r ) ⎞ ⎛ ε h (r ) ⎞⎟⎜⎟ ⎜⎝ Ti (r ) ⎠ ⎝ r [cm] ⎠(3.8)

where Ψb is the surface which bounds the plasma, An, k andg are the parameters which describe the profile features. The
density gradient, changed according to the formation of theX-point, effects on the radial electric field. Then the derivativeof the plasma density is connected with the ∇Ψ isk⎡⎛Ψ ⎞ ⎤⎢n ′ = n(0)g An − ⎜ ⎟ ⎥⎝ Ψb ⎠ ⎥⎢⎣⎦1/ 2The transition from one root to the case of three roots under
the density gradient increase is seen on Fig. 2. If the density
gradient and/or temperature gradient are increased bychanging of flux surfaces through the formation of X-point⎡ ⎛⎤Te ⎞23
X ⎟ + A2 λTi ⎥. (4.3)
2 ⎢ A1 ⎜ λ n − λ Ti −2Ti ⎠1 + 1.5C(1) X ⎢⎣ ⎝⎥⎦Here ap is the plasma radius and the following denotations
are induced (the dot denotes the derivative with respect toradius)

X=ea p ErTe, λn = −a pn′n, λTe = −a p Te ′Te, λTi = −a p Ti ′Ti

(4.4)
∞

A1 =∞

1
1−x−x
dx j x 4 e j , A2 =dx j x 5e j
66

∫b

∫

(4.5)

b1/2

2

1 ⎛ ε ⎞ ⎛T ⎞ ⎛ υ⎞

2

The value C(1) is the magnitude C (xi ) ≡ ⎜ t ⎟ ⎜ e⎟ ⎜⎜ Dhi xi ⎟⎟3 ⎝ ε h⎠ ⎝ Ti ⎠ ⎝νii ap ⎠2taken under xi = 1, where xj = mjV /2Tj and vDhi = cTi εh /eB0r
that can be treated as the drift velocity of ion in the helicalfield. The magnitude C(1) can be expressed in such a manner
3/2Fig. 23301

Neo-classical fluxes of ions and electrons under λ =3.675.

Yakovlev M. et al., Formation of the X-Point in Helical Plasmas upon Injection of High Energy Particles

as (3.9) there is a possibility for the bifurcation of the radial
electric field as shown in Fig. 2. The bifurcation of the electricfield is expected by change in ∇n/n in accordance with themodel [6].

5. The magnetic configurations with Xpoint
The configurations with X-point can be created with thecurrent strongly concentrated and this current can be modelledwith the filament. Such configuration is shown on Fig. 3. The
real for the experimental conditions current distribution on
radii is expected. In that case the configuration with X-point
is changed as shown on Fig. 4.

6. Conclusions
Fig. 3

The X-point formation is expected due to injection ofhigh energy particles in m/n = 1/1 island O-point. In contrast
to X-point from local divertor coils the X-point from local
current in plasma has stronger influence on plasma parameterand could lead to it bifurcation. The analytical treatment forroots of transition for electric field is done in neoclassicalapproximation. The expressions for the radial electric fields
are obtained from the solution of neoclassical non-ambipolar
flux balance equation.

Inner Shift Configuration with X-point created by“filament” current.

References[1] O. Motojima et al., Nucl. Fusion 40, 833 (2000).
[2] S. Inagaki et al., A. Shishkin, J. Plasma Fusion Res.
SERIES 3, 485 (2000).
[3] A.A. Shishkin et al., Research Report NIFS Series,NIFS–211, March 1993.
[4] E.C. Crume et al., Phys. Fluids 31, 11 (1988).
[5] H. Sanuki et al., J. Phys. Soc. Of Japan 69, 445 (2000).
[6] K. Itoh and S. Itoh, Plasma Phys. Contr. Fusion 38, 1(1998).
Fig. 4

Vertically Elongated Configuration with X-point createdby current distributed on r.

4302