J. Plasma Fusion Res. SERIES, Vol. 9 (2010)

Pellet ablation in helical plasmas
Ryuichi ISHIZAKI and Noriyoshi NAKAJIMA
National Institute for Fusion Science, Toki 509-5292, Japan(Received: 20 November 2009 / Accepted: 22 March 2010)

In order not only to clarify the diﬀerence on the motion of a plasmoid created by a pellet injection
between tokamak and helical plasmas but also to obtain the universal understanding on the motion in
torus plasmas, MHD simulations have been carried out in tokamak, vacuum toroidal ﬁeld, RFP-likeand LHD conﬁgurations. It is found that the plasmoids drift in the opposite direction of the curvature
vector in tokamak, vacuum toroidal ﬁeld and RFP-like conﬁgurations, but the plasmoid motions dependon the initial locations of plasmoids in LHD. It is also clariﬁed that there are two main forces acting on
the plasmoid, and the connection length determines which force is dominant on the plasmoid motion.
Keywords: pellet, ablation, MHD, LHD
It is found that plasmoids drift in the opposite direction of the magnetic curvature vector in the tokamak, vacuum toroidal ﬁeld and RFP-like conﬁgurations. On the other hand, it is found that the drift
motions depend on the initial locations of the plasmoids in LHD. The plasmid motion is mainly determined by 1/R force due to toroidal ﬁeld and the forceplpl
due to gradient of BR
along the ﬁeld, where BR
isthe major radius component of the perturbed magnetic ﬁeld induced by the plasmoid. The former force
implies the drift in the opposite direction of the curvature vector which is induced in the tokamak andvacuum toroidal ﬁeld. Since the latter force is dominant in the cases that the plasmoids are located inside
the torus in LHD, the plasmoid motions in LHD areessentially diﬀerent from one in tokamak. It is also
clariﬁed that the connection length determines which
force is dominant on the plasmoid motion.

1. Introduction
Injecting small pellets of frozen hydrogen into
torus plasmas is a proven method of fueling [1]. The
physical processes are divided into the following micro
and macro stages. The micro stage is the ablation ofmass at the pellet surface due to the high temperaturebulk plasma which the pellet encounters. The neutral
gas produced by the ablation is rapidly heated by electrons and ionized to form a high density and low temperature plasma, namely a plasmoid. The macro stage
is the redistribution of the plasmoid by free streaming
along the magnetic ﬁeld lines and by MHD processes
which cause mass ﬂow across ﬂux surfaces. The micro stage is well-understood by an analytic method [2]and numerical simulation [3]. The drift motion of theplasmoid is investigated in the macro stage [4]. Since
the plasmoid drifts to the lower ﬁeld side, the pellet fueling to make the plasmoid approach the coreplasma has succeeded by injecting the pellet from thehigh ﬁeld side in tokamak. On the other hand, sucha good performance has not been obtained yet in theplanar axis heliotron; Large Helical Device (LHD) experiments, even if a pellet has been injected from thehigh ﬁeld side [5]. The purpose of the study is toclarify the diﬀerence on the motion of the plasmoid
between tokamak and helical plasmas.
In order to investigate the motions of the plasmoids in tokamak and LHD, the three dimensional
MHD code has been developed by extending the pellet ablation code (CAP) [3, 6]. Especially, since LHD
has complicated magnetic ﬁeld, the simulations havebeen carried out in three cases in which initial plasmoids are located inside and outside the torus on thehorizontal elongated poloidal cross sections and inside the torus on the vertical elongated poloidal crosssection. In addition, in order to obtain the universal understanding in torus plasmas, MHD simulations
have been also carried out in vacuum toroidal ﬁeld
and RFP-like (Reversed Field Pinch) conﬁgurations.

2. Basic Equations
Since the plasmoid has such a large perturbationthat the linear theory can not be applied, a nonlinear simulation is required to clarify the behavior of
the plasmoid. The drift motion is considered to be a
MHD behavior because the drift speed obtained fromexperimental data [1] is about several % of vA , wherevA is an Alfvén velocity. Thus, the three dimensional
MHD code has been developed by extending the pellet
ablation code (CAP) [3]. The equations used in code
are:

ρ

dρ
dt

=

−ρ∇ · u,dudt=−∇p + J × B��4+νρ ∇ (∇ · u) − ∇ × ω ,
3
�

dp
dt

author’s e-mail: ishizaki@nifs.ac.jp

471

=

(1a)

(1b)

−γp∇ · u + (γ − 1) H + ηJ 2
©2010 by The Japan Society of Plasma
Science and Nuclear Fusion Research

R. Ishizaki and N. Nakajima, Pellet Ablation in Helical Plasmas

��

,

(1c)

∂B
∂t

=

∇ × (u × B − ηJ) ,

(1d)

J

=

∇ × B,(1e)ω=∇ × u.

(1f)

Table 1 Simulation conditions

ρ, B, u and p are normalized by ρ0 , B0 , vA andB02 /μ0 at the magnetic axis, respectively, where μ0 is
the magnetic permeability. γ = 5/3, ν = 10−6 vA L0and η = 10−6 μ0 vA L0 are used as the ratio of thespeciﬁc heats, viscosity and electric resistivity, respectively, where L0 is a characteristic length; 1 m. Heatsource H is given by:
H

=

dq+
dq−
+
,
dl
dl

Case

Configuration

Plasmoidlocation

Poloidalcross section

Curvature
vectorat plasmoid

1
2
3
4
5
6

TokamakVacuum
RFP-like
LHD
LHD
LHD

inside—insideinsideoutsideinside———horizontal
horizontal
verticalnegative
negative
positive
positive
negative
negative

torus on the horizontal elongated poloidal cross section. The one in case 6 is inside the torus on thevertical elongated poloidal cross section.
Figure 1(a) shows a poloidal cross section of atokamak in case 1, in which the contours and colors show the magnetic and plasma pressures, respectively. A circle is an initial plasmoid whose peak values of density and temperature of the plasmoid are1000 times as large as density and 1/1000 times aslarge as temperature of the bulk plasma, respectively.
The plasmoid, whose half width is 0.03, encountersthe electrons with ﬁxed temperature 2 keV and density 1020 m−3 . It is located inside the torus, namely
at the high ﬁeld side. A magnetic curvature vector
where the plasmoid is located is negative in the major
radius direction. Figure 1(b) shows the plasma beta
and safety factor of the tokamak. Figures 2(a) and (b)show the simulation results in the view from the top(2)

where q± is the heat ﬂux model dependent on electron density and temperature in the bulk plasma andthe plasmoid density. l is the distance along the ﬁeld
line. The subscript + (−) refers to the electrons going to the right (left). Then, the heat source can beevaluated on each ﬁeld line. Assuming Maxwellianelectrons incident to the plasmoid, a kinetic treatmentusing a collisional stopping power formula leads to theheat ﬂux model, q± [3] which is used in constructionof the ablation model [2].
In order to investigate the plasmoid motion in
LHD plasmas, an equilibrium obtained by the HINT
code [7] is used as the bulk plasma. Although the
HINT code uses the rotational helical coordinate system, the CAP code uses the cylindrical coordinate system (R, φ, Z) because of preventing numerical instability induced by a locally and extremely large perturbation of the plasmoid, where R is the major radiusand φ is the toroidal angle. In addition, the Cubic Interpolated Psudoparticle (CIP) method is used in thecode as a numerical scheme [8].

(a)

Z

0.5

0.005

1

0

0

-0.5
-0.5

(b)

3. Drift motions of plasmoids in tokamak, vacuum toroidal ﬁeld, RFP-likeand LHD conﬁgurations

1.0 100

0.5

R - R0-2

Pressure42(∇ · u) + ω 2
3

5
post_1224_qprof

β4

5.0 10

3

-3

q

+νρ

�

2

The plasmoid motions are diﬀerent between in
tokamak and LHD experiments. In order to clarify
the diﬀerence and obtain the universal understandingon the plasmoid motions in torus plasmas, six simulations have been carried out as shown in Tab. 1. The
magnetic conﬁgurations of cases 1, 2 and 3 are tokamak, vacuum toroidal ﬁeld and RFP-like ones, respectively. Cases 4, 5 and 6 have been carried out in orderto clarify the dependence on the initial locations of theplasmoids in LHD. Initial locations of the plasmoidsin case 4 and 5 are respectively inside and outside the

1
0.0

0.0

0.2

0.4

ρs

0.6

0.8

1.0

0

Fig. 1 (a) Poloidal cross section in case 1 where R0 = 1.
Contours and colors show the magnetic and plasmapressures, respectively. A circle is an initial plasmoid. (b) Plasma beta β (solid line) and safety
factor q (dashed line) as a function of normalizedminor radius ρs .

472

R. Ishizaki and N. Nakajima, Pellet Ablation in Helical Plasmas

(a)

(a)
φ
R

(b)

(b)

Fig. 2 View from the top of the torus in case 1. Whites
show plasmoid at (a) t = 0 and (b) 5.

-0.5
-0.5

2

0

0

R - R0

0.5

3

-0.2
-0.2

0.5

0

0

0.2

R - R0

1.0

0.5

(b)

Fig. 3 Poloidal cross section in case 2 where R0 = 1. Contours show the magnetic pressure. A circle is an
initial plasmoid.

post_1226_qprof

0.4

β

0.3
0.5

q

0

0.2

(a)

Z

Z

0.5

Pressure

Fig. 4 View from the top of the torus in case 2. Whites
show plasmoid at (a) t = 0 and (b) 5.

0.2
0.1

of the torus at t = 0 and 5, respectively, which is normalized by the Alfvén transit time. It is found thatthe plasmoid expressed by a white sphere is elongatedalong the magnetic ﬁeld with about 25% of the Alfvénvelocity and simultaneously drifts in the direction of
the major radius with about 4% of the Alfvén velocity.
Figure 3 shows a poloidal cross section of a vacuum toroidal ﬁeld in case 2 in which the contours show
the magnetic pressure. A magnetic curvature vector
at the plasmoid is negative in the major radius direction. Figures 4(a) and (b) show simulation results at
t = 0 and 5, respectively. The plasmoid drifts in the
direction of the major radius similarly to the tokamak.
Figure 5(a) shows the poloidal cross section of the
RFP-like conﬁguration in case 3. An initial plasmoid
is located inside the torus. Figure 5(b) shows theplasma beta and safety factor. This conﬁguration is
not a typical RFP because it dose not have the rever-

0.0

0.0

0.2

0.4

ρs

0.6

0.8

1.0

0

Fig. 5 (a) Poloidal cross section in case 3 where R0 = 1.
Contours and colors show the magnetic and plasmapressures, respectively. A circle is an initial plasmoid. (b) Plasma beta β (solid line) and safety
factor q (dashed line) as a function of normalizedminor radius ρs .

sal of the toroidal ﬁeld toward the wall. On the other
hand, it has the features of a typical RFP which are
the toroidal ﬁeld ∼ the poloidal ﬁeld, the plasma beta,β ∼ 1 and the safety factor, q ∼ a/R ∼ 0.1, where aand R are minor and major radii, respectively. Then,
this conﬁguration is called as the RFP-like one here.
Figures 6(a) and (b) show the simulation results at

473

(a)

0.4

4

0

Z

(a)

0.3
0.2
0.1
0

5

Pressure

R. Ishizaki and N. Nakajima, Pellet Ablation in Helical Plasmas

-0.4
-0.8

(b)

-0.4

0

R - R0

0.4

0.8

0.8
0.4

0.2

6

Z

0

0.1

Pressure

0.3

(c)

0

-0.4
-0.8
-0.4

Fig. 6 View from the top of the torus in case 3. Whites
show plasmoid at (a) t = 0 and (b) 2.

0

R - R0

0.4

-2

1.6

4 10

(c)

post_9060_flux

1.4
-2

3 10

1.2

β

1

ι��
��π
��

t = 0 and 2, respectively. The plasmoid is elongatedalong the magnetic ﬁeld and simultaneously drift in
the negative direction of the major radius. Since the
RFP-like conﬁguration has larger poloidal ﬁeld andlarger aspect ratio than the tokamak, the magnetic
curvature is dominated by the poloidal ﬁeld. Then,the curvature vector at the plasmoid is positive in the
major radius direction. It is found that the plasmoids
drift in the opposite direction to the magnetic curvature similarly to cases 1 and 2. The motions in cases 1
and 2 are induced by 1/R force and the motion in case
3 is induced by 1/a force due to the magnetic ﬁeld [9].
The detail is discussed in the following section.
The conﬁguration of LHD in cases 4, 5 and 6 areshown in Fig. 7. Figures 7(a) and (b) show the horizontal elongated poloidal cross section and verticalelongated poloidal cross section, respectively. Initialplasmoids in cases 4 and 5 are shown in Fig. 7(a) and
one in case 6 is shown in Fig. 7(b). The plasmoidsin cases 4 and 6 are located inside the torus, namely
at the left hand side of the magnetic axis, and one incase 5 is located outside the torus, namely at the righthand side of the magnetic axis. The helical plasma has
a saddle point of the magnetic pressure on the poloidal
cross section. Since the plasmoid in case 4 is locatedat the lower ﬁeld side than the saddle point, the curvature vector is positive in the major radius direction.
On the other hand, since the plasmoid in case 5 islocated at the lower ﬁeld side than the saddle point,
the curvature vector is negative. Since the plasmoidin case 6 is located at the higher ﬁeld side than thesaddle point, the curvature vector is negative. Figure 7(c) shows the plasma beta and rotational transform in LHD. The simulation results in cases 4, 5 and

-2

2 10

0.8
-2

1 10

0.6
0

0.4
0.0

0.2

0.4

0.6

0.8

1.0

ρs

Fig. 7 Poloidal cross sections in (a) cases 4 and 5, and(b) case 6 where R0 = 3.82. Contours and colors
show the magnetic and plasma pressures, respectively. Circles are initial plasmoids. (c) Plasma
beta β (solid line) and rotational transform ι/2π
(dashed line) as a function of normalized minor radius ρs .

6 are shown in Figs. 8, 9 and 10, respectively. In case4, the plasmoid slightly drifts back and forth in the
direction of the major radius. In case 5, it drifts inthe direction of the major radius. In case 6, it slightly
drifts but dose not reach the magnetic axis. It is foundthat the results in LHD are not corresponding to onesin cases 1, 2 and 3 in which the plasmoids drift in the
opposite direction to the curvature vector. The reason
is discussed in the following section.

4. Discussion
The force acting on the plasmoid is given by MHD
equations.
��F = −∇ ppl + B eq · B pl + (B pl )2 /2+B eq · ∇B pl + B pl · ∇B eq+B pl · ∇B pl ,(3)

where B eq are the equilibrium magnetic ﬁeld. ppl andB pl are the perturbations of pressure and magnetic474

R. Ishizaki and N. Nakajima, Pellet Ablation in Helical Plasmas

(a)

(a)

(b)

(b)

(c)

(c)

Fig. 8 View from the top of the torus in case 4. Whites
show plasmoid at (a) t = 0, (b) 5 and (c) 10.

Fig. 9 View from the top of the torus in case 5. Whites
show plasmoid at (a) t = 0, (b) 5 and (c) 10.

ﬁeld, respectively, induced by the plasmoid, which are
extremely large values. The ﬁrst term in Eq. (3) represents the force induced by the pressure and magnetic pressure which almost cancel out each other due
to the diamagnetic eﬀect. The other terms represent
the forces dominated by the magnetic curvature. Figures 11(a), (b), (c), (d), (e) and (f) show the temporal
evolutions of the plasmoid accelerations induced bythose forces in cases 1, 2, 3, 4, 5 and 6, respectively.
The red, green, blue and pink lines show the accelerations due to the ﬁrst, second, third and forth terms
in Eq. (3), respectively. The sky blue lines show thetotal accelerations. It is found that the second terms
are the leading one in all cases.
In the next step, the major radius component of
the second term in Eq. (3) is decomposed as follows.
��FR = eR · B eq · ∇B plpl= B eq · ∇BR−

Bφeq Bφpl
R,poral evolutions of the plasmoid accelerations inducedby those forces in cases 1, 2, 3, 4, 5 and 6, respectively. The red and green lines show the ﬁrst andsecond terms in Eq. 4, respectively. The blue lines
show the total accelerations due to FR . In cases 1
and 2, the second terms are dominant. On the other
hand, the ﬁrst terms are dominant in cases 3, 4 and6. In case 5, it is not clear which terms are dominant. The ﬁrst term in Eq. (4) can be approximately
plexpressed by B eq BR/Lc . Lc is the connection length
which depends on the conﬁguration. In the tokamak(case 1), the connection length becomes Lc ∼ πqR.
The second term becomes relatively larger than theﬁrst term in Eq. (4) when q > 1. Since the second
term is always positive due to the diamagnetic eﬀect,
the plasmoid has the positive acceleration. Since themagnetic pressure is constant along the magnetic ﬁeld
in the vacuum toroidal ﬁeld (case 2), Lc becomes inﬁnity. Then, FR is determined by the second term.
In the RFP-like conﬁguration (case 3), the connection
length becomes short because Lc ∼ πqR ∼ πa due
to q ∼ a/R � 1. Then, the ﬁrst term becomes theleading one in case 3. Since the ﬁrst term includes 1/aeﬀect by minor radius which becomes essential in thelarge aspect ratio, the plasmoid motion becomes sim-

(4)

where the cylindrical coordinate (R, φ, Z) is used.
pl
The ﬁrst term in Eq. (4) represents the gradient of BR
along the equilibrium magnetic ﬁeld and the second
term represents 1/R force due to the toroidal ﬁeld [9].
Figures 12(a), (b), (c), (d), (e) and (f) show the tem-475

R. Ishizaki and N. Nakajima, Pellet Ablation in Helical Plasmas(a)Acceleration

(a)

0.1
0.05
0
-0.05
-0.1
0.0

(b)
Acceleration

(b)

1.0

2.0

3.0

4.0

5.0

3.0

4.0

5.0

Time

0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
0.0

(c)

(c)

1.0

2.0

Time

0.1

Acceleration

0.05
0
-0.05
-0.1
-0.15
-0.2
0.0

(d)

0.5

1.0

1.5

2.0

Time

0.03
0.02

Acceleration

Fig. 10View from the top of the torus in case 6. Whites
show plasmoid at (a) t = 0, (b) 5 and (c) 10.

0.01
0
-0.01
-0.02

ilar to the tokamak. The above evaluations of Eq. (4)are corresponding to the results in Figs. 12(a), (b)and (c). In LHD, the connection length depends on
the location of the plasmoid. Figure 13 shows the spatial proﬁle of the equilibrium magnetic pressure along
the ﬁeld line through the plasmoid. Red, green andblue lines are corresponding to cases 4, 5 and 6, respectively, where the plasmoids are located at � = 0.
The connection length in case 5 becomes longer thanones in case 4 and 6 because the deviation of the magnetic pressure around the plasmoid is small in case 5.
When this diﬀerence due to the locations is deﬁned byα(≤ 1), the connection length in LHD is expressed by
Lc = πR/αM , where M is the toroidal pitch number,
α = 1 in cases 4 and 6 and α < 1 in case 5. By using M = 10, α = 1 and R = 3.82 which is the major
radius at the center of the poloidal cross section, theconnection length becomes ∼ 1.2 which agrees withones shown by the red and blue lines in Fig. 13. Sinceplthe ﬁrst term in Eq. 4 is given by αM B eq BR/πR in
LHD, it becomes the leading one because of α = 1 incase 4 and 6. On the other hand, both terms become
dominant because of α < 1 in case 5. Those evaluations are corresponding to the results in Figs. 12(d),(e) and (f).

-0.03
0.0

(e)

2.0

4.0

6.0

8.0

10.0

6.0

8.0

10.0

6.0

8.0

10.0

Time

0.01

Acceleration

0.005
0-0.005-0.01-0.0150.0

Acceleration

(f)

2.0

4.0

Time

0.01
0.005
0
-0.005
-0.01
0.0

2.0

4.0

Time

Fig. 11The accelerations acting on the plasmoids as a function of time in cases (a)1, (b)2, (c)3, (d)4, (e)5 and(f)6. Red, green, blue and pink lines show the accelerations due to the ﬁrst, second, third and forthterms in Eq. (3), respectively. The sky blue lines
show the total accelerations.

476

R. Ishizaki and N. Nakajima, Pellet Ablation in Helical Plasmas
24

0.06
0.05
0.04

Magnetic pressure

Acceleration

(a)

0.03
0.02
0.01
0
-0.01
-0.02
0.0

(b)

1.0

2.0

3.0

4.0

5.0

20
18
16
14
12
-15.0

Time

0.05

22

post_9714-9726-9720_bsqc

-10.0

-5.0

Acceleration

0.03

0.01

1.0

2.0

3.0

4.0

5.0

(a)

Time

0.1

BR1

Acceleration

-0.05

-0.04
0.5

1.0

1.5

-0.06
-2.0

2.0

Time

0.03

(b)

-0.01

Acceleration

-0.5

0.0

0.5

1.0

1.5

2.0

0.5

1.0

1.5

2.0

Length

0.03

0
-0.01

2.0

4.0

6.0

8.0

-0.02

10.0

-0.03
-2.0

Time

0.01

-1.5

-1.0

-0.5

0.0

Length

0.008
0.006
0.004

pl
Fig. 14BR
as a function of the length along the magnetic
ﬁeld line, � in case 4 at (a) t = 0.5 and (c) t = 2.5.
Red, green and blue lines are diﬀerent ﬁeld lines
through the plasmoid.

0.002
0-0.002-0.004-0.006
0.0

2.0

4.0

6.0

8.0

10.0

Time

0.01

There is another essence that the ﬁrst terms in
Eq. (4) clearly have oscillations as shown in Figs. 12(d)and (f). The plasmoid motions in Figs. 8 and 10 are
induced by such oscillations. The oscillations can be
explained as follows. Figure 14 shows the spatial propl
ﬁles of BR
along the magnetic ﬁeld in case 4. The
magnetic ﬁeld lines denoted by red, green and blue are
diﬀerent ones through the plasmoid. The red, green
and blue ones pass through (Rpl − Δ, 0, 0), (Rpl , 0, 0)
and (Rpl + Δ, 0, 0), respectively, where the center ofthe plasmoid is located at (Rpl , 0, 0) in the cylindricalcoordinate and Δ is a tiny value. Figures 15(a) and(b) show the physical images to understand Figs. 14(a)and (b), respectively. When we focus on the pertur-

0.005

Acceleration

-1.0

0.01

0

-0.02

(f)

-1.5

0.02

0.01

BR1

Acceleration

0.02

(e)

0
-0.02

-0.1

-0.03
0.0

0.06

0.02

0

-0.15

(d)

15.0

0.04

0.05

-0.2
0.0

10.0

Fig. 13The equilibrium magnetic pressure, (B eq )2 as afunction of the length along the magnetic ﬁeld line,
�. Red, green and blue lines are ones in cases 4, 5
and 6.

0.02

0

(c)

5.0

Length

0.04

-0.01
0.0

0.0

0
-0.005-0.01-0.015
0.0

2.0

4.0

6.0

8.0

10.0

Time

Fig. 12The accelerations due to B0 · ∇B1 as a function oftime in cases (a)1, (b)2, (c)3, (d)4, (e)5 and (f)6.
Red, green and blue lines show the accelerations
due to the ﬁrst, second and total forces in Eq. (4),respectively.

477

R. Ishizaki and N. Nakajima, Pellet Ablation in Helical Plasmas

R

(a)

dense

Dipole field
B1

tion (0 < t < 3.5) is slightly larger than the negative
acceleration (3.5 < t < 6.5) as shown in Fig. 11(f).
Although the plasmoid motion is quantitatively determined by the summation of all forces, the essenceof the motion is determined by the ﬁrst term in Eq. (4)in cases 4 and 6. In case 5, since the ﬁrst term andsecond term become comparable because the connection length is long as shown in Fig. 13, the plasmoid
has a positive acceleration as shown in Fig. 11(e).

B0sparse

φ

B1
Z

R

(b)

5. Summary

B1

It is veriﬁed by simulations using the CAP codethat the motions of the plasmoids with a high pressure
induced by heat ﬂux are diﬀerent among tokamak,vacuum toroidal ﬁeld, RFP-like and LHD conﬁgurations. The plasmoids drift in the opposite directionof the magnetic curvature vector in the tokamak, vacuum toroidal ﬁeld, RFP-like conﬁgurations. On theother hand, in LHD, the plasmoid slightly drifts backand forth in the major radius direction when it is initially located inside the torus on the horizontal elongated poloidal cross section. It slightly drifts in the
major radius direction but does not reach the magnetic axis when it is initially located inside the torus
on the vertical elongated poloidal cross section. It
drifts in the major radius direction when it is initiallylocated outside the torus on the horizontal elongatedpoloidal cross section. The plasmid motion is mainly
determined by 1/R force due to toroidal ﬁeld and theplforce due to gradient of BR
along the ﬁeld. The former forces imply the drifts in the opposite directionof the curvature vector which are induced in the vacuum toroidal ﬁeld and tokamak. Since the latter forces
are dominant in the cases that the plasmoids are located inside the torus in LHD, the plasmoid motions
in LHD are essentially diﬀerent from ones in tokamak.
It is clariﬁed that the connection length determines
which force is dominant on the plasmoid motion. That
physics might be one inducing the diﬀerence between
experiments in tokamak and LHD. In RFP-like conﬁguration, although the former force is dominant, theplasmoid drifts in the opposite direction of the curvature vector similarly to tokamak due to the minor
radius eﬀect including in the former force.

B0

φ

B1
Z

Fig. 15Physical images around the plasmoid in case 4 at(a) t = 0.5 and (b) t = 2.5.

bation of the magnetic ﬁeld, the dipole ﬁelds are induced around the plasmoid by the diamagnetic eﬀect
as shown in Fig. 15(a). Since the curvature vectorof the equilibrium magnetic ﬁeld is positive in the
major radius direction in case 4, the ﬁeld lines become dense and sparse in the above region (blue line)
and below region (red line), respectively, of the ﬁeldline through the center of the plasmoid as shown in
Fig. 15(a). Then, the magnitude of the above ﬁeldline is stronger than the below one. Those physicalimages are corresponding to Fig. 14(a) in which the
red and blue lines have opposite phases, and the absolute value of the blue line is larger than one of the redline. Subsequently, the deformation of the dipole ﬁelds
is induced by the diﬀerence between the strengths of
the above and below ﬁeld lines as shown in Fig. 15(b).
Therefore, all lines in Fig. 14(b) become same phaseplas the blue line in Fig. 14(a). Since ∂BR/∂� become
negative on all lines, the force acting on the plasmoid
becomes negative in the major radius direction according to the ﬁrst term in Eq. (4). On the other hand, themagnetic ﬁeld has the restoring force due to the tension. Then, the ﬁrst term has oscillation in Fig. 12(d).
Since the negative acceleration (0 < t < 4) is slightly
smaller than the positive one (4 < t < 7.5) in the total acceleration as shown in Fig. 11(d), the plasmoid
drifts back and forth as shown in Fig. 8. In case 6,
the ﬁrst term has oscillation due to the same physics.
Although it continues to be negative at t > 2 as shownin Fig. 12(f), the total acceleration oscillates aroundzero as shown in Fig. 11(f) due to the increment of
the acceleration by the other force. As a result, theplasmoid slightly drifts because the positive accelera-[1] Y. W. Muller et al. Nucl. Fusion, 42, 301 (2002).
[2] P. B. Parks and M. N. Rosenbluth. Phys. Plasmas, 5,1380 (1998).
[3] R. Ishizaki et al. Phys. Plasmas, 11, 4064 (2004).
[4] P. B. Parks et al. Phys. Rev. Lett., 94, 125002 (2005).
[5] R. Sakamoto et al. in proceedings of 29th EPS conference on Plasma Phys. and Control. Fusion.
[6] R. Ishizaki et al. IAEA-CN-149/TH/P3-6 (2006).
[7] K. Harafuji et al. J. Comp. Phys., 81, 169 (1989).
[8] H. Takewaki et al. J. Comput. Phys., 61, 261 (1985).
[9] J. P. Freidberg, Ideal Magnetohydrodynamics (Plenum
Press, New York, 1987).

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