Proceedings of ITC/ISHW2007

Motion of the plasmoid in helical plasmas
Ryuichi ISHIZAKI1) and Noriyoshi NAKAJIMA1)
1)

National Institute for Fusion Science, Toki 509-5292, Japan

In order to clarify the difference on the motion of a plasmoid created by a pellet injection between tokamak
and helical plasmas, the MHD simulation including ablation processes has been carried out. In a straight helicalplasma, the plasmoid drifts to the lower field side in several Alfvén transit times, similarly to tokamak. However,
it is found that the drift direction is subsequently reversed in the case that an initial location of the plasmoid is thehigher field side than one at the magnetic axis. This fact might be one of the reasons why there is the difference
on the motion of the plasmoid between tokamak and helical plasmas.
Keywords: Pellet, Ablation, Drift, Tire tube force, CIP

1Introduction

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 macrostages. The micro stage is the ablation of mass at the pellet
surface due to the high temperature bulk plasma which thepellet encounters. The neutral gas produced by the ablation is rapidly heated by electrons and ionized to form ahigh 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 andby 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 the plasmoid 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 core plasma has
succeeded by injecting the pellet from the high field side intokamak. On the other hand, such a good performance has
not been obtained yet in the planar axis heliotron; Large
Helical Device (LHD) experiments, even if a pellet has
been injected from the high field side [5]. The purposeof the study is to clarify the difference on the motion of theplasmoid between tokamak and helical plasmas.
In order to investigate the motion of the plasmoid, thethree dimensional MHD code including the ablation processes has been developed by extending the pellet ablation
code (CAP) [3]. It is found through the comparison between simulation results and an analytical considerationthat the drift motion to the lower field side in tokamak
is induced by a tire tube force due to the extremely largepressure of the plasmoid and a 1/R force due to the magnetic pressure gradient and curvature in the major radiusdirection [6]. It is also found that the plasmoid dose notdrift when the perturbation of the plasmoid is small. In thestudy, the motion of the plasmoid is investigated in straighthelical plasmas in the cases that the plasmoids are locatedauthor’s e-mail: ishizaki@nifs.ac.jp

at lower and higher field sides than one at the magnetic
axis. In the former case, the plasmoid drifts to the lower
field side, namely to the outside. In the latter case, it drifts
to the lower field side, namely to the inside at first. Subsequently, the drift direction of it is reversed, namely a part ofthe plasmoid dirfts to the higher field side. This fact mightbe one of the reasons why the motion of the plasmoid dose
not depend on the location of the pellet injection so muchin LHD experiments.

2Basic Equations

Since the plasmoid is such a large perturbation that 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 thedrift speed obtained from experimental data [1] is about
0.01 ∼ 1.0vA , where vA is an Alfvén velocity. Thus, thethree dimensional MHD code including the ablation processes has been developed by extending the pellet ablation
code (CAP) [3]. The equations used in code are:ρdρdt=−ρ∇ · u,(1a)

dudt

=

β− ∇p + (∇ × B) × B,
2(1b)

dp
dt

=−γp∇ · u + H,

(1c)

∂B
= ∇ × (u × B)
(1d)∂t
All variables are normalized by ones at the magnetic√
axis, ρ0 , p0 , B0 and vA , where vA = B0 / µ0 ρ0 . γ andβ = 2µ0 p0 /B20 are the ratio of the specific heats and plasmabeta, respectively. Heat source H is given by:
dq+ dq−+.
(2)
dldl
where q± is the heat flux model dependent on electron
density and temperature in the bulk plasma and the plasH=

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3Fig. 2 Contours of the mass flow through the flux surface where
the initial plasmoid is located. (a), (b), (c) and (d) arecorresponding to t = 2.5, 5.0, 7.5, 10.0 τA , respectively.
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Plasmoid simulations in straight helical

When a plasmoid is heated in tokamak plasmas, it is expanding along the magnetic field and simultaneously drifts
to the lower field side due to a tire tube force and a 1/R
force induced by the magnetic field with curvature [6]. In
the study, the motion of the plasmoid is investigated in thestraight helical plasma that consists of a uniform pressure
and a vacuum magnetic field with l = 2 configuration. Figures 1(a) and (b) show the poloidal cross section (u1 -u2plane) at u3 = 0 and the flux surface (u3 -θ plane) throughthe center of an initial plasmoid, respectively, where θ

Mass flow

0.003

3

Mass flow(d)
Poloidal Angle Tan (u /u )moid density. l is the distance along the field line. The
subscript + (−) refers to the right (left)-going electrons.
Then, the heat source can be calculated on each field line.
Assuming Maxwellian electrons incident to the plasmoid,a kinetic treatment using a collisional stopping power formula leads to the heat flux model, q± [3] which is usedin construction of one of the ablation models [2]. The
rotational helical coordinate system (u1 , u2 , u3 ) is used
in which the poloidal cross section (u1 -u2 plane) rotatesalong the straight toroidal direction (u3 ) with same pitch as
the external helical coils [7]. The Cubic Interpolated Psudoparticle (CIP) method is used in the code as a numerical
scheme [8].

2.0’j_7555/pplot1/cim_7555_pplot06.0075’ using 6:7:93.0
Poloidal Angle Tan (u /u )Fig. 1 Initial location of the plasmoid at lower field side than
one at the magnetic axis in (a) the poloidal cross section
at u3 = 0 and (b) the flux surface through the center of theplasmoid in straight helical plasma. The initial location
is denoted by circle. The colors show the contours ofthe magnetic pressures. The lines in (a) show the flux
surfaces.

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Fig. 3 Temporal evolution of mass flow integrated on the flux
surfaces 1, 2, 3, 4 and 5 which are located from the magnetic axis to the outside. The initial plasmoid is locatedon the flux surface 3.

Proceedings of ITC/ISHW2007
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00.2 0.4x(b)Fig. 4 Initial location of the plasmoid at higher field side one
at the magnetic axis in (a) the poloidal cross section atu3 = 0 and (b) the flux surface through the center of theplasmoid in straight helical plasma. The initial location
is denoted by circle. The colors show the contours ofthe magnetic pressures. The lines in (a) show the fluxsurfaces.

is the poloidal angle defined by θ = Tan−1 (u2 /u1 ). The
boundaries at u1 = ±0.8 and u2 = ±0.5 are assumed to be
the perfect conductors in Fig. 1(a). The ones at u3 = ±2.0
are the periodic conditions in Fig. 1(b). The colors show
the contours of the magnetic pressures which is uniform inu3 -direction because the helical coordinate system is used.
The lines show flux surfaces in Fig. 1(a). Initial locationof the plasmoid is denoted by circle, namely it is locatedin the lowest field side in the flux surface. The peak valuesof density and temperature of the plasmoid are 100 times
density and 1/100 times temperature of the bulk plasma,respectively. The plasmoid, whose half width is 0.03, encounters the electrons with fixed temperature 2 keV anddensity 1020 m−3 . Figure 2 shows the contours of the mass
flow through the flux surface in which the initial plasmoid
is located. A negative flow means one toward the magnetic axis. Figures 2(a), (b), (c) and (d) are the results att = 2.5, 5.0, 7.5, 10.0 τA , respectively. When the plasmoid
is heated, the pressure of it increases. It is found in Fig. 2that the plasmoid quickly expands along the magnetic fieldand simultaneously drifts to the outside, namely the lower
field side. The fact is due to a tire tube force and a 1/R
force induced by the magnetic field with curvature simi-

larly to the tokamak. Figure 3 shows the temporal evolution of the mass flow integrated on the flux surfaces 1, 2,3, 4 and 5 which are located from the magnetic axis to theoutside. The initial plasmoid is located on the flux surface
3. The mass flow in the surface 3 increases at first. Subsequently, it decreases after it reaches a peak value and one
in the surface 4 increases because the plasmoid drifts theoutside. One in the surface 5 increases after one in the surface 4 decreases. It is found that the plasmoid integrated
on the flux surface also drift to the outside.
The case that the plasmoid is located at the highest
field side on the flux surface is considered. Figures 4(a)and (b) show the poloidal cross section at u3 = 0 and theflux surface through the center of an initial plasmoid, respectively, similarly to Figs. 1(a) and (b). The conditions
except the initial location of the plamoid are same as Fig. 1.
Figures 5(a), (b), (c) and (d) show the contour of the mass
flow on the flux surface at t = 2.5, 5.0, 7.5, 10.0 µs, respectively. It is found that the plasmoid quickly expands alongthe magnetic field and simultaneously drifts to the inside,
namely the lower field side as shown in Figs. 5(a) and (b).
The mass flow at θ = ±π/2 becomes positive as shown inFig. 5(c) because the direction to the lower field side becomes the one to the outside. Subsequently, the mass flowat θ = 0 becomes positive as shown in Fig. 5(d), namely
the plasmoid drifts to the outside. In other words, it drifts
to the higher field side. Figure 6 shows the temporal evolution of the mass flow integrated on the flux surfaces 1,2, 3, 4 and 5 which are located from the magnetic axis
to the outside. The initial plasmoid is located on the flux
surface 3. The mass flow in the surface 3 becomes negative at first. One in the surface 2 also becomes negative
because the plasmoid drifts to the magnetic axis. Subsequently, the mass flow is reversed at θ = ±π/2 as shown in
Fig. 5(c). In addition, it is also reversed at θ = 0 as shownin Fig. 5(d). The integrated mass flow is thus reversed andbecomes positive. This fact might be one of the reasons
why the motion of the plasmoid dose not depend on the location of the pellet injection so much in LHD experiments.

4Summary and Discussion

It is verified by simulations using the CAP code that theplasmoid with a high pressure induced by heat flux drifts tothe lower field side in spite of the initial location in an earlystage for several Alfvén transit times in a straight helicalplasma. Such the drift is due to a tire tube force comingfrom a extremely large pressure of the plasmoid and 1/R
force of magnetic curvature similarly to tokamak. However, it is found that the drift direction is reversed in thecase that an initial location of the plasmoid is the higher
field side than one at the magnetic axis. This fact might be
one of the reasons why there is the difference on the motionof the plasmoid between tokamak and LHD experiments.
The detail analysis will be needed to clarify the physics.

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In addition, the simulation will be carried out in LHD configuration in order to seek the ways obtaining good performance on fueling.
[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 conferenceon 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).