J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)542–545
000–000

Radial Electric Field Control for Retardation of Radial
Transport of Bounce Ions in the Tandem Mirror
ISHII Kameo, TAKEMURA Yuichiro, KOJIMA Atsushi, HAGISAWA Kazuhisa,
MIYATA Yoshiaki, KATANUMA Isao and CHO Teruji
Plasma Research Center, University of Tsukuba, Ibaraki 305-8577, Japan(Received: 9 December 2003 / Accepted: 26 June 2004)

Abstract
Existence of the bounced ions by the plug potential is essential in order to improve the confinement in thetandem mirror. The anchor cell with non-axisymmetric magnetic field configuration is installed between the plug/barrier cell and the central cell to stabilize the magnetohydrodynamics (MHD) instability. The electro-statically trappedions are bounced at the plug region after passing through the anchor cell from the central cell. The trajectories of thebounce ions are calculated on the assumption that the cross-sectional shape of the magnetic flux tube is slightly
different from the cross-sectional shape of the equi-potential surface at the mirror throats of the anchor cell. Radialdrift is estimated as a function of the discrepancy of the shapes. It is pointed out that control of the radial electric fieldis effective to retard the radial transport.
Keywords:
radial electric field, bounce ion, trajectory, radial transport, tandem mirror

1. Introduction

discrepancy of the shapes between the magnetic flux tube andthe equi-potential surface at the mirror regions of the anchorcell.

In the tandem mirror, axial confinement has been
improved by electrostatic potentials created on both sides ofthe plasma [1,2,3]. The hyperboloidal loss boundaries in thevelocity space were shifted in the direction of the higher
energy side along a velocity component axis parallel to themagnetic field so as to decrease the inflow of plasma into theloss region from the trapped region caused by the Coulomb
collision. We confirmed, from a microscopic point of view,the confining potential created by the microwave injection,that is, from observation of the velocity distribution functionof the end-loss ions [4]. As a subsequent subject of theimprovement of the axial confinement due to the potential
plugging, radial transport across the magnetic field has
attracted interest recently. The radial transport across themagnetic field seems to be induced mainly at the anchor cellhaving a non-axisymmetric magnetic field configuration andat the plug/barrier cell with a potential hill and a potential
dip, and the ion diffusion across the loss boundary seems to
be enhanced by fluctuations excited in the plasma [5].
Investigation of the radial transport of ions bouncing
between magnetic mirrors or electrostatic potentials is usefulfor the optimization of the radial potential profile to improve
the tandem mirror experiment. Here we calculated the ion
trajectories using a direct method and the guiding center
approximation, and paid attention to the radial transport ofthe bounced ions by the plug potential caused by the

2. Magnetic field configuration of thetandem mirror
Two anchor cells with a minimum B magnetic
configuration are located on both sides of an axisymmetriccentral solenoid through the re-circularizing regions in thetandem mirror GAMMA 10, and two axisymmetric plug/barrier cells are connected to the anchor cells as the outboardthermal barrier configuration in order to remove the resonancediffusion [3]. The cross section of the magnetic flux tube iswell approximated by an ellipse, and at the mirror throat of
the anchor cell the eccentricity 1 – b 2e / a 2e of the ellipse isestimated to be 0.999819, where ae and be are a major axis
and a minor axis of the ellipse, respectively. The magnetic
coils and the magnetic field lines are shown together with thelocations of the heating systems in Fig. 1 (a), and the axial
profiles of the magnetic field strength and the schematicelectrostatic potential are shown in Fig. 1 (b). In order toproduce the main plasma and heat the ions, ion-cyclotron
range of frequency (ICRF) waves are injected into the target
plasma in the central cell, which is produced along themagnetic field line from both ends of the machine. The
magnetic field strength is almost the same and the strongest©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: ishii@prc.tsukuba.ac.jp1542

Ishii K. et al., Radial Electric Field Control for Retardation of Radial Transport of Bounce Ions in the Tandem Mirror

Fig. 1(a) Locations of the heating systems.
(b) A dotted line shows a schematic axial profile of the potential.

(~30 kG) at both inner mirror throat (IMT) and outer mirror
throat (OMT) of the plug/barrier cell. Two microwaves are
injected into the plug region (~10 kG) and the thermal barrier
region (~5 kG) of the plug/barrier cell in order to create theplug potential and the thermal barrier potential by carryingout electron-cyclotron resonance heating (ECRH),respectively. Coaxially separated end plates divided into 5
areas are installed in front of the end walls of the vacuum
vessel in order to control the radial potential profile of thecore plasma.

3. Bounce ions
3.1 Ion bounce region
The bounce ions are shown schematically in Fig. 2 (a)with axial profiles of the magnetic field strength and theelectrostatic potential. Some of the ions that have passed
through the mirror throats of the central and anchor cells fromthe central cell are reflected in front of the IMT due to themagnetic mirror effect, and some residual ions with lower
energy than the confining potential and with small pitch
angles, which passed through the IMT, are reflected near theplug region due to the potential hill. A portion of the ionshaving high energy is reflected in the neighborhood of theOMT after passing over the plug potential hill. Here, we
define these ions as an IMT bounce ion, a plug potential (PP)bounce ion and an OMT bounce ion, respectively. Figure 2(b) shows, as an example, the end-loss region and the bounce
regions in which the IMT, PP and OMT bounce ions occupy
in the ε – µ space, where ε and µ are the ion total energy andthe magnetic moment, respectively. The loss region issurrounded with the PP bounce region and the OMT bounce
region in this case, and therefore the ions trapped in thetandem mirror flow mainly into the loss region via the PPFig. 2(a) Bounce ions and axial profiles of the magnetic fieldstrength and electrostatic potential.
(b) Bounce ion regions in the ε – µ space.

and OMT bounce regions. In the case that the machine transit
time of the trapped ions in the PP and OMT bounce regions
is much shorter than the deflection time, the ions go to theend wall as end-loss ions after several bouncing.

3.2 Trajectories of bounce ions
The anchor cells are installed between the central cell2543

Ishii K. et al., Radial Electric Field Control for Retardation of Radial Transport of Bounce Ions in the Tandem Mirror

Fig. 3The initial ion was emitted from the position of a radius
0.03 m on the mid-plane of the central cell. The trappedion was changed to the end-loss ion.

and the plug/barrier cells in order to stabilize the MHD
instability. The eccentricity of the elliptic cross section of themagnetic flux tube is 0.999819 at the mirror throat of theanchor cell. It seems that in the case of high radial electricfield the electrostatic potential is not perfectly constant along
the magnetic field line through the anchor cells between theIMTs of both plug/barrier cells and the equi-potential surface
slightly deforms due to the slight deviation from equilibrium
state.
On the assumption that the shape of the magnetic flux
tube did not necessarily coincide with the equi-potential
surface, we calculated, as an example of radial drift, thetrajectories of the ions that started from the mid-plane of thecentral cell. The point to which special attention should be
paid is the radial drift of the bounce ions due to thediscrepancy of the curvatures in the major axis region of theelliptic cross section at the mirror throat areas of the anchor
cells between the magnetic flux tube and the equi-potential
surface. We assumed that the radial potential profile φ(r) was
axi-symmetric and Gaussian on the mid-plane (z = 0 m) of
the central cell and that the central potential was 300 V, thebarrier potential was 0 V and the plug potential was 1,500 V,
that is, the thermal barrier depth was 300 V and theconfinement potential was 1,200 V. φ(r) = φ0(0)exp[ – (ln(2)/rh(0)2)r2], where φ0(0) is the potential at z = 0 m and r = 0 m
and equals to 300 V, r is a radial coordinate at z = 0 m andrh(0) = 0.07 m. As the cross section of the magnetic flux tube
was well approximated by an ellipse, we assumed, as a
generalized radial potential profile, that the radial potential
profile had two Gaussian types of components in the
directions of the x and y axes. That is, φx(rx(z)) = φ0(z)exp[ – (ln(2)/r xh(z) 2)r x (z) 2], φy(r y(z)) = φ0(z)exp[ – (ln(2)/ryh(z)2)ry(z)2], where rxh(z) = rxhf (z) – [rxhf (z) – rh(0)]Df andryh(z) = ryhf (z) – [ryhf (z) – rh(0)]Df , 2rxh(z) and 2ryh(z) are thefull width at half maximum (FWHM) for each potential
profile at the coordinate z, rxhf (z) and ryhf (z) are distances ofthe magnetic field lines from the z axis, which pass throughFig. 4The loss rate was estimated as a function of thestarting position and the deformation factor.

the points of xyz coordinates (0.07, 0, 0) and (0, 0.07, 0),
respectively, and Df is defined as a deformation factor. In thecase of Df = 0, the equi-potential surface coincides perfectlywith the magnetic flux tube. We directly calculated thetrajectories of the plug potential bounce ions moving under
the influence of the magnetic field and the above electric field,
and found the effect of the deformed radial potential profileon the radial drift as a function of the deformation factor Df .
The trajectories of 9 ions were calculated, which started
from the mid-plane of the central cell with pitch angles of12°, 16°, 20° on the condition of Df = 0.0, 0.05, 0.1 and 0.2.
The initial 9 ions were placed at regular intervals on
semicircles with radiuses of 0.01, 0.03, 0.05, 0.07, 0.09 m,
located on the mid-plane of the central cell, and were emittedwith energies of 0.5 keV and 1.0 keV. As the loss cone angle
is 21.2°, all ions with pitch angle of 12° will be reflected in
front of the plug potential after passing through the IMTregion. Figure 3 shows, as an example, the projectedtrajectories of the transferred ions from the trapped region tothe loss region.

4. Radial transport of ionsThree types of Gaussian potential profiles were used with
the FWHM of 0.08 m, 0.14 m and 0.28 m at the central cell.
The profiles have the maximum electric field of ~5,000,~3,000 and ~1,500 V/m, respectively. We estimated the loss
rate of the ions changed to end-loss ions during one
reciprocation to the initially trapped ions by the confining
potential as a function of the deformation factor and the radiusof the starting circle of the ion position as shown in Fig. 4.
The calculated results indicated that the radial drift was noteffective for the ions moving under the influence of the→→E × B drift in the case of D f = 0.0, and the radial drift3544

Ishii K. et al., Radial Electric Field Control for Retardation of Radial Transport of Bounce Ions in the Tandem Mirror

depended on the increase of the deformation factor and alsostrongly on the radial electric field. The radial drift wasenhanced even in D f = 0.05 which corresponded to thedecrease of 0.07% in the eccentricity of the cross section.
Increase of the area of the core plasma with the flattenedradial electric field, which is confined by the electrostatic
potential, is quite effective for preventing the plug potential
bounce ion from flowing into the loss region and sustaining
the bounce ions.

ions to flow from the bounce region to the loss region. It iseffective to increase the area of the core plasma with theflatted radial electric field for improvement of theconfinement, and therefore the microwave injection for thecreation of the plug potential should be considered from thispoint of view.

References[1] G.I. Dimov, V.V. Zakaidakov and M.E. Kishinevskii, Sov.
J. Plasma Phys. 2, 326 (1976).
[2] T.K. Fower and B.G. Logan, Comments Plasma Phys.
Controll. Fusion 2, 167 (1977).
[3] M. Inutake et al., in Proceedngs of the 3rd TechnicalCommittee Meeting and Workshop (IAEA, Tokyo, 1983)
Vol. 1, p. 429 (Report No. IAEA-TC-392/25, 1983).
[4] K. Ishii et al., Phys. Fluids B 4, No. 12, 3823 (1992).
[5] K. Ishii et al., J. Plasma Fusion Res. 78, 1239 (2002).

5. Conclusions
The trajectories of the bounced ions by the plug potential
were calculated on the assumption that the shape of themagnetic flux tube was slightly different from the shape ofthe equi-potential surface at the mirror throats of the anchor
cell with the non-axisymmetric magnetic field configuration.
The discrepancy enhances the radial transport for the bounce4545