J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)461–464
000–000

Ion Radial Transport due to a Non-Uniform Electrostatic Potential
in a Magnetic Mirror
KATANUMA Isao, SAIMARU Hiroshi, TATEMATSU Yoshinori, SAITO Teruo,
ISHII Kameo and CHO Teruji
Plasma Research Center, University of Tsukuba, Tsukuba 305-8577, Japan(Received: 9 December 2003 / Accepted: 20 March 2004)

Abstract
The density around the thermal barrier in a tandem mirror is sustained to be much lower than that in the central
cell experimentally, which suggests that the ions trapped in the thermal barrier potential escape from there rapidly.
Although the mechanism of radial loss of ions trapped in the axisymmetric end-mirror cells are unknown in thepresent tandem mirror experiment, the electrostatic potential has been measured to be not axisymmetric around theplug region.
Because a numerical calculation reveals that the ion orbits trapped in a non-axisymmetric electrostatic potential(with many high m azimuthal modes around the plug) have a chaotic behavior, the non-axisymmetric electrostaticpotential can be a candidate of a large ion radial loss in the axisymmetric end-mirror cells.
Keywords:
particle orbits, chaotic orbits, non-uniform electrostatic potential, mirror, magnetic mirror

1. Introduction

where z p is the axial position at local maximum ofelectrostatic potential.

Although the tandem mirrors have revealed several
unknown mechanisms on plasma confinement in an openended magnetically confinement system, one of a major
purpose of a tandem mirror to find the mechanism of the plugand thermal barrier potential formations has been solved byourselves theoretically [1,2,3].
Figure 1 shows the numerical results of a Monte-Carlosimulation [1,2,3], where the following electron populationand ion loss are required to obtain the results, i.e., thepopulation of electrons with two temperatures (coldcomponent Tec and warm component Tew) in the central cellfor the deep thermal barrier potential formation, where thestandard notations are used through this paper, a nonMaxwellian electrons with different Te|| and Te⊥ in the plugregion for a plug potential formation, and the rapid loss ofions trapped in the thermal barrier potential for the thermal
barrier potential formation. The parameters adopted in Fig. 1
are not contradictory to the observations in the experimentsof the plug potential formation in the GAMMA10 tandem
mirror [4].
It is seen that the plug potential has a sufficient height
to confine the ions from the central cell because e(φp – φi)/Ti~– 3.2 in Fig. 1(a) and e(φp – φi)/Ti ~– 2.4 in Fig. 1(b), while
ion density gradually decreases from z = zi toward z = zm ,> ni (zp),which clearly satisfies the relation ni (zi) >> ni (zb) ~Fig. 1

Numerically obtained axial profiles of electrostatic
potential and ion density in the plug/thermal barrier(end-mirror) cell. Here (a) is the case τloss /τD = 1.8 × 10-1
and (b) is τloss /τD = 1.8 × 100.

©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: katanuma@prc.tsukuba.ac.jp1461

Katanuma I. et al., Ion Radial Transport due to a Non-Uniform Electrostatic Potential in a Magnetic Mirror

The ion loss time is assumed as τloss /τD = 0.18 in Fig.
1(a) and τloss /τD = 1.8 in Fig. 1(b), where τD is the Coulomb
deflection times defined as τD ≡ mi Ti3/2/( 2 πn0 e4 lnΛii).
Radial loss of ions, which is included in this numericalcalculation as τloss , is necessary for steady state potentialformation. In GAMMA10, the plug/barrier potential structure
can be sustained during 150 msec and no ion filling into thethermal barrier region has been observed [5]. This strongly
suggests the existence of an ion radial loss in the plug/thermal
barrier cell in GAMMA10 experiments.
The open-ended system such as a tandem mirror does
not have any magnetic surfaces, which is a different pointfrom a closed system such as a tokamak. The cross sectionof a magnetic flux tube, therefore, does not necessarily
coincide with the equi-contour surface of an electrostatic
potential, where experimental data on the equi-contour
surfaces will be shown later in this paper.
The shape of a magnetic flux tube and the magnitude ofan electrostatic potential on the flux tube are illustrated inFig. 2, where the GAMMA10 has an axisymmetric magnetic
field in the end-mirror cell in Fig. 2(a). Because a plugpotential is created by electron cyclotron resonance heating(ECRH) mainly the cross section of equi-contour of theelectrostatic potential around z = z p is generally nonaxisymmetric as seen in Fig. 2(b) depending on the radiation
pattern of µ-wave and the absorption rate at the resonancesurface around [5].

Fig. 2Magnetic flux tube of the plug/thermal barrier cell of
GAMMA10 (a), and the magnitude of an electrostatic
potential on the magnetic flux tube (b).

Fig. 3Model of electrostatic potential axial profile used in thecalculation of ion orbits.

2. Ion radial loss in the thermal barrier
regionIn order to investigate an ion radial loss due to a nonaxisymmetric electrostatic potential we adopt a model ofelectrostatic potential profile in Eq. (1), where the axial profile
on axis (ψ = 0) is shown in Fig. 3.

ψ⎡⎤φ (ψ , θ , z ) = ⎢ φp + φ˜cos ( mθ + ξ m ) ⎥ g ( z )
∑ψ0 m
⎣
⎦
⎛× exp ⎜⎜ −⎝⎛ ψ ⎞ψ ⎞⎟ ; for z ≥ z ,⎟ + φ exp ⎜ −Bb⎜ ψ ⎟⎟ψ0⎠0⎠⎝

φ (ψ , θ , z ) = φ i( z − z b )2
( z i − zb )2

⎛exp ⎜⎜ −⎛ ψ ⎞⎟ ;+ φB exp ⎜⎜ −⎟⎝ ψ0 ⎠⎝

orbits in the following [6,7,8]. The half width of electrostatic
potential ψ0 in Eq. (1) is given as ψ0 = 1/2B(z = zb) × 102(Gcm2).
We are interested in the ion radial transport without
Coulomb collisions so that only the ion orbits are traced in
the end-mirror cell of GAMMA10. Ions are distributed at t =0 at (r = 5cm, z = z b) with the same energy of 100eV but
different pitch angles from 0 to π/2.
Figure 4 shows the results of ion orbit calculation when
only m = 3 perturbation is added to the axisymmetric plugpotential in Eq. (1). The ratio φ̃ /φp is the perturbation
amplitude divided by the axisymmetric amplitude ofelectrostatic potential (see Eq. (1) and Fig. 3). A large
amplitude of m = 3 perturbation exists at z = zp in Fig. 4,
where eφB /Ti = 1.5. Figure 4(b) is the Poincare map at z = zbof ion orbits, where each dot represents a locus of test ions at
the position passing through z = zb until t = 5msec. Even ifψ ⎞⎟ψ 0 ⎟⎠for z < z b .

(1)

Here g(z) = –A(z – a)(z – b)2 with a > b, where z = zb = b isthe coordinate at the thermal barrier and z = zp = (2a + b)/3
is the coordinate at the plug, A is a normalization factor to beg(zp) = 1, i.e., A = 33/(4(a – b)3). The flux coordinates (ψ, θ,z) are adopted in Eq. (1), where magnetic field is represented
by B = ∇ψ × ∇θ. A long thin approximation to the magnetic
field lines is assumed to calculate magnetic field lines and its
curvatures at arbitrary point (ψ, θ, z) in order to trace ion
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Katanuma I. et al., Ion Radial Transport due to a Non-Uniform Electrostatic Potential in a Magnetic Mirror

large ion drifts from r = 5cm initial radial position exist, manyions have regular orbits. As long as a single m mode
perturbation is added to the axisymmetric electrostatic
potential around plug region z = zp , almost all the ion orbits
are regular.
On the other hand, the case of ∑ m ⇒ ∑11
m=3 in Eq. (1) isplotted in Fig. 5. Hereφ˜ /φp = 0.08 per each perturbed m mode
was added to the axisymmetric electrostatic potential z ≥ zb .
Equi-contour surfaces of electrostatic potential at z = zp isplotted in Fig. 5(a), where eφB /Ti = 1.5. The small irregular
perturbations are seen in the equi-contour surfaces of potential
in Fig. 5(a). The Poincare map at z = zb of ion orbits in theelectrostatic potential of Fig. 5(a) is plotted in Fig. 5(b) at t =5msec. Under this electrostatic potential profile many ionshave irregular motion, i.e., chaotic orbits.
Ion orbits become chaotic under the existing of manymode perturbations of electrostatic potential around plug as
seen in Fig. 5. The mean square displacement 〈∆r2〉 of ions isdefined asN( ri − r0 )

i =1

N

〈 ∆r 2 〉 ≡ ∑2,Nrii =1

Nr0 ≡ ∑.

Fig. 4Equi-potential surfaces at zp in the case of which an m =3 perturbation is added (a), and a Poincare map at z = zbof ion orbits (b).

Fig. 5Equi-potential surfaces at zp in the case of which theperturbations from m = 3 to m = 11 are added to anaxisymmetric electrostatic potential (a), and a Poincare
map at z = zb of ion orbits (b).

Fig. 6Time evolution of test ion mean square displacements.

(2)

Here ri is the th test ion radial position mapped at z = zb . Ifions have a random walk process, 〈∆r2〉 will be proportional
to time t, in the case of which ion radial diffusion coefficientD can be defined as D ≡ 〈∆r2〉/t.
Although it is known that ions under a chaotic motion
have a sub-diffusion process but not a simple diffusion
process, it is interesting to investigate the ion diffusion underthe electrostatic potential profile such as that in Fig. 5(a). InGAMMA10 we estimate that the ion radial diffusion in the> 5 × 103cm2/sec, because
thermal barrier region is roughly D ~collisional filling time of passing ions with ni ~– 1011/cm3 andTi ~– 100eV into the thermal barrier is about several msec andso the radial diffusion time of ions trapped there iscomparable to the collision filling time in the steady state,〈( 5 cm) 2 〉i.e., D ~– 5×10−3 sec . Now we introduce a target diffusion
coefficient Dtarget = 5 × 103cm2/sec. That is, the purpose here
is to make clear that how much amplitude of electrostatic
perturbation causes an ion radial diffusion of Dtarget .
The radial diffusion coefficient measured by a meansquare displacement of test ions in the electrostatic potentialin Fig. 4(a) is about 102cm2/sec which is much smaller than
Dtarget . The case that many modes perturbation is added to anaxisymmetric electrostatic potential such as in Fig. 5(a),therefore, is calculated hereafter.
Figure 6 displays the ion mean radial displacement in
the electrostatic potential where the perturbations from m = 3
to 11 modes were added to the axisymmetric electrostatic
potential in Eq. (1). Here eφB /Ti = 0.5 is chosen, in whichions drift roughly 2π/9 in the θ-direction per one bounce axial
motion. Many ions with different pitch angles resonate withthe electrostatic perturbation and their motion become chaotic
as seen in Fig. 5(b). It is found that the ion radial diffusion> Dtarget when the ratio φ̃ /φp ~> 0.08 incoefficient satisfies D ~Fig. 6(a).
The electrostatic potential at the thermal barrier z = zb ismeasured by a beam probe in GAMMA10 [5,6], where Fig.
7 is a latest result of GAMMA10. The radial profile of theelectrostatic potential depends strongly on the radiation
pattern of externally injected µ-wave for ECRH at ω = ω ce
fundamental resonance surface around the plug z = zp . By
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Katanuma I. et al., Ion Radial Transport due to a Non-Uniform Electrostatic Potential in a Magnetic Mirror

3. Summary and discussion

comparison of the radial profile in Fig. 7 with that in Fig.
5(a) the high-m modes perturbation in the electrostatic
potential in Fig. 7 can cause the ion radial transport of orderof Dtarget .

The mechanism of ion radial loss in the plug/thermal
barrier cell was proposed. That is, non-axisymmetricelectrostatic potential added to the axisymmetric one leads toa large ion radial transport.

Acknowledgements
The author would like to acknowledge Dr. L. Zheng, Dr.
J. VanDam, Dr. T. Petrosky and Prof. W. Horton for theirhelpful comments and their support in this work.

References

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(2001).
[2] I. Katanuma et al., Phys. Plasmas 9, 3449 (2002).
[3] I. Katanuma et al., Phys. Plasmas 10, 677 (2003).
[4] T. Cho et al., Nucl. Fusion 41, 1161 (2001).
[5] K. Yatsu et al., Nucl. Fusion 41, 613 (2001).
[6] I. Katanuma et al., Phys. Fluids B 2, 994 (1990).
[7] R. Minai et al., J. Phys. Soc. Jpn. 66, 2051 (1997).
[8] R. Minai et al., J. Phys. Soc. Jpn. 67, 876 (1998).

The equi-contour surface of an electrostatic potential at
the thermal barrier z = zb measured by beam probe,
where this figure is reprinted from Fig. 1(b) in Ref. [5].

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