J. Plasma
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)538–541
000–000
J.

Particle Orbits and Radial Loss in theGAMMA10 Tandem Mirror
ITO Toru and KATANUMA Isao
Plasma Research Center, University of Tsukuba, Ibaraki 305-8577,Japan(Received: 17 December 2003 / Accepted: 14 May 2004)

Abstract
Ion orbits trapped in the transition mirror cells with non-axisymmetric magnetic field in GAMMA10 are calculated. It is found that those orbits are deviated from the magnetic flux tube. The results indicate the existence of “radialloss cone” in view of the limiter. The radial drift due to the radial loss cone is examined numerically and analytically.
Keywords:
non-axisymmetric magnetic field, radial transport, tandem mirror, GAMMA10

1. Introductiondψc= − kθ (2 ε − μ B),
dt
qdθ c= kψ (2 ε − μ B),
dt qdz= v||,dt
B = ∇ψ × ∇θ ,k = kψ ∇ψ + kθ∇θ ,1ψ = r2B0,
2The experimental results in the GAMMA10 tandem mirror have demonstrated the improvement of the axial plasma
confinement. So the radial loss can not be disregarded. Therecould be some reasons to cause it. We especially pay attention to the radial transport due to the non-axisymmetric magnetic field in this research. GAMMA10 is essentiallydesigned as an axi-symmetrized tandem mirror. Particles
passing through the mid-plane in the GAMMA10 central cellare expected not to drift for a radial direction. However
GAMMA10 consists of not only central mirror cell but othermirror cells. Especially there are mirror cells called transition
mirror cell, where the magnetic field is not axial symmetry. It
has been reported that the particle trapped in the mirror cell
could be caused large radial drift.[1]We calculate the orbits in GAMMA10 magnetic field
without electrostatic potential and investigate the feature ofradial drift due to the non-axisymmetric magnetic field in thetransition mirror cell precisely. Additionally, we examine theparticle radial loss in the velocity space, and find “radial losscone”.

(1)

where z is the coordinate along the mirror axis and z = 0 at
the central mirror cell mid-plane B is the magnetic field, B0 =B (zc) is the magnetic amplitude at the central mirror cell
mid-plane, c is the light velocity, 2 π ψ is the magneticflux, ε is the energy of particle, μ is the magnetic moment ofparticle, q is the electric charge, r is the radial distance interms of the value at central mirror cell mid-plane. k is thecurvature of magnetic field line, where kψ and kθ are the normal and geodesic curvatures with the following relations,11k̂ ψ − k̂θ cos 2θ,
22k θ = k̂θ ψ sin 2θ,σ ''σ + τ ''τ,k̂ ψ =B0σ ''σ + τ ''τk̂ θ = −,B0x = σ (z)x0,
y = τ (z)y0,kψ =2. Radial drift due to the non-axisymmetricmagnetic field
To illustrate the feature of the particle orbits in the transition mirror cell with non-axisymmetric magnetic field in
GAMMA10, we first consider the ions trapped in the transition mirror cell centered around z = 367cm, anchor mirror
cell centered z = 520cm, another transition mirror cell centered z = 673cm in Fig. 1 respectively.
We adopt the paraxial approximation to the magnetic
field with the flux coordinates (ψ, θ, ζ ), the equations ofmotion are written as(2)©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: itot@prc.tsukuba.ac.jp1538

Ito T. et al., Particle Orbits and Radial Loss in the GAMMA10 Tandem Mirror

Fig. 1 Axial magnetic field profile in the GAMMA10 tandem
mirror. (a) Magnitude of magnetic field. (b) Normal
curvature k̂ ψ and geodesic curvature k̂ θ of magnetic
field line. (c) σ and τ in eq. (2).

Fig. 2 Poincare plot at z = 367cm plane (a), (b), (c) and z =673cm plane (d), (e), (f). The solid line is magnetic
flux tube, and dots are ion orbits which given different initial pitch angle p0 and started on the magnetic
flux tube. (a)(d) p0 = 55. (b)(e) p0 = 65. (c)(f) p0 = 75.

where θ is the azimuthal coordinate, (x, y) are the coordinateof the magnetic field line starting from (x0, y0) at the midplane in the transition mirror cell , the primes denote derivatives with respect to z.
We examine the Poincare plot at the mid-plane in thetransition mirror cell in terms of solving eq. (1) to consider
the ion orbits trapped in each mirror cell that is the transition
mirror cell centered around z = 367cm and z = 673cm.
We set ions on the magnetic flux tube at z = 367cm andz = 673cm corresponding to r = 1cm at the mid-plane in thecentral mirror cell and get off with several azimuthal coordinates θ and pitch angles. Figure 2 shows the Poincare plot at
the mid-plane in the transition mirror cell at z = 367cm (a),(b), (c) and z = 673cm (d), (e), (f). The ions trapped in thismirror cell drift radially and deviate from the magnetic flux
tube significantly because the radial displacement piles up by
those bounce motion in the non-axisymmetric magnetic field.
Besides the orbits turn into hyperbolic or elliptic sensitively
depending on the initial pitch angle. The ions with elliptic
orbit rotate in the azimuthal direction sufficiently, the otherswith hyperbolic orbit do not rotate in the azimuthal direction.
Consequently the ions continue to drift out radially in case of
hyperbolic orbits.
Because it is found that the non-axisymmetric magnetic
field causes the large radial drift, we examine the particle
orbits precisely in the next place.
Integrating eq. (1) through the one bounce motion byusing eq. (2), we obtain the amount of Δψ and Δθ, where Δψ
and Δ θ are the displacement of radial coordinate ψ andazimuthal coordinate θ for one bounce time. Assuming thedisplacement of ψ and θ during the one bounce motion issmall, Δψ and Δθ are described asΔψ = c1( ε , μ ) ψ sin 2 θ ,1Δθ = c2( ε , μ ) + c1( ε , μ ) cos 2 θ ,2c1( ε , μ ) = −c2( ε , μ ) =cdz

∫ q (2ε − μB) kˆ v ,θc||dz∫ 2q (2ε − μB) kˆ v ,ψ(3)||the symbol °∫ carries out the integration along one ion axial
bounce in the mirror cell. Assuming that the ion drift velocity
is much smaller than thermal velocity, the differential equation of ψ (θ ) is described asdψ=dθc1( ε , μ) ψ sin 2 θ1c2( ε , μ) + c1( ε , μ ) cos 2 θ2.

(4)

The solution of eq. (4) isψ (θ ) =A1c2( ε , μ) + c1( ε , μ ) cos 2 θ2,(5)where A = ψ 0 ⎡⎣c2 (ε , μ) + 12 c1 (ε , μ) cos 2θ 0 ⎤⎦ , ψ0 and θ0 are
initial value of ψ and θ. Figure 3 shows the comparison ofion orbits in terms of eq. (5) with the numerical calculation in2539

Ito T. et al., Particle Orbits and Radial Loss in the GAMMA10 Tandem Mirror

regions in the velocity space in Figs. 4(a)-(c), where ions ineach region rotate in different azimuthal direction. We havecalculated the ion orbits trapped in the transition mirror cells
and found that some orbits deviate from the magnetic flux
tube significantly. The existence of these orbits could affectthe negative consequences on the plasma confinement.
Finally we evaluate the orbital loss due to the radial drift in
the non-axisymmetric transition mirror cells in GAMMA10.
We assume that the initial ions are Maxwellian,⎛mi ⎞ 3/2εexp ⎜ −⎟⎝ 2π T ⎠⎝TiiFig. 3 Comparison of numerical calculation (circle) and analytical calculation (dashed line) by eq. (5) about ionorbits.

⎛f (ε ) = ⎜⎞⎟ ,⎠(6)the axial and azimuthal distribution is distributed uniformly,
and the radial number density distribution is assumed as
the case Figs. 2(b) and 2(c). The good agreement in Fig. 3
shows that the ion orbits are described by eq. (5).
Additionally eq. (5) is similar to the description of quadratic
curve in the polar coordinate. c1(ε, μ) in eqs. (3) and (5) has a
non-zero value in the non-axisymmetric transition mirror
cells. Because the coefficient c1(ε, μ) is a function of pitch
angle, i.e., μ /ε, particle orbits has two types of elliptic andhyperbolic depending on the pitch angle of particles.
Because the radial drift of ion depends on the pitch
angle, it is important to investigate the loss-cone region in thevelocity space. We judge the ion radial loss by the radial distance rw in terms of the value at the central mirror cell midplane. We set many ions with different energy and magnetic
moment in the transition mirror cell centered around z =367cm and z = 673cm, pursuit the orbits. We see the ion thatreaches a threshold r w of radial distance as the loss ion,experimentally there are limiters corresponding to rw with rw= 20 ~ 30cm. Figure 4 shows the radial loss of ions in thevelocity space, where (a) is the case of threshold rw = 20cm,(b) is the case of rw = 30cm, (c) is the case of rw = 40cm, (d)is the case of rw = 100cm.
Similar results are obtained about each transition mirror
cell. The lost ions illustrated in Fig. 4(d) correspond to theion drawing hyperbolic orbits. There are two radial loss⎛n (ψ) = n0 exp ⎜ −⎝ψ ⎞⎟ .
ψs ⎠(7)

Here n0 is the number density at ψ, and ψs is the coordinate atmagnetic flux tube corresponds to r = 20cm at the mid-planein the central mirror cell. In these initial conditions, all ionshave different orbits with each other. When an ion reached athreshold rw, we count that the ion is lost radially. So ions areonly lost radially, but are not lost axially. We change thethreshold rw = 10cm, 20cm, 30cm, 100cm respectively. Figure
5 shows the axial density distribution of test ions after sufficient time compared with the azimuthal drift period has
passed. The radial ion loss arises extremely only in the nonaxisymmetric transition mirror cells, (the region of density
dip in Fig. 5) hardly in the axisymmetric mirror cells. We
note that the axial distribution in the transition mirror cells
formed bumps. This corresponds to the radial loss particle inthe velocity space are divided into two in Figs. 4 (a), (b), (c).

3. Summary and discussion
We especially take notice of the mirror cells called transition mirror cell in GAMMA10. The magnetic field is nonaxisymmetric in the mirror cells. The ion orbits trapped in themirror cell are calculated and as illustrated by Fig. 2 that theions trapped in the transition mirror cells drift radially anddeviate largely from the magnetic flux tube. It is confirmedthat the radial drift is caused by the non-axisymmetric magnetic field.
We find that the ion orbits trapped in the non-axisymmetric magnetic field orbits turn into hyperbolic or elliptic by
the pitch angle. The form of ion orbit depends on the pitch

Fig. 4 Distribution of radial loss ions trapped in the transition mirror cells in the velocity space. (a) rw = 20cm.
(b) rw = 30cm. (c) rw = 40cm. (d) rw = 100cm.

Fig. 5 Axial number density of ion in each magnetic flux
tube.

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Ito T. et al., Particle Orbits and Radial Loss in the GAMMA10 Tandem Mirror
2angle.
These ions deviated largely from the magnetic flux tube
could not be trapped radially. So we examine that what ion in
the velocity space is not able to trap radially in the nonaxisymmetric magnetic field transition mirror cells inGAMMA10. We find out the “radial loss cone” illustrated asFig. 3. The ions in this region escape radially.
We neglected the electrostatic potential in this paper. If
the radial electric field exists, some ions can escape from theloss-region due to the E × B drift. So the radial loss would besmaller.
We consider the radial drift velocity vr of ion in the radial loss cone in terms of eq. (1), the radial loss time τloss = rw /_vr is estimated as τloss ~rwrqB0 ∫ turn dzc(2 ε − μ B) ∫ turn kθ dz

time τilling is determined by the relation

2<(10°) > <(90°) >_~. Inτ collision
τ fillingcase of Ti = 1keV and ni = 1012cm-3, τloss and τilling are esti_ 7ms and τilling ~_ 2ms. τilling << τloss is satisfied,mated as τloss ~so the assumption of Maxwellian distribution is valid.

References[1] Md. K. Islam, Y. Nakashima, K. Yatsu and I. Katanuma,
et al., J. Phys. Soc. Jpn. 69 (2000) 2493.
[2] I. Katanuma, Y. Kiwamoto, M. Ichimura, T. Saito andS. Miyoshi, Phys. Fluids B. 5 (1990) 994.
[3] J.A. Byers, Nucl. Fusion 22 (1982) 49.
[4] R. Minai, I. Katanuma and T. Tamano J. Phys. Soc. Jpn
66 (1997) 2051.
[5] R. Minai, I. Katanuma, T. Tamano and K. Yatsu, J.
Phys. Soc. Jpn. 67 (1998) .

by assuming thatdθ~ 0 in the radial loss cone. The loss cone angle due to thedt

radial drift is about 10° in Fig. 4(b), so the collisional filling4541