J. Plasma Fusion Res. SERIES. Vol.5 (2002) 42H27

Physical Properties of Collisionless Pitch Angle Scatteringat X-Points and those Effects on Particle Confinementof Field-Reversed Configuration
JIMBO Shigeaki, TAKAHASHI Toshiki and KONDOH Yoshiomi
Department of Electronic Engineering, Gunma University, Kiryu, Gunma 376-8515, Japan(Received: 12 December 2001 / Accepted: 15 May 2002)

Abstract
The single particle dynamics in the peripheral regions of a Field-Reversed Configuration (FRC) isinvestigated numerically in order to find physical and statistical properties of collisionless pitch angle
scattering at the X-points. The results from the ensemble average of positions for a number of ions clarify
that due to the statistical properties of the scattering the density concentration of higher energy ions isgenerated near the X-points. Estimation of the electrostatic potential along a magnetic line of force is also

carried out. Although the calculation of self-consistent electric fields remains, it is found that thepotential peak is formed in the vicinity of X-points.

Keywords:
FRC, edge plasma, field-null X-point, adiabaticity, electrostatic potential

1. Introduction

speed of end-loss ions, and makes the particle

Field-reversed configuration (FRC) is one of
attractive plasma confinement systems, which has notoroidal field and consequently has a high beta value.
The FRCs have two magnetic field-null points on theconfinement time independent of the separatrix length.
Chiang et al.have calculated the potential profile using
a single particle tracing routine [2]. Though the positionof potential peaks are slightly different between the twoworks, both of those show the electrostatic potential
peak might locate near X-points and influence the axial

geometric axis and separatrix and on the magnetic axis,i.e., X- and O-point respectively. In spite of the longtime effort. however. the dominant mechanism of theparticle transport between the bulk region and outside
the separatrix has not been made clear. Since theexistence of field-null points makes particle motion
complicated, an application of MHD theory to the edge

flow and particle confinement.
The ion motion near X-point, however, becomescomplicated because of a large gradient of the magnetic
field along a line of force and the resultant breaking of
adiabaticity; the magnetic field profile in the peripheral

region is presented in Fig. l. Takahashi et al. studied

plasma fails to explain the behavior of particle transport.

statistically the collisionless pitch angle scattering at thisSteinhauer pointed out that the electrostatic potential
near the X-point and in off-axis region retards the ion
flowing from the midplane and found that the effect of
this potential is the most promising to explain several
anomalous properties in the edge layer of FRC [1]. Hesuggested that the off-axis electrostatic potential peaknear the X-point broadens the edge layer, increases theC o rre sp ondin g aut ho r' se -point [3]. This work suggests the ion loss rate enhancesin contrast with the electron loss rate near the separatrix.
Moreover, the large radial density gradient is created
due to the enhanced axial particle loss, which increases
the cross-field particle loss.
@2002 by The Japan Society of Plasma
Science and Nuclear Fusion Researchmail : j imb o @ nifs. ac. j p424

Jimbo S. et al., Physical Properties of Collisionless Pitch Angle Scattering at X-Points and those Effects on Particle Confinement

where r* and B* are the wall radius and the external
magnetic field, respectively. To check the numericalerror in integration of the equation of motion, two

We investigated numerically the single particle
dynamics for a number of ions in the peripheral region,i.e., in the vi<;inity of the separatrix. Especially, themagnetic moment as an adiabatic invariant is used herefor a measure rtf the collisionless pitch angle scattering.
Further, the calculation results are ensemble-averaged inconstants of motion, i.e., the Hamiltonian H andorder to find the statistical and macroscopic propertiesof the ion motion. The estimated electrostatic potential
profile along a line offorce is also presented.

- H(0)yH(0) and (Ps(t) - P0(0))/Pe(0) can be less thancanonical angular momentum Ps (: mvsr + qY) areobserved. The deviations of those defined as (II(t)10-r0, which is sufficiently small to obtain reliable data.

In the present paper, the initial condition is chosen
by the following way: The Hamiltonian FI and thecanonical angular momentum P6 are chosen at first.
Next, we specify the magnetic moment as the thirdinvariant, and then no degree of freedom remains in thevelocity space. Ions are injected toward the X-point

2. Model Description
Ion motions are traced by numerically solving theequation of motion. Here, we neglect the electric field.
Determination of the self-consistent electric field and its

effect on particle motion are subjects for the futurefrom the guiding center surface. Here, the guiding centerof ions in the peripheral region is described by ve = g,study. The equilibrium fields both inside and outside thewhich is equivalent to Y = Pslq where q is the ioncharge. Thus the canonical angular momentum is aseparatrix are calculated by using the Grad-Shafranov
solver. We obtain the flux function Y discretely in two
dimensional r-.. space, and use the interpolation method
to obtain the rnagnetic fields at an arbitrary position.
The calculated magnetic field profile along the magnetic
line of force (Yl&B* = 1.0 x l0-4) is shown in Fig. l,measure of its location. Since we have already specified
the magnetic moment, the ion is accessible in the limited

region e" I z I Zmin, where 4 is the axial bounce positionof the ion with no parallel velocity component and z.1nis the axial position of the minimum magnetic fieldalong a line of force. For example, zmin = 2.05 in Fig. 1.
Therefore, as an initial position, the axial position israndomly chosen in z" I z I z^in.

3. Results and Discussion
Choosing the initial condition as is described inSec. 2, we trace numerically orbits of ions that are
traveling near the separatrix. Observation of the0.0 0 5 1.0

1.5

2.O 2.5

magnetic moment shows various aspects of thescattering. Although it is an adiabatic invariant, it

3.0z/r,I x t0-5]

5.0
4.O

1.0

z/r_0.0

Fig. 1 The nunrerical equilibrium of FRC. (a) Contour
plot of the flux function Y. (b) The calculated magnetic field profile along the line of force (YlriB*=10-a) in the peripheral region, where r- and B* arethe wall r'adius and the external magnetic field on
the separatrix and midplane. The mirror field is almost the same as 8*.

0500

1500 2000 2500tFig. 2 The evolution of the magnetic moment for an ion
moving in the peripheral region of an FRC. The
time f and the magnetic moment p are normalized

here by the reference quantities 1la. and driB*lm, respectively.
425

Jimbo S. et al., Physical Properties ofCollisionless Pitch Angle Scattering at X-Points and those Effects on Particle Confinement

suffers from an abrupt change at the vicinity of the Xpoint, which is shown in Fig. 2. One can find that thedimensionless quantities into the ones having dimension,

magnetic moment is again conserved after the scatteringin a relatively uniform magnetic field. The transition of
the magnetic moment due to the collisionless scattering
is of great importance, because it affects the end loss ofions. Moreover Takahashi pointed out that the enhanced
end loss increase the radial gradient of particle density;
it also enhances the cross-field diffusive flow. Figure 3
shows the histograms of the magnetic moment after thescattering p. The vertical solid line in Fig. 3 indicates

D-3He/FRC fusion reactor ARTEMIS concept [5] as thewe choose the engineering parameters proposed inreference quantities for normalization. It is found that allvalues for 90 keV ions exceed the initial one. This factsuggests that the axial motions of the higher energy ionsretard at the vicinity of X-point. In other words, theparallel velocity component is converted to theperpendicular one. On the other hand, one can find thatthe values of the magnetic moment 14 for 50 keV ionsare distributed also below the initial value. This can be
explained by the Larmor radius and the characteristic
length of perpendicular variance of the magnetic field
for each field lines. If the Larmor radius is greater thanthe characteristic length, ions suffer from the collisionless pitch angle scattering. Due to the geometric property

the initial value of the magnetic moment for all cases.
The kinetic energies of ions are 50 (black colored), 70(hatched) and 90 keV (white colored), and Pslqf*B* =lO-a. From now on, the quantity Prlqfi,B* is written as
P6 for simplicity. In order to convert the normalized andof cusp-like magnetic field, a low value of magnetic
moment tends to change abruptly to a higher one.

The number of ions along the magnetic line of300

force is presented in Fig. 4. Here, the guiding center ofthe ions is P6 - l0-a and the kinetic energies are I and
100 keV. The quantity S is the distance from thet5n.9

F2crtsS loo

midplane; it is calculated along the line of force. In
order to eliminate an effect of our choice of the initial

150

positions and velocities, a long time calculation isneeded enough to obtain a steady state profile of thenumber of particles in each segment from S to S + AS.
In Fig. 4, the position S is plotted at the instance ta" of-)u0l
0.0

1.0 2.0 3.0 4.0 5.0Pr["10-5]5000 for the 100 keV ions and 40000 for the I keV ions,Fig. 3 The histograms of the magnetic moment after thecollisionless pitch angle scattering p for 50 (blackcolored),70 (hatched) and 90 keV ions (white col-where ro" is the cyclotron frequency on the separatrix
and midplane. Since the 1 keV ions move slowly, ittakes longer calculation time until they form a steadystate density profile. In the case for the 100 keV ions,the steep peak is found near the X-point (S = 2.4). The I
keV ions, however, are observed uniformly in the wholeregion. We can say by Figs. 3 and 4 that the higher
energy ions concentrate near the X-point due to thescattering and the lower energy ions and electrons, i.e.
small-gyroradius particles can exist uniformly along theline of force.

ored). The vertical solid line indicates the initialmagnetic moment (before the scattering) for all
three cases.

80o'F 60

I toot<ev
E lkev

340o420
0The electrostatic potential is estimated by using theBoltzmann relation and the ambipolarity:00.5 l1.5

22.533.54

Q =(kT"lse)ln(nln), n=ni=ne,

where ne and Z" are the density at the midplane and theuniform electron temperature. The density n is calculated by using the results shown in Fig. 4 and weighting

Fig. 4 The density profiles along the guiding center ofthe ion motion Pe= lOn. The position of 100 keV{black) and the 1 keV (white) ions are plotted here.
The quantity 5 is the distance from the midplane
along the line of force. The minimum magneiic
field locates at S = 2.4.

by Maxwell distribution:n(S ) =426|n(H,,5 ) exp (-H, / kT) .

Jimbo S. et at., Physical Properties ofCollisionless Pitch Angle Scattering at X-Points and those Effects on Particle Confinement

where Z1 is the ion temperature. Instead of the integration in energy space, the summation of the obtained results with respect to several energies is carried out here.

potential peak. Inclusion of the obtained electric fields
in the particle tracing routine and the iterative calculation should be necessary.

Assuming T = Ti = 7", we estimate potential profiles andshow them in Fig. 5, where three profiles along theguiding centers (Pe = 10-5, 10-a and 10a) are presented.
The electron temperature is 100 keV here. As the guiding center approaches the separatrix, higher potential
peak is generated; the maximum of the potential is about

4. Conclusions

3.6 kT". On the other hand, the potential peak can not be

breaking process at the field-null X-point on thefound in the case for the Pe = 10-3. Since the averaged
Larmor radius for the 100-keV plasma ions is greaterthan the characteristic length of perpendicular varianceof the magnetic field on the guiding center labeled by P6of l0-3, no ions suffer from the collisionless scattering

the collisionless pitch angle scattering, the magnetic
moment for almost all the higher energy ions moving
toward the mirror point is found to increase, whichsuggests the axial motions of the ions retard at the X-and concentrate at the X-points. This result also suggests

point. As a result, the density concentration is formed atthat the electrostatic potential peak is on the geometric

the X-point. Although the self-consistent calculation
remains, up to 3.6 tQ potential peak has been found in
the present study. In order to settle argument regarding

The single particle dynamics and its ensembleaveraged properties for ions in the peripheral and open-

field region of an FRC have been investigated with theaid of numerical calculation. Especially, the adiabaticity
separatrix and geometric axis have been studied. Due toaxis as Chiang reported [2]. The self-consistent calcula-tion, however. is needed to decide the position of thethe position of the electrostatic potential peak, weconsider that the combined effect of the binary collisionand the self-consistent electric fields is important.

lx l()21-'+-

P -lo't0*o-rn4

Referenceseg*t"=to"2.0tll L.C. Steinhauer, Phys. Fluids 29,3379 (1986).

0.0[2] P.-R. Chiang and M.-Y. Hsiao, Phys. Fluids 84,-2.O

t3l T. Takahashi, Y. Tomita, H. Momota and N.V.

3226 (1992).
Shabrov, Phys. Plasmas 4,4301 (1997).

01234[4] T. Takahashi, Doctor thesis,The Graduate Univer-ssity for Advanced Studies (1997).

Fig. 5 The estimated electrostatic potential profiles along
the magnetic lines of force, The profiles are cal-t5l H.Momota, A.Ishida, Y. Kohzaki, G.H. Miley, S.

culated for the ions with the guiding centers P, of10-5, 104 and 10-3. Here, the potential on the midplane is 0.

Y. Tomita, M. Tsuzewski, Fusion Technol. 21,

Ohi, M. Ohi, M. Ohnishi, K. Sato, L.C. Steinhauer,
2307 (r99 .

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