J. Plasma Fusion Res. SERIES, Vol.4 (2001) 383-386

On the lon Distribution Function around the Plug in aTandem Mirror
KATANUMA Isao*, ITO Toru, SHIOTANI Kensuke, TATEMATSU Yoshinori, ISHII Kameo,SAITO Teruo and YATSU Kiyoshi
Plasma Research Center, University of Tsukuba, Tsukuba 305-8577, Japan(Received: 5 December 2000 / Accepted;27 August 2001)

Abstract
It is shown that the plug potential is formed only with flow ions accelerated by the thermal barrier
potential and non-Maxwellian electrons due to the electron cyclotron resonance heating in the plug/thermal barrier region of a tandem mirror. For the purpose to display the plug potential formation, theMonte-Carlo simulation has been carried out, where the effect of coulomb collisions are included.

Keywords:
plug potential, tandem mirror, thermal barrier, orbit, Monte-Carlo simulation

1. lntroduction

found in refs. [5-7]. Although this code adopts the flux
coordinates in the tandem mirror, only one-dimensionalmotion along axis is followed in this paper. The effectsof coulomb collision are included with help of MonteCarlo method [5].
Figure I shows the plug/barrier mirror cell of theGAMMAl0 tandem mirror. where a half mirror cell is
considered. Ions are input at the inner mirror throat ofWe have shown that the plug potential is formed
with only electron cyclotron resonance heating (ECRH)
around the plug region in the tandem mirror [1-4]. The
essential mechanism for the plug potential formation isbelieved to be made clear in these works. However, theseveral assumptions have been made for the calculationof the plug potential formation. One is that the ion andelectron distribution functions are given a priori at thethermal barrier region. The small angle scattering ofions during its flight from outer mirror throat to thepotential maximum is assumed in the given distribution

the plug/barrier mirror cell, where the magnitude ofmagnetic field at the inner mirror throat is the same asthat at the outer mirror throat z = 3. in GAMMAl0. The
ions input at the inner mirror throat have the velocity
with a random number giving Maxwellian distributionof temperature Ti, and the input ions are accelerated byfunction.
The ion distribution has not been measured in theplug/thermal region yet, so that the comparison of theassumption with the experiments has not been made.
Therefore, the purpose of this paper is that the plugthe thermal barrier potential at the thermal barrier 4 = xo.

Therefore, the distribution function of the input ions at z
= 0 is Maxwellian in the region E- ttBi- eg> 0 and e-

potential is shown to be formed under more realistic

ItBv- eev > 0, and no ions in the other regions in thecondition.

velocity space. Here we use the standard notations
through this paper and e is energy of ion and p is the

2. Numerical Method

magnetic moment. The subscripts i and D represent thequantities at the inner mirror throat and thermal barrier,In this section we briefly show the Monte-Carlosimulation code, where the detailed information will be*

Corresponding author's e-mail: katanuma@ prc.tsukuba.acjp
383@2001 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchKatanuma l. et al., On the Ion Distribution Function around the Plus in a Tandem Mirror

respectively.
The ions, which reflect at the midway in the mirror
cell by the magnetic field and electrostatic potential andthen come back again at z - 0, are assumed to be lost,that is, the ions trapped in the thermal barrier cell are
lost within one transit time. The ions passing throughthe outer mirror throat z = zmare assumed to be lost to
the end-wall. The ions moving in the mirror cell receive
a coulomb collision with the field particles (consist ofions and electrons with density 1011 cm-3, temperature
100eV). Those procedures are repeated until a steadylqt flux ftom lnnerminorthrcatwithn
Magnetic field B(z)Potential (p(z)\----\londensityll(z)Outer Mirror Thrcat

Plug

Fig. 1 Axial profile of magnetic field of GAMMA10,
where a half of the plug/barrier mirror cell isshown. The initial state of ion density profile
determined by their axial positions and initialstate is realized.

The equation of motion is given as

electrostatic potential are also shown.

dz _,,

dt -'tt'vil=

TG- t'Br.-arrD ,(1)

Potential eilj'ls determined of ion axial density in the previous <n?fwhere z is the axial coordinate.

N* Potential (p(z)

The electron's motion is not included in thiscalculation. Instead the electron distribution function is

given in advance and the axial potential profile isdetermined from the ion density and electron
distribution function i.e., modified Boltzmann law isused. In this paper we assume the electron distribution

Old Pol€ntial (P(z)

function as

Thermal Bmierf"= no(+)'''

IOld lon densiry n(z)l-Plug

Outcr Miror Throat

Ter/Ti=90, TentTi=2

TJ,{'r"'Fig. 2 The situation how to determine the steady state*'{- ryp (+-+)',),

electrostatic potential axial profile.

(2)

which gives the modified Boltzmann law i.e.,lr| \.
llr",
l,z.r\ no ln tz) \ek)=--:tr" ( | ;:l*l'-o' \ -4; l r*61 ;:' |
ll
4,,

\-'I

where is shown in Fig. 2.
In Fig. 2 the ion densities "Old Ion density" and"New Ion density" are in the steady state obtained by
the Monte-Carlo simulation in the electrostatic potential"Old Potential" and "New Potential", respectively. Thatis, the Monte-Carlo simulation is carried out until its
steady state of ion density profile is come, in the given
electrostatic potential profile. The electrostatic potential(3))

It is known that the ion acoustic waves appear when eqs.
(1) and (3) are taken into account. However we areinterested in the steady state of the system, so that thewaves are removed by time averaging which is shown in
the following.

are distributed at regular intervals along z-axis. The

in the next step is determined by eq. (3) and the steadystate ion density profile as the "New potential g(z)".
The calculation is carried out in this New electrostatic

electrostatic potential eEQ)lTi = -2.0 at first. The
calculation is started at this initial state. On the other
hand, the electrostatic potential is not changed in timepotential until the new steady state of ion density profile
is realized again. The above procedure is repeated until
the steady state of the electrostatic potential is realized.

The initial condition is shown in Fig. I, where ionsbut only ions' motion is calculated until the steady stateof ion density axial profile is realized. The electrostatic
potential is calculated by eq. (3) and the steady state ion
density profile assuming charge neutrality condition,

3. Numerical Results
As seen in ref. [,3] the plug potential formation
should satisfy the condition that384

Katanuma I. et al.. On the Ion Distribution Function around the Plue in a Tandem Mirrorn,(E* B)=n"(g* Bo) ,dr, _fuu
aBp- aBe 'formation. As mentioned in ref. [3] the discontinuous
electrostatic potential profile will appear around the plug
if no ions come from the outer mirror throat. It isenough to make the electrostatic potential continuous(4)

at the plug.

As long as the electron distribution function is
Maxwellian i.s., 4rr - Z"t in eq. (3), the relation dn"ldBo= 0 is hold. Therefore, the peak position of the density

(a)

r>

profile coincided with the maximum point of thepotential profile, which indicates that the large plug

lon f,ur lrom inprq(z)mhrcrthrcetwfthn:3

potentialis not formed.

r>

0.0The ions are reflected in front ofthe plug potential,tons trc tost

(note bulk ion energy with thermal velocity is muchsmaller than the plug potential) that is, the peak point ofion density profile, i.e., electron density exists in frontof the plug potential peak point.This means that the-z.oThermal Banierz-"lxis 'lA*

Outer MirorThrcato)

electrostatic potential around plug region does not obey
the traditional Boltzmann law.

Therefore, the electron distribution function isfound to be non-Maxwellian. The actual electron

enrBp+eop(plug i..., po@tial ffiimDe)

distribution function should be determined including theeffects of ECRH and other loss processes. But this isdifficult in general so that we include the effect of nonMaxwellian electrons by assuming the electron
distribution function in eq. (2).
Figure 3 shows the plug potential in the steady state
for the case of T.1lTi = 30 and T"nlTi= l. Here the used

e=pB;+eQ1(c)

elBru+ehvll< 0..
ElrBm+4m(aBmiffithM)

parameters are that the electrostatic potential is etp,lTi =FpBp+eh0 at the inner mirror throat, and e{golTi = -2 at thermalbarrier.

(plug i..., potdi&lFigure 3(a) displays the steady state electrostatic
potential and ion density axial profiles. It is seen that the|Mimtrn)

elU3i*uft e4B6+eQ6(itErmiffitlr@)/

peak point of ion density locates in front of theelectrostatic potential maximum. The plug (where theelectrostatic potential is maximum) exists in the midway

from thermal barrier region to the outer mirror throat.
The height of the plug potential is eErlTi = 1.8 in Fig.
3(a). The difference between the ion density axialItiprofile and electrostatic potential profile results from thenon-Maxwellian electron distribution function. A large
majority of ions, the mean velocity of which is aboutithermal velocity, reflects at the region where theelectrostatic potential is slightly larger than that at the-4.4

inner mirror throat in Fig. 3(a).

0.0 Vil,Vr 4.0

Fig. 3 The steady state axial profiles. (a) is the ion

The remarkable feature in Fig. 3(a) is that thedensity and electrostatic potential axial profiles,
where dashed line is the initial ion density profile
in EQI = 0. (b) and (c) are ion velocity space
represented by e and ,u, where broken line is e =pB^ + eq^, and dashed line is e = pBe + ege,continuous electrostatic potential is realized especiallyaround the plug region. The charge neutrality conditionis assumed in the simulation so that the discontinuous
electrostatic potential can appear where the charge
neutrality condition breaks such as sheath potential

respectively. (d) is the ion distribution function

observed in the region 0 < z< 1 cm.
385

Katanuma I. et al., On the Ion Distribution Function around the Plug in a Tandem Mirror

even if the population of those ions from outer mirror
throat is very few.
The Figs. 3(b) and 3(c) plot the ion velocity space
represented by e and p, where v;1 > 0 is in Fig. 3(b) andv;1 < 0 is in Fig. 3(c). The ions are input at the innermirror throat i.e., Maxwellian ions are seen in the regions - ttBi - eet> 0 in Fig. 3(b). On the other hand the ionsin the region e- LBp- egp2 0 and e - pB^- ee^20
are lost at the outer mirror throat. It is seen that theMaxwellian ions exist in the region of loss conein Fig.

Fig.4lon density and electrostatic potential axial
profiles. Here dashed line is initial ion density
profile in dzl = O.

3(b), while very few ions are found there in Fig. 3(c).
The ion distribution function around the thermal
barrier region (0 3 z < 1 cm) is plotted in Fig. 3(d).
There is no ions with small velocity of v11. It is foundthat the perpendicular ion temperature around thethermal barrier region is less than unity which is thetemperature given at the inner mirror throat. The reason
why the perpendicular temperature is small is that theions are accelerated by magnetic field from the innerwhich is inconsistent with the fact that the majority ofions reflects in front of the plug point. The effects of
ECRH is included in the non-Maxwellian electrons.
The previous works for the plug potential formation[1-4] made use of the simplified ion distributionfunction so that the artificial dissipation such as theionization effect was required in order to obtain themirror throat to the thermal barrier and theperpendicular kinetic energy is changed to the parallel

continuous electrostatic potential profile across the plug

kinetic energy.
Figure 4 displays the ion density and electrostatic
potential axial profiles in the steady state in the case thatregion. In this work, however, each ion motion is

the magnetically trapped high energy electrons are exist,

coulomb scattering is included as the effects of

where the modified Boltzmann law is siven as

dissipation by Monte-Carlo simulation but without anyother dissipation such as ionization effects, so that theplug potential formation have been demonstrated in thefollowed numerically in the magnetic mirror cell and ther.,,. l,l. r..,\ Bo rlnlzt-nn(z)[\tp(d=i llr";,_;;; t| ,l;*tt-6lB@\'"{ll-o'I|

more realistic condition.

IJ\-References(s)

Fl I. Katanuma, Y. Kiwamoto, Y. Tatematsu, K. Ishii,
T. Saito, K. Yatsu and T. Tamano, Phys. Plasmas

HereB(z)-B^-,\ = no..,nlrlz)
u.6 x t?b - p- p
Db- Dm3,2218 (1996).

[2] T. Saito, I. Katanuma, T. Aota, L.G. Bruskin, T.

(6)

Cho, et al., Fusion Energy 2,105 (1997).
I.
[3] Katanuma, Y. Kiwamoto, Y. Tatematsu, K. Ishii,
T. Saito, K. Yatsu and T. Tamano, Phys. Plasmas
4,2s32 (1997).
[4] I. Katanuma, Y. Kiwamoto, Y. Tatematsu, K. Ishii,
T. Saito, T. Tamano and K. Yatsu, Phys. Plasmas

In the case including the magnetically trapped electronpopulation, high plug potential eQJTi= 2.4 is realized inFig.4.

4. Summary

5, 1560 (1998).

We have shown by the Monte-Carlo simulation thatthe plug potential can be formed only with the ions[5] I. Katanuma, Y. Kiwamoto, M. Ichimura, T. Saito

flowing from the inner mirror throat and nonMaxwellian electron population. The essential point for
the large plug potential formation is that the electron
distribution function is non-Maxwellian. Otherwise, thedensity maximum point coincides with the plug point,and S. Miyoshi, Phys. Fluids B 2,994 (1990).

t6l R. Minai, I. Katanuma and T. Tamano, J. Phys.
Soc. Jpn. 66,2O5I (1997).

l7l R. Minai, I. Katanuma, T. Tamano and K. Yatsu, J.
Phys. Soc. Jpn.67, 876 (1998).

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