J. Plasma Fusion Res. SERIES, Vol.4 (2001) 578-582

stable sheath Formation in Magnetized Plasma
T9MITA YuKihiro*, TAKAYAMA AriMiChi, TAKAMARU HiSANOTi ANd SATO
National Institute .for

TETSUYA

Fusion science, oroshi-cho 322-6,Toki 509-5292' Japan(Received: 5 December 2000 / Accepted: 27 August 2001)

Abstract

plate was studied by a one
The stable sheath formation in an expanding magnetic field to a divertor
boundary is more
plasma-sheath
dimensional analysis. The requirement for flow velocity of ions at asound speed' Theion
therestricted than that ofthe uniform magnetic field, which should be greater thanplasma radius, which isdifference, however, between both cases is an order of the Debye length toplasma was obtained by
negligibly small. The requirement for ion flow velocity inside a quasi-neutralin the quasi-neutral plasma'taking into account the effects of the plasma source. Without plasma source
ion sound speed' The ion
thegreater
thanthe ion flow velocity at an injection point should be much
mitigates thisconsiderably
source inside the quasi-neutral plasma, such as electron impact ionization,requirement.

Keywords:
plasma sheath, expanding magnetic field, quasi-neutral plasma, divertor

incidence angle of the magnetic field. With the use of a
two dimensional kinetic analysis, Sato [2] clarified thismagnetic presheath is attributed to the ion polarization

1. lntroduction
Formation of the electrostatic potential in a divertor
plasma immersed in a nonuniform magnetic field isinteresting in many fusion devices. In fusion devices themagnetic field expands in the direction of the plate, i'e'drift. The potential formation of a presheath in an open
magnetic field studied analytically and numerically [3]'The effects, however, of nonuniformity of the magnetic
field have not been taken into account for the stable

the magnitude of magnetic field decreases towards theplate. In this configuration ions are accelerated towards

Debye sheath formation. In this paper' we consider acollisionless sheath model between an infinite metal
plate and a quasi-neutral plasma in an expanding
magnetic field towards the plate. One dimensional
kinetic analysis leads to a condition for a stable sheathformation in an expanding magnetic field, which iscompared to the criterion for the stable Debye sheathformation in a uniform magnetic field [4'5]. In a quasi-

the plate due to the gradient of the magnetic fieldstrength, called the mirror force. The bombardment ofaccelerated ions to the plate may cause several severe
problems to fusion plasmas, for example, release oflarge amount of impurities from the divertor plate'
Limited research has been carried out describing
magnetic field effects on various potential formations
and particle and heat fluxes to the divertor plate' The

neutral plasma region outside the Debye sheath with aplasma source, the requirement for ion velocity isplasma-wall interaction in a uniform and obliquemagnetic field to the plate has been studied by means of
ID-PIC numerical simulation [1]. This analysis shows
the formation of a quasi-neutral magnetic presheath
preceding the electrostatic Debye sheath, which scales tothe ion gyroradius at the sound speed and to the*obtained by limiting the formation of a negative gradientof electrostatic Potential.

@2001 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author's e-mail: tomita@ nifs.ac.jp578

Tomita Y. et al., Stable Sheath Formation in Magnetized Plasma

The magnetic field lines connecting two divertor

because of much shorter sheath thickness (- Debyelength ) than the collisional mean-free-path.
The ion distribution function j (2, oz, ?Jr) governing

plates in a SOL (Scrape-Off Layer) region of a toroidal

the steady sheath is determined by the Vlasov equation:

2. Stable Sheath Formation in an
Expanding Magnetic Field

magnetic fusion device consist of a single or multiple
mirror configurations. In a divertor region, the magnetic
field is expanding towards the plate. Figure I shows thedutA,(z,a",ut7 ^strength of the magnetic field between two divertor
plates in the LHD (Large Helical Device) [6]. The
model geometry for a one dimensional analysis is shownin Fig.2, where the divertor plate is located at theposition of z = L, and the plasma-sheath boundary isdenoted by za. The magnitude of magnetic field B(z),*-_#=$.

dt

The kinetic ecuations of an ion are# = -#y, - ##,p dB _ pD, dB
dut-d- _- TIard,- Mard/'

which is directed toward the z direction. decreases to theplate so slowly that the magnetic moment of ions isconserved. Inside the sheath (x6 < z < L), collisions

4----:(2)13)\J'

where the charge and mass are denoted by q (= Ze) andM, respectively. The magnetic moment of an ion p(z) =Mttl 2B(z) is one of the constants of motions because ofthe slow change of the magnetic field. In the case of an
expanding magnetic field (dBldz < 0), the second termof RHS in eq. (2) acts as a mirror acceleration for ions.
The transformation of variables from (2,a.,o1) to (z,e,p)

between plasma and neutral particles are neglected

a:4(l)

dDr

exPanding oitu -.....------.-,

gives

Eo?

.d[k.e.lt)- =-6r'=r,-nur(2,€,lt)9))owhere e is the total energy of an ion (= Mu? t Z + !B(z)+ S[kD and the pr(z,e,p) is the particle velocity towards
the plate, which should be expressed by the newvariables (2,e, p), i.e.

Oi'Elq)po(4)ooozo406080

u,(z,e,lD=W.

length of field line (m)(5)

In a decreasing effective potential energy U (: pB(z) +qQkD, there are no ions with zero or negative velocityu,(z,€,p) < 0. Therefore the Vlasov equation eq. (4)shows the ion distribution function in the (z,e,p) spaceholding constant along the z direction inside the sheath,i.e.fik,e,p) = fikt,e,F) G ft@,tt).This leads to the ion

Fig.1 Magnetic field strength between divertor plates inLHD.

divertorplate

density inside the sheath as:nitZ)=2nBk)n;rrldplaef;e4otW,n6 i,

where U6 is the effective potential energy at the plasmasheath boundary G pB(z) + q[k)).In the vicinity ofthe plasma-sheath boundary 26, ion density is expressedby the expansion A@ = Qk) - Qk) and A,B = B(a) -sheath

B(d,Fig.2 Model and coordinate for one dimensionalanalysis.
579

Tomita Y. et al., Stable Sheath Formation in Magnetized Plasman;(z)=ni6l-t@,")uLQ"t\\'-"'b[The scale length of electrostatic potential is to the orderof the sheath width (- Debye length ). On the otherhand, the scale length of the magnetic field is as large asa minor radius of a toroidal plasma. Since the magnitudeof 76 becomes much smaller than unity, the effect of theexpanding magnetic field on the condition of the stablesheath formation may be negligible.

(7)-(t*!<"'r':>') oM

where ni6 and (..), denote ion density and the averageover the ion energy distribution function at the sheathedge 26, respectively. The electron density, which isassumed to satisfy the Boltzmann relation, is also
expanded around the plasma-sheath boundary:n"(z) = n"a (l - eLQ / T"),The relation between the density gradients ofelectrons and ions at the sheath edge is obtained fromthe charge neutral condition.

(8)

t#

where n"6 (= na) is the electron density at the sheathedge, which is equal to that of ion n;6, and where Z, is.dn,l _ dn"l' d,lu- A,

the electron temperature, which is assumed uniform in
the whole region.
These densities give the Poisson's equation aroundthe plasma-sheath boundary,

tod2L6diflu<';'),.2(, +l@ir ui),)(1s)d(rn B) / dzl--a07az

IThis relation means that the requirement for the stablesheath formation, eq. (10), is equivalent to the.2,. Zr",
M,\u')arequirement that the magnitude of the gradient of ion
density should be smaller than that of electrons at theI^.. lLz..e)=+ltT"(,-!tot? ttt2\.\d(lnB)ldzl'" I -7trr2\urt""/blI

sheath edge [7],IdOldz l.lt'l*l,l = l*l,l'o I(16)

We can easily obtain the condition of a stable ornonoscillatory solution of electrostatic potential at theSince the electron density decreases faster than that ofplasma-sheath edge:

the ion in the decreasing electrostatic potential, ion

21/M@;1i'> l-Tu

density will be larger than that of electrons everywhere
inside the sheath and the stable sheath will be formed.

(10)

andTu <r,

3. lon Velocity Distribution in Ouasi-Neutral(11)

PlasmaThe requirement for ion velocity distribution isinvestigated in a quasi-neutral plasma outside the Debye
sheath (1;, I z I zo see in Fig. 2). Here the ion source

where the factor y6 indicates the effect of nonuniformityof the magnetic field strength asy,=+(l +lqu,,rulyu)W#1, e2)

due to electron impact ionization of neutral atoms is
considered as well as the injected plasma from the upperstream. In a divertor region ionization by electron
impact is effective because of the electron temperatureof 10 - 100 eV. The distribution function of the ion

In the case of a uniform magnetic field (y6 = 0), thecriterion (10) becomes@;\u' > zltM,(13)source is given as:

which is the generalized Bohm criterion [5] for theuniform magnetic field. The decreasing magnetic fieldand electrostatic potential give a positive value ofthe factor y6, which means the average value of(a;2);t has to be larger than that of the generalized

where no is the density of the neutral atoms, (oo); is theBohm criterion eq. (13):rate coefficient of ionization by electron impact and1",-2\-t > ZT"IM >ZT"\"ztb -l_Tu

Mt t2

S(z.ep) = :!!
- n "(z) n,(cu), I u,(z.e.lt) |4nT-'
exp f-Ma" (z,e,p) | 27,1,(r7)( isthe temperature of source ions, which is assumed(r4)

uniform. In this distribution function. the ion velocitv u580

Tomita Y. et al., Stable Sheath Formation in Magnetized Plasma

temperatures of electrons T" and source ions d areassumed uniform inside the plasma region. The thirdand ?, should be expressed by the function of (e'e'p).
This velocity distribution of the source ions makes theion distribution function equal to the Maxwellian

term of RHS of eq. (21), which designates thedistribution [8]. Taking into account the turn of sourceions with negative velocity o. in decreasing effectivepotential, the density ofthe source ions is obtained,

nonuniformity of the magnetic field, is the order of unity
because of the same scale length of the magnetic field
and the electrostatic potential. On the other hand, thesecond term of RHS of eq. (21), which indicates thenx(z) =r;fTnon"o(ou),exp[-qQQ) lT,l,l\l ++at^effects of the generated ion source, is much larger thanunity because of the higher plasma temperaturecompared to that of the source ions. Therefore, theib

I dr' ,*pl@ lr,- e /7")Qk)1, (1S)averaged value of (u;2)7] at the injection point becomesmuch less than the ion sound speed (ZT"|M). In the case,7t,p)where 2"6 is the electron density at the position of @ - 0,of no plasma source (4 = 0), the RHS of eq. (21)and z, is the turning point of an ion with negative u,.
which depends on e and p:

becomes less than unity, which implies that the value of(a;2);l is required to be larger than the ion sound speed
at the injection boundary.

Zt=O '. E) ItB^{,-ttB(2,)-SQk)=0: t) lB^*.

(19)

4. Goncluding Remarks
The stable sheath formation in an expanding

For the ions injected from the upper stream of theboundary Zi,, the ion density is obtained by the samemagnetic field towards the divertor plate was studied bya one dimensional kinetic analysis. The requirement forstable sheath formation is that the averaged ion velocity
at the plasma-sheath boundary must be larger than theion sound speed, which is more restricted than therequirement in a uniform magnetic field (the generalizedmanner stated in the previous section. The local density
gradient of these ions is easily obtained,dn,n'^\z)clz =Bohm criterion [5]). The factor, however, of thedifference between both cases is as small as the order of(20),- ,-,_ ni,\z)

q d0

T AZ

the ratio of the Debye length to the plasma radius. The
requirement for the ion velocity inside the plasma wasobtained from the quasi-neutral condition of the plasma.
If there is no plasma source in the quasi-neutral plasma,,

the ion flow velocity at the injection point z;n into theplasma should be much larger than the ion sound speed.
The ion source inside the plasma, such as ionization ofwhere n,n and n;" are the injected ion and source ion
density, respectively, and (..)* denotes the average overthe ion velocity distribution function at the plasma

injection boundary z;n. Electron density, which isneutral particles due to electron impact, considerably

assumed to satisfy the Boltzmann relation, and thecharge neutrality condition in this region give therelation between the ion flow velocity and the gradients

reduces this requirement, where the required ion flow
velocity becomes lower than the ion sound speed. Futurestudies in this area of research should include otherof electrostatic potential and the magnetic field:

phenomena, for example Coulomb collision and charge
exchange process.

ZT" 1^._2174 \u z lin

Acknowledgment

r+6"1ft_l-1-4 - ( I -41 2lt,.+ la"r u',7,^)

The authors express their appreciation to Dr. K.
Sato for useful discussions.

, (2r)

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K. Sato, H. Katayama, and F. Miyawaki, Contrib.
Plasma Phys. 34, 133 (1994).

581

Tomita Y. et al., Stable Sheath Formation in Masnetized plasma

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