Effects of Fast lons and an External Inductive Electric Field

on the Neoclassical Parallel Flow, Current and Rotation

in General Toroidal Systems

Noriyoshi Nakajima and Masao Okamoto

Natior~al Institute for Fusior~ Science, Nagoya 464-01.

(Received April 15, 1992/Revised Manuscript Received May 23, 1992)

I

Abstract

Effects of external momentum sources, i.e., fast ions produced by the neutral beam

injection and an external inductive electric field, on the neoclassical ion parallel fiow, cur-

rent, and rotation are analytically investigated for a simple plasma in general toroidal

systems. It is shown that the contribution of the external sources to the ion parallel flow

becomes large as the collision frequency of thermal ions increases because of the momen-

tum conservation of Coulomb collisions and sharply decreasing viscosity coefiicients with

collision frequency. As a result, the beam-driven parallel flow of thermal ions becomes

comparable to that of electrons in the Pfirsh-Schluter collisionality regime, whereas in the

~7~, 1/v or banana regime it is smaller than that of electrons by the order of me/mi (me

and mi are electron and ion masses). This beam-driven ion parallel flow can not produce

a large beam-driven current because of the cancellation with electron parallel flow, but

produces a large toroidal rotation of ions. As both electron and ions approach the Pfrsh-

Schluter collisionality regime the contribution of thermodynamical forces becomes negli-

gibly small and the large toroidal rotation of ions is predominated by the beam-driven

component in the non-axisymmetric conflguration with large helical ripples.

Keywords: neoclassical theory, non-axisymmetric system, fast ions, external inductive field, parallel

flow, neoclassical current, plateau collisionality regime, Pfirsh-Schluter collisionality

regime,

1. Introduction

In recent experiments of Compact Helical System (CHS) heliotron/torsatron devicel) with

tangential neutral beam injection, plasma rotations have been measured over a wide range of densities and magnetic fleld ripples2). In CHS central helical ripples along the magnetic axis

can vary according to the outward or inward shift of the plasma. If R ･ = 90-95 cm, where a***

R*.i~ is the major radius of the magnetic axis in the vacuum fleld, the central helical ripples

are negligibly small and they increase sharply if R . exceeds 95 cm. In the vicinity of the "*~~

region where the helical ripples are vanishingly small the system can be thought to be nearly

axisymmetric. It is reported in ref. 2 that the profile of the toroidal rotation is dominated by

an anomalous shear viscosity when the field ripples are weak near the axis, while the parallel

viscosity is found to be dominant when the ripples are strong enough. The observed parallel

viscosities have been concluded to agree with the neoclassical predictions within a factor of

46

研究論文　　　Effects　of　Fast　Ions　and　an　Extemal　Inductive　Electric　Field　on　the

Neoclassical　Parallel　Flow，Current　and　Rotation　in　General　Toroidal　Systems中島，岡本

three．In　these　experiments　both　electrons　and　ions　were　in　the　plateau　collisionality　regime．

Plasma　rotations　or　flows　are　determined　by　the　ba！ance　between　external　momentum　sources

（or　input　torque）an（1（iamping　forces　of　viscosity　an（i　friction．Neoclassical　theories　inclu（1ing

momentum　sources　have　been　extensively　developed　for　axisymmetric　tokamaks3）．Recently

the　authors　have　developed　the　neoclassical　theory　for　the　parallel　force　balance　to　investigate

the　flow，current，and　rotation　of　a　multispecies　plasma　with　external　momentum　sources　ingeneral　toroidal　systems　including　both　axisymmetric　and　non－axisymmetric　toroidal　devices4）．

As　extemal　momentum　sources　fast　ions　produced　by　the　tangential　neutral　beam　inlection

an（i　an　external　inductive　electric且e1（i　have　been　consi（iere（i．　In　this　ref。4，neoclassical　expres－

sions　obtained　so　far　in　each　collisionality　regime（ref．5in　the1／レand　Pfirsh－Schl貢ter　regimes

and　refs．6and7in　the　plateau　regime）have　been　unified　in　terms　of　the　geometric　factor．

General　formula　common　to　all　collisionalities　have　been　obtained　for　neoclassical　flow　and

consequently　for　parallel　current　an（1plasma　rotation．　This　exten（ie（1neoclassical　theory　has

reveale（i　that　in　a（1（lition　to　the　conventional　pressure（iriven　neoclassical　current　（bootstrap

current）the　parallel　current　generated．directly　by　the　ra母iaユelectric　field，which　dose　not　exist

in　the　axisymmetric　system，exists　in　the　nonaxisymmetric　system4）．Rotations　of　bulk　ions

and　impurity　ions　have　been　investigated　for　a　plasma　in　the　g』eneral　toroidal　system　based　on

ref．4and　the　difference　between　the　two　has　been　clarified8）．

　　　　In　the　present　paper，we　examine　the　effects　of　external　momentum　sources’on　the　neo－

classical　para．11el　flow，current，and　rotation　applying　the　extended　neoclassical　theory4）to　a

simple　electron．ion　plasma　in　general　toroidal　systems。We　consider　fast　ions　produced　by　the

tangential　neutral　beam　injection　and　an　inductive　electric　fleld　as　the　extemal　momentum

sources．　Equations　for　parallel　flow，current，and　rotation　are　analytically　investigated　by

using　the　m＆ss　ratio　expansion　of　frictiOn　and　viscosity　coe伍cients．We　concentrate　our

attention　on　the　high　collisionality　regime　taking　into　account　the　operation　conditions　of　the

above　mentioned　CHS　experiment．In　the　neoclassical　theory，the　parallel　particle　Hows　are

determined　by　the　parallel　momentum　and　heat　flux　balance　equations，i．e．，the　balance　among

the　friction，viscosity　and　extemal　momentum　sources．The　friction　matrix　consisting　of

friction　coe伍cients　is　not　regular　because　the　Coulomb　collision　conserves　the　total　momenta．

The　viscosity　matrix　consisting　of　viscosity　coe伍cients　approaches　the　singular　matrix　as　the

collision　frequency　becomes　large　because　the　viscosity　coef丑cients　become　very　small　in　the

high　collisionality　regime．　As　a　result　of　it，the　contribution　of　the　extemal　momentum

so’urces　to　the　parallel　flow　may　change　drastically　according　to　the　collisionality　regime．

　　　　As　is　well　known，when　both　ions　and　electrons　are　in　the1／レor　banana　collisionality

regime，the　ion　parallel　flow　driven　by　the　extemal　momentum　sources　is　smaller　than　that　of

electr・nsbyε≡爾when窺ean伽iaremasses・felectr・nsandi・ns，respectively．Ofcourse，the　momentum　transfer　to　ions　from　the　extemal　momentum　sources　is　larger　thanthat　to　electrons　byε一1．Therefore，we　see　tha，t　the　light　electrons　carry　the　current　and　the

heavy　ions　carry　the　momentum　in　the　banana　regime．It　will　be　shown，however，that　if　elec－

trons　and　ions　are　in　the　Pfirsh－Sch1茸ter　collisionality　regime，the　contribution　of　fast　ions

becomes　so　large　that　the　beam－driven　parallel　flow　of　ions　becomes　the　same　order　of　that　of

electrons．Accordingly，the　momentum　transfer　to　ions　from　the　fa、st　ions　is　larger　tha、n　that

to　electrons　byε一2．This　beam－driven　ion　parallel　flow　can　not　produce　a　lafge　beam－driven

current，because　it　is　canceled　out　by　beam－driven　electron　parallel　flow．As　a．result　of　it，the

neoclassical　parallel　current　reduces　to　the　usuaユclassical　expression，while　it　can　contribute

to　the　rotation　of　ions　significεしntly．It　will　be　also　shown　that　in　the　non－axisymmetric　to－

roidal　system　with　la、rge　helical　ripples　the　contribution　of　the　thermodynamical　forces　to　the

parallel　flow　is　negligibly　sma、ll　and　the　ion　parallel　flow　and　toroidal　rotation　are　predomi－

nated　by　the　beam－driven　component　or　the　extemal　momentum　sources．However，it　will　be

remarked　that　if　the　axisymmetry　is　partially　recovered　in　some　region　as　is　the　case　of　CHS

47

I

f*~~~~~~~~f*~t ~~68~~~~1~~ 1992~~7~

under an experimental condition, the thermodynamical contribution to the ion flow must be

included. A different case where electrons and ions exist in different collisionality regimes

will be discussed analytically and the similar results will be obtained.

The organization of this paper is as follows. In S2 parallel force balances are given and

the singular behavior of friction and viscosity matrices in the high collisionality regime is

indicated. The parallel flow, current, and rotation are derived for a simple plasma in the gen-

eral toroidal system. These are analytically investigated in three special cases, placing em-

phasis on the high collisionality regime in S3. Section 4 is devoted to conclusion a,nd discus-

sion. In Appendix I the expressions of flux-surface-averaged parallel frictions and viscosities

used in S2 are given. Appendix 2 gives the deflnitions of quantities in Eq. (2). The friction

and viscosity coefficients are given in Appendix 3.

2. Parallel Flow. Current. and Rotation Including External Momentum Sources for a Simple

Plasma

Neoclassical theories for the parallel flow, current, and rotation have been extended to a multispecies plasma in general toroidal systems4). This extended theory includes the external

moinentum sources of fast ions produced by the tangential neutral beam injection and an ex-ternal inductive electric fleld. In this ref. 4 general expressions, w:hich are common in all colli-

sionality regime (11,,, plateau, and Pfrsh-Schlilter regimes), have been derived for the parallel

flow, current, and rotation in terms of the geometric factor. In this section we apply this

theory to a simple plasma consisting of electrons and one species ions.

First we concider asymptotic behaviors of viscous and frictional matrices. To determine the flux-surface-averaged parallel flow <1~ul!'> and heat flux <L1q//'>, the flux-surface-averaged

parallel momentum and heat flux balance equations are used3~8).

<B F~1> n.e.<B E(A)> <B F.fl> r ･ l <B V n > (1) L ' ' "]=L ' l+1 ' l + . <B F.2> <B F.f2> L o J' -<B 'V･@ > - ･ - -where H. and @~ are the stress and heat stress tensors, respectively. F~j and F.fj(j = I or 2) are frictions of species a with thermal sp~cies and fast ions, respectively. E(A) indicates the

external inductive electric field. The expressions of the flux-surface-averaged parallel quanti-

ties can be found in ref. 4 and repeated in Appendix 1.

Substitution of Eqs. (A. 1) and (A. 4)-(A. 5) in Appendix I into Eq. (1) gives the follow-

ing linear algebraic equation for X:

/1 (X+ G) = LtX + F + EA , (2) where X is the parallel flow and parallel heat flux to be solved, /1 and Lt indicate the viscous

and frictional matrices, which are constructed in terms of viscosity and friction coefiicients,

respectively. G, Ff and EA express the thermodynamical force, the friction with fast ions,

and the external inductive electric fleld, respectively. The definitioh of each quantity is given

in Appendix 2. The solutions to Eq. (2) are given by

X=(Lt- /1)~1/lG -(Lt-p)~1 (F + E ) (3) We can see from Ep. (3) that the parallel flow and heat flux consist of two terms: one is due

to the thermodynamical force and the other is due to the external momentum sources. The

friction force conserves the total momenta as shown by Eq. (A. 3) in Appendix l, which is expressed in terms of Lt as follows:

l Lt I = O . (4) As is mentioned in Appendix 1, the viscosity coefiicients decrease so rapidly as the collision

frequency increases that they become small in the plateau regime and very small in the Pfrsh-

Schluter regime. From this property and the total momentum conservation given by Eq. (4), we see, in the limit of high collisionality regime (plateau or Pfirsh-Schluter regime),

48

l

'l~l /UPFO

Effects of Fast lons and an External Inductive Electric Field on the Neoclassical Parallel Flow , Current and Rotation in General Toroidal Systems

~F ~~' ~~~

p-O, (Lt-p)~1_oo - flnite. (5) , (Lt-/t)~1!e

Therefore, in these high collisionality regimes, the effects of external momentum sources (the

second term on the right-hand side of Eq. (3)) may become significant. The parallel viscosity

and heat visosity given by Eq. (A. 5) are calculated by using the gyro-averaged distribution

function. There is another type of viscosity, i.e., gyroviscosity, which is obtained by using

the gyrovarying distribution function. The gyroviscosity dose not depend on the collisionality

regime. The order of the gyroviscosity of particle spcies a is smaller than the parallel viscosi-

ty by the order of O (p./L) for p~/L << 1, where p~, and L are the Larmor radius of particle

species a, and the characteristic length of the system, respectively, hence the significancy of

the effects of the external momentum sources may not change in the high collisionality regime

even if the gyroviscosity is included.

To clarify the effects of the external sources on the parallel flow, current, and rotation,

let us consider a simple electron-ion plasma. Assurning T* - Ti and using

n*m*

Tii i J~ Z:ff (6) T** 1 <<1, e= nim C::

as the expansion parameter, we calculate the elements of matrices Lt and p up to the order of

e. Substitution of this result into Eq. (3) gives the parallel flows for electrons and ions, the

parallel current, and the poloidal and toroidal rotations for the simple plasma. We use the Boozer coordinates (c, e, ~, where ip is defined as the toroidal flux divided by 27T, e and ~

are, respectively, the poloidal and toroidal angles. For the simple plasrna the parallel flows

for electrons and ions are given by, respectively,

<Bu//'> = <GBS>. {(L{1) + Li{)) (- eP_~. + ip ) + (L{~) + L{~)) T. }

e

- (1 + Li~) + Lll)) <GBS>i{ eP_ni. ' (1) Ti } + ip + L34 eZ*ff

+(1 + L{~)+ Lil)) Zf _ fL1 + (1 + ~:)(~)3 Zfnf <~u//f> } { J * N(1) Z*ff Z 33 n

ex (1 + L{1)) {A(No~+ A(N1~} m:i <BE;f)>, (7)

> fTa)( P a) T <Bu//i> =<GBs ' I~ll ~~ e~ + ip')tL32 e'}

f P. (o) T. - (1 + L{1)) <GBS>ile~ + c' + L34 eZ,ff }

(1) Z i v 3 Zfnf } * +{L f _(1+L{}))Z 1+ 1+ ~ )(~~) JN(1) <J~u//f~ f[ ( f 33

11 Z*ff uf n

co) (1) eT.i <BE;f)> (8) + Z*ffLll N33 m*

The parallel current is given by

<~J//> = en. {<Bu//i> - <Bu//*>} + <BJ//f>

= L{~){(<GBs>i - <GBS>.) en ip' + <GBS>,p: + <GBS>iP; + <G > L(o)n T'}

* Bsi 34 i i (Lj~) + L(1) _ L!~)) <GBS>. n. T: 12

+ Z mi ) ( _: ) I N;~)} <~J//f> { + Llt)Zf [1 + (1 + v3 1 - (1 + L{~)) Z.:f mf u

co) (1) e21~*T*i + {(1 + L{})) (A(No~ + A(N1~) + Z.ff L11 N33 } <~E;f)>. (9)

m* The toroidal and poloidal rotations are given by, respectively,

49

I

~~~!-~f*~~ ~~68~~~~1~~, 1992~l~ 7 ~j

1 1 r I dP. + dip ) <u V~>= J+t:1 <Bu//'>- J+tl ~ e.n. dip dip ' (10)

c J / I dP. + dc ) <~.･V0>= J+tl <Bu//'>+ J+tl ~e.n. dep dip ' (11) In Eqs. (7) to (11) transport coefficients are given by

L(o) _ {/1.1(l~~ /1.3) - ~ (~~~ - ~.2)} ~ 11 ~ *2

D*2 L(1)= D N(1)

33 , 11

1 L(o) _ {//.21~~ = ~.3 ~~~} ~~ , 12 ~

1 12 ~ {1~~ I~~ (l~~)2} ~.2 ~ N(1) L(1) _ ~ 33 ,

1 L;04) = ~;*2 ~i2 Dn

1 (1) L(1) _ {/1.2 Ll~~l~~ (l~~)2 J - ~ ~~ L~.1 ~.3 - (~.2)2 J} ~~ N33 , 32 -

Di2

N(1)= Dn e, (12) 33

1 A(No~ = ~{~ {~~~ - ~.3} D.1 '

A(Nl~ ~ ~ I (1) .. *. -= - 111 I12 /1.2 D.1 N33 ,

D =D*1-D Na) *2 33 ,

" /1.1) (~ 22 - /1.3) (l~~ - ~.2)2 , D*1=(~ -- '* 11

D*2 - L~ 11 ~22 - (~~~)2] ~.1 ~ ~~ [/1.1 /1.3 (ll.2) l - .* .. -Dn = ~ ~n (~;~ - ~i3) - (~i2)2

Di2 = ~;*2 - ~i3 ,

and tr, 27cJ and 21TI are the rotational transform, the total poloidal current outside a flux surface, and the total toroidal current inside a flux surface, respectively. <GBS>. and <GBS>i are

geometric factors for electrons and ions4,5) reflecting the breaking of axisymmetry and their

expressions can be found in ref. 4. The normalized friction coefiicients l~~ (i, j = 1, 2) and l;.2

are given in Appendix 3, which do not depend on the collisionality. p*k and ~ik (k = 1, 2, 3)

are the normalized viscosity coefficients of electrons and ions, respectively, the asymptotic

forms of which in the 1/,, or banana, plateau, and Pfirsh-Schltiter regime are also given in

Appendix 3. In the 1/v collisionality regime, the normalized viscosity coefficients are propor-

tional to ftlf. where ft is the fraction of trapped particles and f. = I - ft' In the plateau

regime, they are proportional to ~~/~pL Where ~. and ~pL are the mean free path of species a

and the characteristic length of the magnetic field in the plateau regime,' respectively. In the Pflrsh-Schltiter regime, the normalized viscosity coef~:cients are proportional to (~./~ps)2

where ~ps Is the characteristic length of the magnetic field in the Pfirsh-Schltiter regime. As

the mean free path is inversely proportional to the collision frequency (~. oc 1/v~~), the nor-

malized viscosity coefiicients appearing in Eq. (12) decrease sharply with increasing collision

frequency. Note that teams with superscript (1) are proportional to e.

3. Special Limiting Cases

To understand effects of the collisionality on the parallel flows due to the external mo-

mentum sources, we consider analytically three asymptotic cases, i.e., 1) the case where both

electrons and ions exist in the 1/~, or banana collisionality regime, which is realized by ~.k, ~ik

>> e, 2) the case where both electrons and ions exist in the Pflrsh-Schltiter regime, which is

50

l

~~~~7i~-E-m~^~~ Effects of Fast lons and an External Inductive Electric Field on the ~~ ~~. ~] ~~c Neoclassical Parallel Flow , Current and Rotation in General Toroidal Systems

realized by p*k, ~ik - e, and 3) the case where electrons and ions exist in the 1/v and Pfirsh-

Schluter regime, respectively, which is realized by ~*k >> ~ik -e. Note that for the plateau

collisionality regime we need different ordering with respect to ~*k and ~ik. In this section,

however, we exclude such a case.

At flrst, we consider the situation where both electrons and ions exist in the 1/,) or ba-

nana collisionality regime (~.k, ~ik>> e). As is seen from Eqs. (7) and (8) thermodynamical

forces and external momentum sources (fast ions and the external inductive field) drive the

flows of electrons and ions. By retaining the leading terms for each driving term, following

equations are obtained:

{ (d) ( P. A,~ T(o) T.l <1~u//'>= <GBS>. L11 ~~ en. +Y/+ ~l2 e f

- (1+ L{~)) <GBS>i{ eP_ni + ip + L34 eZ.ff } * ' co) Ti

+ (1 + Ljt)) ZZ~ff Z_nfnf <Bu//f>

- A(o) eT.i <~E;f)>, (13) NC m*

f Pi . (o) Ti <Bu//i>=- <GBS>ilen + ip +L34 eZ,ff}

+N(1) D.2 Z mf)(~) l} Zfnf <Bu//f> { f_ mi v 3 Zf L1 + (1 + 33 D*1 Z*ff n* u

+Z (o) a) eT,i <~E;f)> (14) *ffL11 N33 m*

<BJ//> = Ljl) {(<GBs>i - <GBS>.) en.c + <GBS>,p. + <GBS>iP. + <GBS>iLco)n T }

* 34 i i Lco) - <G BS>* n* T. 12

{ Z <BJ//f> } + I -(1+L{~)) Z,ff

+ A(o) e2n.~.i <BE;~)>, (15) NC m*

where L11 ~ {/1.1(~~; - ~.3) - ~.2(Zj~ /1.2)} D}1 (o) _

1 (o) _ {/1.2~~~ - ~.3~~~} D.1 L12 ~

L;~) = ~;~~i2 1 (16) Dil '

Di2 N(1) = Dn e,

33

l 1 A(No~ lj~ {l~~ - ~.3f D.1 '

D.1' D.2, Dn' and Di2 are the same as those m Eq (12) In Eq (16) N;~) rs smaller than

others by the order of e. Comparing Eq. (13) with Eq. (14), we see that the parallel flow of

electrons due to fast ions and the external inductive fleld is larger than that of ions by the

order of e. As shown in Eq. (15), the beam-driven current and Spitzer current are predomi-

nated by light electrons. However, it should be noted that the parallel momentum given to electrons, m* <~u//*>, from both fast ions and the external inductive field is smaller than that

of ions, mi <Bu//i>, by the order of e. Thus, we have usual results.

Secondly, Iet us consider the case wheire both electrons and ions exist in the Pflrsh-

Schltiter regime, i.e., ~ek ~ ~ik - 8. In this case, Eqs. (7)-(9) become as follows:

51

I

~~:A* ~~~~~tt 1992~~ 7 ~

P: . ' <Bu { co) T' } >==<GBS>i e~ +c + L34 eZ.ff //*

Z Zf nf } * + { f _ L ( i )( ~' ) I Na) <~u//f> Z.ff Zf 1+ 1+ ~f vf 3 33 n

-A(o) eT.i <JBE;f)>, (17) NC m* (o) T; <GBS>1 ePn~i + ip' +L

=- .{ . } <1~u//i> 34 eZ.ff m v 3 Zn m ) (~) I (1) f f <J~u//f> - Zf Ll + (1 + i

N 33 n* u

+ Z.ffL{~)N;~) e_mT:i <BE;f)> (18) <1~J//> = {1 - Zf (o) e2n.T.i <BE;f)>, }

<BJ//f> + A ' (19) Z*ff NC m* where

- 1 L(o) = {~.1~~~ - ~.2 l~~} -D.1 ' 11

L(o) _ I/i2

34 ~il '

N(1) _ e (20) 33 ~u 1

A(No~ l~~~~~ -D.1 '

D.1 = ~ 11 ~22 - (~~~)2 ** **

In Eq. (20), Li~) and N~l3) become the order of 8 and the order of unity, respectively, because

~ek ~ ~ik - e. Equation (19) shows that the parallel neoclassical current is reduced to the

usual classical parallel current, i.e., the classical beam-driven current with electron return

current effect -Zf/Z*ff and the classical Spitzer current. However, as is clear from Eqs. (17)

and (18), the parallel flows of both electrons and ions are driven by the thermodynamical forces, fast ions, and the external electric fleld. Since the parallel flows driven by the thermo-

dynamical forces are the same for electrons and ions in the leading order, the parallel current

due to the thermodynarnical force vanishes. The ion flow due to fast ions are canceled by that

of electrons in Eq. (19). Thus, the momentum transferred from fast ions to thermal ions is larger than that to electrons by the order of e~2. Hence, although the expression of the paral-

lel current is reduced to the classical one, neoclassical effects can be seen in the plasma rota-

tion. The toroidal and poloidal rotations of ions are expressed using Eqs. (10), (11) and (18)

as follows:

<u V~>=- J<GBS>iN+tl I dPi + dc _ J<GBS>iNL;04) I dTi

t(J+tl) en. dip dip ~(J+tD eZ.ff dip

Zf m v 3 Zn J+11 m )(~1) J (1) f f <Bu//f>, (21) - L1 + (1 + ' N 33 n* uf

(o) I dTi 1 dPi + dc _ J<GBS>iNL34 J ( ) [1-<GBS>iN] en, dep dip <~. ･ Ve> = J +tl eZ*ff dip * J + tl

Zn ~rZ mi v. 3 N(1) f f <~u//f>, _ f L1 +(1 + mf ) (~u'~'f ) 1 33 n. (22) J +tl

where <GBS>iN E <GBs>i/<GBs>T ~ I (<GBS>T is the geometric factor for the axis~mmetric toroi-

dal systems4)) and we have neglected the term due to the external inductive electric field, be-

cause Lj?) is the order of c. Because V~ - ec/R and Ve - ee/r where e~ and e6 are the unit

vectors in the toroidal and poloidal directions, respectively, and R and r are the maior and

52

I

~~*~t~"--~-･^~ Effects of Fast lons and an External Inductive Electric Field on the

Neoclassical Parallel Flow , Current and Rotation in General Toroidal Systems ~~ i~, ~l~

minor radii, respectively, R <~i ' V~> and r <~i ' V0> have the physical meanings as the toroidal

and poloidal rotations, respectively. The ion parallel flow due to fast ions in the Pfirsh-Schltiter regime (Eq. (18)) is larger than that in the 1/v regime (Eq. (14)) by the order of e~1.

Thus, the ion toroidal and poloidal rotations due to fast ions become large as the ion collision

frequency increases. Let us consider the two limiting cases in the non-axisymmetric toroidal

system. In the non-axisymmetric toroidal system, the helical ripples due to non-axisymmetry make the normalized geometric factor <GBS>*N Signiflcantly small4) and the total toroidal cur-

rent 21TI inside a flux surface is very small compared with the total poloidal current 27TJ out-

side a flux surface. In such the case, the toroidal rotation due to fast ions is predominant and

Eq. (21) reduces to the following equation:

<~i'~~> - RZf )( _:) I Z n <Bu//f>. (23) v3 mi :: L1 + (1 + (1) f f N 33 n* mf J u

In this case the thermodynamical contribution vanishes and the magnetic field dependence of the ion toroidal rotation can be seen only through the term of N(313) as follows:

<~i '~~>0c-N(1) oc I (24) 33 ~il '

where ~n in the Pfirsh-Schluter regime is given in Eq. (A. 12) in Appendix 3. Thus, the ion to-"

roidal rotation is inversely proportional to the ion viscosity coefiicient ~n in this limit. On

the other hand, the poloidal rotation of ions reduces to

<~i '~6> - I dPi + dip~ _ ~rZf ) (~~) J Z n <Bu!!f>. (25) m v3 ( [1 + (1 + ' (1) f f N r 33 n * ~ en. dc dip ) J m' uf

In this case, the thermodynamical contribution rernains. In CHSl) central helical ripples along the magnetic axis can vary according to the out-

ward or inward shift of the plasma If R . = 90-95 cm, where Ra*i* is the major radius of **rs

the magnetic axis in the vacuum fleld, the central helical ripples are negligibly small and they

increase sharply if R . exceeds 95 cm. In the vicinity of the region where the helical ripples "**~

are vanishingly small the system can be thought to be nearly axisymmetric and <GBS>iN C:: 1.

In such the case Eqs. (21) and (22) become (o)

<u e >c~-~( I dPi + dc _ 34 1 dTi RL )

t ¥ en* dep dip t eZ*ff dep

RZ v 3 Zn J ) (=: ) I (1) f f <Bu//f>, (26) mi _ f [1+(1+ N 33 n* mf u

+rZf v 3 Z n <~i ･~6> ::: - rLco) I dT )(=: ) I (1) f f <1~u//f>. (27) m ' [1 + (1 + ' N 34 eZ*ff dep ~ 33 n* m J u As the gradients of the ion pressure and temperature, and electrostatic potential with respect

to c, i.e., dPi/dip, dTi/dip, and dc/dip do not vanish on the magnetic axis, the thermodynam-

ical contribution to toroidal and poloidal rotations of ions should not be neglected in the

non-axisymmetric toroidal system with axisymmetry near the magnetic axis. Note that what is mentioned on the toroidal rotation is applicable to the parallel flow.

Finally, we consider the situation where electrons exist in the l/v collisionality regime

and ions exist in the Pflrsh-Schltiter regime. This situation is realized by setting ~*k >> ~ik - e.

In this case, the expressions of Eqs. (7)-(12) do not change, except for

L(o) _ /li2 _ e (28) N(1)

34 ~ ~il ' ~il ' 33

Note that the coef~cients including N(313) must be evaluated using N(313) in Eq. (28). In the case

where ions exist in the high collisionality regime (pik - e), the terms with superscript (1) be-

come the same order as the terms with superscipt (O). Thus, the ion parallel flow due to fast

ions and the external inductive field becomes the same order as that of electrons. As a result

53

1992~~ 7 ~

of it, the mornenturn gained by ions is larger than that of electrons by the order of e~2. This

result still holds when the direct interaction between fast ions and thermal ions is negligible, i.e., (v*/~~f)3 << 1.

I

4. Conclusion and Discussion

The effects of the external sources, i.e., fast ions due to the neutral beam injection and

the external inductive electric fleld, on the neoclassical flow, current, and ion toroidal rotation

are examined intensively. When both electrons and ions exist in the 1/v or banana collision-

ality regime, Iight electrons carry the current attributed to fast ions and the external induc-

tive field, while the heavy ions carry the momentum transferred from fast ions. It should be

noted that the parallel ion flow due to the fast ions is smaller than the parallel electron flow

' ~J~~ due to fast rons by the order of m./mi . As the ion collision frequency increases, however,

the situation becomes complicated, the reason of which comes from the momentum conserva-tion of the Coulomb collision and the property of viscosity coefflcients d,ecreasing sharply with

increasing collision frequency. Especially, when both electrons and ions exist in the Pfrsh-

Schluter regime, the parallel neoclassical current reduces to the usual classical current. How-

ever, the parallel ion flow due to fast ions becomes the same order as that of electrons, and

the momentum transferred from fast ions to thermal ions is larger than that to electrons by the order of mi/m. . As a result of it, the ion toroidal rotation is predominated by the paral-

lel flow due to fast ions and the contribution of the thermodynamical forces is negligible in

the nonaxisymmetric toroidal configuration with the signiflcantly small normalized geometric

factor or large helical ripples. Thus, the ion toroidal rotation is inversely proportional to the

ion viscosity coeffic_ ient. If axisymmetry is recovered near the magnetic axis by shifting the

plasma as is the case of CHS under an operation condition, the contribution of the thermo-

dynamical forces remains there.

The results which we have obtained for the contribution of the external momentum sources to the parallel flow, current, and rotation in the Pflrsh-Schluter collisionality regime

may be qualitatively applicable even to the plateau regirne if the quantities in the Pfirsh-

Schltiter regime are replaced by those in the plateau regime, because viscosity coef~icients in

the plateau regime are enough small compared with 1/v or banana regime.

Appendix I . Expressions for the Flux-Surface-Averaged Parallel Ouantities in Eq. (1)

The flux-surface-averaged parallel friction <B ' F.1> and the heat friction <B ' F.2> with thermal species are given by3)

N "b *b <~u//b> ll J [ <B ' F.1> _~ r 5 <~q//b> (A. 1) In 112 - <B ' F.2> I ~b=0 L 12"Ib 12"2b - ~ pb

where l,~,P (i, j = 1, 2, a, b O, , , N) are the fnctron coefficrents between specles a and b (a O

for electrons and a ~ O for ion species). Since the friction coefiicients are independent of both

the magnetic fleld B and the collisionality regime, the expressions in Eq. (A. 1) for parallel

friction forces in terms of the parallel flow and heat flux are the same between axisymmetric

and non-axisymmetric toroidal systerns in all collisonality regimes. The friction coefiicients l,~j~ (i, j = 1, 2, a, b = O -N) have the following relations:

li",b = Ib~ (A. 2) ji

N

~ I~b=0 (i=1 2) (A.3) b=0 *1 ' ' N which ensure the total momentum conservation of the Coulomb collision operator, i.e., b~0F.1

The flux-surface-averaged parallel friction and head friction with fast ions are given by3)

54

I

~~*~t~~~~~~o^~~ Effects of Fast lons and an External Inductive Electric Field on the

Neoclassical Parallel Flow , Current and Rotation in General Toroidal Systems ~'~~' ~I~~

<B ' Fofl> n.m I l Z{ nf = *[ L_ I -<B ' Fof2> T･･ 3 J <Bu//f> for electrons, n

v3 ~ (A. 4) n a e2~ 1 1 f ) * ( * <B ' Fafl> = m + m <B'Fofl> for ron specres il3 N nbe2b

f~ b=1 mb

where v// 3 Ic m N nbe;mf ., S v3 ffd3v 1 v3 ~ ~/~ * ~ ' -3

~ 4 m 3 ~ fv//ffd3v f b=1 n*mb T*' ~ The quantities with subscript f denote those which belong to fast ions, v* is the critical veloc-

ity at which the drag exerted by the electrons on the fast ion beam is equal to that of back-ground ions on the fast ion bea~n. Here the conditions vT. >> I ~fl >> vT. for a = I - N have

been. assumed where ~f is the beam velocity, vT* the thermal velocity of electrons an~ vT~ the

thermal velocity of a species ipns. The heat friction of thermal ions with fast ions has been

negl ected.

The flux-surface-averaged parallel viscosity and the parallel heat viscosity have the fol-

lowing general form

( P. .¥ <Bu//'> + <GBS>~ ~ e.n~ + c ) <B ' V ･ n.>] _ Lll.1 /1.2] [

･ J, L (A. 5) - 2 <Bq//~> <GBS>. Tea

-<B 'V ･e) > _ . ~ /1*2 Ila3 _ _ 5 P where l/~j (j = I - 3) are viscosity coefficients and <GBS>~ is the geometric factor, and the prime

denotes d/dV with V being the volume. The viscosity coefiicients ll.j have the geometrical

dependences of the magnetic fleld and depend on the collisionality regime. The magnitude of

them sharply deceases as the collision frequency inceases without regard to the existence of

axisymmetry. However, the gepmetric factor <GBS>. is deeply affected by whether the axi-

symmetry exists or not. In the axisymmetric toroidal system it has the com~non expression to all particle species in all collisionality regimes, which is expressed <GBS>T4). In the non-axi-

symmetric toroidal systems, however, it depends on the collisionality regime where the parti-

cle species a exists and is smaller than the axisymmetric one, i.e., <GBS>* ~ <GBS>T . The mag-

nitude of it roughly decreases as the collision frequency increases. The asymptotic forms for

<GBS>a in each collisionality regime can be seen in ref. 4.

Appendix 2. Definitions of Ouantities in Eq. (2)

The quantities in Eq. (2) are defied as follows for (a, b = O -N)

Ft :

Ff :

EA :

V

F ~~: <B ' F~1>, t 2a+1

F ~~ <B ' Fafl>, f 2*+1 EA 2a+1 ~~ eana <B ' E(A)>,

V2~+1 <B ' V ' na>,

r Pa ' ) G:G2a+1~~<G > +ip , BS a¥ eana

X : X2a+1 ~~ <Bu//a>,

ab ab Lab In ll2 ~ J L , ab ab 121 122

Loo

Lro Lt ~

LN O

F ~~ - <B ' Fa2>, t 2a+2

= B 'Faf2>6ao' F ~~ < f 2a+ 2

E ~ O, A 2a+2

V2a+2 ~ ~ <B ' V ' ea>,

T~ G2a+2 ~ <GBS>a ea '

X2a+2 2 <~ql'a> = ~ ~ Pa

Lou . ON L L11 IN L

LNI ' LNN

(A. 6)

(A. 7)

55

I

~i~pl*~A~~~~~ft ~i~68~~~~1~~ 1992~~7~I

/1 o

/1*1 /1*2 /ll ~~ J ~~ L , /1* /1*2 /l~~ (A. 8) ll

/1 N

6~o I for a = O, = O for a ~ O.

Appendix 3. Friction and Viscosity Coefficients

In this Appendix we give friction and viscosity coefricients for a sirnple plasma consisting

of electrons and one species ions. The normalized friction coefficients up to the order of e are

given by

-** -.. - 3 l =-Z*ff' l 11 12 ~ ~ ~ Z*ff' (A. 9) 13 ' I~;=- ~/~+ 4 Z ~ ~/~+ 15 vTi - ( ) =- ) , ( "' 122

*f f. 2Z*ff vT* '

where 'vT* and vTi are thermal velocities of electrons and ions, respectively, therefore v lv - Z:ff e.

The viscosity coefiicients up to the order of e are:

for the 1/v or banana collisionality regime

f t

~.1 = { ~~T L In (1 + ~f~) + c~}, f .

_ f /1.2= : { (A. 10) 5 3 ~} , 2 ~~: - ~ In(1+ ~1~)+ ~~ c

f _ _ ft f39 ~/~- 245 In(1+~f~)+ 13 c.}, /1 ~3 ~ f, 1 8 4

for the plateau collisionality regime

A ~. _ _ ~~ 13 ~.1 = ~pL 2, ~~2 =- ~pL ' (A. 11) /1*3- ~pL 2 '

for the Pflrsh-Schltiter collisionality regime

~~ 2 205 ~/~/32 + c. 5114 ft I = ~ps 267/40 + c. 903/80 + c~ 36/5 '

~. 2 331 ~/~~/32 + c~57/2 -= ( ) !l~2 - ~ps (A. 12) 267/40 + c. 903/80 + c~ 36/5 '

~. 2 1045 ~f~/32 + c.81 /1 3 = ~ps 267/40 + c~903/80 + c~36/5 '

where c*= Z*ff for elecsrons (a = e) and O for ions (a = i). The fraction of trapped particles

ft and untrapped particles f., the mean free path ~. of the particle species a, and the charac-

teristic length of the magnetic field in the plateau regime ~pL and in the Pfirsh-Schltter regime

~ps are given by

3 <~2> 1 1 - J[ l ft= I T B~.. o I - ~ B ~d~,

~"*

f=1-ft' ~ =T v . ~ ~* T~, 1 _ <DT (t/lp + /It) I _ 3 <(u'VB)2> (A. 13) ~pL <~2> ' ~ps2 ~ 2 <L:2> '

where r.. is the Braginskii collision time: T!~ = 4/(3 ~~) ･ 4lrn~ e~ In A/(m~v3T.),

56

~~*~t~"~~'~'F~^)~ Effects of Fast lons and an External Inductive Electric Field on the

Neoclassical Parallel Flow , Current and Rotation in General Toroidal Systems ~~;, ~l~~

~/~ rn'V1~ (tm + n) J. 13~~ sin [27z: (mOH + n~H)] 2 ~(~) ~ (A. 14) t/Lp + /Lt = -T ~~0, ~~0 Itm + n

B~~ is the spectrum of ~ in the Hamada coordinates, i.e., ~= ~~~~~~ cos[27c (m6H + n~H)].

The spectrum of 1~ in the Hamada coordinates is easily obtained from the Boozer coordinates as shown in ref. 4.

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T.

Y.

l

57