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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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l
57