Proceedings of ITC/ISHW2007

Collisional transport of multi-ion-species plasmas
in general non-symmetric toroidal configurations
S.Nishimura, H.Sugama, Y.Nakamura aaNational Institute for Fusion Science, Oroshi-cho 322-6, Toki, 509-5292, Japan
Graduate School of Energy Science, Kyoto University, Gokasho, Uji, 611-0011, JapanA previous formulation of the neoclassical transport in helical/stellarator devices based on the moment equation
approach is extended to allow the poloidal and toroidal variation of the densities and temperatures of na/na, Ta/Ta< B/B. Since the transport of impurities with high collisionalities (so-called Pfirsch-Schlüter diffusions which areseparated in our previous works) is determined by the local parallel force balance before the flux-surface averaging
including these variations na, Ta an important purpose of this extension is to study radial profiles of the impurity
density under the self-consistent ambipolar radial electric field Er in plasmas containing electrons and main ionscorresponding to the collisionless (1/, , or banana) regimes or the plateau regime, and impurity ions in thePfirsch-Schlüter regime. The Legendre-Laguerre expansion with orders of l=0,1 and j=0,1,2 is used for this local
momentum balance to include the energy scattering collisions and the effect of the radial electric field in
non-symmetric toroidal plasmas.
Keywords: neoclassical transport, multi-ion-species, impurity transport, moment equation approach,drift kinetic equation, non-symmetric toroidal plasmas

1. Introduction
Recently, various types of high-density operations arestudied in helical/stellarator devices [1,2] and theneoclassical processes on the impurity transport in thesehigh-density conditions also attract much attention [1,3].
Although codes based on the so-called moment equation
approach [4] are used for this kind of studies inaxisymmetrictokamaks,codes
tohandlemulti-ion-species plasmas in the helical/stellarator
devices are still under developments. Even though amethod to obtain the neoclassical transport matrix ingeneralmulti-ion-species
plasmasingeneralnon-symmetric toroidal configurations had been shown[5], it handled only a part relating to the flux surface
averaged
part
of
the
momentumbalanceB•• aeanaBE//=BF//a1. We had not shown any
method to handle the poloidally and toroidally varying
part (local structure) of the momentum balance and flows
nau//a, q//a before the flux-surface averaging. In thismomentum balance, the densities na and temperatures Taof each particle species (a = e, H+, D +, T+, He+, He2+, …)
are not flux surface quantities. Although the poloidal andtoroidal variations of the potential are small because of aconstraint by the total energy conservation to minimize
the Joule loss J•E, the density and temperature
perturbations na, Ta are not limited by this constraint.
Only the total pressure pa=naTa can be the flux surface
quantity in the MHD equilibrium. In a present study, we
extend the stellarator moment equation approach [5] toauthor’s e-mail: nishimura.shin@lhd.nifs.ac.jp

allow the poloidal and toroidal variation of the densities
and temperatures of na/na, Ta/Ta < dB/B. The theory forthe “neoclassical transport of impurities”[6] in general
toroidal plasmas including full parts of collisional
diffusions and neoclassical parallel flows is completed bythis extension. Since the parallel force balance including
the variation na, Ta determines the transport of
impurities with highcollisionalities (so-called
Pfirsch-Schlüter diffusions which are separated in ourprevious works), an important purpose of this extension is
to study radial profiles of the impurity density(accumulation or shielded “hole”) under the
self-consistent ambipolar radial electric field Er in
plasmas containing electrons and main ionscorresponding to the collisionless (1/, , or banana)regimes or the plateau regime, and impurity ions in thePfirsch-Schlüter(P-S) regime.

2. Moment Equations
Although the moment equations for the poloidally andtoroidally varying part must be derived from the
Vlasov-Fokker-Planck equation [7], we will report detailsof this derivation in separated articles. Also on the flux
surface averaged part of momentum balance, which isbasically unchanged from Ref.[5] even in the extension of
the theory except replacing u//a by nau//a/na, the details
will not be described here. We show here only the
essential part of the results on the poloidally andtoroidally varying part. By taking the Legendre-LaguerreProceedings of ITC/ISHW2007
moments of the orders of l=0,1 and j=0,1,2 of a part offollowing the tokamak P-S transport theory [6]. An
the linearized drift kinetic equation (LDKE) after
important purpose of this approximation is to include theseparating Xa as a source of the flux surface averagedenergy scattering collision effects [6] and also the EBpart of the momentum balance [5], we obtain following
drift effect in Eq.(1), which is peculiar to non-symmetricequations. Hereafter, notations in Ref.[5] are followed.
toroidal configurations. These effects were not included
Furthermore we concentrate in this paper only in the local
previous formulations for the multi-ion-species plasmas
momentum balance in which the flux surface averaged
in Refs.[8,9] based on the 13M approximation. Eq.(1)components of flows and forces are subtracted. The
includes u//(rigid)b• corresponding to the parallel velocityof the moving frame in which the adiabatic invariant
particle and momentum conservation laws, which are theμ  ma v2 /2B
andparallelparticlevelocity
Legendre-Laguerre moments of the LDKE with orders of
1/2v = ±v (1  μ B/w ) , where w  ma v 2 /2 , are defined
l=0 and j=0,1,2, are given by[10,11]. In general non-symmetric toroidal plasmas this na u a / na
1 0 0velocity is given bypa  0


0

2q a

0 i 
5 pa

35
 na u a2 / na
8

5
2

0

u(rigid)  Ta (j=0)

.
n

1
0
0
na a1



s  B3(j=1)+ na  1
0  cEs+ u(rigid)b i  Ta1


2

B2
 Ta (j=2)

3 15
na1

1 
 na
2 8
0
0

ab
   0 e11b ab 0 e11

(j=0)
 nb1
Tb / nb

(j=1)
0 
Tb1
ab  n (j=2) T / n
e22  b1bb0

H 2(symmetric) =

1
5
2
0

 Ta (j=0) 
na1

0
 na
5
(j=1)
bi  Ta1
2
 Ta (j=2)
35
na1

 na
8


ab
 l11
 Fa1 ab=  Fa2    l21bab l31
 Fa3

ab
l12
ab
l22
ab
l32

. (2)



ab   nbu b / nb
l13
2q b
ab 
l23

5 pb
ab 
l33

 nbu b2 / nbab
In Eqs.(1)-(2), the energy scattering coefficients ei jaband the friction coefficients li j are calculated by the
ij
ij
ij
ij
coefficients M ab , N ab , Pab and Qab which are listed inEqs.(4.8)-(4.17),(5.21),(5.22) and (6.6)-(6.12) in Ref.[6].
In contrast to the flux surface averaged part of the
momentum balance in Ref [5] using the 13M
approximation, we included the Laguerre order of j=2

author’s e-mail: nishimura.shin@lhd.nifs.ac.jp(3)

' L   ' N

In
these
cases,
H2,( u(rigid)b +cEs sB/B2) • become(1)
The parallel force balances, which are theLegendre-Laguerre moments of the LDKE with orders of
l=0 and j=0,1,2, are given by.

Here, H2 is a constant on the flux surface used in Ref.[12].
This term vanish in symmetric configurations as follows
and thus the present theory automatically includes therigid rotation of the symmetric plasmas [4,6]. In
symmetric configurations, there are only Fourier modes(m,n) of B and distribution functions satisfying
' m +  ' n
' L +  ' N .
(4)= const 
' m   ' n Ta (  pa /s + ea na   /s )c15= s  Bi 2 
pa  Ta /s
ea
2
B 

0



1

na  00
BcEs  ' B   ' B
V'
+
H2
2
2  ' '4 2
B

' L +  ' N,
' L   ' Nu(rigid)u(rigid) =,and

BcEs B L + B N
B2  ' L   ' N

B L + B N

 
'
+'   
' L   ' N
cEs
s  B i =u(rigid)b + cEs
2 
2

 
B
g
B 

 B
 B

  

(5)This operator vanishes for the “symmetric” Fouriercomponents of cos(m n) and sin(m n) satisfying
Eq.(4).
In Eq.(2), we do not include the poloidally andtoroidally varying part of the parallel electric field
determined by the charge neutrality [13]. A reason of it isthat we cannot forbid an existence of l=0, j3components in the electron distribution functions and in
the electron force balance since we assume here caseswith sufficiently high electron temperatures [1-3] giving
long mean free paths of the electrons (eev Te>>Lc) even if
the collisionality of the ions including the impurities may
correspond to the P-S regime. When we allow theexistence of l=0, j3 components in the electron
distribution, the problem described by Eqs.(1)-(3) is not
closed. In these cases with sufficiently electron
temperatures, however, we can close this problem by the
charge neutrality without the parallel electric field insteadL11HH-(L11HH+L12HH)-L11HN=-L11NH-(L11HN+L12HN)
L11He-(L11He+L12He)
P-S_diffusion(without E )r
L11NH+L12NH
L11NN
L11NN+L12NN
L11Ne-(L11Ne+L12Ne)-L11Ne
L11Ne+L12Ne

PS ab

e

(L ) /n [m/s/(keV/m) ]Proceedings of ITC/ISHW2007
of including the higher order Laguerre terms. Although
this procedure for the moment closure will be reported ina separated article, the conclusion of it is only replacing
100
ea
ae
the friction coefficients l3 j = l j 3 in Eq.(2) by other
ea
ae
10-1
newly defined friction coefficients 3 j =  j 3 . In Eq.(1),the resulting P-S current under this charge neutrality,
10-2
which should be a function only of total pressure gradient,
is independent of the density and temperature
10-3
perturbations na, Ta of individual species. By the
10-4
momentum conservation of the friction forces, the totalpressure perturbation given by Eq.(2) satisfies
10-5(j =0)(j =1) 
PS

(the
total
=
p
=
T
n
+
n
T
0
 a1   a a1
a a1

a
10-6
pressureisaa flux surface quantity).
0.001

0.01

0.1
20

and thermodynamic forces corresponding to radial
gradients of the pressures and temperatures Xa1, Xa2 also
are defined in Eq.(10) in Ref.[5]. The Onsager symmetry
PS ab(LPS )ba
j i = (L )i j is satisfied by the symmetric relations
ba
ab
ba
ab
of the collision coefficients e j i = ei j and l j i = li j [6],PS
and the intrinsic ambipolar condition of a also is
satisfied by the momentum conservation of the friction.
Because of the stellarator symmetry B(, )=B(, ),these coefficients are even functions of the radial electricfield strength Er. Figure.1 shows an example of theresults. In this example, following Refs.[5,12], themagnetic field assumed there is that with B=B0[1tcosB +h cos(LB N B)], L=2, N=10, B0=1T,  '=0.15Tm, '=0.4Tm, B =0, and B =4Tm. The contained ion
assumed here is a mixture of protons (H+) and fully
ionized neon (Ne10+), which is used for the charge
exchange spectroscopic measurements and the impurity
transport studies in the Large Helical Device (LHD) [3],with an ion density ratio corresponding to Zeff=5.74, andthe assumed temperatures are Te= Ti=1keV. With theseassumptions, a dependence of the diffusion coefficientson the density in a range of ne51020cm3 (up to the“SDC” [2] density regime) is investigated here. The mean
free path of electron-electron collision is v Teee=28.3m
corresponding to the plateau regime even at203
ne=510 cm .

author’s e-mail: nishimura.shin@lhd.nifs.ac.jp

100

PS ab

e

(L ) /n [m/s/(keV/m) ]

ab 

 
(LPS )ab (LPS )12  aPS  Xb1  . (6)c U F a1=    PS 11
 PSab
PS ab  Xeb  qa /Taa U F a2
(L )21 (L )22  b2
 ( , ) is defined in Appendix A in Ref.[5],The function U

10

ne [10 m ]3. Numerical Examples
The problem described by Eqs.(1)-(3) including the
ea
ae
ea
ae
replacement of l3 j = l j 3 by 3 j =  j 3 can be solved by
a Fourier expansion method in the Boozer coordinates. In
this section, we show a numerical example of the solution.
By solving this problem, we can obtain the P-S diffusion
coefficients defined by [5, 9],

1
3

10-1

L11HH
-(L11HH+L12HH)-L11HN=-L11NH-(L11HN+L12HN)
L11He-(L11He+L12He) P-S_diffusion(with E =5kV/m)r
L11NH+L12NH
L11NN
L11NN+L12NN
L11Ne-(L11Ne+L12Ne)-L11Ne
L11Ne+L12Ne

10-2
10-3
10-4
10-5
10-6
0.001

0.01

0.1
20

1

10

3

ne [10 m ]Fig.1 The impurity (Ne10+) diffusion coefficients inab(a) cases with energy scattering collision effects ei j
and without the radial electric filed (Er=0), (b) caseswith both of the energy scattering collisions and afinite radial electric field of Er =5kV/m.
The collisionless limit of ne=1017m3 in (a) coincides
with the 13M approximation [8,9] without both effects.
The particle species are denoted by e (a,b=e), H(a,b=H+), and N (a,b=Ne10+) in these figures.

Since Eq.(6) includes full non-diagonal coupling terms
between particles species, there are 21 P-S diffusion
coefficients even in this simple 2-ion-species model. We
show here only coefficients relating to ion particle
diffusions (i.e., a=H+, Ne10+ and b=e, H+, Ne10+), since a
main application area of the P-S transport is the impurity
transport studies in high-density operations. It is wellknown that neoclassical theory without the temperature
gradient terms predicts a “pessimistic” impurity
accumulation but the temperature gradient terms prevent
it. Although the final goal of the impurity transport
studies is determining the steady-state impurity density
profile including not only aPS but also thebanana-plateau and ripple diffusion fluxes abn and theProceedings of ITC/ISHW2007
classical diffusion fluxes a , such a calculation isrotation”, it also should be noted that the basic idea of
this theory is applicable to tokamaks with the rotation
complicated. We show here only comparison of thePS abdamping due to the symmetry-breaking by MHD
density gradient terms (L )11 and the temperatureabab+ (LPS )12
in aPS as thegradient terms (LPS )11
activities, and so on [15]. It also should be noted that thecomponent tests. In Fig.1(a) without the radial electric
plasma rotation assumed in Eq.(3), resulting density
field (Er=0), we can see the effects of the energy
perturbations in Eqs.(1)-(2), and the assumed
scattering collisions by deviations from pure ne scaling
electro-static potential being a flux surface quantity are
given by the 13M-moment approximation in Refs.[8,9]. It
consistent with previous experimental results.[16-17]
enlarges the negative value of (L11HN+L12HN) andAcknowledgements
positive value of (L11NH+L12NH). It also reduces the
The authors thank Drs. Noriyoshi Nakajima, Kimitaka
positive value of L11HH at ne>1020m3. These arefavorable tendencies in viewpoint of the impurity control.
Itoh, Katsumi Ida, Mikirou Yoshinuma, Dave Mikkelsen,
We can see also an increase and an invert of ion
Mike Zarnstorff, Don Spong, Steve Hirshman, Masa Ono,
diffusions driven by electron temperature gradient in thisMasayoshi Taguchi, and Rainer Burhenn for suggestions
high-density range, which indicate an importance of theand discussions relating to this topic. The National
electron temperature controls in for the impurity controls.
Institute for Fusion Science (NIFS) coordinated research
The finite radial electric field effects with Er=5keV/m on
programs NIFS07KNXN103 and NIFS06KUHL007 also
these diffusion coefficients are only changes of order of
support this work.
unity as in Fig.1 (b) since Eqs.(1)-(3) make the P-S
current independent of Er. Because of this characteristic
References
the relative flow velocities between particle species[1] R.Burhenn, et al., Fusion Sci.Technol. 46, 115 (2004)
u//au//b determining the friction forces F//a1 areinsensitive even when the absolute values of the flow[2] N.Ohyabu, T.Marisaki, S.Masuzaki, et al.,velocities of individual species are largely changed by the
Phys.Rev.Lett.97, 055002 (2006)[3] Y.Nakamura, Y.Takeiri, R.Kumazawa, et al,EB term in Eq.(1). Nevertheless, there are
non-negligible effects in view point of impurity density
Nucl.Fusion 43, 219 (2003)
profiles determined by balances of temperature gradient[4] W.A.Houlberg, K.C.Shaing, S.P.Hirshman, andterms and the density gradient terms. The aforementioned
M.C.Zarnstorff, Phys.Plasmas 4, 3230 (1997)
increases of (L11HN+L12HN) and (L11NH+L12NH) are[5] H.Sugama and S.Nishimura, Phys.Plasmas 9, 4637 (2002)
enhanced by a radial electric field effect at ne~10^17m-3.
[6] S.P.Hirshman and D.J.Sigmar, Nucl.Fusion 21,1079 (1981)
The radial electric field effect at ne>1020m3 is[7] R.D.Hazeltine and J.D.Meiss, Plasma Confinement
complicated. These coefficients are reduced by the radial(Addison-Wesley, CA, 1992), p.196
electric field at ne>1020m3 and thus the ion temperature[8] M.Taguchi, Phys.Fluids B 4, 3638 (1992)
gradient is not effective for the impurity control in this[9] H.Sugama and W.Horton, Phys.Plasmas 3, 304 (1996)high density limit with finite radial electric fields. In this[10] R.D.Hazeltine and A.A.Ware, Plasma Phys. 20, 673(1978);condition, controls of density and temperature profiles of
R.D.Hazeltine and F.L.Hinton, Phys.Plasmas 12, 102506
the electron will be more important.
(2005)
cl

4. Concluding Remarks
By adding this poloidally and toroidally varying partof the momentum equations to a previously formulation
handling the flux-surface averaged part [5], thedevelopment of the “non-symmetric version of
NCLASS” [12] based on the concept of “stellarator
moment equation approach” toward a future integrated
simulation system [14] is almost completed in viewpointof the “basic framework”. (Although there are still manyremaining technical problems on the viscosity
coefficients in the flux-surface averaged part, this topic is
discussed in another presentation in this conference
P2-017). In non-symmetric configurations, a dependencePSPSof the P-S diffusions a , qa and the accompanied na,
Ta on the Er is predicted. Since a cause of thisdependence on the Er is the viscous damping of the “rigid
author’s e-mail: nishimura.shin@lhd.nifs.ac.jp[11] F.L.Hinton and S.K.Wong, Phys.Fluids 28, 3082 (1985);
H.Sugama and W.Horton, Phys.Plasmas 4, 2215 (1997)[12] S.Nishimura, H.Sugama, and Y.Nakamura,
Fusion Sci.Technol.51, 61 (2007)[13] R.D.Hazeltine and F.L.Hinton, Phys.Fluids 16,1883 (1973)[14] Y.Nakamura, M.Yokoyama, N.Nakajima, et al.,Fusion Sci.Technol. 50, 457 (2006)[15] W.Zhu, S.A.Sabbagh, R.E.Bell, et al.,Phys.Rev.Lett.96, 22002 (2006)[16] K.Ida, T.Minami, et al., Phys.Rev.Lett. 86, 3040 (2001);
M.Yoshinuma, K.Ida, et al., in 16th International Stellarator
Workshop (Toki, Japan, Oct.15-19, 2007)[17] S.Nishimura, K.Ida, M.Osakabe, et al., Phys.Plasmas 7,
437 (2000)[18] K.Ida, A.Fujisawa, H.Iguchi, et al.,Phys.Plasmas 8,1(2001)