1. science
2. physics

PHYSICS OF PLASMAS
VOLUME 9, NUMBER 11
NOVEMBER 2002
How to calculate the neoclassical viscosity, diffusion, and current
coefficients in general toroidal plasmas
H. Sugamaa)
National Institute for Fusion Science, Toki, Gifu 509-5292, Japan
and Graduate University for Advanced Studies, Toki, Gifu 509-5292, Japan
S. Nishimura
National Institute for Fusion Science, Toki, Gifu 509-5292, Japan
共Received 4 April 2002; accepted 14 August 2002兲
A novel method to obtain the full neoclassical transport matrix for general toroidal plasmas by using
the solution of the linearized drift kinetic equation with the pitch-angle-scattering collision operator
is shown. In this method, the neoclassical coefficients for both poloidal and toroidal viscosities in
toroidal helical systems can be obtained, and the neoclassical transport coefficients for the radial
particle and heat fluxes and the bootstrap current with the nondiagonal coupling between
unlike-species particles are derived from combining the viscosity-flow relations, the friction-flow
relations, and the parallel momentum balance equations. Since the collisional momentum
conservation is properly retained, the well-known intrinsic ambipolar condition of the neoclassical
particle fluxes in symmetric systems is recovered. Thus, these resultant neoclassical diffusion and
viscosity coefficients are applicable to evaluating accurately how the neoclassical transport in
quasi-symmetric toroidal systems deviates from that in exactly symmetric systems. © 2002
American Institute of Physics. 关DOI: 10.1063/1.1512917兴
I. INTRODUCTION
sical transport in helical systems often employs numerical
methods.15–23 The Drift Kinetic Equation Solver 共DKES兲19,20
is one of powerful numerical codes to directly solve the drift
kinetic equation. However, we should note that, even in such
numerical calculations, approximated collision operators
such as the pitch-angle-scattering 共or Lorentz兲 collision
model are generally used instead of the full Landau collision
term.24 By using this collision model, perturbed distribution
functions of unlike species and of different kinetic energies
can be solved independently, and therefore the neoclassical
transport coefficients can quickly be calculated. However,
since such simple collision models neglect the field particle
collision part and break the collisional momentum conservation, the resultant transport coefficients neither contain the
nondiagonal part connecting fluxes and forces of unlike species, nor recover the well-known intrinsic ambipolarity of the
radial particle fluxes in the symmetric limit.1–3 These errors
seem to be a serious problem, especially when using the
numerical results to show how the neoclassical transport in
designed quasi-symmetric configurations differ from that in
exactly symmetric systems. In the present work, it is shown
how to obtain the neoclassical transport coefficients in general toroidal systems including the coupling effects between
unlike-species particles as well as the collisional momentum
conservation.
Here, we follow the basic idea of the moment method by
Hirshman and Sigmar2 that, in order to derive the neoclassical transport coefficients, the fluid momentum balance equations and the friction-flow relations, in which the collisional
momentum conservation is already taken into account, are
used together with the viscosity-flow relations obtained from
the solution of the drift kinetic equation. Since the test particle portion of the collision operator dominates over the field
Neoclassical transport theory1–3 describes diffusion processes caused by binary Coulomb collisions between charged
particles in magnetically confined plasmas. In most fusion
plasma experiments, observed particle and heat fluxes across
magnetic surfaces are dominated not by neoclassical transport but by turbulent or anomalous transport,4 although the
neoclassical transport theory is still useful for predicting
transport fluxes tangential to magnetic surfaces such as poloidal and toroidal flows and bootstrap currents. Especially
for nonaxisymmetric systems, neoclassical analyses are important because neoclassical transport fluxes due to particles
trapped in helical ripples5– 8 are expected to be significantly
large for high temperature and play a key role in determining
the radial electric field under the ambipolar-diffusion
condition.9 Recently, quasi-symmetric toroidal systems such
as quasi-axisymmetric systems are attracting much attention
as an advanced concept of helical devices, in which the neoclassical ripple transport and the neoclassical viscosity
against flows in the direction of symmetry are nearly suppressed by optimizing the helical configuration so as to make
the magnetic field strength independent of a certain symmetry coordinate.10–12 Thus, there are many demands for accurate and fast calculation of neoclassical quantities including
the particle and heat diffusivities, the bootstrap-current coefficients, and the viscosity coefficients for the poloidal and
toroidal flows.
The neoclassical transport coefficients are obtained from
solution of the drift kinetic equation.13,14 Because of complexity of the magnetic geometry, calculation of the neoclasa兲
Electronic mail: [email protected]
1070-664X/2002/9(11)/4637/17/$19.00
4637
© 2002 American Institute of Physics
Downloaded 03 Mar 2009 to 133.75.139.172. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
4638
Phys. Plasmas, Vol. 9, No. 11, November 2002
particle portion for the l⫽2 spherical harmonic perturbations
of the distribution functions,2 it is more accurate to use the
solution of the drift kinetic equation with the pitch-anglescattering collision model for derivation of the neoclassical
viscosity coefficients than for direct calculation of the neoclassical particle and thermal diffusivities, which are significantly influenced by neglect of the field particle portion. In
tokamaks, the neoclassical toroidal viscosity vanishes due to
the axisymmetry, and analytical expressions of the viscosityflow relations are obtained for any collisionality in the
Pfirsch–Schlu¨ter, plateau, and banana regimes.2,3 Analytical
calculations of the the parallel viscosity coefficient in finiteaspect-ratio tokamaks are shown to be in good agreement
with numerical results.25,26 In general toroidal systems with
no symmetry, we need to calculate viscosities in both poloidal and toroidal directions, and these viscosity coefficients
are analytically derived for the Pfirsch–Schlu¨ter and plateau
regimes.27,28 However, for the banana regime, analytical formulas are given only for the parallel viscosities.5,29–31 In
order to accurately calculate both poloidal and toroidal viscosity coefficients in toroidal helical systems for lowcollisionality regimes, we need to make use of numerical
solution of the drift kinetic solution as effectively as possible, and the present paper shows how to do that.
Taguchi also showed another method to calculate the
neoclassical transport coefficients in nonaxisymmetric multispecies plasmas.32 He ingeniously used a momentum conserving collision operator and its self-adjoint property to derive the particle and heat diffusivities and the bootstrap
current coefficient. In addition to these transport coefficients,
our method gives a useful recipe to obtain the neoclassical
viscosity coefficients, which play an important role in determining plasma rotation profiles. Since our work follows a
line of the moment method, it is more transparently connected or applicable to past theoretical studies of neoclassical
transport in nonaxisymmetric systems5,27–31,33 which are also
based on the moment method. Furthermore, in the present
study, the validity of our procedures is satisfactorily verified
by numerical examples, in which our results are compared
with analytical formulas on the parallel viscosity, the ripple
transport coefficient, and the geometrical factor of the bootstrap current in various collision frequency regimes. In this
sense, our work is a generalization of previous comparative
studies between numerical and analytical evaluations of neoclassical coefficients of viscosities and other fluxes in
tokamaks25,26 to the case of nonaxisymmetric systems.
The rest of this work is organized as follows. In Sec. II,
we derive the linearized drift kinetic equation for the distribution functions with l⫽1 spherical harmonic part subtracted, based on which two types of conjugate pairs of neoclassical fluxes and forces are specified. When we take the
parallel flows and the radial gradients as thermodynamic
forces, the parallel viscosities and the radial transport are
regarded as fluxes conjugate to those forces. We can also
consider the poloidal and toroidal viscosities to be fluxes
conjugate to the poloidal and toroidal flows as forces. The
inner product of these fluxes and forces represents the entropy production rate associated with the neoclassical transport processes.33,34 In Sec. III, it is shown that the conjugate
H. Sugama and S. Nishimura
pairs of fluxes and forces defined in Sec. II are related to
each other by the Onsager-symmetric matrices.35 The poloidal and toroidal viscosity coefficients are included as elements in one of the matrices. We find how to calculate these
matrices by using their relations to the monoenergetic diffusion tensor obtained as an output of commonly used numerical codes such as the DKES. Once these Onsager-symmetric
matrices are derived, all neoclassical transport coefficients
for the radial particle and heat fluxes and the bootstrap current are immediately obtained. In Sec. IV, numerical examples of these procedures are shown and compared with
several analytical predictions. Conclusions are given in Sec.
V. For readers’ convenience, useful formulas and relations
for the Boozer36 and Hamada37 coordinates, the poloidal and
toroidal viscosity coefficients, and other neoclassical transport coefficients are written in Appendices A, B, and C, respectively. Also, the case of symmetric systems is described
in Appendix D. Finally, Appendix E shows how to treat effects of the E⫻B drift on the neoclassical transport coefficients.
II. CONJUGATE PAIRS OF NEOCLASSICAL FLUXES
AND FORCES
In general toroidal configurations, the magnetic field is
written in terms of the flux coordinates (s, ␪ , ␨ ) as
B⫽ ␺ ⬘ ⵜs⫻ⵜ ␪ ⫹ ␹ ⬘ ⵜ ␨ ⫻ⵜs⫽B s ⵜs⫹B ␪ ⵜ ␪ ⫹B ␨ ⵜ ␨ ,
共1兲
where ␪ and ␨ represent the poloidal and toroidal angles,
respectively, s is an arbitrary label of a flux surface. The
poloidal and toroidal fluxes are given by 2 ␲␹ (s)
⫽(2 ␲ ) ⫺1 兰 V(s) d 3 xB•ⵜ ␪ and 2 ␲ ␺ (s)⫽(2 ␲ ) ⫺1 兰 V(s) d 3 xB
•ⵜ ␨ , respectively, where V(s) is the volume enclosed by the
flux surface with the label s. The derivative with respect to s
is denoted by ⬘⫽d/ds so that ␺ ⬘ ⫽d ␺ /ds and ␹ ⬘ ⫽d ␹ /ds.
The covariant radial, poloidal, and toroidal components
of the magnetic field B are written as B s ⬅B• ⳵ x/ ⳵ s
⬅ 冑gB•(ⵜ ␪ ⫻ⵜ ␨ ), B ␪ ⬅B• ⳵ x/ ⳵ ␪ ⬅ 冑gB•(ⵜ ␨ ⫻ⵜs), and
respectively,
where
B ␨ ⬅B• ⳵ x/ ⳵␨ ⬅ 冑gB•(ⵜs⫻ⵜ ␪ ),
冑g⬅ 关 ⵜs•(ⵜ ␪ ⫻ⵜ ␨ ) 兴 ⫺1 represents the Jacobian for the coordinates (s, ␪ , ␨ ). Here, we may regard (s, ␪ , ␨ ) as either
Boozer,36 Hamada37 coordinates, or arbitrary other flux coordinates. Useful formulas for the Boozer and Hamada coordinates are written in Appendix A, where it is also shown that
the symmetry condition for the magnetic field strength in the
Boozer coordinates is equivalent to that in the Hamada coordinates.
The distribution function for the particle species a with
the mass m a and the charge e a is written as
冋
f a ⫽ f aM 1⫹
ea
Ta
冕 冉
l
dl
B2
BE 储 ⫺ 2 具 BE 储 典
B
具B 典
冊册
⫹ f a1 ,
共2兲
where the local Maxwellian distribution function is repre⫺3
exp(⫺x2a) with the equilisented by f aM ⬅ ␲ ⫺3/2n a v Ta
brium density n a , the temperature T a , the thermal velocity
v Ta ⬅(2T a /m a ) 1/2, and the normalized velocity x a ⬅ v / v Ta .
Here, E 储 ⬅E•b (b⬅B/B: the unit vector tangential to
the magnetic field兲 is the parallel electric field, 兰 l dl denotes the integral along the magnetic field line, and
Downloaded 03 Mar 2009 to 133.75.139.172. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
Phys. Plasmas, Vol. 9, No. 11, November 2002
How to calculate the neoclassical viscosity, diffusion . . .
具•典⬅养d ␪ 养d ␨ 冑g•/V ⬘ with V ⬘ ⬅养d ␪ 养d ␨ 冑g represents the
flux surface average. The neoclassical transport is caused by
the deviation f a1 from the local Maxwellian. We should note
that the drift kinetic theory is concerned with the gyrophaseaveraged part of the distribution function and that f a1 is regarded as a gyrophase-averaged function in the present work.
The linearized drift kinetic equation is given by
V 储 f a1 ⫺C La 共 f a1 兲 ⫽⫺vda •ⵜ f aM ⫹
ea
具 BE 储 典
f , 共3兲
vB
T a 储 具 B 2 典 aM
where the operator V 储 ⬅ v 储 b•ⵜ and the guiding center drift
velocity vda ⫽(c/e a B)b⫻(m a v 2储 b•ⵜb⫹ ␮ ⵜB⫹e a ⵜ⌽) are
used, and f a1 and f aM are regarded as functions of the phasespace variables (x,E, ␮ ) 共x: the particle’s position, E
⬅ 12 m a v 2 ⫹e a ⌽: The particle’s energy; ␮ ⬅m a v⬜ 2/2B: the
magnetic moment兲. Here, the linearized collision operator C La
is defined by2
C La 共 f a1 兲 ⫽
兺b 关 C ab共 f a1 , f bM 兲 ⫹C ab共 f aM , f b1 兲兴 ,
共4兲
where C ab represents the Landau collision operator for collisions between the species a and b.
Hereafter, we use (x, v , ␰ ) ( ␰ ⬅ v 储 / v ) as the phase-space
variables instead of (x,E, ␮ ). Then, the collisionless orbit
operator V 储 is represented by
⳵
1
V 储 ⫽ v ␰ b•ⵜ⫺ v共 1⫺ ␰ 2 兲共 b•ⵜ ln B 兲 ,
2
⳵␰
⭓2. Integrating Eq. 共3兲 multiplied by 1 and 21 m a v 2 over the
velocity space, we obtain the incompressibility conditions
共8兲
ⵜ•ua ⫽ⵜ•qa ⫽0,
where ua ⫽u 储 a b⫹u⬜a and qa ⫽q 储 a b⫹q⬜a with the diamagnetic perpendicular flows
u⬜a ⫽
⳵⌽
⳵Ta
1 ⳵pa
⫺e a
, X a2 ⬅⫺
,
共10兲
na ⳵s
⳵s
⳵s
respectively, where the pressure p a ⬅n a T a , the temperature
T a , and the electrostatic potential ⌽ are flux surface functions independent of ␪ and ␨. 共Exactly speaking, ⵜ•qa ⫽0 is
valid to the lowest order of the small mass ratio m e /m i
Ⰶ1.) Integrating the incompressibility conditions in Eq. 共8兲
gives the local parallel flows as
X a1 ⬅⫺
u 储a⫽
B•ⵜ
兺 F (l)共 x, v , ␰ 兲 ,
2l⫹1
2
冕
1
⫺1
d ␩ P l 共 ␩ 兲 F 共 x, v , ␩ 兲 .
3
2
冕
1
⫺1
冉 冊
˜
U
1
⫽B⫻ⵜs•ⵜ 2 ,
B
B
˜ 典 ⫽0.
具 BU
共12兲
˜ is associated with the Pfirsch–
As shown later in Eq. 共20兲, U
Schlu¨ter fluxes and its specific expressions are written in
Eqs. 共A4兲 and 共A8兲.
Now, let us define g a by
(l⫽1)
.
g a ⫽ f a1 ⫺ f a1
共6兲
(l⫽1)
The l⫽1 Legendre component f a1
of the distribution
function f a1 is associated with the parallel flows and is ex(x 2a ) 关 L (3/2)
(x 2a )
panded by the Laguerre polynomials L (3/2)
j
0
5
(x 2a )⫽ 2 ⫺x 2a ,... 兴 as
⫽1,L (3/2)
1
(l⫽1)
f a1
⬅共 v储 /v 兲
冉冊
共11兲
共13兲
The neoclassical viscosities which we are concerned with are
derived from the l⫽2 component included in g a . Substituting Eq. 共13兲 into Eq. 共3兲, we obtain
l⫽0
F (l) 共 x, v , ␰ 兲 ⫽ P l 共 ␰ 兲
cX a1
具 u 储aB 典
˜U ,
B⫹
2
ea
具B 典
2 具 q 储aB 典
cX a2
2
˜U ,
q 储a⫽
B⫹
2
5pa
5 pa 具B 典
ea
˜ is given as a solution of
where U
⬁
F 共 x, v , ␰ 兲 ⫽
cX a1
ⵜs⫻b,
e aB
q⬜a 5 cX a2
ⵜs⫻b.
共9兲
⫽
pa
2 e aB
Here, the thermodynamic forces X a1 and X a2 are defined by
共5兲
where the second term in the right-hand side is related to the
magnetic mirror force. The E⫻B and magnetic drifts are not
included in V 储 . A more general case including the E⫻B drift
operator is treated in Appendix E. Let us consider the expansion of an arbitrary function F(x, v , ␰ ) by the Legendre polynomials P l ( ␰ ) 关 P 0 ( ␰ )⫽1,P 1 ( ␰ )⫽ ␰ , P 2 ( ␰ )⫽ 23 ␰ 2 ⫺ 21 ,... 兴 as
4639
⫹H (l⫽2)
,
V 储 g a ⫺C La 共 g a 兲 ⫽H (l⫽1)
a
a
where the l⫽1 and l⫽2 Legendre component terms in the
right-hand side are written as
⫽
H (l⫽1)
a
ea
具 BE 储 典
(l⫽1)
f ⫹C La 共 f a1
兲
v 储B
Ta
具 B 2 典 aM
H (l⫽2)
⫽
a
f aM
2 具 q 储aB 典 2 5
具 u 储aB 典
␴ Ua
⫹
x a⫺
2
Ta
5 p a 具 B 2典
2
具B 典
and
d 共 v 储 / v 兲共 v 储 / v 兲 f a1
冋 冉 冊 册
2
5 2 q 储a
⫽
␰ x a u 储 a ⫹ x 2a ⫺
f ⫹ f (l⫽1,j⭓2) ,
2 5 p a aM a1
v Ta
⫽
共15兲
冉 冊冎
冋 再
再 冉 冊冎册
冉 冊冎
冋 再
冉 冊冎册
再
⫹ ␴ Xa X a1 ⫹X a2 x 2a ⫺
共7兲
where the coefficients of the first and second Laguerre polynomial components are given in terms of the parallel velocity
3
u 储 a ⬅n ⫺1
a 兰 d v f a1 v 储 and the parallel heat flow q 储 a
(l⫽1,j⭓2)
3
⬅T a 兰 d v f a1 v 储 (x 2a ⫺ 52 ), respectively, and f a1
denotes
the sum of the jth Laguerre polynomial components with j
共14兲
5
2
f aM V ⬘ ␴ Pa
2
5
具 u ␪a 典 ⫹
具 q ␪ 典 x 2a ⫺
Ta 4␲2 ␹⬘
5pa a
2
⫹
␴ Ta
2
5
具 u a␨ 典 ⫹
具 q ␨a 典 x 2a ⫺
␺⬘
5pa
2
,
共16兲
Downloaded 03 Mar 2009 to 133.75.139.172. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
4640
Phys. Plasmas, Vol. 9, No. 11, November 2002
H. Sugama and S. Nishimura
respectively. In deriving Eqs. 共14兲–共16兲, Eq. 共11兲 is used and
(l⫽1,j⭓2)
in Eq. 共7兲 is neglected by employing the thirteenf a1
moment 共13M兲 approximation.3 By including the high-order
parallel flow variables corresponding to the j⫽2,3,... La(l⫽1)
guerre polynomial components of f a1
, the formulation
presented in this work can be extended straightforwardly to
the cases with the higher-order 共21M, 29M,...兲
approximations.3 In Eq. 共16兲, u ␪a ⫽ua •ⵜ ␪ (u a␨ ⫽ua •ⵜ ␨ ) and
q a␪ ⫽qa •ⵜ ␪ (q ␨a ⫽qa •ⵜ ␨ ) are contravariant poloidal 共toroidal兲 components of the flows, and ␴ Ua , ␴ Xa , ␴ Pa , and ␴ Ta
are defined by
␴ Ua ⫽⫺m a v 2 P 2 共 ␰ 兲 B•ⵜ ln B⫽⫺V 储 共 m a v ␰ B 兲 ,
␴ Xa ⫽⫺ v 2 P 2 共 ␰ 兲
⫽⫺ v 2 P 2 共 ␰ 兲
冉
冊
B
ⵜs⫻b
˜ b⫹
U
•ⵜ ln B
⍀a
B
˜兲
b•ⵜ 共 BU
,
2⍀ a
共17兲
␴ Pa ⫽⫺m a v 2 P 2 共 ␰ 兲 BP •ⵜ ln B,
␴ Ta ⫽⫺m a v P 2 共 ␰ 兲 BT •ⵜ ln B,
2
respectively, where BP ⬅ ␹ ⬘ ⵜ ␨ H ⫻ⵜs and BT ⬅ ␺ ⬘ ⵜs⫻ⵜ ␪ H
are the poloidal and toroidal magnetic fields, respectively,
represented by the Hamada coordinates (s, ␪ H , ␨ H ) 共the subscript H is added to the angle variables whenever the Hamada coordinates should be used兲 and ⍀ a ⬅e a B/(m a c) is the
gyrofrequency. Then, we find that the parallel, poloidal, and
toroidal neoclassical viscosities are written in terms of ␴ Ua ,
␴ Pa , and ␴ Ta as
具 B• 共 ⵜ• ␲a 兲 典 ⫽
具 B• 共 ⵜ•⌰a 兲 典 ⫽
冓冕
冓冕
具 BP • 共 ⵜ•⌰a 兲 典 ⫽
具 BT • 共 ⵜ• ␲a 兲 典 ⫽
具 BT • 共 ⵜ•⌰a 兲 典 ⫽
d 3 v g a ␴ Ua x 2a ⫺
冓冕
冓冕
冓冕
5
2
,
d 3 v g a ␴ Pa ,
d 3 v g a ␴ Pa
5
x 2a ⫺
2
,
共18兲
d 3 v g a ␴ Ta ,
d 3 v g a ␴ Ta x 2a ⫺
5
2
,
␲a ⬅ 兰 d 3 v m a ( v 2储 ⫺ 12 v⬜2 ) f a1 (bb⫺ 31 I)
and
⌰a
where
3
⬅ 兰 d v m a ( v 2储 ⫺ 12 v⬜2 )(x 2a ⫺ 25 ) f a1 (bb⫺ 31 I). We also note that
the neoclassical radial particle and heat fluxes are written in
terms of ␴ Xa as
⌫ a⫽
qa
⫽
Ta
⫽
冓冕
冓冕
冓冕
冔 冓冕
冉 冊冔
冉 冊冔
d 3 v g a vda •ⵜs ⫽
d 3 v g a vda •ⵜs x 2a ⫺
d
3
v g a ␴ Xa x 2a ⫺
5
2
冔
d 3 v g a ␴ Xa ⫹⌫ PS
a ,
5
2
⫹
q PS
a
Ta
,
c
˜ F 储 a1 典 ,
具U
ea
q PS
a
Ta
⫽⫺
c
˜ F 储 a2 典 ,
具U
ea
共20兲
respectively, with the parallel friction forces
F 储 a1 ⫽
F 储 a2 ⫽
冕
冕
d 3 v m a v 储 C La 共 f a1 兲 ,
冉 冊
d 3 v m a v 储 x 2a ⫺
5 L
C 共 f 兲.
2 a a1
共21兲
As shown in Refs. 29 and 33, 具 兰 d 3 v g a ␴ Xa 典 ⫽⌫ a ⫺⌫ PS
a
2
PS
bn
3
⬅⌫ bn
a and T a 具 兰 d v g a ␴ Xa (x a ⫺5/2) 典 ⫽q a ⫺q a ⬅q a can be
written as the sum of banana-plateau and nonaxisymmetric
parts. Multiplying Eq. 共14兲 by g a / f aM , integrating it in the
velocity space, taking its flux surface average, and using Eqs.
共16兲, 共18兲, and 共19兲, we can express the flux-surfaceaveraged entropy production rate33,34 S˙ a associated with g a
per unit volume by the inner product of conjugate pairs of
fluxes and forces as
T a S˙ a ⫽⫺T a
冓冕
d 3v
⫽ 具 B• 共 ⵜ• ␲a 兲 典
⫻
ga L
C 共g 兲
f aM a a
冔
2
具 u 储aB 典
⫹ 具 B• 共 ⵜ•⌰a 兲 典
2
5p a
具B 典
q bn
具 q 储aB 典
a
bn
⫹⌫
X
⫹
X
a a1
T a a2
具 B 2典
冋
V⬘
具 u a␪ 典
⫽ 2 具 BP • 共 ⵜ• ␲a 兲 典
4␲
␹⬘
d 3 v g a ␴ Ua ,
冓冕
具 BP • 共 ⵜ• ␲a 兲 典 ⫽
冔
冉 冊冔
冔
冉 冊冔
冔
冉 冊冔
⌫ PS
a ⫽⫺
共19兲
PS
¨
where ⌫ PS
a and q a are the Pfirsch–Schluter 共PS兲 radial particle and heat fluxes defined by
⫹ 具 BP • 共 ⵜ•⌰a 兲 典
2 具 q ␪a 典
具 u ␨a 典
⫹ 具 BT • 共 ⵜ• ␲a 兲 典
5pa ␹⬘
␺⬘
⫹ 具 BT • 共 ⵜ•⌰a 兲 典
2 具 q a␨ 典
.
5pa ␺⬘
册
共22兲
We find from Eq. 共22兲 that the parallel viscosities 具 B•(ⵜ
• ␲a ) 典 and 具 B•(ⵜ•⌰a ) 典 are transport fluxes conjugate to the
parallel flows 具 u 储 a B 典 / 具 B 2 典 and (2/5p a ) 具 q 储 a B 典 / 具 B 2 典 as
forces, respectively, and that the radial neoclassical fluxes
bn
⌫ bn
a and q a /T a are conjugate to the radial gradient forces
X a1 and X a2 , respectively. Also, as another choice, the poloidal viscosities 关 具 BP •(ⵜ• ␲a ) 典 , 具 BP •(ⵜ•⌰a ) 典 兴 and the toroidal viscosities 关 具 BT •(ⵜ• ␲a ) 典 , 具 BT •(ⵜ•⌰a ) 典 兴 can be regarded as transport fluxes conjugate to the poloidal flows
and the toroidal flows
关 具 u ␪a 典 / ␹ ⬘ ,(2/5p a ) 具 q ␪a 典 / ␹ ⬘ 兴
关 具 u ␨a 典 / ␺ ⬘ ,(2/5p a ) 具 q a␨ 典 / ␺ ⬘ 兴 , respectively. Now, our main
concern is how to obtain the transport matrices which connect these conjugate pairs of fluxes and forces. This is explained in the next section.
Here, we consider the l⫽1 Legendre component term
H (l⫽1)
, which also makes a significant contribution to the
a
solution g a of Eq. 共14兲 especially in the weakly collisional
⫹1
g a d ␰ ⫽0. Substituting Eq. 共7兲
regime in order to insure 兰 ⫺1
into Eq. 共15兲 and using Eq. 共11兲 and the rotational symmetry
of the collision operator C La , we can write H (l⫽1)
in the
a
following form:
Downloaded 03 Mar 2009 to 133.75.139.172. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
Phys. Plasmas, Vol. 9, No. 11, November 2002
H (l⫽1)
⫽ f aM
a
a
m a␯ D
Ta
˜ ␥a兲,
v ␰ 共 B ␣ a ⫹U
How to calculate the neoclassical viscosity, diffusion . . .
共23兲
where ␣ a and ␥ a are functions of (s, v ) and are independent
of ( ␪ , ␨ , ␰ ). We find in the next section that ␣ a is written in
terms of the parallel flows and the radial gradient forces and
that ␥ a is unnecessary for calculation of neoclassical transport coefficients.
III. RESPONSE FUNCTIONS AND TRANSPORT
COEFFICIENTS
Hereafter, as an approximation of the linearized collision
operator in Eq. 共14兲, we use the pitch-angle-scattering operator defined by
C PAS
a ⬅
where 具 • 典 denotes the flux surface average. With respect to
are found to be
this inner product, the operators V 储 and C PAS
a
antisymmetric and symmetric, respectively,
共 V 储 F,G 兲 ⫽⫺ 共 F,V 储 G 兲 ,
⳵
⳵
共 1⫺ ␰ 2 兲 ,
2 ⳵␰
⳵␰
兺b
a 2
2
3 m a T a ␯ D x a 具 B 典 ⫺ 共 ␴ Ua ,G Ua 兲
共24兲
3 冑␲ ⫺1 ⫺3
␶ ab x a H共 x b 兲 ,
4
共25兲
⫺1
3
⬅4 ␲ n b e 2a e 2b ln ⌳/(m2avTa
) (ln ⌳: The
where (3 冑␲ /4) ␶ ab
Coulomb
logarithm兲
and
H(x)⬅ 关 (2x 2 ⫺1)⌽(x)
⫹x⌽ ⬘ (x) 兴 /(2x 2 ) 关 ⌽(x)⬅2 ␲ ⫺1/2兰 x0 exp(⫺t2)dt: The error
in Eq. 共14兲 is considered to
function兴. The use of C PAS
a
be a better approximation than that in Eq. 共3兲 from the viewpoint of the momentum conservation because generally
but
we have 兰 d 3 v m a v 储 C La ( f a1 )⫽ 兰 d 3 v m a v 储 C PAS
a ( f a1 )
(l⫽1)
(l⫽1)
兰 d 3 v m a v 储 C La ( f a1 ⫺ f a1
)⫽0⫽ 兰 d 3 v m a v 储 C PAS
a ( f a1 ⫺f a1 ).
Then, the formal solution of Eq. 共14兲 with the source terms
given by Eqs. 共16兲 and 共23兲 is written as
g a⫽
冉 冊冎
冋 再
再 冉 冊冎
冕 册
f aM
2 具 q 储aB 典 2 5
具 u 储aB 典
G Ua
⫹
x a⫺
2
Ta
5p a 具 B 2 典
2
具B 典
⫹G Xa X a1 ⫹X a2
a
⫹m a ␯ D
␥a
l
5
x 2a ⫺
2
共26兲
冋 册冋 册
G Ua
␴ Ua
⫽
.
G Xa
␴ Xa
In deriving Eq. 共26兲, we have also used the relation
a
共 V 储 ⫺C PAS
a 兲共 G Ua ⫹m a v ␰ B 兲 ⫽m a ␯ D v ␰ B.
共28兲
We can easily find that the last term including ␥ a in the
right-hand side of Eq. 共26兲, which is independent of ␰, makes
no contribution to flows, viscosities, and radial transport
fluxes.
Here, we define the inner product (F,G) for arbitrary
functions F( ␪ , ␨ , ␰ ) and G( ␪ , ␨ , ␰ ) by
共 F,G 兲 ⬅
1
2
冕
1
⫺1
d ␰ 具 FG 典 ,
冉 冊冎册
共29兲
共30兲
共 ␴ Ua ,G Ua 兲
冉 冊冎
5
2
⫹ 共 ␴ Ua ,G Xa 兲
共31兲
.
Substituting again Eq. 共31兲 into Eq. 共26兲, g a is rewritten as
g a⫽
冉 冊冎
冋 再
冕 册
再 冉 冊冎
f aM
2 具 q 储aB 典 2 5
具 u 储aB 典
GUa
⫹
x a⫺
2
Ta
5 p a 具 B 2典
2
具B 典
⫹GXa X a1 ⫹X a2 x 2a ⫺
5
2
a
⫹m a ␯ D
␥a
l
˜ dl ,
U
共32兲
where GUa and GXa represent the responses of the distribution function g a to the parallel flow 具 u 储 a B 典 / 具 B 2 典 and the
radial gradient force X a1 , respectively, which are defined by
冋
3 共 ␴ Ua ,G Ua 兲
a 2
2m a T a ␯ D
x a具 B 2典
⫻ G Ua ⫹
GXa ⫽G Xa ⫹
共27兲
冋
2 具 q 储aB 典 2 5
具 u 储aB 典
⫹
x a⫺
2
5 p a 具 B 2典
2
具B 典
⫻ X a1 ⫹X a2 x 2a ⫺
冋
⫹ ␣ a 共 G Ua ⫹m a v ␰ B 兲
˜ dl ,
U
再
再
GUa ⫽ 1⫺
where G Ua and G Xa are defined as solutions of
共 V 储 ⫺C PAS
a 兲
1
␣ a⫽ 2
a
given by2
with the energy-dependent collision frequency ␯ D
␯ Da ⬅
PAS
共 C PAS
a F,G 兲 ⫽ 共 F,C a G 兲 .
Then, substituting Eq. 共26兲 into the relation ( v ␰ B,C PAS
a g a)
⫽0, which is derived from the definition of g a in Eq. 共13兲,
we can represent ␣ a by a linear combination of the parallel
flows and the radial gradient forces as
⫻
␯ Da
4641
册
⫺1
册
3 共 ␴ Ua ,G Ua 兲
m v␰B ,
a 2
2m a T a ␯ D
x a具 B 2典 a
3 共 ␴ Ua ,G Xa 兲
a 2
2m a T a ␯ D
x a具 B 2典
⫻ 共 G Ua ⫹m a v ␰ B 兲 .
冋
1⫺
3 共 ␴ Ua ,G Ua 兲
a 2
2m a T a ␯ D
x a具 B 2典
册
⫺1
共33兲
Using Eq. 共30兲 and the definitions of ␴’s, G’s, and G’s, we
can prove the following relations:
共 ␴ Ua ,G Xa 兲 ⫽ 共 ␴ Xa ,G Ua 兲 ,
共 ␴ Ua ,GXa 兲 ⫽ 共 ␴ Xa ,GUa 兲 ,
共34兲
which are associated with the Onsager symmetry33–35 of the
transport coefficients.
Substituting Eq. 共32兲 into Eqs. 共18兲 and 共19兲, we obtain
the linear relations between the conjugate pairs of
bn
2
关具B•(ⵜ• ␲a ) 典 , 具 B•(ⵜ•⌰a ) 典 ,⌫ bn
a ,q a /T a 兴 and 关 具 u 储 a B 典 /具 B 典 ,
2
(2/5 p a ) 具 q 储 a B 典 / 具 B 典 ,X a1 ,X a2 ] as
Downloaded 03 Mar 2009 to 133.75.139.172. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
4642
Phys. Plasmas, Vol. 9, No. 11, November 2002
冋 册冋
具 B• 共 ⵜ• ␲a 兲 典
具 B• 共 ⵜ•⌰a 兲 典
⌫ bn
a
bn
q a /T a
⫽
M a1
M a2
N a1
N a2
M a2
M a3
N a2
N a3
N a1
N a2
L a1
L a2
N a2
N a3
L a2
L a3
冋 册
册 冋 册冋
M a1 P P
具 BP • 共 ⵜ• ␲a 兲 典
M a2 P P
具 BP • 共 ⵜ•⌰a 兲 典
⫽
具 BT • 共 ⵜ• ␲a 兲 典
M a1 PT
具 BT • 共 ⵜ•⌰a 兲 典
M
具 u 储aB 典 / 具 B 典
2
2
具 q 储aB 典 / 具 B 2典
⫻ 5p a
.
X a1
共35兲
X a2
Here, the coefficients M a j , N a j , and L a j ( j⫽1,2,3) in the
Onsager-symmetric matrix are written in the form of the energy integral
关 M a j ,N a j ,L a j 兴 ⫽n a
冕
2
冑␲
⬁
0
dK 冑Ke
⫺K
冉 冊
5
K⫺
2
⫽
N a共 K 兲 ⫽
共36兲
⫽
冋
册
冋
册
3 共 ␴ Ua ,G Ua 兲
1
共 ␴ ,G 兲 1⫺
a
T a Ua Ua
2m a T a ␯ D
共 K 兲K 具 B 2典
1
共 ␴ ,G 兲
T a Xa Ua
1
3 共 ␴ Ua ,G Ua 兲
共 ␴ Xa ,G Ua 兲 1⫺
a
Ta
2m a T a ␯ D
共 K 兲K 具 B 2典
⫺1
,
⫺1
,
共37兲
L a共 K 兲 ⫽
⫽
1
共 ␴ ,G 兲
T a Xa Xa
1
3 共 ␴ Xa ,G Ua 兲 2
共 ␴ Xa ,G Xa 兲 ⫹
a
Ta
2m a T 2a ␯ D
共 K 兲K 具 B 2典
冋
⫻ 1⫺
3 共 ␴ Ua ,G Ua 兲
a
2m a T a ␯ D
共 K 兲K 具 B 2典
册
M a2 P P
M a1 PT
M a2 PT
M a3 P P
M a2 PT
M a3 PT
M a2 PT
M a1TT
M a2TT
M a3 PT
M a2TT
M a3TT
冤 冥
具 u ␪a 典 / ␹ ⬘
2
具 q ␪典 / ␹ ⬘
5pa a
具 u a␨ 典 / ␺ ⬘
册
共38兲
.
2
具q␨典/␺⬘
5pa a
Here, the Onsager-symmetric poloidal and toroidal viscosity
coefficients M a j P P , M a j PT , and M a jTT ( j⫽1,2,3) are also
written in the form of the energy integral
2
冑␲
冕
⬁
0
dK 冑Ke ⫺K
冉 冊
5
K⫺
2
j⫺1
⫻ 关M aPP共K兲,M aPT共K兲,M aTT共K兲兴,
where M a (K), N a (K), and L a (K) represent contributions of
monoenergetic particles with K⬅x 2a ⬅m a v 2 /2T a to M a1 ,
N a1 , and L a1 , respectively, which are given by the inner
products of the source terms ␴’s and the response functions
G’s as
1
共 ␴ ,G 兲
T a Ua Ua
⫻
a2 PT
关 M a j P P ,M a j PT ,M a jTT 兴 ⫽n a
j⫺1
⫻ 关 M a 共 K 兲 ,N a 共 K 兲 ,L a 共 K 兲兴 ,
M a共 K 兲 ⫽
H. Sugama and S. Nishimura
⫺1
.
In the same way, we obtain the linear relations between
the conjugate pairs of 关 具 BP •(ⵜ• ␲a ) 典 , 具 BP •(ⵜ•⌰a ) 典 , 具 BT
•(ⵜ• ␲a ) 典 , 具 BT •(ⵜ•⌰a ) 典 兴
and
关 具 u a␪ 典 /␹ ⬘ ,(2/5p a ) 具 q a␪ 典 /
␨
␨
␹ ⬘ , 具 u a 典 /␺ ⬘ ,(2/5p a ) 具 q a 典 / ␺ ⬘ ,] as
共39兲
where M a P P (K), M a PT (K), and M TT (K) represent contributions of monoenergetic particles to M a j P P , M a j PT , and
M a jTT , respectively, which are given in terms of M a (K),
N a (K), and L a (K) as shown by Eq. 共B5兲 in Appendix B.
Now, we need the solutions G Ua and G Xa of Eq. 共27兲 in
order to obtain the monoenergetic coefficients
关 M a (K),N a (K),L a (K) 兴 in Eq. 共37兲. Since, in the DKES19,20
and other numerical codes for the neoclassical transport coefficients, the drift kinetic equation to be solved is not Eq.
共14兲 but Eq. 共3兲 with the pitch-angle-scattering operator, they
appear at first to be irrelevant to calculation of the
(G Ua ,G Xa ) and 关 M a (K),N a (K),L a (K) 兴 . However, in fact,
these codes can be made use of to obtain them as shown in
the following.
When solving Eq. 共3兲 by the DKES, the right-hand side
of Eq. 共3兲 are written as a linear combinations of the source
⫹
20
terms ␴ ⫹
1 and ␴ 3 defined by
2
1
v
␴⫹
1⫹ P 2 共 ␰ 兲 b⫻ⵜ ln B•ⵜs,
1 ⬅⫺
3⍀ a
2
冋
␴⫹
3 ⬅
v2
册
P 2 共 ␰ 兲 B•ⵜ ln B⫽V 储
冉 冊
B
共40兲
v␰ ,
␯ Da
which are associated with the radial particle flux and the
bootstrap current, respectively. Here, it is noted that, since
we have neglected vE•ⵜ f a1 (vE⬅cE⫻B/B 2 ) in Eq. 共3兲, ␴ ⫹
3
defined in Eq. 共40兲 corresponds to ␴ ⫹
3 (E s ⫽0) in Rij and
Hirshman.20 This definition of ␴ ⫹
3 based on Ref. 20 differs
from that in Ref. 19 in that the former excludes the Spitzer
flow part from the total parallel flow. Effects of vE•ⵜ f a1
such as nonlinear E s -dependences of the neoclassical transport can be included by the procedures described in Appendix E. It is useful to find that ␴ Ua and G Ua are directly
⫹
⫺
related to ␴ ⫹
3 and (F 3 ⫹F 3 ), respectively, by
␯ Da
␴ Ua ⫽⫺m a ␯ Da ␴ ⫹
3 ,
a
⫺
G Ua ⫽⫺m a ␯ D
共F⫹
3 ⫹F 3 兲 ,
共41兲
Downloaded 03 Mar 2009 to 133.75.139.172. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
Phys. Plasmas, Vol. 9, No. 11, November 2002
How to calculate the neoclassical viscosity, diffusion . . .
⫹
and that ␴ Xa and G Xa are written in terms of ␴ ⫹
1 and (F 1
⫺
⫹F 1 ), respectively, as
␴ Xa ⫽⫺ ␴ ⫹
1 ⫺V 储
冉
a
˜ 2 典 of L a (K)
and plateau regimes, the term (B 2 v 2 ␯ D
/3⍀ 2a ) 具 U
in Eq. 共43兲, which corresponds to the Pfirsch–Schlu¨ter-flux
part, is negligibly small.
Now, we have found that numerical solvers such as the
DKES can be utilized to calculate the coefficients
关 M a (K),N a (K),L a (K) 兴 by using Eq. 共43兲. Once
关 M a (K),N a (K),L a (K) 兴 are given, the monoenergetic poloidal and toroidal viscosity coefficients 关 M a P P (K),
M a PT (K),M aTT (K)] are immediately derived from Eq. 共B5兲
in Appendix B, and the energy-integrated coefficients
(M a j ,N a j ,L a j ) and (M a j P P ,M a j PT ,M a jTT ) are obtained by
Eqs. 共36兲 and 共39兲, respectively. Then, all the neoclassical
transport coefficients for radial fluxes and parallel currents
can be calculated from (M a j ,N a j ,L a j ) as shown in Appendix C. It should be noted that the parallel momentum balance
equations and the friction-flow relations with collisional momentum conservation are used to derive the neoclassical
transport coefficients in Appendix C. Therefore, these coefficients include the coupling effects between unlike-species
particles as well as they recover the intrinsic ambipolarity of
the radial particle fluxes in the symmetric limit. These properties are not obtained by only solving the drift kinetic equation 共3兲 without the field particle collision term
C ab ( f aM , f b1 ). For the symmetric case, M a (K), N a (K), and
L a (K) are proportionally related to each other as shown by
Eq. 共D4兲 in Appendix D.
a
In the Pfirsch–Schlu¨ter regime, 关 ␯ D
(K)Ⰷ v Ta 冑K/L c ,
(L c : The characteristic length of magnetic ripples along the
a
(K)
field line兲兴, the plateau regime 关v Ta 冑K/L c Ⰷ ␯ D
3/2
Ⰷ( ␦ B/B) v Ta 冑K/L c , ( ␦ B: The field strength variation in
a
(K)
the magnetic ripples兲兴, and the banana regime 关 ␯ D
3/2
Ⰶ( ␦ B/B) v Ta 冑K/L c 兴 , the monoenergetic coefficients
M a (K) and N a (K) associated with the parallel viscosities
can analytically be given by33
冊
B
˜ ,
v␰U
⍀a
⫺
G Xa ⫽⫺ 共 F ⫹
1 ⫹F 1 兲 ⫺
␯a B
B
˜⫹ D
v␰U
⍀a
⍀a
冕
l
共42兲
˜ dl.
U
⫺
⫹
⫺
Here, (F ⫹
1 ⫹F 1 ) and (F 3 ⫹F 3 ) represent the response
⫹
functions associated with the source terms ␴ ⫹
1 and ␴ 3 for
20
the case of E s ⫽0 in Rij and Hirshman, where the superscripts ⫹ and ⫺ denote the even and odd parts with respect
to ␰, respectively. Then, substituting Eqs. 共41兲 and 共42兲 into
Eq. 共37兲, we have
M a共 K 兲 ⫽
m 2a
Ta
a
关␯D
共 K 兲兴 2 D 33共 K 兲
冋
冋
1⫺
a
3m a ␯ D
共 K 兲 D 33共 K 兲
2T a K 具 B 2 典
N a共 K 兲 ⫽
a
3m a ␯ D
共 K 兲 D 33共 K 兲
ma a
␯ D 共 K 兲 D 13共 K 兲 1⫺
Ta
2T a K 具 B 2 典
L a共 K 兲 ⫽
a
B 2v 2␯ D
1
˜ 2典
D 11共 K 兲 ⫺
具U
Ta
3⍀ 2a
冉
⫹
册
册
⫺1
,
⫺1
, 共43兲
a
3m a ␯ D
共 K 兲关 D 13共 K 兲兴 2
2T a K 具 B 2 典
冋
⫻ 1⫺
a
3m a ␯ D
共 K 兲 D 33共 K 兲
2T a K 具 B 2 典
册 冊
⫺1
,
where
D jk 共 K 兲 ⬅
1
2
冕
1
⫺1
⫹
d␰具␴⫹
j Fk 典
共44兲
共 j,k⫽1,3 兲 ,
represent the transport coefficients for monoenergetic particles which can be obtained as an output of the DKES20 共for
the case of E s ⫽0). For collision frequencies in the banana
M a共 K 兲 ⫽
冦
2
m v 2 ␶ 具 共 B•ⵜ ln B 兲 2 典 K 2 关 ␶ aa ␯ Ta 共 K 兲兴 ⫺1
5 a Ta aa
1
␲ m a v Ta 具 B 2 典 1/2共 4 ␲ 2 /V ⬘ 兲
4
兺
(m,n)⫽(0,0)
2
a
m ␶ ⫺1 共 f / f 兲 具 B 2 典 K 关 ␶ aa ␯ D
共 K 兲兴
3 a aa t c
⫽m a v Ta K 3/2⫻
and
N a 共 K 兲 ⫽⫺
冉
cG (BS)
a
e a具 B 2典
冦
共 Pfirsch–Schlu¨ter兲
冊
兩 ␤ mn 兩 2 兩 m ␹ ⬘ ⫺n ␺ ⬘ 兩 K 3/2
共 Pfirsch–Schlu¨ter兲
1
␲ 具 B 2 典 1/2共 4 ␲ 2 /V ⬘ 兲
4
兩 ␤ mn 兩 2 兩 m ␹ ⬘ ⫺n ␺ ⬘ 兩
兺
(m,n)⫽(0,0)
2
a
共 f / f 兲具 B 2典 关 ␯ D
共 K 兲 / v兴
3 t c
M a共 K 兲 ,
共 plateau兲
共 banana兲
2
具 共 B•ⵜ ln B 兲 2 典 关 ␯ Ta 共 K 兲 / v兴 ⫺1
5
冉
4643
冊
共 plateau兲
共45兲
共 banana兲
共46兲
Downloaded 03 Mar 2009 to 133.75.139.172. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
4644
Phys. Plasmas, Vol. 9, No. 11, November 2002
H. Sugama and S. Nishimura
respectively. Here, G (BS)
is a flux-surface function, which represents the geometrical factor associated with the bootstrap
a
current5,29–31,33,38 关see the paragraph after Eq. 共49兲兴, and is determined by the magnetic configuration as33,38
G共aBS兲
再冉
(4␲2/V⬘)具(B•ⵜ ln B) 2 典⫺1 [B (Boozer)
具(⳵ ln B/ ⳵ ␪ H )(B•ⵜ ln B) 典 ⫺B (Boozer)
具 共⳵ ln B/ ⳵␨ H )(B•ⵜ ln B) 典 ]
␨
␪
⫽
兺 兩 ␤ mn兩 2兩 m ␹ ⬘ ⫺n ␺ ⬘兩
(m,n)⫽(0,0)
冊
⫺1
B
兺
(m,n)⫽(0,0)
冊
␤ mn exp关 i 共 m ␪ ⫺n ␨ 兲兴 ,
共48兲
where we should note that the existence of the plateau regime requires 兩 ␤ mn 兩 Ⰶ1 and that it does not make a significant difference which of the flux-coordinate systems (s, ␪ , ␨ )
is used to calculate ␤ mn for the plateau regime. If all the
particles in the velocity space are dominantly contained in
either of the Pfirsch–Schlu¨ter, plateau, and banana regimes,
we obtain from Eqs. 共35兲, 共36兲, and 共46兲,
L a共 K 兲 ⫽
⫽
c 2 M a共 K 兲
e 2a 具 B 2 典 2
m ac 2
e 2a
再冉
v Ta K
冋 具具
册
冋
M a1
B• 共 ⵜ• ␲a 兲 典
⫽ 具 B 2 典 ⫺1
B• 共 ⵜ•⌰a 兲 典
M a2
⫺1
useful. In Eq. 共45兲, f t ⬅1⫺ f c and f c ⬅ 43 具 B 2 典 兰 0 maxd␭ ␭/
具(1⫺␭B)1/2) 典 represent the fractions of trapped and circulating particles, respectively, and ␯ Ta (K) for the Pfirsch–
a
⫹ ␯ Ea ⬅(3 冑␲ /
Schlu¨ter regime is given by2 ␯ Ta ⬅3 ␯ D
⫺1
3
4)兺b␶ab 关兵⌽(xb)⫺3G(xb)其/xa⫹4(Ta /Tb)(1⫹mb/m a )G(x b ) / x a 兴
with G(x)⬅ 关 ⌽(x)⫺x⌽ ⬘ (x) 兴 /(2x 2 ). Thus, in order to correctly reproduce the viscosity coefficients for the Pfirsch–
a
Schlu¨ter regime, we should replace ␯ D
with ␯ Ta /3 when using
the pitch-angle-scattering operator in Eq. 共24兲 for that collisional region. In Eqs. 共45兲 and 共47兲, ␤ mn for the plateau
regime are the coefficients in the Fourier expansion of B:
冉
⫻
冉冋
M a2
M a3
具 u 储aB 典
册
兺 兩 ␤ mn兩 2兩 m ␹ ⬘ ⫺n ␺ ⬘兩
(m,n)⫽(0,0)
再
冊
⫺1
兺
册
冋 册
c X a1
2
⫺G (BS)
a
e a X a2
具q B典
5 p a 储a
冊
. 共49兲
As shown by substituting Eq. 共49兲 into the parallel momentum balance equations 关see Eq. 共C1兲 in Appendix C兴, G (BS)
a
represents the geometrical factor which enters the coefficients relating the parallel flows to the thermodynamic
forces. Also, it is directly confirmed from Eqs. 共46兲, 共C5兲,
ap共C8兲 and 共C11兲–共C13兲 that the geometrical factor G (BS)
a
pears in the neoclassical transport coefficients for the bootstrap current as well as in the nondiagonal coefficients connecting the electrons’ fluxes 共forces兲 with the ions’ forces
共fluxes兲. For symmetric systems described in Appendix D,
defined by Eq. 共46兲 is indepenthe geometrical factor G (BS)
a
dent of the collision frequency 关see Eq. 共D5兲兴 and, therefore,
Eq. 共49兲 is always satisfied. For example, G (BS)
a
⫽B ␨(Boozer) / ␹ ⬘ in the axisymmetric case. However, for nonaxisymmetric systems, Eq. 共49兲 is not generally valid 共except
for the limiting collision frequency regimes兲, and therefore,
the two independent 2⫻2 matrices 关 M a j 兴 and 关 N a j 兴 obtained from the energy integral in Eq. 共36兲 should be used
instead for relating the parallel viscosities to the parallel
flows and to the radial gradient forces.
We can analytically express the monoenergetic coefficient L a (K) for the Pfirsch–Schlu¨ter and plateau regimes
as33
共 4 ␲ 2 /V ⬘ 兲 2 具 共 B•ⵜ ln B 兲 2 典 ⫺1 具 关 B ␨(Boozer) 共 ⳵ ln B/ ⳵ ␪ H 兲 ⫺B (Boozer)
共 ⳵ ln B/ ⳵␨ H 兲兴 2 典
␪
3/2
共47兲
(plateau)
兲共 m ␹ ⬘ ⫺n ␺ ⬘ )/ 兩 m ␹ ⬘ ⫺n ␺ ⬘ 兩
兺 兩 ␤ mn兩 2共 mB ␨(Boozer) ⫺nB (Boozer)
␪
(m,n)⫽(0,0)
and the analytical expression of G (BS)
for the banana regime
a
for
is given in Refs. 5 and 29–31. When we evaluate G (BS)
a
the Pfirsch–Schlu¨ter regime given by Eq. 共47兲, Eq. 共A11兲 is
B⫽B 0 1⫹
共 Pfirsch–Schlu¨ter)
(m,n)⫽(0,0)
共 Pfirsch–Schlu¨ter兲
兩 ␤ mn 兩 2 共 mB ␨(Boozer) ⫺nB (Boozer)
兲 2 / 兩 m ␹ ⬘ ⫺n ␺ ⬘ 兩
␪
2
5
共 4 ␲ 2 /V ⬘兲 2 具B 2 典 ⫺2 具 关B (Boozer)
共⳵ ln B/ ⳵ ␪ H 兲 ⫺B ␪(Boozer) 共⳵ ln B/ ⳵␨ H 兲兴2 典关␯ Ta 共 K 兲 / v兴 ⫺1
␨
1
4
␲ 具 B 2 典 ⫺3/2共 4 ␲ 2 /V ⬘ 兲
⫺nB (Boozer)
兲 2 / 兩 m ␹ ⬘ ⫺n ␺ ⬘ 兩
兺 兩 ␤ mn兩 2共 mB (Boozer)
␨
␪
(m,n)⫽(0,0)
共 plateau兲
共 Pfirsch–Schlu¨ter兲
共 plateau兲
,
共50兲
Downloaded 03 Mar 2009 to 133.75.139.172. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
Phys. Plasmas, Vol. 9, No. 11, November 2002
How to calculate the neoclassical viscosity, diffusion . . .
which shows that, for these regimes, L a (K) has the same
dependence on the collision frequency and the energy as
M a (K).
It is well-known that, for nonsymmetric systems, the
centers of trapped-particle orbits move across magnetic surfaces and cause the neoclassical ripple transport in the
weakly collisional sub-regime 共so-called 1/␯ regime兲.5– 8 The
bounce-averaged part of the distribution function 具 f a1 典 b
⬅(养 f a1 dl/ v 储 )/(养dl/ v 储 ) makes no contribution to the parallel viscosities5,29 and consequently to M a (K) and N a (K),
while it contributes dominantly to the radial particle and heat
fluxes and to L a (K) in the 1/␯ regime. Using the analytical
solution of the bounce-averaged drift kinetic equation by
a
(K) 兴 as
Shaing and Hokin,7 we obtain L a (K) 关 ⬀1/␯ D
L a共 K 兲 ⫽
⫽
1
4& ␲ 2 T a
1
冉 冊
m ac
e a␺ ⬘
m ac 2
2
2& ␲ 2 e a
2
4
␶ aa
v Ta
v Ta K 3/2
K2
␶ aa ␯ Da 共 K 兲
␯)
G (1/
a
␯)
G (1/
a
a
共 ␺ ⬘ 兲2关 ␯ D
共 K 兲 / v兴
共 for the 1/␯ regime兲 , 共51兲
␯)
represents the geometrical factor for the neowhere G (1/
a
classical ripple transport defined by7
␯)
G (1/
⫽
a
冕
2␲
0
冋 冉 冊
冉 冊册
3/2
d␪ ⑀H
G1
⫹G 3
⳵⑀H
⳵␪
⳵⑀T
⳵␪
2
⫺2G 2
冉 冊冉 冊
⳵⑀T
⳵␪
⳵⑀H
⳵␪
2
2
,
共52兲
with G 1 ⫽16/9, G 2 ⫽16/15, and G 3 ⫽0.684 for the magnetic
field
strength
B⫽B 0 关 1⫹ ⑀ T (s, ␪ )⫹ ⑀ H (s, ␪ )cos(l␪
⫺n␨)兴 ( 兩 ⑀ T 兩 Ⰶ1,兩 ⑀ H 兩 Ⰶ1). Here, the safety factor q(s)
⬅ ␺ ⬘ / ␹ ⬘ is assumed to satisfy nq(s)Ⰷl. For this case, the
a
1/2
(K)/ ⑀ H Ⰶ ⑀ H
1/␯ regime is defined by 具 ␪˙ 典 b Ⰶ ␯ D
v Ta /(R/n),
where R denotes the major radius of the torus and 具 ␪˙ 典 b represents the bounce-averaged poloidal angular velocity of helically trapped particles. 关Note that, in the present study using Eq. 共3兲 as the basic equation, we do not treat the case of
a
(K)/ ⑀ H (r: the minor radius of the
cE r /(rB 0 )⬃ 具 ␪˙ 典 b ⭓ ␯ D
a
a
) and N a (⬀ ␯ D
) make
torus兲兴. In the 1/␯ regime, M a (⬀ ␯ D
little contribution to the radial transport fluxes so that Eq.
bn
共35兲 gives ⌫ bn
a ⯝L a1 X a1 ⫹L a2 X a2 and q a /T a ⯝L a2 X a1
⫹L a3 X a2 , in which dependence on X b1 and X b2 with b⫽a
are negligible. This fact justifies conventional calculations of
the neoclassical ripple transport using the pitch-anglescattering collision model,7,8 in which the collisional momentum conservation and the nondiagonal coupling between
unlike-species particles are not taken into account. However,
in general, we should use all elements M a j , N a j , and L a j in
Eq. 共35兲 to calculate the total neoclassical transport fluxes,
especially when the magnitude of the banana-plateau transport induced by the parallel viscosity is comparable to or
larger than that of the ripple transport as is the case in quasisymmetric systems.10–12
4645
IV. NUMERICAL EXAMPLES
Here, in order to illustrate the validity of the procedures
described in the previous sections, we present numerical results for the simple nonsymmetric system, in which the magnetic field strength is given by
B⫽B 0 共 s 兲关 1⫺ ⑀ t 共 s 兲 cos ␪ B ⫺ ⑀ h 共 s 兲 cos共 l ␪ B ⫺n ␨ B 兲兴 . 共53兲
The mean minor radius of the flux surface is used for the
radial coordinate s. For simplicity, we consider a single flux
surface of a large-aspect-ratio torus with the minor radius s
⫽0.4 m and the major radius R⫽4 m. Then, parameters
used for numerical calculations are determined as B 0 ⫽1 T,
⑀ t ⫽0.1, 0⭐ ⑀ h ⭐0.1, ␺ ⬘ ⫽0.4 T•m, ␹ ⬘ ⫽0.15 T•m (q
⫽4 T•m, B ␪(Boozer) ⫽0 T•m 共no
⬅ ␺ ⬘ / ␹ ⬘ ⫽2.6667), B (Boozer)
␨
net toroidal current兲, l⫽2, and n⫽10 共corresponding to the
Large Helical Device39兲. Using these parameters and Eq.
共53兲, we can calculate 具 B 2 典 ⫽4 ␲ 2 /( 兰 20 ␲ d ␪ B 兰 20 ␲ d ␨ B B ⫺2 ) and
V ⬘ ⬅4 ␲ 2 ( ␺ ⬘ B ␨(Boozer) ⫹ ␹ ⬘ B ␪(Boozer) )/ 具 B 2 典 . Hereafter, subscripts representing particle species are omitted.
The
monoenergetic
diffusion
coefficients
关 D 11(K),D 13(K),D 33(K) 兴 are obtained by using the DKES.
* ⬅D 11(K)/ 关 21 v T (B v T /⍀) 2 K 3/2兴 , D 13
*
Figure 1 shows D 11
1
1
1/2
*
⬅D 13(K)/ 关 2 v T (B v T /⍀)K 兴 , and D 33⬅D 33(K)/( 2 v T K )
as a function of ␯ D / v for ⑀ h ⫽0, 0.005, 0.01, 0.02,
0.05, and 0.1. Substituting these monoenergetic diffusion coefficients into Eq. 共43兲 and using Eq. 共B5兲 give other
monoenergetic coefficients 关 M (K),N(K),L(K) 兴 and
关 M P P (K),M PT (K),M TT (K) 兴 , which are illustrated in Figs.
2–5.
Figure 2 shows M * ⬅M (K)/(m v T K 3/2) as a function of
* as
␯ D / v . Here, M * is written in terms of D 33
M *⫽
共 ␯ D / v 兲2D *
33
* / 具 B 2典
1⫺ 32 共 ␯ D / v 兲 D 33
共54兲
.
In Fig. 2, dotted curves with open circles and solid lines
* in Fig.
represent M * obtained from numerical results of D 33
1 and from the analytical formulas in Eq. 共45兲, respectively.
When the formula for the Pfirsch–Schlu¨ter regime given by
Eq. 共45兲 is used in Fig. 2, ␯ T is replaced with 3 ␯ D . However,
as mentioned after Eq. 共47兲, the correct functional form of
␯ T (K) should be taken into account when we calculate the
energy-integrated viscosity coefficients. We can see an excellent agreement between the numerical and analytical results
except for transition regions between the banana, plateau,
and Pfirsch–Schlu¨ter regimes. A simple rational
approximation,2 which smoothly connects the three analytical expressions, would be useful for this case.
Figure 3 shows L * ⬅L(K)/ 关 21 ( v T /T)(B v T /⍀) 2 K 3/2兴 as
* and
a function of ␯ D / v . Here, L * is given in terms of D 11
* by
D 13
˜ 2典 ⫹
* ⫺ 32 共 ␯ D / v 兲 具 U
L * ⫽D 11
3
2
* 兲2/ 具 B 2典
共 ␯ D / v 兲共 D 13
* / 具 B 2典
1⫺ 32 共 ␯ D / v 兲 D 33
.
共55兲
Downloaded 03 Mar 2009 to 133.75.139.172. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
4646
Phys. Plasmas, Vol. 9, No. 11, November 2002
H. Sugama and S. Nishimura
FIG. 2. M * ⬅M (K)/(m v T K 3/2) as a function of ␯ D / v for ⑀ h ⫽0, 0.005,
0.01, 0.02, 0.05, and 0.1. Dotted curves with open circles and solid lines
represent M * obtained from numerical results of D *
33 in Fig. 1 and from the
analytical formulas in Eq. 共45兲, respectively.
In the same way as in Fig. 2, dotted curves with open circles
and solid lines in Fig. 3 represent L * obtained from numeri* , D 13
* , and D 33
* in Fig. 1 and from the
cal results of D 11
analytical formulas in Eqs. 共50兲 and 共51兲, respectively. We
see that, in the 1/␯ regime with ⑀ h ⫽0.005 and 0.01, numerically obtained L * are significantly smaller than the analytical
predictions. This is because, for such small ⑀ h ’s, the fraction
of helically trapped particles are overestimated by the analytical formula in Eq. 共51兲, where the lowest-order guidingcenter motion is regarded as a toroidal one instead of a parallel one under the condition of nqⰇl. Recently, an
improved formulation of the neoclassical ripple transport has
been given by Beidler and Maaßberg.8
We plot the geometrical factor for the bootstrap current
G (BS) ⬅⫺(e 具 B 2 典 /c)N(K)/M (K) 关see Eq. 共46兲兴 instead of
N(K) as a function of ␯ D / v in Fig. 4. Here, G (BS) is written
* and D 33
* as
in terms of D 13
G (BS) ⫽⫺
FIG.
1.
1
关 2 v T (B v T
1
2 3/2
D*
11⬅D 11(K)/ 关 2 v T (B v T /⍀) K 兴
* ⬅D 33(K)/( 21 v T K 1/2)
D 33
共a兲,
D*
13⬅D 13(K)/
/⍀)K 兴 共b兲, and
共c兲 as a function of
␯ D / v for ⑀ h ⫽0, 0.005, 0.01, 0.02, 0.05, and 0.1 obtained by using the
DKES.
*
具 B 2 典 D 13
.
*
共 ␯ D / v 兲 D 33
共56兲
In Fig. 4, dotted curves with open circles represent G (BS)
* and D 33
* in Fig. 1.
obtained from numerical results of D 13
The axisymmetric case with ⑀ h ⫽0 is given by the constant,
G (BS) ⫽B (Boozer) / ␹ ⬘ ⫽26.667. Analytical results given by Eq.
共47兲 for the Pfirsch–Schlu¨ter and plateau regimes are represented by thick line segments, which are in good agreement
with the numerical results, although the latter do not show
clear constancy in the plateau regime.
* 兴 ⬅ 关 M P P (K),
Figure 5 shows 关 M *
P P ,⫺M *
PT ,M TT
⫺M PT(K),M TT(K)兴/关(4␲2/V⬘)mvT(␺⬘␹⬘)2K3/2兴 as a function of
␯ D / v . For ⑀ h ⭐0.02, M TT takes small negative values
around the plateau regime, which are not plotted in Fig. 5. As
⑀ h increases in the Pfirsch–Schlu¨ter and plateau regimes, the
magnitude of the viscosity coefficients M PT and M TT increases more rapidly than M P P . It is also seen that, in
the 1/␯ regime, M P P ⯝⫺M PT ⯝M TT ⬀1/␯ D , which reflects
Downloaded 03 Mar 2009 to 133.75.139.172. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
Phys. Plasmas, Vol. 9, No. 11, November 2002
1
FIG. 3. L * ⬅L(K)/ 关 2 ( v T /T)(B v T /⍀) 2 K 3/2兴 as a function of ␯ D / v for ⑀ h
⫽0, 0.005, 0.01, 0.02, 0.05, and 0.1. Dotted curves with circles and solid
lines represent L * obtained from numerical results of D *
11 , D *
13 , and D *
33 in
Fig. 1 and from the analytical formulas in Eqs. 共50兲 and 共51兲, respectively.
from the fact that the parallel viscosity 具 B•(ⵜ• ␲a ) 典
⫽ 具 BP •(ⵜ• ␲a ) 典 ⫹ 具 BT •(ⵜ• ␲a ) 典 (⬀ ␯ D ) is much smaller than
the viscosities in other directions (⬀1/␯ D ).
From the results shown above, it is confirmed that all
neoclassical coefficients for the viscosities, the bananaplateau and nonsymmetric radial transport fluxes, and the
geometrical factor associated with the bootstrap current are
obtained straightforwardly by using our method.
V. CONCLUSIONS
In the present paper, we have presented two types of
Onsager-symmetric matrices: One of them, with the elements
How to calculate the neoclassical viscosity, diffusion . . .
4647
FIG. 5. Poloidal and toroidal viscosity coefficients as a function of ␯ D / v for
⑀ h ⫽0, 0.005, 0.01, 0.02, 0.05, and 0.1. Curves with circles, crosses, and
* , respectively.
triangles represent M *P PT , ⫺M *PT , and M TT
(M a j ,N a j ,L a j ), relates the parallel viscosities and the radial
fluxes to the parallel flows and the radial-gradient forces as
in Eq. 共35兲, and the other, represented by
(M a j P P ,M a j PT ,L a jTT ), connects the poloidal and toroidal
viscosities to the poloidal and toroidal flows as in Eq. 共38兲.
We have shown that the matrix elements (M a j ,N a j ,L a j ) can
be obtained readily from the output of commonly used numerical codes such as the DKES and that the poloidal and
toroidal viscosity coefficients (M a j P P ,M a j PT ,L a jTT ) can be
derived directly from (M a j ,N a j ,L a j ). Using the matrix elements (M a j ,N a j ,L a j ) in the parallel momentum balance
equations combined with the friction-flow relations yields
the neoclassical transport coefficients for the radial particle
and heat fluxes and the bootstrap current, which include the
coupling effects between unlike-species particles as well as
the intrinsic ambipolarity of the radial particle fluxes in the
symmetric case. These procedures for accurate calculation of
neoclassical viscosity and transport coefficients, the validity
of which has been verified by numerical examples, are considered to be useful especially when evaluating how these
neoclassical coefficients in quasi-symmetric toroidal systems
such as quasi-axisymmetric systems deviate from those in
exactly symmetric systems.
ACKNOWLEDGMENTS
FIG. 4. The geometrical factor for the bootstrap current G (BS) as a function
of ␯ D / v for ⑀ h ⫽0, 0.005, 0.01, 0.02, 0.05, and 0.1. Dotted curves with open
circles represent G (BS) obtained from numerical results of D *
13 and D *
33 in
Fig. 1. The axisymmetric case with ⑀ h ⫽0 is given by the constant, G (BS)
⫽B (Boozer) / ␹ ⬘ ⫽26.667. Analytical results given by Eq. 共47兲 for the Pfirsch–
Schlu¨ter and plateau regimes are represented by thick line segments.
The authors thank Dr. S. P. Hirshman for providing the
revised version of the DKES code. Useful discussions with
Dr. J. Nu¨hrenberg, Dr. S. Okamura, Dr. K. Matsuoka, Dr. N.
Nakajima, and Dr. M. Yokoyama on neoclassical transport in
quasi-symmetric systems are acknowledged.
This work is supported in part by the Grant-in-Aid from
the Japanese Ministry of Education, Science, Sports, and
Culture, No. 12680497.
APPENDIX A: BOOZER AND HAMADA COORDINATES
We consider general toroidal configurations, in which
the magnetic field B is written as in Eq. 共1兲 of Sec. II. In the
Downloaded 03 Mar 2009 to 133.75.139.172. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
4648
Phys. Plasmas, Vol. 9, No. 11, November 2002
H. Sugama and S. Nishimura
Boozer36 coordinates (s, ␪ B , ␨ B ), the covariant poloidal and
toroidal components of the magnetic field B are flux-surface
functions given by
⳵x 2
B ␪(Boozer) ⬅B•
⫽ I 共 s 兲,
⳵␪B c T
B ␨(Boozer) ⬅B•
which is rewritten as
冉
共A1兲
V ⬘共 s 兲 具 B 2典
⫽
.
4␲2 B2
4␲2
␹ ⬘共 s 兲 ,
V ⬘共 s 兲
␨
⬅B•ⵜ ␨ H ⫽
B (Hamada)
4␲2
␺ ⬘共 s 兲 ,
V ⬘共 s 兲
共A2兲
冉
冉
冊
共A3兲
冊
共A4兲
⬅B•( ⳵ x/ ⳵ ␪ H ) and B (Hamada)
⬅B•( ⳵ x/ ⳵␨ H ).
where B (Hamada)
␪
␨
(Hamada)
and
We should also note that 具 B ␪
典 ⫽B (Boozer)
␪
(Hamada)
(Boozer)
.
具B␨
典 ⫽B ␨
The transformation from the Boozer to Hamada
coordinates40 are written in terms of the generating function
G as
␨ H ⫽ ␨ B ⫹ ␺ ⬘ G 共 s, ␪ B , ␨ B 兲 .
共A5兲
Here, the generating function G(s, ␪ B , ␨ B ) is periodic in ␪ B
and ␨ B and satisfies the magnetic differential equation
B•ⵜG⫽
1
冑g H
⫺
1
冑g B
,
⳵G
⳵G
1
B ␨(Hamada)
⫺B ␪(Hamada)
.
B
⳵␪H
⳵␨ H
冉
冊
共A8兲
⳵x
⳵x
⳵G
冑g B,
⫽
⫹
⳵␪B ⳵␪H ⳵␪B H
⳵x
⳵x
⳵G
冑g B,
⫽
⫹
⳵␨ B ⳵␨ H ⳵␨ B H
共A9兲
⳵ G 具 B 2典 ⳵ G
⫽ 2
,
⳵␨ B
B ⳵␨ H
共A6兲
共A10兲
where the partial derivatives ⳵ / ⳵ ␪ H and ⳵ / ⳵␨ H are taken with
the Hamada coordinates (s, ␪ H , ␨ H ) used as the independent
variables. Using Eqs. 共A9兲 and 共A10兲, we have
B (Boozer)
␨
冉
⳵ ln B
⳵ ln B
⫺B ␪(Boozer)
⳵␪H
⳵␨ H
⫽ B ␨(Hamada) ⫹
冉
冊
⳵ G ⳵ ln B
V⬘
具 B 2典
4␲2
⳵␨ H ⳵ ␪ H
⫺ B (Hamada)
⫹
␪
B B (Hamada)
具 B (Hamada)
典
␪
␪
⫽⫺
⫺
,
2
2
␺⬘
B
具B 典
␪ H ⫽ ␪ B ⫹ ␹ ⬘ G 共 s, ␪ B , ␨ B 兲 ,
⫽
⳵ G 具 B 2典 ⳵ G
⫽ 2
,
⳵␪B
B ⳵␪H
respectively. Here, the poloidal and toroidal fluxes are given
and
2 ␲ ␺ (s)
by
2 ␲␹ (s)⫽(2 ␲ ) ⫺1 兰 V(s) d 3 xB•ⵜ ␪
⫽(2 ␲ ) ⫺1 兰 V(s) d 3 xB•ⵜ ␨ , respectively, V(s) represents the
volume enclosed by the flux surface with the label s, and the
derivative with respect to s is denoted by ⬘⫽d/ds. Then, we
find that, using the Hamada coordinates, Eq. 共12兲 is easily
solved to yield
B B (Hamada)
具 B ␨(Hamada) 典
␨
⫺
2
␹⬘
B
具 B 2典
冊
⳵G
⳵G
B
B (Boozer)
⫺B (Boozer)
2
␨
␪
⳵␪B
⳵␨ B
具B 典
and
V ⬘共 s 兲
冑g H ⬅ 关 ⵜs• 共 ⵜ ␪ H ⫻ⵜ ␨ H 兲兴 ⫺1 ⫽ 2 ,
4␲
˜⫽
U
冉
⫽
From Eq. 共A5兲, we obtain
Next, in the Hamada37 coordinates (s, ␪ H , ␨ H ), the contravariant poloidal and toroidal components of the magnetic
field B and the Jacobian 冑g H are flux-surface functions written as
␪
B (Hamada)
⬅B•ⵜ ␪ H ⫽
V⬘
B⫻ⵜs•ⵜG
4 ␲ 2B
˜⫽
U
Here, the poloidal and toroidal currents are defined by
I dP (s)⬅ 兰 S d (s) B•dS and I T (s)⬅ 兰 S T (s) B•dS, respectively,
P
where S dP (s) represents the part of a ␪ ⫽constant surface that
lies outside the flux surface with the label s and S T (s) is the
part of a ␨ ⫽constant surface that lies inside the flux surface.
The Jacobian for the Boozer coordinates is given by
冑g B ⬅ 关 ⵜs• 共 ⵜ ␪ B ⫻ⵜ ␨ B 兲兴
共A7兲
˜ is related to G
Comparing Eq. 共12兲 with 共A7兲, we find that U
by
⳵x 2 d
⫽ I 共 s 兲.
⳵␨ B c P
⫺1
冊
⳵
⳵
具 B 2典
␺⬘
⫹␹⬘
G⫽ 2 ⫺1.
⳵␨ B
⳵␪B
B
冊
⳵ G ⳵ ln B
V⬘
2
2 具B 典
4␲
⳵ ␪ H ⳵␨ H
冉
⫽
V⬘
V⬘
具 B 2 典 ⵜG
2 共 ⵜs⫻ⵜ ln B 兲 • B⫹
4␲
4␲2
⫽
B2
具 B 2典
冉
冋冉
B ␨(Boozer) ⫹
⫺ B (Boozer)
⫹
␪
冊
冊
⳵ G ⳵ ln B
V⬘
具 B 2典
4␲2
⳵␨ B ⳵ ␪ B
冊 册
⳵ G ⳵ ln B
V⬘
2
,
2 具B 典
4␲
⳵ ␪ B ⳵␨ B
共A11兲
and L a for the
which is useful when evaluating G (BS)
a
Pfirsch–Schlu¨ter regime 关see Eqs. 共47兲 and 共50兲兴. We also
find from Eq. 共A7兲 that
B•ⵜ
2 ⳵ ln B
⳵G
⫽⫺
,
⳵␪B
冑g H ⳵ ␪ B
B•ⵜ
⳵G
2 ⳵ ln B
⫽⫺
.
⳵␨ B
冑g H ⳵␨ B
共A12兲
Downloaded 03 Mar 2009 to 133.75.139.172. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
Phys. Plasmas, Vol. 9, No. 11, November 2002
How to calculate the neoclassical viscosity, diffusion . . .
Here, let us assume that c 1 ⳵ B/ ⳵ ␪ B ⫹c 2 ⳵ B/ ⳵␨ B ⫽0,
where c 1 and c 2 are constants and (c 1 ,c 2 )⫽(0,0). This condition implies that the magnetic field strength is written as
B⫽B(s,c 2 ␪ B ⫺c 1 ␨ B ), and it is satisfied approximately in
quasi-symmetric systems, where the neoclassical ripple
transport is suppressed. The axisymmetric, poloidally symmetric, and helically symmetric cases correspond to c 1 ⫽0,
c 2 ⫽0, and c 1 •c 2 ⫽0, respectively. Under this symmetry condition c 1 ⳵ B/ ⳵ ␪ B ⫹c 2 ⳵ B/ ⳵␨ B ⫽0, we find from Eq. 共A12兲
that c 1 ⳵ G/ ⳵ ␪ B ⫹c 2 ⳵ G/ ⳵␨ B is a flux surface function, and
therefore, c 1 ⳵ G/ ⳵ ␪ B ⫹c 2 ⳵ G/ ⳵␨ B ⫽ 具 c 1 ⳵ G/ ⳵ ␪ B ⫹c 2 ⳵ G/ ⳵␨ B 典
⫽0. Then, we also obtain c 1 ⳵ x/ ⳵ ␪ B ⫹c 2 ⳵ x/ ⳵␨ B ⫽c 1 ⳵ x/ ⳵ ␪ H
⫹c 2 ⳵ x/ ⳵␨ H and c 1 ⳵ G/ ⳵ ␪ H ⫹c 2 ⳵ G/ ⳵␨ H ⫽0 from Eqs. 共A9兲
and 共A10兲, respectively. Consequently, we have c 1 ⳵ B/ ⳵ ␪ H
⫹c 2 ⳵ B/ ⳵␨ H ⫽c 1 ⳵ B/ ⳵ ␪ B ⫹c 2 ⳵ B/ ⳵␨ B ⫽0.
Inversely,
if
c 1 ⳵ B/ ⳵ ␪ H ⫹c 2 ⳵ B/ ⳵␨ H ⫽0
is
assumed,
c 1 ⳵ B/ ⳵ ␪ B
⫹c 2 ⳵ B/ ⳵␨ B ⫽0 is concluded. The equivalent conditions described above are summarized as
where the flux-surface-averaged poloidal and toroidal flows
in the left-hand side do not depend on what flux coordinates
(s, ␪ , ␨ ) are chosen. From Eq. 共17兲, we obtain
冋 册
␴ Pa 4 ␲
⫽
␴ Ta
V⬘
⇔ c1
冋
共A13兲
Thus, either Boozer or Hamada coordinates can be used to
describe the symmetry condition for the magnetic field
strength to suppress the neoclassical ripple transport.
⫽
冋
册 冋
⫻
冋
2
具 q ␪典 / ␹ ⬘
5p a a
2
具q␨典/␺⬘
5p a a
冋具
册
册
冋
册
⫽
冋
4␲ 1
V⬘ 1
冋
2
具 q 储aB 典 / 具 B 2典
5p
⫻
a
X a2
册
,
册
共B1兲
冋
␺ ⬘ B (Boozer)
/ 具 B 2典
␨
册
⫺
ea
␺ ⬘␹ ⬘
c
ea
␺ ⬘␹ ⬘
c
具 B• 共 ⵜ• ␲a 兲 典 具 B• 共 ⵜ•⌰a 兲 典
⌫ bn
a
M a j PT
M a j PT
M a jTT
4␲
V⬘
q bn
a /T a
册
册
冋 册
␴ Ua
.
␴ Xa
共B2兲
册
共B3兲
.
⫻
冋
冋
册
␹ ⬘ B (Boozer)
/ 具 B 2典
␪
␺ ⬘ B (Boozer)
/ 具 B 2典
␨
␹ ⬘ B (Boozer)
/ 具 B 2典
␪
⫺
⫺
ea
␺ ⬘␹ ⬘
c
ea
␺ ⬘␹ ⬘
c
册冋
␺ ⬘ B ␨(Boozer) / 具 B 2 典
ea
␺ ⬘␹ ⬘
c
ea
␺ ⬘␹ ⬘
c
Maj
Naj
Naj
Laj
册
,
册
共B4兲
and correspondingly those between the monoenergetic coefficients 关 M a (K),N a (K),L a (K) 兴 and 关 M a P P (K),M a PT (K),
M aTT (K)],
M a P P共 K 兲
M a PT 共 K 兲
M a PT 共 K 兲
M aTT 共 K 兲
4␲
V⬘
2
⫺cB (Boozer)
/ 共 e a␹ ⬘具 B 2典 兲
␨
cB (Boozer)
/ 共 e a␺ ⬘具 B 2典 兲
␪
ea
␺ ⬘␹ ⬘
c
ea
␺ ⬘␹ ⬘
c
␺ ⬘ B ␨(Boozer) / 具 B 2 典
␹ ⬘ B (Boozer)
/ 具 B 2典
␪
M a jPP
⫽
2
冋
2
冋
u 储aB 典 / 具 B 典
,
X a1
2
⫺
Using Eqs. 共35兲, 共38兲, 共B1兲, and 共B3兲, we obtain the relations
between the poloidal and toroidal viscosity coefficients
(M a j P P ,M a j PT ,M a jTT ) and the coefficients (M a j ,N a j ,L a j )
for the parallel viscosities and the radial fluxes
APPENDIX B: POLOIDAL AND TOROIDAL VISCOSITY
COEFFICIENTS
(Boozer)
/ 共 e a␹ ⬘具 B 2典 兲
具 u ␪a 典 / ␹ ⬘ 4 ␲ 2 1 ⫺cB ␨
⫽
V⬘ 1
具 u ␨a 典 / ␺ ⬘
cB (Boozer)
/ 共 e a␺ ⬘具 B 2典 兲
␪
4␲
V⬘
2
⫽
The poloidal and toroidal flows can be linearly related to
the parallel flows and the radial gradient forces as
冋
␹ ⬘ B (Boozer)
/ 具 B 2典
␪
具 BP • 共 ⵜ• ␲a 兲 典 具 BP • 共 ⵜ•⌰a 兲 典
具 BT • 共 ⵜ• ␲a 兲 典 具 BT • 共 ⵜ•⌰a 兲 典
⫻
⳵x
⳵x
⳵x
⳵x
⫹c 2
⫽c 1
⫹c 2
.
⳵␪B
⳵␨ B
⳵␪H
⳵␨ H
2
Then, we find from Eqs. 共18兲, 共19兲, and 共B2兲 that the poloidal
and toroidal viscosities are written in terms of the parallel
viscosities and the radial fluxes as
⳵B
⳵B
⳵B
⳵B
⫹c 2
⫽0 ⇔ c 1
⫹c 2
⫽0
c1
⳵␪B
⳵␨ B
⳵␪H
⳵␨ H
⳵G
⳵G
⳵G
⳵G
⫹c 2
⫽0 ⇔ c 1
⫹c 2
⫽0
⇔ c1
⳵␪B
⳵␨ B
⳵␪H
⳵␨ H
4649
⫻
⫻
冋
冋
册
␹ ⬘ B (Boozer)
/ 具 B 2典
␪
␺ ⬘ B (Boozer)
/ 具 B 2典
␨
M a共 K 兲
N a共 K 兲
N a共 K 兲
L a共 K 兲
冋
册
␹ ⬘ B (Boozer)
/ 具 B 2典
␪
⫺
ea
␺ ⬘␹ ⬘
c
⫺
ea
␺ ⬘␹ ⬘
c
ea
␺ ⬘␹ ⬘
c
册
␺ ⬘ B ␨(Boozer) / 具 B 2 典
ea
␺ ⬘␹ ⬘
c
册
.
共B5兲
Downloaded 03 Mar 2009 to 133.75.139.172. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
4650
Phys. Plasmas, Vol. 9, No. 11, November 2002
H. Sugama and S. Nishimura
APPENDIX C: NEOCLASSICAL TRANSPORT
COEFFICIENTS FOR RADIAL FLUXES
AND PARALLEL CURRENTS
Integrating Eq. 共3兲 multiplied by m a v 储 and
m a v 储 (m a v 2 /2T a ⫺5/2) and taking the flux surface average
give the parallel momentum balance equations
具 B• 共 ⵜ• ␲a 兲 典 ⫺n a e a 具 BE 储 典 ⫽ 具 BF 储 a1 典 ,
具 B• 共 ⵜ•⌰a 兲 典 ⫽ 具 BF 储 a2 典 .
共C1兲
The parallel friction forces F 储 a1 ⬅ 兰 d
and
F 储 a2 ⬅ 兰 d 3 v m a v 储 (m a v 2 /2T a ⫺5/2)C La ( f a1 ) in the right-hand
side of Eq. 共C1兲 are related to the parallel flows u 储 a and q 储 a
by the friction-flow relations 共in the 13M approximation兲
3
冋
册 冋
ab
ab
l 11
⫺l 12
具 BF 储 a1 典
⫽兺
ab
ab
具 BF 储 a2 典
⫺l 21
l 22
b
册冋
v m a v 储 C La ( f a1 )
具 Bu 储 b 典
册
共C2兲
2
,
具 Bq 储 b 典
5p b
where the coefficients l ab
jk are defined by Eq. 共4.4兲 in Hirshba
man and Sigmar,2 and satisfy the conditions l ab
jk ⫽l k j and
ab
兺 a l 1k ⫽0, which are derived from the self-adjointness
and the momentum conservation property of the linearized
collision operator, respectively. The parallel viscosities
具 B•(ⵜ• ␲a ) 典 and 具 B•(ⵜ•⌰a ) 典 in the left-hand side of Eq.
共C1兲 are written by Eq. 共35兲 in terms of the parallel flows
具 Bu 储 a 典 and 具 Bq 储 a 典 . Then, combining Eqs. 共35兲, 共C1兲, and
共C2兲, we obtain
兺
b⫽e
冉 冋
␦ ab M a1
具 B 2 典 M a2
⫽⫺
⫹
⯝⫺
冋
冋
冋
M a2
M a3
ae
l 11
ae
⫺l 12
ae
⫺l 21
ae
l 22
冉 冋
⫽⫺
⫹
冋
M e3
N e2
兺
a⫽e
冋
册冋
2
具 Bq 储 e 典
5p e
册冋 册
M e2
N e1
ab
l 22
具 Bu 储 e 典
N a2 X a1
N a3 X a2
and
1 M e1
具 B 2 典 M e2
ab
⫺l 21
册冋 册 冋
N a2
N a2
⫺
ab
⫺l 12
册冊冋
N a2 X a1
n a e a 具 BE 储 典
⫹
0
N a3 X a2
N a1
N a1
册冋
ab
l 11
册
ee
⫺l 12
ee
⫺l 21
ee
l 22
N e2 X e1
n e e 具 BE 储 典
⫺
0
N e3 X e2
ea
⫺l 12
ea
⫺l 21
ea
l 22
册冋
具 Bu 储 a 典
册
冋 册冋
ee
L 11
ee
L 12
ei
L 11
ei
L 12
e
L 1E
ee
L 21
ee
L 22
ei
L 21
ei
L 22
e
L 2E
ie
L 11
ie
L 12
ii
L 11
ii
L 12
i
L 1E
ie
L 21
ie
L 22
ii
L 21
ii
L 22
i
L 2E
e
L E1
e
L E2
i
L E1
i
L E2
L EE
册冋 册
X e1
X e2
X i1 ,
X i2
XE
共C5兲
where the force X E associated with the parallel electric field
is denoted by
共C6兲
J EBS
for ion species a 共 ⫽e 兲
ee
l 11
ea
l 11
2
具 Bq 储 b 典
5p b
⌫ bn
e
q bn
e /T e
⌫ bn
⫽
i
bn
q i /T i
J EBS
X E ⬅ 具 BE 储 典 / 具 B 2 典 1/2,
册冋
册冊冋
册冋 册 冋 册
⫺
册
具 Bu 储 b 典
O关 (m e /m a ) 1/2兴 . Then, neglecting these O关 (m e /m a ) 1/2兴
terms in Eq. 共C3兲, the lowest-order parallel flows 具 Bu 储 a 典 and
具 Bq 储 a 典 for ion species a(⫽e) can be expressed as a linear
combination of the thermodynamic forces X b1 and X b2 (b
⫽e), and these expressions are substituted into Eq. 共C4兲 in
order to write the electron parallel flows 具 Bu 储 e 典 and 具 Bq 储 e 典
in terms of the thermodynamic forces X e1 , X e2 , X b1 , X b2
(b⫽e), and 具 BE 储 典 . Substituting these expressions of 具 Bu 储 e 典
and 具 Bq 储 e 典 in turn into Eq. 共C3兲, the parallel ion flows
具 Bu 储 a 典 and 具 Bq 储 a 典 (a⫽e) of the next order can be given in
terms of the ion and electron thermodynamic forces. Once
the relations of the parallel flows to the thermodynamic
forces are obtained for all species a, substituting them into
Eqs. 共35兲 and 共B1兲 and using Eq. 共B3兲 yield the expressions
bn
of the radial neoclassical fluxes 关 ⌫ bn
a ,q a 兴 , the parallel, poloidal, and toroidal viscosities 关 具 B•(ⵜ• ␲ a ) 典 , 具 B•(ⵜ
•⌰ a ) 典 , 具 BP •(ⵜ• ␲ a ) 典 , 具 BP •(ⵜ•⌰ a ) 典 , 具 BT •(ⵜ• ␲ a ) 典 , 具 BT •(ⵜ
and the poloidal and toroidal flows
•⌰ a ) 典 兴 ,
关 具 u ␪a 典 , 具 q ␪a 典 , 具 u a␨ 典 , 具 q ␨a 典 兴 , in terms of the thermodynamic
forces 关 X e1 ,X e2 ,X b1 ,X b2 (b⫽e), 具 BE 储 典 兴 .
Applying the procedures described above to the case of a
toroidal plasma consisting of electrons and a single species
of ions, we can derive the following transport equations for
the neoclassical radial fluxes of particles and heat and the
neoclassical parallel electric current 共bootstrap current兲
具 Bu 储 e 典
2
具 Bq 储 e 典
5p e
册
2
.
具 Bq 储 a 典
5p a
共C3兲
is defined by the difference
and the bootstrap current
between the total parallel electric current J E and the classical
parallel electric current J Ecl ,
J EBS⬅J E ⫺J Ecl⬅n e e 具 B 共 u 储 i ⫺u 储 e 兲 典 / 具 B 2 典 1/2⫺ ␴ SX E ,
共C7兲
␴S
with
the
classical
Spitzer
conductivity
e
e ˆe
e 2
⬅(n e e 2 ␶ ee /m e ) ˆl 22
/ 关 ˆl 11
l 22⫺( ˆl 12
) 兴 . Here, the dimensionless
ˆe
friction coefficients ˆl ai j ⬅⫺( ␶ aa /n a m a )l aa
i j are given by l 11
3
13
e
e
i
⫽Z i , ˆl 12⫽ 2 Z i , ˆl 22⫽&⫹ 4 Z i , and ˆl 22⫽& with the ion
charge number Z i , and high-order terms with respect to
(m e /m i ) 1/2 are neglected. Defining the 2⫻2 matrices for
electrons and ions (a⫽e,i) by
册
La ⬅
共C4兲
Here, general cases of multispecies of ions are considered.
We should note that m e /m a Ⰶ1 for ion species a and that the
parallel electric field term and the ion–electron friction term
in Eq. 共C3兲, are smaller than the other terms by a factor of
冋
Na ⬅
⌳ e⬅
L a1
L a2
L a2
L a3
冋
册
␶ aa N a1
n a m a N a2
冋
Ma ⬅
,
N a2
N a3
e
ˆl 11
e
⫺ ˆl 12
e
⫺ ˆl 12
e
ˆl 22
册
,
冋
M a1
␶ aa
2
n a m a 具 B 典 M a2
E11⬅
冋 册
1
0
0
0
册 冋 册
,
⌳ i⬅
0
0
0
i
ˆl 22
,
M a2
M a3
册
,
共C8兲
,
Downloaded 03 Mar 2009 to 133.75.139.172. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
Phys. Plasmas, Vol. 9, No. 11, November 2002
How to calculate the neoclassical viscosity, diffusion . . .
the transport coefficients in Eq. 共C5兲 are explicitly given by
冋
aa
L 11
aa
L 12
aa
L 21
aa
L 22
册
⫽La ⫺
⫹ ␦ ai
冋
n am a
N 共 Ma ⫹⌳ a 兲 ⫺1 Na
␶ aa 具 B 2 典 a
⫺1
⫹⌳ ⫺1
E11共 Mi ⫹⌳ i 兲 ⫺1 Ni ,
e 兲
ei
L 11
ei
L 12
ei
L 21
ei
L 22
册冋
⫽
ie
L 11
ie
L 21
ie
L 12
ie
L 22
共C9兲
册
⫹⌳ i 兲 ⫺1 Ni ,
共C10兲
n ee
关 1 0 兴共 Me ⫹⌳ e 兲 ⫺1 Ne ,
具 B 2 典 1/2
共C11兲
n ee
关 1 0 兴共 Me ⫹⌳ e 兲 ⫺1 Me E11
具 B 2 典 1/2
⫻ 共 Mi ⫹⌳ i 兲 ⫺1 Ni ,
冋册
n e e 2 ␶ ee
1
⫺1
L EE ⫽⫺
关 1 0 兴 兵 ⌳ ⫺1
其 0 .
e ⫺ 共 Me ⫹⌳ e 兲
me
共C12兲
共C13兲
In the right-hand side of Eq. 共C9兲, the term with ␦ ai (⬅1 for
a⫽i, 0 for a⫽e), which is of O关 (m e /m i ) 1/2兴 , is kept in
order to reproduce the intrinsic ambipolar particle fluxes ⌫ i
⫽Z ⫺1
i ⌫ e in the symmetric case 共see Appendix D兲. It should
be noted that the transport coefficients given in Eqs. 共C9兲–
共C13兲 satisfy the Onsager relations
ba
L ab
jk ⫽L k j ,
L ajE ⫽⫺L aE j
c2
c1
c2
c1
M a j P P⫹
M a j PT ⫽ M a j PT ⫹
M
⫽0.
␹⬘
␺⬘
␹⬘
␺ ⬘ a jTT
⌫ bp
a ⫽
i
i
i
i
L E2
L 2E
关 L E1
兴 ⫽⫺ 关 L 1E
兴
⫽⫺
c1
c2
具 B • 共 ⵜ•⌰a 兲 典 ⫹ 具 BT • 共 ⵜ•⌰a 兲 典 ⫽0,
␹⬘ P
␺⬘
共D2兲
The expressions for the banana-plateau particle and heat
fluxes for the symmetric case in terms of the parallel viscosities are derived from Eqs. 共18兲, 共19兲, and 共D1兲 as
neme
⫽⫺
N 共 Me ⫹⌳ e 兲 ⫺1 ⌳ e E11共 Mi
␶ee具B2典 e
e
e
e
e
L E2
L 2E
关 L E1
兴 ⫽⫺ 关 L 1E
兴⫽
c1
c2
具 BP • 共 ⵜ• ␲a 兲 典 ⫹ 具 BT • 共 ⵜ• ␲a 兲 典
␹⬘
␺⬘
⫽
n em e
N 共 Mi ⫹⌳ i 兲 ⫺1 E11共 M ⫺1
e
␶ ee 具 B 2 典 i
4651
共 a,b⫽e,i; j,k⫽1,2 兲 . 共C14兲
q bp
a
Ta
⫽
⫹c 2 B ␨(Boozer) 兲
c 共 c 1 B (Boozer)
␪
e a 共 c 1 ␺ ⬘ ⫺c 2 ␹ ⬘ 兲 具 B 2 典
c 共 c 1 B ␪(Boozer) ⫹c 2 B ␨(Boozer) 兲
e a 共 c 1 ␺ ⬘ ⫺c 2 ␹ ⬘ 兲 具 B 2 典
具 B• 共 ⵜ• ␲a 兲 典 ,
具 B• 共 ⵜ•⌰a 兲 典 .
共D3兲
Using Eq. 共D3兲 and the parallel momentum balance in Eq.
共C1兲 with the charge neutrality condition 兺 a n a e a ⫽0, we obtain the well-known intrinsic ambipolarity condition that, in
the symmetric case, 兺 a e a ⌫ bp
a ⫽0 is satisfied for arbitrary values of the thermodynamic forces.
Equations 共36兲, 共37兲, 共46兲, and 共D1兲 give the relations of
and
the
coefficients
关 M a j ,N a j ,L a j 兴
关 M a (K),
N a (K),L a (K)],
N a j L a j N a共 K 兲 L a共 K 兲
⫽
⫽
⫽
M a j N a j M a共 K 兲 N a共 K 兲
⫽
c 共 c 1 B ␪(Boozer) ⫹c 2 B (Boozer)
兲
␨
e a 共 c 1 ␺ ⬘ ⫺c 2 ␹ ⬘ 兲 具 B 2 典
,
共D4兲
,
and the geometric factor G (BS)
a
⫽
G (BS)
a
c 1 B ␪(Boozer) ⫹c 2 B (Boozer)
␨
⫺c 1 ␺ ⬘ ⫹c 2 ␹ ⬘
共D5兲
,
for the symmetric case.
APPENDIX D: SYMMETRIC CASE
APPENDIX E: EFFECTS OF THE EÃB DRIFT
Here, we consider the symmetric case, in which
c 1 ⳵ B/ ⳵ ␪ B ⫹c 2 ⳵ B/ ⳵␨ B ⫽0 holds. It should be recalled that
the axisymmetric, poloidally symmetric, and helically symmetric cases correspond to c 1 ⫽0, c 2 ⫽0, and c 1 •c 2 ⫽0, respectively. As shown in Eq. 共A13兲, this case is also described
by c 1 ⳵ B/ ⳵ ␪ H ⫹c 2 ⳵ B/ ⳵␨ H ⫽0. Then, Eqs. 共17兲 and 共B2兲 yield
In the left-hand side of Eq. 共3兲, the collisionless orbit
operator V 储 contains only the part of particles’ parallel motion because other drift motions are neglected as higher-order
terms in the gyroradius expansion. Here, in order to consider
additional effects of the E⫻B drift on the neoclassical transport coefficients, we use the drift kinetic equation given by
c1
␴ Pa
␴ Ta
⫹c 2
⫽0,
␹⬘
␺⬘
c 1 B ␪(Boozer) ⫹c 2 B (Boozer)
␨
具 B 2典
c 1 B ␪(Boozer) ⫹c 2 B (Boozer)
␨
具 B 2典
V f a1 ⫺C La 共 f a1 兲 ⫽⫺vda •ⵜ f aM ⫹
ea
␴ Ua ⫹ 共 ⫺c 1 ␺ ⬘ ⫹c 2 ␹ ⬘ 兲 ␴ Xa ⫽0,
c
共D1兲
G Ua ⫹ 共 ⫺c 1 ␺ ⬘ ⫹c 2 ␹ ⬘ 兲
ea
G ⫽0.
c Xa
Thus, we find from Eq. 共18兲 that the viscosities and the viscosity coefficients associated with the symmetry direction
vanish
ea
具 BE 储 典
f ,
v 储B
Ta
具 B 2 典 aM
共E1兲
where the operator V⬅V 储 ⫹V E consists of the parallel motion part V 储 given by Eq. 共5兲 and the E⫻B drift part V E
defined by
V E ⬅vE •ⵜ⬅
cE s
ⵜs⫻B•ⵜ,
具 B 2典
共E2兲
with ⵜ taken for ( v , ␰ ) being fixed. The E⫻B drift operator
V E given by Eq. 共E2兲 has the same form as employed in the
DKES19,20 and by Taguchi.32 Here, following Taguchi,32 we
Downloaded 03 Mar 2009 to 133.75.139.172. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
4652
Phys. Plasmas, Vol. 9, No. 11, November 2002
H. Sugama and S. Nishimura
where it appears. 关Note that, by doing this replacement in Eq.
⫹
共40兲, definitions of both ␴ ⫹
1 and ␴ 3 coincide with those in
20
Rij and Hirshman even for E s ⫽0.] Also, Eqs. 共46兲 and
共54兲–共56兲 are available. Thus, using these formulas, we can
calculate dependence of the neoclassical coefficients on the
radial electric field. Figure 6 shows the normalized monoenergetic neoclassical viscosity coefficients M *
PP , M *
PT , and
* as a function of cE s / v , which are numerically obtained
M TT
in the same way as in Fig. 5. Here, ⑀ h ⫽0.05 and ␯ D / v ⫽3
⫻10⫺6 are used while other parameters are the same as in
Sec. IV. These parameters correspond to the 1/␯ regime for
the case of E s ⫽0. In Fig. 6, M P P ⯝⫺M PT ⯝M TT and their
reduction with increasing cE s / v are clearly seen. The
E s -dependent neoclassical transport coefficients for radial
fluxes and parallel currents can also be calculated in the same
way as in Appendix C.
FIG. 6. Poloidal and toroidal viscosity coefficients as a function of cE s / v
for ⑀ h ⫽0.05 and ␯ D / v ⫽3⫻10⫺6 . Other parameters are the same as in Sec.
IV. Curves with circles, crosses, and triangles represent M *P P , ⫺M *PT , and
M *P P , respectively.
neglect effects of vE •ⵜ 兵 兰 d 3 v f a1 关 1,( 21 m a v 2 ⫺ 52 T a ) 兴 其 in the
density and energy balance equations, and accordingly assume that the incompressibility conditions in Eq. 共8兲 and the
expressions for the local parallel flows in Eq. 共11兲 are still
valid. Then, using Eq. 共13兲, Eq. 共E1兲 is rewritten as
⫹
Vg a ⫺C La 共 g a 兲 ⫽H ⫺
a ⫹H a .
H⫺
a
共E3兲
is equal to
given by Eq. 共15兲 and H ⫹
Here,
a is
written in the same expression as in the right-hand side of
Eq. 共16兲 if we note that ␴ Ua , ␴ Xa , ␴ Pa , and ␴ Ta now include the E⫻B terms and are redefined by
H (l⫽1)
a
␴ Ua ⫽⫺m a v 2 P 2 共 ␰ 兲 B•ⵜ ln B⫺V E 共 m a v ␰ B 兲
⫽⫺V 共 m a v ␰ B 兲 ,
␴ Xa ⫽⫺ v 2 P 2 共 ␰ 兲
冉
冊
˜兲
b•ⵜ 共 BU
B
˜ ,
⫺V E
v␰U
2⍀ a
⍀a
␴ Pa ⫽⫺m a v 2 P 2 共 ␰ 兲 BP •ⵜ ln B⫺V E 共 m a v ␰ BP •b兲 ,
共E4兲
␴ Ta ⫽⫺m a v 2 P 2 共 ␰ 兲 BT •ⵜ ln B⫺V E 共 m a v ␰ BP •b兲 ,
⫺
respectively. The superscripts ⫹ and ⫺ in H ⫹
a and H a represent the symmetric and anti-symmetric parts with respect
to the transformation ( ␰ ,E s )→(⫺ ␰ ,⫺E s ), respectively. In
this appendix, we also assume the stellarator symmetry,
B(s, ␪ , ␨ )⫽B(s,⫺ ␪ ,⫺ ␨ ), which is satisfied by practically
all helical devices. Then, all neoclassical transport coefficients are even functions of E s .
Using what we have noted above, we can show that,
even if E⫻B drift term is included, Eqs. 共18兲–共44兲 in Secs.
II and III are still valid by replacing V 储 with V⬅V 储 ⫹V E
F. L. Hinton and R. D. Hazeltine, Rev. Mod. Phys. 48, 239 共1976兲.
S. P. Hirshman and D. J. Sigmar, Nucl. Fusion 21, 1079 共1981兲.
3
R. Balescu, Transport Processes in Plasmas 共2 volumes兲 共North-Holland,
Amsterdam, 1988兲.
4
W. Horton, Rev. Mod. Phys. 71, 735 共1999兲.
5
M. Wakatani, Stellarator and Heliotron Devices 共Oxford University Press,
Oxford, 1998兲, Chap. 7.
6
L. M. Kovrizhnykh, Nucl. Fusion 24, 851 共1984兲.
7
K. C. Shaing and S. A. Hokin, Phys. Fluids 26, 2136 共1983兲.
8
C. D. Beidler and H. Maaßberg, Plasma Phys. Controlled Fusion 43, 1131
共2001兲.
9
D. E. Hastings, W. A. Houlberg, and K. C. Shaing, Nucl. Fusion 25, 445
共1985兲.
10
J. Nu¨hrenberg and R. Zille, Phys. Lett. A 129, 113 共1988兲.
11
P. Garabedian and L.-P. Ku, Phys. Plasmas 6, 645 共1999兲.
12
S. Okamura, K. Matsuoka, S. Nishimura et al., Nucl. Fusion 41, 1865
共2001兲.
13
R. D. Hazeltine, Phys. Plasmas 15, 77 共1973兲.
14
R. D. Hazeltine and J. D. Meiss, Plasma Confinement 共Addison-Wesley,
Redwood City, CA, 1992兲, p. 113.
15
A. H. Boozer and G. Kuo-Petravic, Phys. Fluids 24, 851 共1981兲.
16
R. H. Fowler, J. A. Rome, and J. F. Lyon, Phys. Fluids 28, 338 共1985兲.
17
H. E. Mynick and W. N. G. Hitchon, Nucl. Fusion 26, 425 共1986兲.
18
C. D. Beidler, W. N. G. Hitchon, and J. L. Shoet, J. Comput. Phys. 72, 220
共1987兲.
19
S. P. Hirshman, K. C. Shaing, W. I. van Rij, C. O. Beasley, Jr., and E. C.
Crume, Jr., Phys. Fluids 29, 2951 共1986兲.
20
W. I. van Rij and S. P. Hirshman, Phys. Fluids B 1, 563 共1989兲.
21
J. Todoroki, J. Phys. Soc. Jpn. 59, 2758 共1990兲.
22
Y. Ogawa, T. Amano, N. Nakajima, N. Ohyabu, K. Yamazaki, S. P. Hirshman, W. I. van Rij, and K. C. Shaing, Nucl. Fusion 32, 119 共1992兲.
23
R. Kanno, N. Nakajima, H. Sugama, M. Okamoto, and Y. Ogawa, Nucl.
Fusion 37, 1463 共1997兲.
24
See, for example, p. 164 in Ref. 14.
25
E. C. Crume, Jr., C. O. Beasley, S. P. Hirshman, and W. I. van Rij, Phys.
Fluids 30, 1152 共1987兲.
26
K. C. Shaing, M. Yokoyama, M. Wakatani, and C. T. Hsu, Phys. Plasmas
3, 965 共1996兲.
27
K. C. Shaing, S. P. Hirshman, and J. D. Callen, Phys. Fluids 29, 521
共1986兲.
28
M. Coronado and H. Wobig, Phys. Fluids 29, 527 共1986兲.
29
K. C. Shaing and J. D. Callen, Phys. Fluids 26, 3315 共1983兲.
30
K. C. Shaing, S. P. Hirshman, and J. S. Tolliver, Phys. Fluids 29, 2548
共1986兲.
31
N. Nakajima and M. Okamoto, J. Phys. Soc. Jpn. 61, 833 共1992兲.
32
M. Taguchi, Phys. Fluids B 4, 3638 共1992兲.
1
2
Downloaded 03 Mar 2009 to 133.75.139.172. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp
Phys. Plasmas, Vol. 9, No. 11, November 2002
H. Sugama and W. Horton, Phys. Plasmas 3, 304 共1996兲.
H. Sugama, M. Okamoto, W. Horton, and M. Wakatani, Phys. Plasmas 3,
2379 共1996兲.
35
S. R. de Groot and P. Mazur, Nonequilibrium Thermodynamics 共NorthHolland, Amsterdam, 1962兲, Chap. 4.
36
A. H. Boozer, Phys. Fluids 26, 1288 共1983兲.
How to calculate the neoclassical viscosity, diffusion . . .
4653
S. Hamada, Nucl. Fusion 2, 23 共1962兲.
K. Y. Watanabe, N. Nakajima, M. Okamoto, K. Yamazaki, Y. Nakamura,
and M. Wakatani, Nucl. Fusion 35, 335 共1995兲.
39
O. Motojima, H. Yamada, A. Komori et al., Phys. Plasmas 6, 1843 共1999兲.
40
N. Nakajima, J. Todoroki, and M. Okamoto, J. Plasma Fusion Res. 68, 395
共1992兲.
33
37
34
38
Downloaded 03 Mar 2009 to 133.75.139.172. Redistribution subject to AIP license or copyright; see http://pop.aip.org/pop/copyright.jsp