Using PHP format - CRPP - EPFL

PHYSICS OF PLASMAS

VOLUME 7, NUMBER 4

APRIL 2000

Neoclassical transport coefficients for general axisymmetric equilibria in the banana regime C. Angionia) and O. Sauter Centre de Recherches en Physique des Plasmas, Association EURATOM-Confe´de´ration Suisse, Ecole Polytechnique Fe´de´rale de Lausanne, 1015 Lausanne, Switzerland

共Received 6 October 1999; accepted 3 January 2000兲 Using the standard approach of neoclassical theory, a set of relatively simple kinetic equations has been obtained, suited for an implementation in a numerical code to compute a related set of distribution functions. The transport coefficients are then expressed by simple integrals of these functions and they can be easily computed numerically. The code CQL3D 关R. W. Harvey and M. G. McCoy, in Proceedings of IAEA Technical Committee Meeting on Advances in Simulation and Modeling of Thermonuclear Plasmas, Montreal, 1992 共International Atomic Energy Agency, Vienna, 1993兲, pp. 489–526兴, which uses the full collision operator and considers the realistic axisymmetric configuration of the magnetic surfaces, has been modified to solve the bounce-averaged version of these equations. The coefficients have then been computed for a wide variety of equilibrium parameters, high-lighting interesting features of the influence of geometry at small aspect ratio. Differences with the most recent formulas for the ion neoclassical heat conductivity are pointed out. A set of formulas, which fit the code results, is obtained to easily evaluate all the neoclassical transport coefficients in the banana regime, at all aspect ratios, in general axisymmetric equilibria. This work extends to all the other transport coefficients, at least in the banana regime, the work of Sauter et al. 关O. Sauter, C. Angioni, and Y. R. Lin-Liu, Phys. Plasmas 6, 2834 共1999兲兴 which evaluates the neoclassical conductivity and all the bootstrap current coefficients. Formulas for arbitrary collisionality regime are proposed, obtained combining our results for the banana regime with the results of Hinton and Hazeltine 关F. L. Hinton and R. D. Hazeltine, Rev. Mod. Phys. 48, 239 共1976兲兴, adapted for small aspect ratio. © 2000 American Institute of Physics. 关S1070-664X共00兲02404-6兴

I. INTRODUCTION

method was also used to compute the like–particle collisions contribution on the viscosity matrix,4 and these results were applied in Ref. 10 to compute the bootstrap current coefficients at low aspect ratio. When compared with Ref. 2, these results are in very good agreement in general, but present a non-negligible error on the bootstrap current coefficients in which the contribution given by the like–particle collision operator is particularly important.2 For all the electron perpendicular transport coefficients the only formulas available at small aspect ratio are those in Ref. 11, valid in the banana regime, which use the analytical values of the transport coefficients at ⑀ ⫽1 and the values at large aspect ratio of Ref. 12 to obtain a set of formulas with a linear interpolation between these two limits, which should be valid also at small aspect ratio. In the more recent investigations on the ion thermal conductivity,5,6 the intermediate aspect ratio corrections show a difference with the results of Ref. 11 of almost a factor of 2. In this sense a complete investigation of the small aspect ratio corrections for all the neoclassical transport coefficients, taking into account the full collision operator, is necessary. It is well known that the neoclassical theory can not completely explain the perpendicular transport in tokamaks, however a precise computation is useful in order to allow a correct evaluation of the anomalous contribution by means of the comparison with the experimental data. This is becoming even more important with the recent improved

Recent improvements in neoclassical transport theory are almost completely dedicated to parallel transport. In particular, in Ref. 2, we have computed the neoclassical conductivity and all the bootstrap current coefficients, taking into account the full collision operator and including the advection parallel to the magnetic field, considering the realistic axisymmetric magnetic configuration of the flux surface. We have given relatively simple formulas valid for general axisymmetric equilibria and arbitrary collisionality regimes. For the other transport coefficients, improvements have been done only on the ion thermal conductivity in the banana regime4–6 and for various collision frequencies.7 Important recent new results have been presented in Ref. 8, solving a set of multispecies fluid equations: this enables one to compute the neoclassical conductivity, the bootstrap current and the particle and heat neoclassical fluxes, for arbitrary collisionality and aspect ratio. However, a set of equations must be solved in a specific code, which is not practically suited for numerical implementation in one-dimensional tokamak transport modelling codes. Moreover, all these latter results4–8 use an approximate collision operator, usually following the expansion method of Hirshman and Sigmar.9 This a兲

Author to whom correspondence should be addressed. Electronic mail: [email protected]

1070-664X/2000/7(4)/1224/11/$17.00

1224

© 2000 American Institute of Physics

Downloaded 31 Aug 2001 to 128.178.128.176. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp

Phys. Plasmas, Vol. 7, No. 4, April 2000

Neoclassical transport coefficients for general . . .

confinement modes of operation, with internal transport barriers and relatively small anomalous transport. In Sec. II we describe the approach to obtain the linear drift-kinetic equations suitable for implementation in a Fokker–Planck code and the expressions to compute the transport coefficient as simple integrals of the distribution functions. The related bounce-averaged equations in the banana regime are then obtained, and the Lorentz model is investigated analytically. In Sec. III we show the numerical results for the banana regime, computed with the Fokker– Planck code CQL3D,1 which solves the linearized drift kinetic bounce-averaged equation with the full collision operator and considering the realistic axisymmetric configuration of the magnetic surfaces. Some benchmarks are considered to validate the results, and the comparison with some previous numerical and analytical results is shown. In Sec. IV we give a set of formulas which fit our numerical results and allow to easily evaluate all the neoclassical transport coefficients in general axisymmetric equilibria for arbitrary aspect ratio and ion charge in the banana regime. Combined formulas for arbitrary collisionality regime are then proposed in the last subsection.

A. Transport coefficients

冉 冊 d␺ d␳

Qe d␺ Te d␳

冓 冔冓 冔 j 储B j 储SB ⫺ Te Te

⫺

I共 ␺ 兲具E B典n e * 具 B 2典

⫽

冋

e L11

e L12

e L13

e L14

e L21 e L31 e L41

e L22 e L32 e L42

e L23 e L33 e L43

e L24 e L34 e L44

册

冉 冊 1 ⳵pe 1 ⳵pi ⫹ pe ⳵␺ pe ⳵␺

⫻

1 ⳵Te Te ⳵␺

具E 储B典 具 B 2典

,

共1a兲

冉 冊 冋 册冉 冊 具 j 储 Ri B 典 Ti

Qi d␺ Ti d␳

⫽

i L12 i L22

Te A , T i e4

B e4 ⫽I 共 ␺ 兲 n e A i1 .

共2兲

The kinetic expressions of the conjugated neoclassical thermodynamic fluxes can be written in the following form: B en ⫽

冓冕

冉

dv共 v 储 bˆ•ⵜ ␥ en 兲 G e ⫺

兺n ␥ en A en f e0

冊冔

, 共3兲

n⫽1,2,3,4,

冓冕

冔

dv␤ in 共 G i ⫺ ␥ i2 A i2 f i0 兲 ,

⫺

具 E *B 典 具 B 2典

1 ⳵Ti Ti ⳵␺

.

␥ e1 ⫽

I 共 ␺ 兲v 储 , ⍀e

␥ e2 ⫽ ␥ e1

冉

␥ e3 ⫽

v 储 f se B, f e0

␥ e4 ⫽ ␥ e1

B2 , 具 B 2典

␤ i1 ⫽

q iv 储 B, Ti

n⫽1,2,

共4兲

␥ i2 ⫽

I 共 ␺ 兲v 储 v 2 5 . 2 ⫺ ⍀i 2 v Ti

v2

2 ⫺ v Te

冊

5 , 2

␤ i2 ⫽ v 储 bˆ•ⵜ 共 ␥ i2 兲 ,

冉

冊

The distribution functions G e and G i are related to the firstorder perturbations, f e1 and f i1 , by means of two suitable transformations. The first one follows Eqs. 共5.42兲 and 共5.43兲 of Ref. 3, leading to the functions H e and H i . Then the second one, simply G e ⫽H e ⫹⌺ n ␥ en A en f e0 and G i ⫽H i ⫹ ␥ i2 A i2 f i0 , allows to write the Linearized Drift Kinetic Fokker–Planck equations, Ref. 3, Eqs. 共5.21兲–共5.24兲, in the following simple form: l v 储 bˆ•ⵜG e ⫺C e0 共 G e 兲 ⫽⫺

q e K i共 ␺ 兲 2 具B 典 n iT e I共 ␺ 兲

i L11 i L21

B i1 ⫽⫺I 共 ␺ 兲 n e

in which we have introduced the functions

Our approach follows the standard neoclassical theory, in particular the one of Ref. 3. The flux surface averaged thermodynamic forces and their conjugated neoclassical fluxes are chosen as follows: ⌫e

A en and A in and the left-hand side vectors are their conjugated neoclassical fluxes, referred as B en and B in . ⌫ ␴ and Q ␴ are the particle and heat fluxes of species ␴, j 储 is the total parallel electric current, j 储 S is the Spitzer current, j 储 Ri is the ion contribution to the so-called ‘‘return current,’’ E ⬟E 储 * ⫹F i 储 (q i n i ) ⫺1 is the ‘‘effective electric field’’ and F i 储 is the friction force between ions and electrons. We have called ␳ a generic radial coordinate and 具 典 denotes the flux surface average. The function K i ( ␺ ), of the magnetic poloidal flux ␺, is related to the flux surface average 具 j 储 Ri B 典 by the equation: 具 j 储 Ri B 典 ⫽q i K i ( ␺ ) 具 B 2 典 . Note that ion and electron forces and fluxes are mutually dependent

B in ⫽

II. KINETIC THEORY

1225

兺n C e0l 共 ␥ en f e0 兲 A en ,

v 储 bˆ•ⵜG i ⫺C lii 共 G i 兲 ⫽⫺ ␤ i1 f i0 A i1 ⫺C lii 共 ␥ i2 f i0 兲 A i2 ,

共1b兲

The right-hand side vectors of Eqs. 共1a兲 and 共1b兲 are, respectively, the electron and ion forces, which will be referred as

共5兲 共6兲

l and C lii are, respecin which the collision operators C e0 tively, defined in Eqs. 共4.45兲 and 共1.19兲 of Ref. 3. Note that these equations are particularly useful, as all the coefficients of the thermodynamic forces in the source terms can be evaluated analytically13 l ⫺C e0 共 ␥ e1 f e0 兲 ⫽Z i ␯ e0 共 v 兲 ␥ e1 f e0 ,

Downloaded 31 Aug 2001 to 128.178.128.176. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp

1226

Phys. Plasmas, Vol. 7, No. 4, April 2000

C. Angioni and O. Sauter

l ⫺C e0 共 ␥ e2 f e0 兲 ⫽Z i ␯ e0 共 v 兲 ␥ e2 f e0

⫺ ␯ e0 共 v 兲 h l ⫺C e0 共 ␥ e3 f e0 兲 ⫽

冉 冊

v ␥ f , v Te e1 e0

q ev 储B f , T e e0

共7兲

numerically is the distribution function g en or g in . Note that in the banana regime the first term, computed analytically, gives directly the value of the transport coefficient at ⑀ ⫽1, when all the particles are trapped; the second term gives the reduction of transport due to the presence of passing particles. In Appendix A we show that Eqs. 共11兲 and 共12兲 satisfy the Onsager relations of symmetry as expected.

l ⫺C e0 共 ␥ e4 f e0 兲 ⫽Z i ␯ e0 共 v 兲 ␥ e4 f e0 ,

⫺C lii 共 ␥ i2 f e0 兲 ⫽⫺ ␯ i0 共 v 兲 h

冉 冊

v I 共 ␺ 兲v 储 f i0 , ⍀i v Ti

B. Banana regime: bounce-averaged equations

When the collision frequency ␯ ei is much smaller than the bounce frequency ␯ b , the distribution functions g en can be expanded as follows:

with

␯ e0 共 v 兲 ⫽

3 ␯ ei 共 v 兲 3 冑␲ v Te ⫽ , Z 4Z i ␶ e v 3

␯ i0 共 v 兲 ⫽

3 3 冑␲ v Ti

2& ␶ i v

3

,

0 ⫹ g en ⫽g en

h 共 x 兲 ⫽ 共 10⫺4x 2 兲 erf共 x 兲 ⫺10x erf⬘ 共 x 兲 , and, according to definitions given in Ref. 3 n i e 4 ln ⌳ 1 4 ⫽ 冑2 ␲ 1/2 3/2 , ␶e 3 me Te

n i Z 4i e 4 ln ⌳ 1 4 ⫽ 冑␲ 3/2 . ␶i 3 m 1/2 i Ti

共8兲

n⫽1,2,3,4.

冕 冕

共10a兲

v 储 bˆ•ⵜg i2 ⫺C lii 共 g i2 兲 ⫽⫺C lii 共 ␥ i2 f i0 兲 .

共10b兲

冓冕

⫺

i L1n ⫽

i L22 ⫽

冓冕

冓冕

i L21 ⫽⫺

l dv␥ em C e0 共 ␥ en f e0 兲

冓冕

冔

dv

2

,

␲

⫺␲

d␪p

B l C 共 g 0 兲 ⫽S en , 兩 v 储 兩 e0 en

d␪p

B l 0 C 共 g 兲 ⫽S in , 兩 v 储 兩 ii in

n⫽1,2,3,4,

共13a兲

n⫽1,2,

共13b兲

␥ S en ⬟2 ␲␴ 具 BCen 共 v ,B 兲 典 f e0 ,

n⫽1,2,3,4,

共13c兲

S i1 ⬟2 ␲␴

q i具 B 2典 f i0 , Ti

␥ S i2 ⬟2 ␲␴ 具 BCi2 共 v ,B 兲 典 f i0 ,

共13d兲

冔

C␴␥ n 共 v ,B 兲 ⬟C ␴l 共 ␥ ␴ n f ␴ 0 兲 / 共 v 储 f ␴ 0 兲 . n,m⫽1,2,3,4, 共11兲

n⫽1,2,

冔 冔 冓冕

g i1 l C 共␥ f 兲 , f i0 ii i2 i0

dv␥ i2 C lii 共 ␥ i2 f i0 兲 ⫺

␯ ei ␯b

where ␴ ⫽ v 储 / 兩 v 储 兩 and where we have introduced the set of functions C␴␥ n ( v ,B), defined as follows:

g em l dv C 共␥ f 兲 , f e0 e0 en e0

dvg in ␤ i1 ,

冓冕

冔

冋冉 冊 册

with

The kinetic definitions of the thermodynamic fluxes, Eqs. 共3兲 and 共4兲, written in terms of the functions g ␴ n , allow to readily identify the transport coefficients, introduced in Eq. 共1兲 e Lmn ⫽

␲

⫺␲

共9兲

v 储 bˆ•ⵜg i1 ⫺C lii 共 g i1 兲 ⫽⫺ ␤ i1 f i0 ,

␯ ei 1 g ⫹O ␯ b en

and analogously for the ion distribution functions. A somewhat standard derivation3,12 shows that the functions g ␴0 n are independent of the poloidal angle ␪ p , and that they are zero in the trapped particle region of velocity space. In the passing particle region, the functions g ␴0 n satisfy the following bounce-averaged equations:

Introducing the set of functions g ␴ n , in such a way that G ␴ ⫽⌺ n g ␴ n A ␴ n , Eqs. 共5兲 and 共6兲 can be linearly decoupled l l v 储 bˆ•ⵜg en ⫺C e0 共 g en 兲 ⫽⫺C e0 共 ␥ en f e0 兲 ,

冉 冊

共12兲 dv

冔

g i2 l C 共␥ f 兲 . f i0 ii i2 i0

In this way we have obtained a simple set of expressions to compute all the neoclassical transport coefficients, once we have solved the drift-kinetic equations, Eqs. 共9兲 and 共10兲, to obtain the distribution functions g en and g in . We see that every expression is composed of the sum of two terms: The first one is an integral that can be computed analytically and that, for some coefficients, is identically zero; the second one has an integrand in which the only term to be computed

The analytical expressions of these functions can easily ␥ ( v ,B) 典 be obtained from Eqs. 共7兲. Note that 具 BCe4 0 0 ␥ ⫽ 具 BCe1 ( v ,B) 典 , so that the functions g e1 and g e4 solve the same equation in the banana regime; in particular it follows that L44⫽L14 : This is a consequence of our choice of thermodynamic forces and fluxes. We see that at ⑀ ⫽1, when all 0 are the particles are trapped, the distribution functions g en zero everywhere, and the first terms in the expressions for the transport coefficients, Eqs. 共11兲 and 共12兲, give directly the entire coefficient. The code CQL3D has been modified to solve Eqs. 共13兲 in general axisymmetric equilibria and with the full collision operator.

C. Lorentz model

For the Lorentz gas model, Z i Ⰷ1, the set of Eqs. 共13a兲 is solved analytically.3,14 In fact, as collisions between electrons can be neglected, the collision operator can be approximated by the pitch-angle scattering operator:

Downloaded 31 Aug 2001 to 128.178.128.176. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp

Phys. Plasmas, Vol. 7, No. 4, April 2000

l C e0 ⫽ ␯ ei 共 v 兲 L⬟ ␯ ei 共 v 兲

Neoclassical transport coefficients for general . . .

1 ⳵ ⳵ 共 1⫺ ␰ 2 兲 2 ⳵␰ ⳵␰

and

␰⬟

1227

v储 . v

The solutions of Eq. 共13a兲 in this approximation can be written in the following form: v ␥ 共 v ,B 兲 典 f e0 具 BCen 2 ␯ ei 共 v 兲

0 ⫽⫺ ␴ g en

⫻

冕

␭c

␭

d␭ ⬘ H 共 ␭ c ⫺␭ 兲 , 具 共 1⫺␭ ⬘ B 兲 1/2典

共14兲

where ␭⬟(1⫺ ␰ 2 )/B, ␭ c ⬟1/B max and H(x) is the Heaviside function. Introducing Eq. 共14兲 in the expressions for the coefficients Eqs. 共11兲, all the electron transport coefficients can be written as integrals in the absolute value of velocity v e Lmn ⫽

4␲ 3

冕

⬁

v 4d v

0

冓

␥ em v储

␥ ⫻ 具 BCen 典 f e0 ,

冔

␥ Cen f e0 ⫹ ␲

冕␯ ⬁

0

v 4d v ␥ 具 BC em 典 ei 共 v 兲

n,m⫽1,2,3,4.

The integrals of v can easily be computed analytically, and the results have been compared with Ref. 3, Eqs. 共5.121兲, 共5.124兲–共5.130兲, finding a complete agreement. As the Lorentz model coefficients will be used not only as a benchmark for the results of CQL3D, but also to analyze the results of the code with the full collision operator and in different axisymmetric configurations, we report all the electron transport coefficients, which can be written in the following simple form: e ⫽⫺0.5Ld 关 B 20 具 B ⫺2 典 兴 f dt , L11

共15a兲

e ⫽0.75Ld 关 B 20 具 B ⫺2 典 兴 f dt , L21

共15b兲

e ⫽⫺ 138 Ld 关 B 20 具 B ⫺2 典 兴 f dt , L22 e e ⫽L34 ⫽⫺Lb f t , L31 e ⫽⫺ L33

e L32 ⫽0,

共15c兲

32 L 关 B ⫺2 具 B 2 典 兴 f t , 3␲ ␴ 0

共15d兲

e ⫽0.75Ld 关 B 20 具 B 2 典 ⫺1 兴 f t . L42

Ld ⫽

冉 冊

2 n e ␳ ep d␺ ␶e d␳

2

,

Lb ⫽I 共 ␺ 兲 n e ,

L␴ ⫽

n e q 2e ␶ e m eT e

B 20 , 共16兲

and where we have introduced two definitions for the trapped fraction f dt ⫽1⫺ 43 具 B ⫺2 典 ⫺1 I␭ , I␭ ⫽

冕

␭c

0

␭ ⬘ d␭ ⬘ . 具 共 1⫺␭ ⬘ B 兲 1/2典

␳ ␴ p共 ␳ 兲 ⫽

冑2m ␴ T ␴ v T␴ ⫽ , 兩 ⍀ ␴ p 兩 兩 q ␴ 兩 B p0 共 ␳ 兲

f t ⫽1⫺ 43 具 B 2 典 I␭ ,

共17兲

The second one, f t , is the usual definition for the trapped particle fraction.4 Note that the integral I␭ can easily be evaluated using the formulas in Ref. 15. The poloidal gyroradius ␳ ␴ p of species ␴ is given by

共18兲

like in Ref. 3, Eq. 共5.122兲, where the poloidal magnetic field B po ( ␳ ) is defined by B po ⬟(d ␺ /d ␳ )B 0 ( ␺ )/I( ␺ ), and B 0 ( ␺ ) is an arbitrarily chosen function introduced to normalize the magnetic field on a given flux surface. Note that the flux surface averaged integrals I 11 , I 13 , and I 33 which appear in the results of Ref. 3 can be reduced to only the two trapped fractions, Eq. 共17兲, with the following relations: I 11⫽ 34 关 B 20 具 B ⫺2 典 兴 f dt ,

e e ⫽L44 ⫽⫺0.5Ld 关 B 20 具 B 2 典 ⫺1 兴 f t , L41

with

FIG. 1. Transport coefficients Le1n and Le2n for an almost cylindrical equilibrium, computed by CQL3D in the approximation of the Lorentz model 共circles兲, and compared with the analytical results, Eq. 共15兲.

I 13⫽ 34 f t ,

2 I 33⫽ 34 关 B ⫺2 0 具B 典兴 f t . 共19兲

We see, therefore, that all the coefficients in the Lorentz model depend essentially on f dt and f t . We shall show in the next Sections that in the general case this property remains true, namely that all the equilibrium effects on the neoclassical transport coefficients are functions of only these two trapped fractions. III. NUMERICAL RESULTS A. Benchmarks

As the Lorentz model gives an analytical solution, it can be used as a first benchmark for the numerical results. In Fig. e e and L2n relative to 1 we show the transport coefficients L1n the sources S e1 and S e2 , Eqs. 共13兲, computed by CQL3D in the approximation of the Lorentz model: very good agreement is obtained for all ⑀. The coefficients in the Figure, e , are plotted normalized by the relative facindicated by L mn tors Ld or Lb given in Eq. 共16兲. This normalization for the electron transport coefficients is also kept in Figs. 2 and 4,

Downloaded 31 Aug 2001 to 128.178.128.176. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp

1228

Phys. Plasmas, Vol. 7, No. 4, April 2000

C. Angioni and O. Sauter

FIG. 2. Onsager symmetry is correctly respected by the numerical results. 共a兲 The transport coefficient Le13 共symbols兲, plotted vs f t , is well aligned with the formula 共14a兲 of Ref. 1 for the bootstrap current coefficient Le31 共solid line兲. The different symbols refer to different equilibria in all the figures. Note that the results given by the different equilibria are perfectly overlapped, as they are plotted vs f t . 共b兲 Transport coefficient Le12 共symbols兲 and Le12 共solid lines兲 computed with four different equilibria, plotted vs ⑀ 1/2. Note that when the complete coefficients are plotted vs the inverse aspect ratio, a strong dependence on equilibria appears at small aspect ratio.

and analogously in Fig. 3 for the ion transport coefficients. The complete definition of a set of dimensionless coefficients will be given in the next section. When the full collision operator is used, the symmetry of the transport matrix gives a second benchmark of the numerical results: indeed, each off-diagonal coefficient can be computed in two different ways, Lnm and Lmn . Note that we have already computed2 the neoclassical resistivity and the bootstrap current coefficients L3n , n⫽1,2,4. In Fig. 2共a兲 we e , computed solving the plot the results for the coefficient L13 kinetic equation with the source S e1 , Eqs. 共13a兲 and 共13c兲, n⫽1: they are perfectly aligned with the solid line given by Eq. 共14a兲 of Ref. 2, which fits the code results for the boote , hence computed with the strap current coefficient L31 source S e3 , Eqs. 共13a兲 and 共13c兲, n⫽3. The exact relations between the bootstrap current coefficients defined in Ref. 2 and the transport coefficients defined in this paper will be presented in the next section. In Fig. 2共b兲 we plot the two e e and L21 , computed considering four differcoefficients L12 ent equilibria, as shown in Fig. 1 of Ref. 2, and whose main specifications and related symbols, full or open, used in all e , obthe figures, are given in Table I. The coefficient L12 TABLE I. Equilibria specifications and related symbols used in the figures. Symbol

R mag 关m兴

R geo 关m兴

a 关m兴

k

␦

䊊 䊐 䉯 䉰

2.13 8.44 1.67 0.92

2.13 8.00 1.22 0.88

1.925 2.750 0.875 0.252

1.0 1.8 3.0 2.5

0 0.32 0.56 ⫺0.65

FIG. 3. The ion heat conductivity, transport coefficient Li22 , computed by CQL3D with an almost cylindrical equilibrium 共solid circles兲, divided by the flux surface average B 20 具 B ⫺2 典 , plotted vs the trapped particle fraction f dt and compared with formulas of Ref. 3, CH 82 共dashed–dotted line兲, Ref. 4, T 88 共dashed line兲, and Ref. 9 共modified兲, HHR 73 共solid line兲.

tained solving Eqs. 共13a兲 and 共13c兲, n⫽1, is plotted with e symbols, the coefficient L21 , obtained solving Eqs. 共13a兲 and 共13c兲, n⫽2, is plotted with solid lines: we find a very good agreement between the two coefficients, within 1% for ⑀ ⭓0.1. We see also that the behavior of the transport coefficient strongly depends on the equilibrium at small aspect ratio. Previous formulas, which give the transport coefficients with an expansion in powers of ⑀ 1/2, are correct only for almost cylindrical equilibria and are of practical interest in general equilibria only for ⑀ ⬍0.1: this must be taken into account when comparing with our results. It indicates that for each transport coefficient an appropriate geometrical pae e and L13 in Fig. 2共a兲, needs to be used rameter, like f t for L31 instead of ⑀, as it will be shown in the next subsection. B. Comparison with previous results and behavior at small aspect ratio

As we have said in Sec. I, the most recent investigations on perpendicular neoclassical transport were dedicated only to the ion thermal conductivity.5,6 In Fig. 3 we compare our i , obtained with an almost cyresults for the coefficient L22 lindrical equilibrium, with the results of Refs. 5, 6, and 11. As mentioned at the end of the previous paragraph, a correct geometrical parameter must be chosen to plot a given transport coefficient. As it can be inferred from the Lorentz model e , Eq. 共15b兲, and as it results for the electron coefficient L22 will be presented later, the ion heat conductivity is a function of the trapped fraction f dt , Eq. 共17兲. We find good agreement with the most recent formulas of Refs. 5 and 6. These results enable to finally resolve the discrepancy between the formulas given in Refs. 5 and 6, obtained with approximated collision operators. It turns out that the results with the full collision operator, CQL3D, are in between the previous re-

Downloaded 31 Aug 2001 to 128.178.128.176. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp

Phys. Plasmas, Vol. 7, No. 4, April 2000

FIG. 4. The transport coefficient Le22 , main contribution to the electron heat conductivity, computed by CQL3D with four different equilibria: 共a兲 The complete coefficient is plotted vs ⑀ 1/2, a strong dependence on the different equilibria appears at small aspect ratio; 共b兲 the coefficient is divided by an appropriate flux surface average B 20 具 B ⫺2 典 , and plotted vs the correct geometrical parameter, f td , which allows to perfectly align all the points of the different equilibria.

sults. Note that the plotted formula of Ref. 11 has been modified, keeping in the expression for the transport coefficient the flux surface average of the magnetic field, which were correctly computed in the reference to obtain the limit at ⑀⫽1, but then not taken into account in the final formulas. When different axisymmetric equilibria are considered in the numerical calculations, the transport coefficients present particular features of the influence of geometry at small aspect ratio, as already highlighted in Fig. 2共b兲. In Fig. e 4 we also show the electron coefficient L22 , computed with e is the four different equilibria of Table I. In Fig. 4共a兲, L22 1/2 plotted versus ⑀ : we see differences up to 30% already at e ⑀⫽0.15. In Fig. 4共b兲 we show the same coefficient L22 di2 ⫺2 vided by the flux surface average (B 0 具 B 典 ) and plotted versus the trapped particle fraction f dt , defined in Eq. 共17兲, as suggested by the results of the Lorentz model: the points are well-aligned at all f dt , i.e., at all ⑀. The same behavior is obtained for all the other transport coefficients: the coefficients must be normalized by a suitable flux surface average and a correct geometrical parameter must be used to encapsulate the effects of the different equilibria. Note that the definition of a new trapped particle fraction, f dt , Eq. 共17兲, is effectively necessary, as suggested by the Lorentz model, to correctly describe the geometrical behavior of certain coefficients, in particular all the particle and heat conductivities, for which the usual one, f t , turns out to be inadequate. From Fig. 4 it is also evident that, when plotted versus the correct parameter, the transport coefficients turn out to be very simple functions, almost proportional to the appropriate trapped fraction. This explains the relatively simple formulas of Ref. 2 for the electrical conductivity and the bootstrap current coefficients, and will be completely presented in the next section and shown in Fig. 5, for all the perpendicular

Neoclassical transport coefficients for general . . .

1229

␴ FIG. 5. Computed values of the dimensionless transport coefficients Kmn 共symbols兲, compared with the fitting formulas, Eqs. 共23兲 and 共24兲 共solid lines兲. 共a兲 Coefficients Ke11 plotted vs f dt 共solid symbols兲, and Ke14 plotted vs f t 共open symbols兲. 共b兲 Coefficients Ke12 and Ke21 , plotted vs f td 共solid symbols兲, and Ke24 plotted vs f t 共open symbols兲. 共c兲 Coefficient Ke22 plotted vs f dt 共solid symbols兲. 共d兲 Coefficient Ki22 plotted vs f dt 共solid symbols兲.

transport coefficients. Hence the study of the effects of plasma shape on the neoclassical transport can be simply obtained considering the dependence of the trapped fraction on plasma elongation and triangularity at a given aspect ratio. Both the two given expressions for the trapped fraction, f t and f dt , Eq. 共17兲, turn out to be almost independent of elongation, and increasing when decreasing triangularity. In this sense, in the neoclassical transport, at a given value of the aspect ratio, a highly triangular plasma shape is favorable for confinement, as shown in Figs. 2共b兲 and 4共a兲 共symbol 䉯兲, but not favorable for driving bootstrap current. IV. TRANSPORT COEFFICIENTS FORMULAS A. Analytical fits to the numerical results for the banana regime

Considering the results of the previous section, we can introduce a set of dimensionless electron and ion transport ␴ coefficients Kmn e e Lnm ⫽Ld 关 B 20 具 B ⫺2 典 兴 Knm 共 f dt 兲 , e e ⫽Lb Kn3 Ln3 共 f t兲,

n,m⫽1,2,

共20b兲

n⫽1,2,4,

e e ⫽Ld 关 B 20 具 B 2 典 ⫺1 兴 Kn4 Ln4 共 f t兲,

共20a兲

n⫽1,2,

共20c兲

e e 2 ⫽L␴ 关 B ⫺2 L33 0 具 B 典 兴 K33共 f t 兲 ,

共20d兲

i i 2 ⫽L␴i 关 B ⫺2 L11 0 具 B 典 兴 K11共 f t 兲 ,

共20e兲

i i ⫽Lib K12 L12 共 f t兲,

共20f兲

Downloaded 31 Aug 2001 to 128.178.128.176. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp

1230

Phys. Plasmas, Vol. 7, No. 4, April 2000

C. Angioni and O. Sauter

i i L22 ⫽Lid 关 B 20 具 B ⫺2 典 兴 K22 共 f dt 兲 ,

共20g兲

and analogously all their symmetrics, where Ld , Lb , and L␴ are defined by Eq. 共16兲; the ion normalization factors, Lid , Lib , and L␴i , are defined as follows: Lid ⫽

冉 冊

n i ␳ 2ip d ␺ ␶i d␳

2

,

Lib ⫽I 共 ␺ 兲 n i ,

L␴i ⫽

n i q 2i ␶ i m iT i

B 20 . 共21兲

The dimensionless coefficients are functions of only one suitable geometrical parameter, i.e., a trapped fraction, which completely encapsulates the effects of the various equilibria: ␴ can be in this way the code results for the coefficients Kmn fitted in terms of the appropriate trapped fraction, f t or f dt , as they perfectly overlap, regardless the equilibrium considered in the calculation, even highly noncircular and at small aspect ratio. Note that only in this way relatively simple formulas valid in general axisymmetric equilibria and at all aspect ratios can be given. We have already computed the neoclassical conductivity and the bootstrap current coefficients, in Ref. 2, solving the same kinetic equation of Eq. 共13a兲, n⫽3, and computing the transport coefficients with the same integrals given by Eq. 共11兲, n⫽3, and Eq. 共12兲, n ⫽2, m⫽1, which were first obtained using an adjoint formalism13 adapted to calculate only the bootstrap current: the general kinetic results of Sec. II show, however, that the adjoint formulation is not necessary. The relations between e and the neoclassical conductivthe transport coefficient L33 ity ␴ neo , Eq. 共13a兲 in Ref. 2, and between the dimensionless e , m⫽1,2,4, and the bootstrap current coefcoefficients K3m bs ficients L3m , Eqs. 共14兲 and 共15兲 in Ref. 2, read e ⫽ L33

␴ neo ⫺ ␴ Sptz 2 具B 典, Te

e bs K3m ⫽⫺L3m .

共22兲

i The coefficient K12 is related to the coefficient ␣, Eqs. 共17a兲 and 共17b兲 in Ref. 2, by Eq. 共25兲. 关Note: ⫺0.315 should be replaced by ⫹0.315 in Eq. 共17b兲.兴 We have run the code CQL3D with different equilibria and for the electron coefficients we have also varied the ion charge, to obtain the dependence on the effective charge Z. Our idea is that, at least for the electron transport coefficients, an effective charge approximation for multispecies cases should still be valid: collisions between electrons and main ions, or between electrons and impurity ions are almost of the same kind, involving basically the pitch-angle scattering. In any case, the comparison with the results of multispecies codes8 should enable one to determine the correct form of Z, instead of the usual definition of Z eff , to be used in our formulas, as already mentioned in Ref. 2. For the ions, the presence of one heavy impurity species leads to collisions between main ions and impurity ions which involve basically the pitch-angle scattering, and which are completely different from like–particle collisions. In this case, as shown in Ref. 16, the thermal conductivity computed as Z eff times the pure ion conductivity is underestimated. Using the results of Ref. 16, which uses the large aspect ratio limit of Ref. 17, we have generali , to include ized our formula for the transport coefficient L22 the effect of a single heavy impurity species in the Pfirsch– Schlu¨ter regime. We have also adapted the formula for the

bootstrap current coefficient ␣ in the banana regime, Eq. 共17a兲 in Ref. 2, to include the same effect, using the large aspect ratio limit of Ref. 17, and noting that at ⑀ ⫽1 not only the pure plasma coefficient, but also the impurity contribution must be equal to zero. The analytical fits to the results of CQL3D for all the transport coefficients not already computed in Ref. 2, valid in the banana regime, for arbitrary trapped fraction and Z, and the modified formula for the bootstrap current coefficient ␣, read as follows: e K11 共 f dt 兲 ⫽⫺0.5F 11共 f dt 兲 ,

共23a兲

e K12 共 f dt 兲 ⫽0.75F 12共 f dt 兲 ,

共23b兲

e e ⫽⫺0.5F 11共 f t 兲 , K14 共 f t 兲 ⫽K44

共23c兲

e K22 共 f dt 兲 ⫽⫺

冉

冊

13 & ⫹ F 共 f d兲, 8 2Z 22 t

共23d兲

e K24 共 f t 兲 ⫽0.75F 12共 f t 兲 ,

共23e兲

i i K22 共 f dt 兲 ⫽⫺F 22 共 f dt 兲 ,

共23f兲

冉

F 11共 X 兲 ⬟ 1⫹ ⫹

冉

1.6 0.6 X 3⫺ X 4, Z⫹0.5 Z⫹0.5

F 12共 X 兲 ⬟ 1⫹ ⫹

冉

冊

0.9 1.9 X2 X⫺ Z⫹0.5 Z⫹0.5

冊

0.6 0.95 2 X X⫺ Z⫹0.5 Z⫹0.5

0.3 0.05 4 X 3⫹ X , Z⫹0.5 Z⫹0.5

F 22共 X 兲 ⬟ 1⫺

共24a兲

冊

0.11 0.08 2 0.03 3 X ⫹ X , X⫹ Z⫹0.5 Z⫹0.5 Z⫹0.5

共24b兲

共24c兲

i F 22 共 X 兲 ⬟ 共 1⫺0.55兲共 1⫹1.54␣ I 兲 X

⫹ 共 0.75X 2 ⫺0.7X 3 ⫹0.5X 4 兲共 1⫹2.92␣ I 兲 ,

␣ 共 f t 兲 ⫽⫺K12i ⫽⫺

共24d兲

1⫺ f t 0.62⫹1.5␣ I , 0.53⫹ ␣ I 1⫺0.22 f t ⫺0.19f 2t 共25兲

where ␣ I ⫽n I Z I2 /n i Z 2i is the usual impurity strength parameter, and index I refers to the ion impurity species. The factorizations used in Eqs. 共23兲 and 共24兲 are such that the Lorentz limit (Z→⬁), the low ( f t →0) and the large aspect ratio ( f t →1) are easily recovered. Moreover the functions e e and K14 , as well F i j have values within 关0,1兴. Note that K11 e e as K12 and K24 have the same functional dependence on their respective trapped fraction. These relations can be considered as the extension to a general axisymmetric equilibrium at all aspect ratios of Eqs. 共6.28兲–共6.30兲 and Eq. 共6.47兲 in i , which is Ref. 3. We have also computed the coefficient L11 usually not considered, following the weak-coupling approximation, which neglects the force A i1 . For completeness, we give also the fit to the code results for the transport i coefficient K11 i K11 共 f t 兲 ⫽ 共 0.11⫹1.7f t ⫺1.25f 2t ⫹0.44f 3t 兲 ⫺1 ⫺1.

Downloaded 31 Aug 2001 to 128.178.128.176. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp

Phys. Plasmas, Vol. 7, No. 4, April 2000

Neoclassical transport coefficients for general . . .

1231

Note that Eq. 共2兲 allows to reduce the number of independent thermodynamic forces from 6 to 4, hence with only 4 conjugated thermodynamic fluxes. Taking the first 3 electron forces and the second ion force, whose conjugated fluxes have more direct physical meaning and more direct application in the fluid transport equations, the relations which connect fluxes with forces read as follows:

再

3

B en ⫽

兺 m⫽1

e Lnm ⫺

冎

i e L11 Ln4 1 Ti A 2 F Z i T e 关 I 共 ␺ 兲 n i 兴 2 em

i 1 T i e L12 ⫺ L A , F Z i T e n4 I 共 ␺ 兲 n i i2

n⫽1,2,3,

共26a兲

冎

共26b兲

3

i 1 L21 Le A B i2 ⫽ F I 共 ␺ 兲 n i m⫽1 4m em

兺

⫹

再

i L22 ⫺

i i L12 L21 1 Ti Le A , F Z 2i T e 关 I 共 ␺ 兲 n i 兴 2 44 i2

Ti

i e L11 L44

where F⬟1⫹

Z 21 T e 关 I 共 ␺ 兲 n i 兴 2

,

and Z i is the main ion charge number. The condition for the validity of the weak-coupling approximation is given by Eq. 共5.86兲 in Ref. 3, and is simply F⫺1Ⰶ1. Introducing the ␴ , this relation reads, consisdimensionless coefficients Kmn tently with the estimate given in Ref. 3, Table IV

冉 冊冉 冊

2& m e Zi mi

1/2

Ti Te

3/2 e i K14 K11 Ⰶ1.

共27兲

e i K11 turns out to be smaller The absolute value of the term K14 then 0.25, which confirms the validity of the weak coupling approximation in the banana regime. In this way Eq. 共26兲 are reduced to: 3

B en ⫽

B i2 ⫽

T

Li

i 12 e e Lnm A em ⫺ Ln4 A , 兺 Z T I ␺ 兲 n i i2 共 m⫽1 i e

i L21

共28a兲

3

兺

I 共 ␺ 兲 n i m⫽1

再

n⫽1,2,3,

i ⫹ L22 ⫺

e L4m A em

Ti

i i L12 L21

Z 2i T e 关 I 共 ␺ 兲 n i 兴 2

冎

e L44 A i2 .

共28b兲

In Fig. 5 we compare the code results for the dimensionless ␴ 共symbols兲 with the algebraic fortransport coefficients Knm mulas, Eqs. 共23兲 and 共24兲, which fit the data, 共solid lines兲. B. Combined formulas for all collisionality regimes

In order to compute the neoclassical transport coefficients at arbitrary collisionality regime, the nonbounceaveraged kinetic equations, Eqs. 共9兲 and 共10兲, must be solved. This, as already mentioned, has been done in Ref. 2, to compute the neoclassical resitivity and all the bootstrap current coefficients, using the code CQLP, which includes the advection parallel to the magnetic field, without any as-

FIG. 6. Dependence on collisionality for the transport coefficients ␴ neo , 共a兲 and Le31 , 共b兲, for different values of the trapped fraction f t , as given by Ref. 1 共solid lines兲, by Ref. 12, with the value at ␯ e * ⫽0, banana limit, corrected with the results of Ref. 1 共dashed lines兲, and still by Ref. 12, with also the collisional parameter rescaled by Eq. 共29兲 共dashed–dotted lines兲.

sumption on the ratio between the collision frequency and the bounce frequency. In order to strictly compare only the dependence on collisionality, in Fig. 6 we have plotted formulas of Ref. 2 共solid lines兲 and those of Ref. 3, Sec. VI F 共dashed lines兲, in which we have replaced the banana limit with the correct results of the code CQL3D. The neoclassical resistivity is shown in Fig. 6共a兲 and the bootstrap current e coefficient L31 in Fig. 6共b兲, for three values of the trapped fraction. At low aspect ratio there is a very good agreement, which falls down at larger values of the trapped fraction. This comes from the main approximation adopted to compute the banana-plateau regime, in Ref. 18, which neglects the energy scattering in the like-particle collision operator and which underestimates the neoclassical transport at low aspect ratio.7 However, for both the neoclassical resistivity e , and also for the and the bootstrap current coefficient L31 e coefficient L32 not shown here, the Ref. 3 formulas go down to zero at smaller values of ␯ e with respect to the rigorous * results of Ref. 2, with approximately the same behavior. When the collisional parameter ␯ ␴ , defined in Ref. 2, Eq. * 共18兲, is rescaled in terms of the trapped fraction with the simple transformation

␯␴ * , ␯␴f ⫽ * 1⫹7 f 2t

共29兲

Ref. 3 formulas allow an agreement within 20% for all the bootstrap current coefficients and the neoclassical resistivity 共dashed–dotted lines兲, comparing with Ref. 2. Hence, following the idea of Ref. 5, in which a formula valid for all collisionality regimes for the ion heat conductivity is obtained connecting a new banana limit, valid also at small aspect ratio, with the collisional dependence of Ref. 3, we propose to combine formulas of Ref. 3, Sec. VI F, adapted to

Downloaded 31 Aug 2001 to 128.178.128.176. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp

1232

Phys. Plasmas, Vol. 7, No. 4, April 2000

C. Angioni and O. Sauter

small aspect ratio, with the results of this paper in the limit at e ␯ ␴ ⫽0. The electron transport coefficients Kmn , m,n⫽1,2 * can be computed at arbitrary collisionality regimes as follows: e e K11 , 共 f dt , ␯ e 兲 ⫽H11

e e e K12 ⫺ 25 H11 , 共 f dt , ␯ e 兲 ⫽H12

*

*

e e e e ⫺5H12 ⫹ 254 H11 , K22 共 f dt , ␯ e 兲 ⫽H22 * e Hmn 共 f dt , ␯ e 兲 ⫽

*

共30a兲

e共 0 兲 d Hmn 共 f t , ␯ e ⫽0 兲

1⫹a mn 共 Z 兲 ␯ 1/2 ef

*

V. CONCLUSION

* ⫹b mn 共 Z 兲 ␯ e f 3

⫺

*

6

d mn 共 Z 兲 ␯ e f f dt 共 1⫹ f dt 兲

*

3

6

1⫹c mn 共 Z 兲 ␯ e f f dt 共 1⫹ f dt 兲

F PS ,

*

共30b兲 e(0) d Hmn ( f t , ␯ e ⫽0) *

where the banana limit coefficients readily evaluated using Eqs. 共23兲 and 共24兲, with e共 0 兲 e ⫽K11 H11 共 f dt 兲 ,

can be

e共 0 兲 e e H12 ⫽K12 共 f dt 兲 ⫹ 25 K11 共 f dt 兲 ,

共30c兲

e共 0 兲 e e e H22 ⫽K22 共 f dt 兲 ⫹5K12 共 f dt 兲 ⫺ 254 K11 共 f dt 兲 . e The coefficients K4n are given by e e , K41 共 f t , ␯ e 兲 ⫽H41

*

e e e K42 ⫺ 25 H41 , 共30d兲 共 f t , ␯ e 兲 ⫽H42

*

冋

e共 0 兲 H4n 共 f t , ␯ e ⫽0 兲 * 1⫹a 1n 共 Z 兲 ␯ 1/2 e f ⫹b 1n 共 Z 兲 ␯ e f *

e共 0 兲 e H41 ⫽K41 共 f t兲,

e共 0 兲 e e H42 ⫽K42 共 f t 兲 ⫹ 25 K41 共 f t兲,

e H4n 共 f t ,␯e 兲⫽

*

* d 1n 共 Z 兲 ␯ e f f 3t 共 1⫹0.8f 3t 兲 * ⫺ F共4兲 1⫹c 1n 共 Z 兲 ␯ e f f 3t 共 1⫹0.8f 3t 兲 PS * 1 ⫻ 共30e兲 1⫹ ␯ 2e f f 12 * t

册

where analogously 共30f兲

and with F PS ⫽1⫺

1

具 B 2 典具 B ⫺2 典

,

4兲 F 共PS ⫽ 具 B ⫺2 典具 B 2 典 ⫺1.

共30g兲

i is given by The ion thermal conductivity K22 i K22 共 f dt , ␯ i 兲 ⫽ *

i K22 共 f dt 兲

1⫹a 2 ␮ 1/2 i f ⫹b 2 ␮ i f

*

⫺

*

3 6 d 2 ␮ i f f dt 共 1⫹ f dt 兲 * 3

i 共30h兲 for the ion thermal conductivity K22 includes the effects of a single heavy impurity species in the Pfirsch– Schlu¨ter regime, according to Ref. 16, using the modified ion collisionality parameter ␮ i f and the factor H P , which take * into account the enhancement of main ion thermal transport due to the presence of the impurity species.16

6

1⫹c 2 ␮ i f f dt 共 1⫹ f dt 兲

H P F PS , 共30h兲

We have presented an approach for the neoclassical transport theory which allows to obtain simple equations suited for implementation in numerical codes in order to compute all the neoclassical transport coefficients. The code CQL3D, solving the bounce-averaged linearized drift-kinetic Fokker–Planck equation with the full collision operator, has been modified to calculate all these coefficients at all aspect ratios of various axisymmetric equilibria in the banana regime. We have shown that the limits at large and unit aspect ratio are correctly respected by the numerical results, as also the Onsager symmetry of the nondiagonal transport coefficients. Investigating the dependence of the coefficients on geometry parameters, we have shown that appropriate definitions of trapped fractions are required in order to encapsulate all the geometry effects in a single variable. In this way, a set of simple formulas can be obtained from the numerical results and allow the evaluation of any transport coefficient for every axisymmetric equilibrium and at all aspect ratios. Our formula for the ion thermal conductivity is in good agreement with the most recent evaluations of this coefficient,5,6 which, however, do not use the full collision operator, with errors at finite aspect ratio of about 10% by excess and by defect, respectively. For all the other perpendicular transport coefficients, in particular the electron thermal conductivity, our formulas are the only existing to date and to our knowledge, computed for general axisymmetric equilibria taking into account finite aspect ratio effects. The transport coefficients formulas, which fit the numerical results in the banana regime, are given by Eqs. 共20兲–共24兲 of Sec. IV A. Extension of this work is to compute the transport coefficients at all collisionality regimes: note that this has already been done for the neoclassical conductivity and the bootstrap current coefficients in Ref. 2. These results, compared with the ones of Refs. 3 and 5, have motivated us to propose combined formulas for all the other transport coefficients, valid for arbitrary collisionality regime, needed to correctly evaluate these coefficients over the whole plasma minor radius. The thermodynamic fluxes

*

with16 ␮ i f ⫽ ␯ i f (1⫹1.54␣ I ) and H P ⫽1⫹1.33␣ I (1 * * ⫹0.60␣ I )/(1⫹1.79␣ I ). The coefficients a mn (Z), b mn (Z), c mn (Z), and d mn (Z) are given in Appendix B, obtained by interpolation of the data given in Ref. 3, Table III for the electron coefficients and below Eq. 共6.133兲 for the ion coefficient. The dependence on ⑀ of the plateau-collisional terms of the formulas of Ref. 3, first computed in Ref. 19, has been rescaled on f t or f dt . Finite aspect ratio effects in these terms have been taken into account, like in Ref. 5, introducing the complete expression of the Pfirsch–Schlu¨ter geometrical factor, by means of F PS and F (4) PS , Eq. 共30g兲. Note that Eq.

B e1 ⫽⌫ e B e3 ⫽

d␺ , d␳

具 j 储B典 Te

⫺

B e2 ⫽

具 j 储SB 典 Te

,

Qe d␺ , Te d␳ B i2 ⫽

Qi d␺ , Ti d␳

where ⌫ e is the perpendicular electron particle flux, Q e is the electron perpendicular heat flux, j 储 and j 储 S are the parallel electric current and the Spitzer current, and Q i is the ion perpendicular heat flux, are given by Eqs. 共28兲, in the weak coupling approximation, whose validity is confirmed by Eq. 共27兲. Equations 共28兲 can be reordered, and the thermody-

Downloaded 31 Aug 2001 to 128.178.128.176. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp

Phys. Plasmas, Vol. 7, No. 4, April 2000

Neoclassical transport coefficients for general . . .

namic fluxes can be expressed directly in terms of the electron and ion temperature and density perpendicular gradients and the parallel electric field

⳵ ln n e ⳵ ln T e e e e ⫹ 共 Ln1 ⫹Ln2 B en ⫽Ln1 兲 ⳵␺ ⳵␺ ⫹

n⫽1,2,3, ⫽

冋

冉

1⫺R pe ⳵ ln n i EB e 具 储 典 ⫹L43 R pe ⳵␺ 具B典2

冊

1⫺R pe ␣ 2 e ⳵ ln T i i ⫹ L22 ⫹ L , R pe Z i 44 ⳵ ␺ where R pe ⬟p e / p and Z i is the main ion charge number. The ␴ , for general axiperpendicular transport coefficients Lmn symmetric equilibria and arbitrary collisionality regime, are given by Eqs. 共30兲 of Sec. IV B in terms of the trapped fractions f t or f dt , Eq. 共17兲, the collisionality parameter ␯ * and the effective charge number Z. The neoclassical conduce and ␣, are tivity and the bootstrap current coefficients, L3n connected with formulas of Ref. 2 by Eqs. 共22兲 and 共25兲.

We are grateful to Y.R. Lin-Liu and J. Vaclavik for interesting discussions. One of the authors, C.A., would also like to thank P. Helander for useful discussions. This work was partly supported by the Swiss National Science Foundation.

⫽

冔 冔

l e共 1 兲 dv␥ en C e0 . 共 ␥ em f e0 兲 ⬟Lnm

dv

dv

g em l C 共␥ f 兲 f e0 e0 en e0 ⫺ g em

f e0

冔

l C e0 共 ␥ en f e0 兲 ⫺

⫺ g em

冔

⫹ l ⫺ ⫺C e0 兲兴 关v 储 bˆ•ⵜg en 共 g en

冔

dv

⫽⫺

dv

1 ⫹ ⫺ ⫺ l ⫺ 兩 v 储 兩 bˆ•ⵜg en ⫺g em C e0 兲兴 关 ⫺g em 共 g en f e0

⫽⫺

dv

1 ⫹ l ⫹ ⫺ l ⫺ C e0 C e0 兲 ⫺g em 兲兴 关 ⫺g em 共 g en 共 g en f e0

⫽

dv

f e0

冔

冔

冔

1 g Cl 共g 兲 , f e0 em e0 en

which is a symmetric expression, using the self-adjointness of the collision operator. 共Note that we have used in this derivation the fact that the operator ⫺ v 储 bˆ•ⵜ is the adjoint of the operator v 储 bˆ•ⵜ.) Hence we can conclude that

⫽

冓冕 冓冕

冔 冔

dv

1 g Cl 共g 兲 f e0 em e0 en

dv

1 e共 2 兲 g C l 共 g 兲 ⫽Lnm , f e0 en e0 em

共A3兲

e . A somewhich shows the symmetry of the coefficients Lmn what analogous calculation can be performed for the ion coi i and L21 , which shows that the two given exefficients L12 pressions, Eq. 共12兲, satisfy the following relation:

共A4兲

consistently with the result in Ref. 3, Eq. 共5.99兲.

The expressions for the transport coefficients given by Eq. 共11兲 for the electrons and by Eq. 共12兲 for the ions satisfy the Onsager relations of symmetry, as expected.3 We begin with the electron case. In Eq. 共11兲, the first term, e(1) l ⬟ 具 兰 dv␥ em C e0 ( ␥ en f e0 ) 典 , is symmetric directly from Lmn the self-adjointness of the collision operator. Hence l dv␥ em C e0 共 ␥ en f e0 兲

冓冕 冓冕 冓冕 冓冕 冓冕 冓冕

i i ⫽⫺L21 , L12

APPENDIX A: ONSAGER SYMMETRY OF THE TRANSPORT COEFFICIENTS

冓冕 冓冕

共A2兲

⫽⫺

e共 2 兲 Lmn ⫽

ACKNOWLEDGMENTS

e共 1 兲 Lmn ⬟

l ⫹ ⫺ 兩 v 储 兩 bˆ•ⵜg en ⫺C e0 兲 ⫽0 共 g en

e共 2 兲 ⬟ Lmn

⳵ ln n e ⳵ ln T e Qi d␺ e e e ⫽ ␣ L41 ⫹ 共 L41 ⫹L42 B i2 ⫽ 兲 Ti d␳ ⳵␺ ⳵␺ e ⫹L41

l ⫺ l ⫹ 兩 v 储 兩 bˆ•ⵜg en ⫺C e0 兲 ⫽⫺C e0 共 g en 共 ␥ en f e0 兲 ,

l l l 关as C e0 ( ␥ en f e0 ) ⫺ ⫽C e0 ( ␥ en f e0 ) and C e0 ( ␥ en f e0 ) ⫹ ⫽0, n⫽1,2,3,4兴 we can perform the following derivation:

1⫺R pe e ⳵ ln n i 1⫺R pe e e Ln1 ⫹ 兲 共 Ln1 ⫹ ␣ Ln4 R pe ⳵␺ R pe

⳵ ln T i EB e 具 储 典 ⫻ ⫹Ln3 , ⳵␺ 具 B 2典

1233

共A1兲

e(2) l ⬟ 具 兰 dvg em / f e0 C e0 ( ␥ en f e0 ) 典 , we For the second term, Lmn shall rewrite it in a symmetric form. Introducing the following notation,3 for a generic function f (v), f ⫹ ⬟ 21 关 f ( ␴ ⫽⫹1)⫹ f ( ␴ ⫽⫺1) 兴 is its even part in ␴ ⫽ v 储 / 兩 v 储 兩 and f ⫺ ⬟ 21 关 f ( ␴ ⫽⫹1)⫺ f ( ␴ ⫽⫺1) 兴 is the odd part, so that

APPENDIX B: COEFFICIENTS FOR THE COMBINED FORMULAS OF SEC. IV B

The coefficients a mn (Z), b mn (Z), c mn (Z), and d mn (Z) for the electron transport coefficients are defined as follows: a 11共 Z 兲 ⫽

1⫹3Z , 0.77⫹1.22Z

a 12共 Z 兲 ⫽

0.72⫹0.42Z , 1⫹0.5Z

b 11共 Z 兲 ⫽

1⫹1.1Z , 1.37Z

b 22共 Z 兲 ⫽

Z , ⫺3⫹5.32Z

共B1a兲 a 22共 Z 兲 ⫽0.46,

b 12共 Z 兲 ⫽

1⫹Z , 2.99Z 共B1b兲

Downloaded 31 Aug 2001 to 128.178.128.176. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp

1234

Phys. Plasmas, Vol. 7, No. 4, April 2000

c 11共 Z 兲 ⫽

0.1⫹0.34Z , 1.65Z

c 12共 Z 兲 ⫽

6

0.27⫹0.4Z , 1⫹3Z 共B1c兲

0.22⫹0.55Z , c 22共 Z 兲 ⫽ ⫺1⫹7Z d 11共 Z 兲 ⫽

C. Angioni and O. Sauter

0.23Z , ⫺1⫹3.85Z

0.22⫹0.38Z d 12共 Z 兲 ⫽ , 1⫹6.1Z

0.25⫹0.05Z d 22共 Z 兲 ⫽ . 1⫹0.82Z

共B1d兲

For the ion thermal conductivity, the coefficients are a 2 ⫽1.03,

b 2 ⫽0.31,

c 2 ⫽0.22,

d 2 ⫽0.175.

共B2兲

O. Sauter, C. Angioni, and Y. R. Lin-Liu, Phys. Plasmas 6, 2834 共1999兲. S. P. Hirshman and D. J. Sigmar, Nucl. Fusion 21, 1079 共1981兲. 3 C. S. Chang and F. L. Hinton, Phys. Fluids 25, 1493 共1982兲. 4 M. Taguchi, Plasma Phys. Controlled Fusion 30, 1897 共1988兲. 5 C. Bolton and A. A. Ware, Phys. Fluids 26, 459 共1983兲. 1

2

W. A. Houlberg, K. C. Shaing, S. P. Hirshman, and M. C. Zarnstorff, Phys. Plasmas 4, 3230 共1997兲. 7 S. P. Hirshman and D. J. Sigmar, Phys. Fluids 19, 1532 共1976兲. 8 S. P. Hirshman, Phys. Fluids 31, 3150 共1988兲. 9 R. D. Hazeltine, F. L. Hinton, and M. N. Rosenbluth, Phys. Fluids 16, 1645 共1973兲. 10 M. N. Rosenbluth, R. D. Hazeltine, and F. L. Hinton, Phys. Fluids 15, 116 共1972兲. 11 R. W. Harvey and M. G. McCoy, in Proceedings of International Atomic Energy Agency Technical Committee Meeting on Advances in Simulation and Modeling of Thermonuclear Plasmas, Montreal, 1992 共International Atomic Energy Agency, Vienna, 1993兲, pp. 489–526; J. Killeen, G. D. Kerbel, M. G. McCoy, and A. A. Mirin, Computational Methods for Kinetic Models of Magnetically Confined Plasmas 共Springer-Verlag, New York, 1986兲. 12 F. L. Hinton and R. D. Hazeltine, Rev. Mod. Phys. 48, 239 共1976兲. 13 Y. R. Lin-Liu, private communication 共see Ref. 29 in Ref. 8兲. 14 P. H. Rutherford, Phys. Fluids 13, 482 共1970兲. 15 Y. R. Lin-Liu and R. L. Miller, Phys. Plasmas 2, 1666 共1995兲. 16 C. S. Chang and F. L. Hinton, Phys. Fluids 29, 1493 共1986兲. 17 S. P. Hirshman, Phys. Fluids 19, 155 共1976兲. 18 F. L. Hinton and M. N. Rosenbluth, Phys. Fluids 16, 836 共1973兲. 19 J. M. Rawls, M. S. Chu, and F. L. Hinton, Phys. Fluids 18, 1160 共1975兲.

Downloaded 31 Aug 2001 to 128.178.128.176. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/pop/popcr.jsp