J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)233–236
000–000

Effect of Magnetic Axis Shift on Neoclassical Transport
in Helical TorusSANUKI Heiji and TODOROKI Jiro
National Institute for Fusion Science, Toki 509-5292, Japan(Received: 9 December 2003 / Accepted: 4 February 2004)

Abstract
Neoclassical transport for large helical device (LHD) configurations is studied by solving the bounce-averaged
Fokker-Planck equation. Numerical code employed in the present paper (CHD1) is much faster and more efficientthan existing transport codes. Effects of the magnetic axis shift on the mono-energetic transport coefficients are studied
in detail for the LHD configurations, revealing that a strong inward shift of the magnetic axis can reduce remarkably
the neoclassical ripple transport.
Keywords:
neoclassical transport, bounce averaged kinetic equation, magnetic axis shift, LHD

1. Introduction

averaged orbit theory mentioned above. The reduction ofneoclassical transport, particularly in the low collisionality
region is one of the most important issues for any reactor
scale heliotron devices. In the previous paper [10], the effectof radial electric field on neoclassical transport has also beenanalyzed in detail when a boundary layer and/or resonanceare present. Furthermore, great efforts have been undertaken
to optimize magnetic configurations associated with thereduction of neoclassical transport [7,11]. In this paper, westudy the neoclassical transport in typical configurations suchas inward-shifted configuration of LHD device by solving thebounce averaged Fokker-Planck equation (CHD1). In thepresent study the magnetic field are calculated by using theMAGN code for fixed coil currents. Then, the transportcoefficients are evaluated for a realistic magnetic field relevantto the operation parameters of the LHD experiment.
The bounce-averaged Fokker-Planck equation together
with the numerical code (CHD1) is presented in Section 2. In
section 3, the numerical results of the transport for the LHD
configuration are presented. The last section is devoted to thesummary and discussions.

During more than two decades, a number of analyticaland numerical approaches have been developed to study theneoclassical transport in toroidal helical systems. Analyticaldescriptions of neoclassical transport are effective because
they give clear physical insight associated with varioustransport and/or loss mechanisms. However, the derivation of
the most of these analytical formulae are originally based ona number of assumptions such as so-called frequency
ordering, which leads somewhat disconnected expressions forthe transport coefficients in different collisionality regimes.
On the other hand, lots of numerical methods have been
discussed to avoid some of these assumptions and todetermine the transport coefficients over the wide range ofcollisionality regime. Among these studies there are theMonte-Calro simulations [1,2], the methods based on thebounce-averaged Fokker-Planck equation, namely, FPSTEL[3], FLOCS [4], and numerical code by solving the drift
kinetic equation, DKES [5]. A general solution of the rippleaveraged kinetic equation; GSRAKE is also developed [6].
The GSRAKE makes it possible to analyze the cases
including the contributions due to E × B and ∇B drifts without
the assumption, Ω E×B >> Ω ∇B.
An analytic representation based on the longitudinal
adiabatic invariant for the general magnetic configuration
instead of a simple model magnetic field frequently employed,
has been presented in a convenient form for the purpose ofnumerical calculation in realistic magnetic configurations [7].
In the previous studies [8,9], a neoclassical transport in ahelical torus based on the bounce averaged Fokker-Planck
equation has been developed by using the generalized bounce-

2. Bounce-averaged kinetic equation
Considerable efforts have been paid on the bounceaveraged drift kinetic equation to describe the neoclassicalparticle and energy transport in toroidal helical systems. Sinceboth fast and slow drift motions are involved in the toroidal
helical systems, some time average is frequently employed in
the formalism. The time average is usually performed overthe periodic bounce motion of particle trapped within local©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: sanuki@nifs.ac.jp1233

Sanuki H. et al., Effect of Magnetic Axis Shift on Neoclassical Transport in Helical Torus

ripples of helical magnetic field. Recently, the definition oftime average is extended to so-called “ripple average” inwhich the average is taken over a characteristic time forparticles to traverse a local ripple regardless of whether aparticles bounces or not. To discuss this formalism, we here
employ the Boozer’s coordinates (ψ, ϑ , φ), ψ being thetoroidal flux. We consider the helical torus with toroidal
period N and the rotational transform per period is assumed
to be small, ι /N << 1. Namely, ϑ and all functions of thisvariable are considered constant with respect to the time
average and the average is carried out along φ . The motionof charged particles in case of this assumption is describedby the longitudinal adiabatic invariant

J ( E, µ , ψ , ϑ ) =12π

Bφrelations,±c

where h (ϑ ) = (h (ϑ) + h (–ϑ))/2 and h (ϑ ) = (h± (ϑ) –h± (–ϑ))/2. We note that D11 and D13 correspond to the particle
transport coefficient and the bootstrap current, respectively.

φThe explicit adiabatic invariant forms for ripple trapped andpassing particles are given in Refs. [7,10]. Hence, we canwrite the averaged kinetic equation in the form

1 ∂J ∂f 1 ∂J ∂f∂f−+ σeVφe ∂ϑ ∂ψ e ∂ψ ∂ϑ∂E∫= C ( f , f ) ≡ C( f , f )Bdφ ,v//(2)–where C is the bounce averaged collision operator. Here, thecontribution from the toroidal loop voltage Vφ is also includedin Eq. (2) and σ = ±1 corresponds to the sign of the parallel
velocity for passing particles.
We now put f = f0(K, ψ) [1 + h(K, λ, σ, ψ, ϑ)], where K
is the kinetic energy, µ is the magnetic moment and f0 is thelocal Maxwellian with density n(ψ) and temperature T(ψ), andintroduce the even part h+ and the odd part h– of h withrespect to the sign of parallel velocity, where h+ ≡ [h(σ = 1)+ h(σ = –1)]/2 and h– ≡ [h(σ = 1) – h(σ = –1)]/2. Then, thelinearized equation can be written in the following form

Ch + + Hh − + Dh + = S + ,Ch − + Hh + + Dh − = S − , (λ < λc )(3)

Ch + + Dh + = S + , h − = 0, (λ > λc ),

(4)

with

S+ =eVφ1 ∂J ∂ ln f0, S− =,e ∂ϑ ∂ψTC = ν ( v)
H =ι∂ ⎛∂⎞⎜λ M ⎟ , M =⎝∂λ∂λ ⎠∫

mv // Bφdφ ,(5)∂dΦ E
∂J ∂, D = ωE, ωE =.
∂ϑdψ
∂K ∂ϑ

(6)

B±±S

In the previous paper [7], the dependence of the particle
confinement on the magnetic axis shift, the shaping ofmagnetic surface and the pitch modulation of helical windings
has been discussed for a LHD geometry. It is recently shownin [2] that the diffusion coefficients of the inward shifted
configuration are about three to ten times smaller than thoseof standard configuration of LHD. In this paper, we used amagnetic field calculated by the MAGN code for the LHD
parameters. We here consider three configurations withinwardly shifted magnetic axis from the center characterizedby ∆ (= 0 cm, –10 cm, –20 cm). The radial profiles of theFourier spectrum of magnetic field in the magnetic coordinate
are shown in Fig. 1 for typical three configurations (0 cm,–10 cm, –20 cm). As was shown in Fig. 1, the magnetic fieldcomponents such as εhl (ρ), εt (ρ), εhl–1(ρ), εhl+1(ρ) and εhl –2 (ρ)
change the radial profiles by shifting the magnetic axis
inwardly. In particular, the components with (1,10) and (3,10)change both its strength and polarity, and it may reduce
effectively the helical ripple strength, leading a significant
reduction of neoclassical transport. Typical results for thecollisionality dependence of the diffusion coefficient (D11) andthe bootstrap current (D 13) in the case of the LHD three
configurations with inwardly shifted magnetic axis are plottedin Fig. 2 (a) and Fig. 2 (b) for fixed value of normalized
electric field, RωE /v. The parameters used in the calculationsare ρ (= r/a) = 0.5, B0 = 3 T, ι = 0.48, εt = 0.074, and εh =0.059, and RωE /v = 1.0 × 10-3. Here, we note that the results
for the case of ∆ = –20 cm as shown in Fig. 2 corresponds to
the results for RωE /v = 1.0 × 10-3 in Fig. 1 of Ref. [10]. It isfound from Fig. 2 (a) that the diffusion coefficient decreases
as the magnetic axis is shifted inwardly. The result for ∆ =–20 cm is smaller about 4 times than the result for ∆ = 0 cm
in the 1/υ region but the difference between these three cases
becomes smaller in the further low collisionality regime, socalled υ -regime. As was discussed in Refs. [2] and [10], theeffect of radial electric field dominates the transport more thanthe effect of magnetic axis shift provided the electric field islarge enough. Also, the local maximum of the diffusion
coefficient is shifted toward lower collisionality regime as ∆increased. Figure 2 (b) indicates that the bootstrap current
does not monotonically increases as the collision frequency
decreases and it saturates with the local maximum at somecollision frequency (at the transition). This transition point
sensitively depends on the electric field strength and it is also
shifted toward low collisionality region as ∆ increased. The
bootstrap current increases in the plateau and 1/ν regions but
decreases in the low collisionality regime so called ν-regimeas ∆ increased. It should be noted that the peaked profile ofφ//±3. Effect of magnetic axis shift∫ P dφ, P = mv B + eψ . (1)φ∫D11 ∝ ( S − c, h − c ), D13 ∝ ( S + S, h + c ), ( a, b) = abdλ , (7)Here, only the pitch angle scattering term, which is dominantparticularly in the low collisionality region is retained in thecollision operator. Also, the ϑ -dependent part of ∂ J/∂ K is kept
in the D operator for the sake of the conservation law. The
numerical scheme to solve Eqs. (3) and (4) has been discussed
in the previous paper [7]. After solving Eqs. (3) and (4), thetransport coefficients are represented by the following2234

Sanuki H. et al., Effect of Magnetic Axis Shift on Neoclassical Transport in Helical Torus

Fig. 2Fig. 1

Collisionality dependence of the diffusion coefficient(D11) [2(a)] and the bootstrap current (D13), [2(b)] for ∆ =0 cm, –10 cm and –20 cm.

dimensional space (λ, ϑ). As for the numerical scheme, thefunction in the ϑ-direction is expressed in terms of Fourier
expansion, and the meshes with variable distances areconcentrated in the boundary layer. So, the structure of thesolution near the boundary layer and the case of helical and/or toroidal resonances can be well analyzed as shown inRef. [10]. A symmetric band matrix solver is used. Then, thecomputation time is dramatically reduced, particularly, in thelow collisionality region compared with the case of DKES
code. On the basis of this improvement of the numerical
scheme, the calculation can be extendable over wide range ofcollision frequency, especially in the very low collisionality
region as shown in Fig. 2, compared with the previous studies[2,11]. Although the relation Ω E×B >> Ω ∇B is assumed in theDKES code, there is no such limitation in CHD1 code, which
is discussed, in the present study. Detailed and realistic
discussions associated with the effects of the magnetic axis
shift and the electric field on the transport of the LHD
configuration in low collisionality region will be discussed in

Radial profiles of magnetic field spectra in vacuum configurations. (a) ∆ = 0 cm , (b) –10 cm, and (c) –20 cm.
Here, epsh, epst, epsh-, epsh+ and epsh-- indicate εh (ρ),εt (ρ), εhl –1 (ρ), εhl +1 (ρ) and εhl –2 (ρ), respectively.

D13 around the maximum as indicated by the solid line inFig. 2(b) contains somewhat ambiguity because there are noexplicit data around the transition point in this figure.

4. Summary and discussions
The theory and numerical code solving the bounce
averaged Fokker-Planck equation has been developed in two3235

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a separate paper. In finite beta plasmas, the deformation of
the magnetic surface due to the Pfirsch-Schluter current may
degradate the particle confinement. Discussions on thisproblem await further investigations.
The authors acknowledge Mr. T. Nakagawa of theMatsushita Electric Industrial Co., Ltd. for his continuouscontributions to numerical works.

References[1] A.H. Boozer and G. Kuo-Petravic, Phys. Fluids 23, 851(1981).
[2] A. Wakasa et al., J. Plasma Fusion Res. Sries 4, 408(2001).
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