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 efficient than 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

SANUKI  Heiji

TODOROKI  Jiro

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 efficient than 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, LH

Sanuki Heiji

Todoroki Jiro

CiteSeerX

Effect of Magnetic Axis Shift on Neoclassical Transport in Helical Torus

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 efficient than 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.journal articl

SANUKI,  Heiji

TODOROKI,  Jiro

NIFS-Repository (National Institute for Fusion Science)

©2004 by The Japan Society of PlasmaScience and Nuclear Fusion ResearchJ. Plasma Fusion Res. SERIES, Vol. 6 (2004) 000–0001Effect of Magnetic Axis Shift on Neoclassical Transportin Helical TorusSANUKI Heiji and TODOROKI JiroNational Institute for Fusion Science, Toki 509-5292, Japan(Received: 9 December 2003 / Accepted: 4 February 2004)AbstractNeoclassical transport for large helical device (LHD) configurations is studied by solving the bounce-averagedFokker-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 studiedin detail for the LHD configurations, revealing that a strong inward shift of the magnetic axis can reduce remarkablythe neoclassical ripple transport.Keywords:neoclassical transport, bounce averaged kinetic equation, magnetic axis shift, LHD1. IntroductionDuring 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 becausethey give clear physical insight associated with varioustransport and/or loss mechanisms. However, the derivation ofthe most of these analytical formulae are originally based ona number of assumptions such as so-called frequencyordering, which leads somewhat disconnected expressions forthe transport coefficients in different collisionality regimes.On the other hand, lots of numerical methods have beendiscussed 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 driftkinetic equation, DKES [5]. A general solution of the ripple-averaged kinetic equation; GSRAKE is also developed [6].The GSRAKE makes it possible to analyze the casesincluding the contributions due to E × B and ∇B drifts withoutthe assumption, ΩE×B >> Ω∇B. An analytic representation based on the longitudinaladiabatic invariant for the general magnetic configurationinstead 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-Planckequation has been developed by using the generalized bounce-averaged orbit theory mentioned above. The reduction ofneoclassical transport, particularly in the low collisionalityregion is one of the most important issues for any reactorscale 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 undertakento 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 togetherwith the numerical code (CHD1) is presented in Section 2. Insection 3, the numerical results of the transport for the LHDconfiguration are presented. The last section is devoted to thesummary and discussions.2. Bounce-averaged kinetic equationConsiderable efforts have been paid on the bounce-averaged drift kinetic equation to describe the neoclassicalparticle and energy transport in toroidal helical systems. Sinceboth fast and slow drift motions are involved in the toroidalhelical systems, some time average is frequently employed inthe formalism. The time average is usually performed overthe periodic bounce motion of particle trapped within localCorresponding author’s e-mail: sanuki@nifs.ac.jp233J. Plas a Fusion es. SERIES, Vol. 6 (2004) 233–236Sanuki H. et al., Effect of Magnetic Axis Shift on Neoclassical Transport in Helical Torus2ripples 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 hereemploy the Boozer’s coordinates (ψ, ϑ, φ), ψ being thetoroidal flux. We consider the helical torus with toroidalperiod N and the rotational transform per period is assumedto be small, ι /N << 1. Namely, ϑ and all functions of thisvariable are considered constant with respect to the timeaverage and the average is carried out along φ. The motionof charged particles in case of this assumption is describedby the longitudinal adiabatic invariantJ E P d P mvBBe( , , , ) ,  .//µ ψ ϑ π φ ψφ φφφ= = +∫12 (1)The explicit adiabatic invariant forms for ripple trapped andpassing particles are given in Refs. [7,10]. Hence, we canwrite the averaged kinetic equation in the form1 1eJ feJ feV fEC f f C f f Bvd∂∂∂∂ −∂∂∂∂ +∂∂= ≡ ∫ϑ ψ ψ ϑσφφ( , ) ( , ) ,//(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 parallelvelocity for passing particles.We now put f = f0(K, ψ) [1 + h(K, λ, σ, ψ, ϑ)], where Kis 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 formCh Hh Dh SCh Hh Dh S c+ − + +− + − −+ + =+ + = <,( ) ,  λ λ (3)Ch Dh S h c+ + + −+ = = >, , ( ),    0 λ λ (4)withSeJ f S eVTC M Mm BBd+ −=∂∂∂∂ ==∂∂∂∂⎛⎝⎜⎞⎠⎟ = ∫1 0ϑ ψν λ λ λ φφφln,  ,  ( ) ,  ,//v v (5)H D JKddE EE=∂∂ =∂∂∂∂ =ι ϑ ω ϑ ω ψ, ,Φ. (6)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 keptin the D operator for the sake of the conservation law. Thenumerical scheme to solve Eqs. (3) and (4) has been discussedin the previous paper [7]. After solving Eqs. (3) and (4), thetransport coefficients are represented by the followingrelations,D S h D S h a b abdc c S c11 13∝ ∝ =− − + + ∫( , ), ( , ), ( , ) ,λ (7)where h±c(ϑ) = (h±(ϑ) + h±(–ϑ))/2 and h±S(ϑ) = (h±(ϑ) –h±(–ϑ))/2. We note that D11 and D13 correspond to the particletransport coefficient and the bootstrap current, respectively.3. Effect of magnetic axis shiftIn the previous paper [7], the dependence of the particleconfinement on the magnetic axis shift, the shaping ofmagnetic surface and the pitch modulation of helical windingshas been discussed for a LHD geometry. It is recently shownin [2] that the diffusion coefficients of the inward shiftedconfiguration 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 LHDparameters. 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 coordinateare 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 axisinwardly. In particular, the components with (1,10) and (3,10)change both its strength and polarity, and it may reduceeffectively the helical ripple strength, leading a significantreduction of neoclassical transport. Typical results for thecollisionality dependence of the diffusion coefficient (D11) andthe bootstrap current (D13) in the case of the LHD threeconfigurations with inwardly shifted magnetic axis are plottedin Fig. 2 (a) and Fig. 2 (b) for fixed value of normalizedelectric 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 resultsfor the case of ∆ = –20 cm as shown in Fig. 2 corresponds tothe results for RωE/v = 1.0 × 10-3 in Fig. 1 of Ref. [10]. It isfound from Fig. 2 (a) that the diffusion coefficient decreasesas the magnetic axis is shifted inwardly. The result for ∆ =–20 cm is smaller about 4 times than the result for ∆ = 0 cmin the 1/υ region but the difference between these three casesbecomes 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 diffusioncoefficient is shifted toward lower collisionality regime as ∆increased. Figure 2 (b) indicates that the bootstrap currentdoes not monotonically increases as the collision frequencydecreases and it saturates with the local maximum at somecollision frequency (at the transition). This transition pointsensitively depends on the electric field strength and it is alsoshifted toward low collisionality region as ∆ increased. Thebootstrap current increases in the plateau and 1/ν regions butdecreases in the low collisionality regime so called ν-regimeas ∆ increased. It should be noted that the peaked profile of234Sanuki H. et al., Effect of Magnetic Axis Shift on Neoclassical Transport in Helical Torus3D13 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 discussionsThe theory and numerical code solving the bounceaveraged Fokker-Planck equation has been developed in twodimensional space (λ, ϑ). As for the numerical scheme, thefunction in the ϑ-direction is expressed in terms of Fourierexpansion, 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 DKEScode. On the basis of this improvement of the numericalscheme, the calculation can be extendable over wide range ofcollision frequency, especially in the very low collisionalityregion as shown in Fig. 2, compared with the previous studies[2,11]. Although the relation ΩE×B >> Ω∇B is assumed in theDKES code, there is no such limitation in CHD1 code, whichis discussed, in the present study. Detailed and realisticdiscussions associated with the effects of the magnetic axisshift and the electric field on the transport of the LHDconfiguration in low collisionality region will be discussed inFig. 1 Radial profiles of magnetic field spectra in vacuum con-figurations. (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.Fig. 2 Collisionality dependence of the diffusion coefficient(D11) [2(a)] and the bootstrap current (D13), [2(b)] for ∆ =0 cm, –10 cm and –20 cm.2 5Sanuki H. et al., Effect of Magnetic Axis Shift on Neoclassical Transport in Helical Torus4a separate paper. In finite beta plasmas, the deformation ofthe magnetic surface due to the Pfirsch-Schluter current maydegradate 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).[3] H.K. Mynick and W.N.G. Hitchon, Nucl. Fusion 26, 425(1986).[4] W.D. D’haeseleer, W.N.G. Hitchon and J.L. Shohet, J.Comput. Phys. 95, 117 (1991).[5] S.P. Hirshman, K.C. Shaing, W.I. Van Rij et al., Phys.Fluids 29, 2951 (1986).[6] C.D. Beidler and W.D. D’haeseleer, Plasma Phys.Control Fusion 37, 463 (1995).[7] J. Todoroki, J. Phys. Soc. Jpn. 59, 2758 (1990).[8] J. Todoroki and H. Sanuki, Proc. of the InternationalConference on Plasma Physics (ICPP96) in Nagoya(1996) pp298-pp301.[9] H. Sanuki, J. Todoroki and T. Nakagawa, J. Accel.Plasma Res. 5, 53 (2000).[10] H. Sanuki, J. Todoroki and T. nakagawa, J. PlasmaFusion Res. Sries 4, 403 (2001).[11] S. Murakami et al., Nucl. Fusion 42, L19 (2002).236

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Effect of Magnetic Axis Shift on Neoclassical Transport in Helical Torus

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