J. Todoroki

M. Kurata

National Institute for Fusion Science (NIFS-Repository)

§3. The Analysis of Magnetic Island Using Generalized Magnetic Coordinates Kurata, M. (Dept. of Energy Engineering and Science, Graduate School of Engineering, Nagoya Univ.), Todoroki, J. The generalized magnetic coordinates (GMC) 1,2,3) is constructed for a simple analytic helical field involving magnetic islands. The GMC are curvilinear coordinates (~, TJ,S) in which the magnetic field is expressed as B = V'P (~,TJ,S) x V S + H ~ (~,TJ)V ~ x VTJ, and the GMC are constructed on the condition that H ~ == -fi B~ does not depend on the periodical toroidal angle S, -fi being the Jacobian. We shall write 'P as the S-dependent part and S -independent part, 'P(~,TJ,S) = ~(~,TJ,S) + 'P(~,TJ), 'P (~,TJ ) = f \{1 (~,TJ,S) dS, where \}I (~, TJ) is the averaged magnetic surface. The GMC system separates the perturbation \}I, which breaks the nested good magnetic surfaces, from the magnetic field. In order to analyze magnetic islands, the residue is obtained by using the amplitude of the Fourier mode components of the perturbation separated in the coordinates resonating with the rotational transform on the rational surfaces. Introducing the curvilinear coordinates (I" ' e, S), such that \}I = \}I (ljI), and the magnetic lines of force without the perturbation \}I are straight in the e - S plane. For the sake of simplicity, we choose ljI so that d\}I/ dljl = l(ljI); ljI is the toroidal flux and 1 is the rotational transform. The pertu~:ng radial mag(n~ :d ~ i~)p~/ess~ as , H (ljI,e,S)= ----- 1 jH dS, d~ dry dTJ d~ which is expanded in Fourier series as m.1l We consider the linear approximation of the field line in the neighborhood of the rational surface ljI 0 of rotational transform 1 = no I mo . If we introduce the variable ¢ = moe - noS, the equations of the field line can be written in the Hamilton's form, and the residue4,S) R is expressed as R = Sin'( m;OJ ) , (I) where the proper frequency (J) is given by 178 2 I /I I m = mol (lfIo)U (¢o)' U (¢o) = 0, and the potential U is represented as U (¢) = - ~ ~ Hk: k (ljIo) exp(ik¢). ~k IIIn , 1In The residue calculated by the eq. (1) is compared with the value obtained by tracing the field line. These comparisons show very close agreement on both the O-point and the X-point. The distribution of Fourier amplitudes IH ;,,,1 of (m,n) = (5,1), (6,1), (7,1) and (12,2) with respect to the rotational transform 1 is shown in Fig. 1. On the rational surface of 1 = 1 I 5, 1 I 7, the fundamental resonance mode is the largest of all components and there are large magnetic islands. On the rational surface of 1 = 1 / 6, the secondary resonance mode is larger than the fundamental mode and there are small magnetic islands of (12,2) mode. The non-resonant modes are broadly distributed, but they do not contribute to the residue. The residue of the tangent map at a fixed point is calculated by using the Fourier component of the perturbation field decomposed by constructing the coordinates. The magnetic island can be described by the resonant mode of the Fourier component of the perturbation on the averaged magnetic surface. II) "C .~ P.. EI « ... . ~ ::s 0 ~ .... 0 II) = ] .D « 10-4 lO-S 10-6 10-7 10-8 II ~ ~ II II S -. -VI 01 -.l 0Cl \0 Inverse of Rotational Transform t ---6 - (m ,n)=(6, 1) ..•.. (m,n)=(7,1) .. .. ,. ... (m ,n)=(12,2) Fig.I. Distribution of some Fourier amplitudes IH ,: ,,1 with respect to the inverse of l. Reference 1) Kurata, M. and Todoroki, 1. : 1. Plasma Fusion Res. SERIES, Vol. I (1998) 491-494. 2) Kurata, M. and Todoroki, 1. : NIFS-PROC-40 (1999) 9-18. 3) Kurata, M. and Todoroki, 1. : NIFS-PROC-46 (2000) 11-20. 4) J.R. Cary and J.D. Hanson, Phys. Fluids 27(4) (1984) 767-769. 

The Analysis of Magnetic Island Using Generalized Magnetic Coordinates

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