第１回核燃焼プラズマ統合コード 研究会（ 京都大学）
2003/07/31-08/01

TASK コード によるアルヴェン固有モード の解析
京大工：福山内容•目 的• アルヴェン固有モード 解析• まとめ淳目的• ト カマクにおける多様なアルヴェン固有モード◦ CAE : 圧縮性アルヴェン固有モード（ 高周波数，球状トカマク）◦ EAE : Ellipticity-induced Alfvén eigenmode （ 楕円断面）◦ TAE : Toroidicity-induced Alfvén eigenmode （トーラスプラズマ）◦ GAE : Global Alfvén eigenmode （ 円柱プラズマ）◦ EPM, RSAE, BAE : Energetic Particle Driven mode （ 高速イオン ）◦ DAE : Drift-Alfvén eigenmode （ド リフト波との結合）• 負磁気シア配位におけるアルヴェン固有モード◦ TAE 安定化の予測：アルヴェン共鳴による吸収の増加（ Nova-K ）◦ 周波数が大幅に変化する固有モード の存在（ JT-60U, JET ）アルヴェン固有モード の線形安定性解析• 電磁流体的解析（ 理想，抵抗性）• 運動論的効果を含めた電磁流体的解析（ 摂動的）
◦ 固有関数は電磁流体方程式から．運動論的効果を含めて成長率を評価．
• 運動論的解析（ 電子の質量，イオンの有限半径，ド リフト運動等）◦ PENN code (Jaun, Alfvén Lab)◦ TASK/WM (Fukuyama)• バルーニング展開（ 高 n モード ）◦ HINST (Gorelenkov, Cheng)◦ 2D-WKB (Vlad, Chen, Zonka)• ３次元波動伝播解析コード TASK/WM◦ MHD 平衡解析から得られる磁気面座標◦ マクスウェル方程式の境界値問題．プラズマの応答は誘電率テンソル◦ ポロイダル方向とトロイダル方向にはフーリエモード 展開，径方向には差分◦ 一定振幅の励起に対して振幅が最大となる複素固有周波数を求める．

磁気面座標• 磁気面座標（ 非直交座標系）◦ 小半径方向：ポロイダル磁束 ψ◦ ポロイダル方向：θ
◦ トロイダル方向：ϕR• 反変基本ベクトルe1 = ∇ψ,e2 = ∇θ,ψZ

e3 = ∇ϕR0を用いると，電界ベクトルはE = E1e1 + E2e2 + E3e3

• J : ヤコビアン• g : 計量係数

11=J= 1 2e · e × e3 ∇ψ · ∇θ × ∇ϕ
共変基本ベクトル ei ≡ ∂r/∂xi を用いて gij = ei · ejϕθ波動方程式• 波動電界 E に対する定常マクスウェル方程式（ 角周波数 ω ，光速 c ）ω2 ↔
∇ × ∇ × E = 2 ² · E + i ωµ0j extc↔

◦ ² : 誘電率テンソル［ 有限温度効果（サイクロトロン減衰，ランダウ減衰）］◦ j ext : アンテナ電流• 非直交座標系における波動方程式（ 半径方向成分）

·½ µ¶
µ¶µ¶¾
∂g
∂E∂Eg
∂E∂Eg
∂E∂E1313232133321(∇ × ∇ × E)1 =−+−+−J ∂x2 J ∂x2
∂x3
J ∂x3
∂x1
J ∂x1
∂x2
½ µ¶
µ¶
µ¶¾¸
∂
g21 ∂E3 ∂E2
g22 ∂E1 ∂E3g23 ∂E2 ∂E1− 3−+−+−∂xJ ∂x2
∂x3
J ∂x3
∂x1
J ∂x1
∂x2

◦ (x1, x2, x3) はそれぞれ座標 (ψ, θ, ϕ) を表す．
◦ ポロイダル成分とトロイダル成分も同様

プラズマ応答↔

• 誘電率テンソル ² は，通常静磁界を z 軸とする規格化直交座標系で記述• 局所的な規格化直交座標系 ês =∇ψ,|∇ψ|êb = êh × êψ ,êh =B0
|B 0|

↔

• 座標変換行列： µ





EsE1
↔



 E2  = µ  Eb EhE3

1dppc2g12 + c3g13 

1111↔
gJg

pµ≡


11c3 J g
c2g22 + c3g23 
 0p0−c2J g 11 c2g32 + c3g33

c2 = B θ /B, c2 = B φ/B
d = c2(g23g12 − g22g31) + c3(g33g12 − g32g31)
g 11 = (g22g33 − g23g32)/J 2• 非直交座標系における誘電テンソル
↔

↔

↔

↔−1

² = µ · ² sbh · µフーリエ展開• ポロイダル方向とトロイダル方向については，フーリエ級数により展開• 電界，マクスウェル方程式の左辺，形状および媒質の変化を表す係数をそれぞれX
E(ψ, θ, ϕ) =Emn(ψ)ei(mθ+nϕ)mn

G(ψ, θ, ϕ) =X

Glk (ψ)ei(lθ+kNpϕ)

lk

J(∇ × ∇ × E) = G(ψ, θ, ϕ)E(ψ, θ, ϕ) =X00

[J(∇ × ∇ × E)]m0n0 ei(m θ+n ϕ)

m0 n0• モード 間結合（ Nh : トロイダル方向のヘリカルコイル回転数）
モード 番号
トロイダル方向 ポロイダル方向
電界ベクトル Enm
媒質変化
kNhlJ(∇ × ∇ × E)
n0m0
関係式n0 = n + kNhm0 = m + l

平行方向の波数↔

• プラズマ誘電テンソル ² (ψ, θ, ϕ, kkm00n00 ) は，00 00

プラズマ分散関数 Z[(ω − N ωcs)/kkm n vTs] を通して，磁力線方向の波数m00 ,n00に依存する．
kk00 00∂∂∂∂
B
kkm ,n = −iêh·∇ = −iêh·(∇θ +∇ϕ ) = −iêh·(e2 +e3 ) = m00∂θ∂ϕ∂θ∂ϕ↔

• 誘電テンソル ² sbh に対応する電界のフーリェ成分
↔

↔−1

↔

↔

↔−1(J ² · E) = J g · µ · ² sbh · µ · E im0`3`2`1mn0k3k2k1nしたがってi1m = m + `1 + `22m0 = m + `1 + `2 + `200

1n = n + k1 + k2
2n 0 = n + k1 + k2 + k300

θ|B|+n00 B

ϕ|B|並列処理による TASK/WM の高速化
• 計算機クラスターによる並列処理（ Fortran77 + MPI ライブラリ）◦ 並列処理による係数行列計算の高速化◦ 並列処理による行列方程式解法の高速化◦ 係数配列の分散記憶による大規模化• 経過時間のプロセッサ数依存性（ 16 モード，径方向 50 点）
Elapsed Time [sec]2800

Dispersed Gauss(PARA)ILDUBCG(PARA, BLK)ILDUCGS(PARA, BLK)ILDUBCGSTAB(PARA, BLK)
Reduced Gaussi(PARA, BLK)ILDUBCG(PARA, BLK, APP)
ILDUCGS(PARA, BLK, APP)
ILDUBCGSTAB(PARA, BLK, APP)

2400
2000
1600
1200
800
4000

123456

Number of Processor78正磁気シア：TAE• q(ρ) = q0 + (qa − q0)ρ2, q0 = 1, qa = 2• 平坦密度分布>?567 9:;<=複素周波数空間における振幅等高線*+**1

fr = 81.95 kHz
fi = −20.32 Hz*+**201*+,.

*+,*+***

0*+**2*+,/0*+**1

2*+*,*+*-*+*.

34567 9:;<=*+*/

Analysis of TAE in Reversed Shear Configuration
qmin Dependence of Eigenmode Frequency*11Assumed q profile

Real partq23 56789

qaq0qmin;>?. @ABC-1,1DBC/1.10

ρmin11*+,

Plasma Parameters

*+-.

.+.

.+/.+,.+-

0:;<=,<>?+ @ABC

Imag part*.,45 789:R0 3 ma 1m
B0 3 Tne (0) 1020 m−3
T (0) 3 keV
q(0) 3q(a) 5ρmin 0.5
n 1
Flat density profile;>?0 @ABC

ρDBC*-,,<>?3 @ABC*-.,*+,,-/0-/1++/++/2;<5=+/0+/13

6E040739

q5

5.8s
6.2s
6.6s

4
32(a)

7.0s
030.2

0.4

qmin

2.521.5
1r/a

0.6

0.8

1(b)

ne(x10 19m-3)

(c)
NNBI

(d)

FIG. 1: (a) Temporal evolution of the q-profile measured with MSE. (b) Temporal evolution of qmin ,(c) line averaged electron density, (d) a typical behavior of frequency spectrum of the n = 1 AE.
Broken lines denote estimated frequency from the RSAE model normalized by observed frequency
at q = 2.5.

probes separated by about 40 degree in the toroidal direction is larger than 0.9. the onlyn = 1 AE is observed because βh is relatively low (βh < 0.2%). AEs with n > 1 are observed
if βh increases. The large upward/downward frequency sweeping from t = 6.0–6.5 s cannot
be explained by the change of VA because the electron density changed only less than 5%.
In the period t = 6.5–6.8 s the n = 1 AE frequency saturates. After t ∼6.8 s the n = 1 AE
frequency decreases due to the increase of electron density.
The observed AE amplitude are enhanced during t ≈ 6.65−6.85 s when the AE frequency
is saturated as shown in Fig. 1(d). To investigate the dependence of mode amplitude on theq-profile change, we show the mode amplitude versus qmin for three shots in Fig. 2. When
NNBI was injected the values of qmin are about 3.0, 2.8 and 2.6 for the shots of E40739,
E40744 and E40743, respectively. For all these cases the n = 1 mode amplitude is largest4負磁気シア配位における固有モード 構造
qmin = 2.4

qmin = 2.5

qmin = 2.6*+,-./0:*+,-./01*+,-./0;

3 56789

3 56789

3 56789

Alfvén resonance*+,-./01

Higher freq.

23.1405 789*+,-./0=+.;<+.;/

Lower freq.

23.=/0> 789*+,-./04

23.1105 789+.;<+.;/:

TAEs*+,-./0=23.==0< 789+.;<+.;/+.;/+.;<::*+,-./01

222:
23.1105 789*+,-./04

23.<?0= 789+.;/+.;<+.;<::

Double TAE

RSAE

E40744-5fluctuation (x10 T)(20ms averaged)

rms of magnetic

1.5
1E40743

E407390.502RSAE

TAE
2.2

2.4

2.6
qmin

2.8

3FIG. 2: Dependence of AE magnetic fluctuation amplitude on qmin . The n = 1 AE amplitude isenhanced in the range of 2.4 < qmin < 2.7.
543210qmin=2.8q(a)minq=2.5(b)min=2.3(c)

TAE

5

RSAE

00
0.5(A.U.)10
FrequencyρFrequency(A.U. )

0.07

0.5
ρ

10

0.5
ρ
(d)

1

HRSAE(2)HRSAE(3)
TAE(2,3)
TAE(3,4)04

LRSAE(4)
3.5

3qmin

LRSAE(3)2.52FIG. 3: Schematic drawing of the n = 1 spectrum of AEs in the RS plasma with (a) qmin = 2.8,(b) qmin = 2.5 and (b) qmin = 2.3. Panel (d) shows the frequencies of RSAE and TAE as a functionof qmin . The bold lines denote the frequencies of the n = 1 AEs and thin lines are for the n = 2AEs. Numbers in the parentheses are poloidal mode numbers.

in the range 2.4 < qmin < 2.7, which is independent of the time length after NNB injection.
To understand the large and rapid frequency sweeping and its subsequent saturation, we
propose a model of RSAE and its transition to TAE as qmin decreases. Fig. 3 shows threeshear Alfvén continuous spectra, frequency vs. r/a, in RS plasmas with (a) qmin = 2.8, (b)qmin = 2.5 and (c) qmin = 2.3. In any cases, n = 1, q(0) = 3 at the plasma center, andq(a) = 5 at the plasma edge. The two gaps around the qmin region shown in Fig. 3(c) are5

Frequency (A.U.)

0.05
7956 883 4

0.04
40.03

75 67
98 29 7654 98387

628

91

0.02
0.01
2.4

2.3

2.2

2.1
2q

1.91.81.7

min

FIG. 5: (a) Typical temporal evolution of AE frequency changes observed in a JT-60U ICRF
heated RS plasma [6]. (b) The frequencies of n= 1–9 LRSAEs (solid upward-sloping lines) andTAEs (horizontal aligned marks) as a function of the qmin decrement.

assume that qmin decreases in time as it does. If the modes in the experiment just trace
the Alfvén dispersion relation, which is ωA /VA = |k// (r)| = |n − m/q(r)|/R, at q = qmin ,and qmin decrease, the frequency would increase in time when k// (r0 ) < 0 and decreasein time when k// (r0 ) > 0. However, the modes with downward sweeping are suppressedin the experiment. They interpret no observation of downward frequency sweeping after
upward frequency sweeping by effect of energetic particles. They assert that the only way
they have found to explain the asymmetry is to describe the fast particle response in a
nonperturbative manner. On the other hand, from our RSAE model, there are inherently
TAEs rather than modes with downward sweeping after upward sweeping, i.e., AE transits
from LRSAE to TAE. We don’t observe TAE after RSAE when βh is not sufficiently large,
because damping rate of TAE is larger than that of LRSAE. When βh is sufficiently high,TAE can be destabilized in the qmin transition range as observed in our NNB RS plasma
9高速イオン励起モード

CBA

Im f [MHz]f [MHz]

A
B
C

ψ1/2

Re f [MHz]m=–1

Eθ: CEθ: B

Eθ: A
m=–1

m=–1
m=–2

ψ1/2

ψ1/2

ψ1/2

(fr, fi) = (41.8, 8.1) kHz (fr, fi) = (57.7, 3.6) kHz (fr, fi) = (58.0, −6.0) kHz

高速イオンによる RSAE の励起 (qmin = 2.6)*+,-./04

23.<?0= 789*+,-./04

nF = 0 m-3• Without EP

23.==0< 789

0.004+.;/+.;<
fi [MHz]

0.002

0.000

-0.002+.;<-0.0040.03

0.04

fr = 38.0 kHz
fi = 160.2 Hz:Eθ

fr = 55.3 kHz
fi = 75.7 Hz

0.002

fi [MHz]−3

m=20.004

0.000-0.002m=3-0.004

0.03

0.04

0.05

0.06

0.07

fr [MHz]ρρnF = 1 × 1017 m-3
fr = 37.2 kHz
fi = 1858.6 Hz

m=3Eθ

fr = 55.3 kHz
fi = 271.6 Hz

0.002

fi [MHz]−3

m=20.004

Eθ• With EP
1 × 10 m
360 keV
0.5 m

0.07

Eθ

m=33 × 10 m
360 keV
0.5 m

170.06nF = 3 × 1016 m-3• With EP
160.05fr [MHz]:0.000-0.002

m=3-0.004

0.03

ρ

0.04

0.05

fr [MHz]0.060.07ρまとめ• 負磁気シア配位においては，TAE はより安定となるが，周波数ギャップの上下端近傍に固有周波数をもつ固有モード RSAE が存在できる．
• RSAE は安全係数の最小値付近に局在した GAE とみなすこともできる．
• RSAE の存在範囲は おおよそ ` + 0.5 . qmin . ` + 0.7 である．
• 高速イオンによるアルヴェン固有モード の不安定化を取り入れ，周波数ギャップよりも低い周波数領域の固有モード が不安定化される条件を調べた．
• 進行中◦ 高速イオンによる RSAE の不安定化◦ ＭＨＤモード の解析（ 阿久津）— 圧力勾配や電流を取り入れた流体的誘電率テンソル— 空間不均一性を取り入れたジャイロ運動論的誘電率テンソル— 有限軌道幅効果を取り入れた積分形誘電率テンソル

E40744-5fluctuation (x10 T)(20ms averaged)

rms of magnetic

1.5
1E40743

E407390.502RSAE

TAE
2.2

2.4

2.6
qmin

2.8

3FIG. 2: Dependence of AE magnetic fluctuation amplitude on qmin . The n = 1 AE amplitude isenhanced in the range of 2.4 < qmin < 2.7.
543210qmin=2.8q(a)minq=2.5(b)min=2.3(c)

TAE

5RSAE

000.5(A.U.)10Frequency
ρFrequency(A.U. )

0.07

0.5ρ

100.5
ρ
(d)

1

HRSAE(2)HRSAE(3)
TAE(2,3)
TAE(3,4)04

LRSAE(4)
3.5

3qmin

LRSAE(3)2.5

2FIG. 3: Schematic drawing of the n = 1 spectrum of AEs in the RS plasma with (a) qmin = 2.8,(b) qmin = 2.5 and (b) qmin = 2.3. Panel (d) shows the frequencies of RSAE and TAE as a functionof qmin . The bold lines denote the frequencies of the n = 1 AEs and thin lines are for the n = 2AEs. Numbers in the parentheses are poloidal mode numbers.

in the range 2.4 < qmin < 2.7, which is independent of the time length after NNB injection.
To understand the large and rapid frequency sweeping and its subsequent saturation, we
propose a model of RSAE and its transition to TAE as qmin decreases. Fig. 3 shows threeshear Alfvén continuous spectra, frequency vs. r/a, in RS plasmas with (a) qmin = 2.8, (b)qmin = 2.5 and (c) qmin = 2.3. In any cases, n = 1, q(0) = 3 at the plasma center, andq(a) = 5 at the plasma edge. The two gaps around the qmin region shown in Fig. 3(c) are5