J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)164–168
000–000

Full Wave Analysis of MHD Modes Using Multi-Fluid
Dielectric Tensor in an Inhomogeneous Plasma
AKUTSU Taku and FUKUYAMA Atsushi
Department of Nuclear Engineering, Kyoto University, Kyoto 606-8501, Japan(Received: 9 December 2003 / Accepted: 26 February 2004)

Abstract
Analysis of linear stability of plasmas by solving Maxwell’s equations using dielectric tensor is described.
Dielectric tensor is derived from multi-fluid equations taking the electron inertia, the pressure gradient, the parallel
current, and the collisional friction into account. Numerical calculations using a newly developed numerical code
TASK/WA illustrate the effect of pressure gradient on the TAE (Toroidicity-induced Alfvén Eigenmode) [1] and thepressure dependence of the internal kink mode.
Keywords:
dielectric tensor, wave analysis, multi-fluid, TAE, internal kink mode

1. Introduction

flux, poloidal angle, and toroidal angle respectively. Other
notations are standard. Then, we obtain eigen functions andeigen frequencies by searching for the complex frequencies
which make the wave electric field maximum for a given
external current jext proportional to the plasma density [3].

Macroscopic instabilities of plasmas are important
subjects in fusion research and have been extensively studied
by use of magnetohydrodynamics. In a high temperatureplasma, however, kinetic effects play a vital role in stabilizing
and destabilizing plasmas. In most of previous analyses,
kinetic effects are involved perturbatively. To take the kinetic
effects into account non-perturbatively, we solve Maxwell’sequations using dielectric tensor which expresses the linear
response of plasmas. Dielectric tensor can be derived for
different physics models such as multi-fluid model, kinetic
model (without drift motion), drift kinetic model, drift kinetic
model taking account of particle orbit, and so on.
In this paper, as a first step, we employ a dielectric tensor
derived from the collisional multi-fluid equations taking theelectron inertia, the pressure gradient, and the parallel current
into account. The numerical results for tokamak plasmas
illustrate the effect of pressure gradient on the TAE and thepressure dependence of the internal kink mode [2].

3. Derivation of dielectric tensor frommulti-fluid equations
We derive a dielectric tensor by linearizing the multifluid equations (equations of continuity, motion, and state):∂nj+ ∇ ⋅ (n j u j ) = 0∂tm j n j ∂ + u j ⋅ ∇ u j = –∇
∇ Pj
∂t

↔

↔

(1)

i ∇⋅jextω ε0(2)∇⋅A = 0(3)(5)

d P n –γ = 0jj
dt(6)

where m j, qj, nj, uj, Pj are the mass, charge, density, fluid
velocity and pressure of the particle species j, E is the electricfield, B is the magnetic field, and 〈σ v〉jk is the collision rate
between the particle species j and k. The multi-fluid equations
are solved with respect to the first order particle flux Γj1 =↔
n j0uj1 + nj1u j0, and a conductivity tensor σ is derived by↔
substituting 1 to the equation j1 = Σj qjΓ1 = Σj σ j · E. Then,dielectric tensor is calculated from

We solve Maxwell’s equations for the scalar φ and vector
potential A in the flux coordinates (α,β,ϕ)∇ ⋅ ε ⋅ ∇φ – i ω ∇ ⋅ ε ⋅ A =+ q j n j (E + u j × B ) – n j Σ σ υ j k (u j – u k )k≠j

2. Full wave analysis of eigenmode2 ↔
↔
– ∇ × ∇ × A + iω2 ε ⋅ ∇φ + ω2 ε ⋅ A = –µ 0 jext
cc(4)

↔
↔
↔
ε = I + Σ ωεi σ j .

j0(7)

↔

where ε is the dielectric tensor, α, β and ϕ are the toroidal
Corresponding author’s e-mail: akutsu@p-grp.nucleng.kyoto-u.ac.jp1164©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchAkutsu T. and Fukuyama A., Full Wave Analysis of MHD Modes Using Multi-Fluid Dielectric Tensor in an Inhomogeneous Plasma

For simplicity, we define local orthogonal coordinates,
derive the dielectric tensor on them, and transform it for theflux coordinates. The definitions of the local orthogonal
coordinates arex = ∇α / | ∇α | , z = B 0 / B0 , y = z × x

4. Numerical calculations
We have developed a new numerical code TASK/WA tosolve Maxwell’s equations. TASK/WA uses the finite
difference method in the α direction, and Fourier
decomposition in the β and ϕ directions. In the presentcalculations, a circular cross section (R – R0 = ρ cos β, Z = ρsin β, ρ = α /α max ) is assumed, and the density profile,temperature profile, and q profile are given by(8)and transformation is expressed by
↔

↔

↔

↔

ε flux = µ ⋅ ε xyz ⋅ µ –1(9)wherex ⋅ ∇α x ⋅ ∇β x ⋅ ∇ϕy ⋅ ∇α y ⋅ ∇β y ⋅ ∇ϕz ⋅ ∇α z ⋅ ∇β z ⋅ ∇ϕ

↔

µ=.

(10)

↔
The conductivity tensor j,xyz of each particle species inσlocal orthogonal coordinates satisfies following relations.
↔

↔

σ i , xyz↔

↔

σ e, xyz

I X ie↔

↔

X ei I=i q 2j n j0 ↔
↔
1
⋅× ×m j C j I + ω u j0 k↔

u k↔
↔
q ↔
X j k = – in j0 qk C j ⋅ I – j0ω

(13)

γ Tj
↔
↔
Mj
⋅ I +Cj =kj′ k ⋅ Mj
(ω j′ + ivj )m j ω Wj

q = q0 + (qa – q0)ρ 2(25)

We studied the effects of the pressure gradient, parallel
current, and collisions on TAE and other low frequency modeswith q0 = 1 and qa = 2. Figure 1(a) ~ (d) are contour plots of[log(∫ |E|2 dV)]–1 as a function of complex frequency f, whereE = –∇φ + iω A is the wave electric field to be integrated
over the plasma volume. The darker the color becomes, thelarger the wave electric field is. Figure 1(a) is a referencecase (without the pressure gradient, parallel current, andcollisions). The hollow point of f ~– 0.165 MHz is TAE. With
the pressure gradient, new two branches appears in the regionof Re( f ) < 0.1 MHz (Fig. 1(b)). Because the effects ofcontinuum damping and radiative damping are not includedin the dielectric tensor, the damping rate of TAE withoutpressure gradient, parallel current, and collisions is nearly 0.
And with pressure gradient, the mode distribution issymmetric with respect to the real axis. With pressure gradient

(14)

γ Tj
↔
Wj = ω ′j + iv j – m ω k ⋅ Mj ⋅ k j′

(15)

↔

↔

j↔

Mj = δ –12(ω j′ + iv j )(24)

4.1 Analysis in the frequency range of TAEand the lower(12)συ j kj

Tj = (Tj0 – Tja) (1 – ρ 2) + Tja

(11)

↔

σ 0e

where the subscripts i and e denote ion and electron,
respectively, andσ 0j =(23)Other parameters are as follows: toroidal mode number n =1, B0 = 3 T, R0 = 3 m, a = 1 m, ne0 = 1020 m–3, nea = 1010 m–3,
Te0 = 3 keV, Tea = 1 eV, 〈σ v〉ei ∝ Z 2Tj–3/2.

↔

σ 0i

nj = (nj0 – nja) (1 – ρ 2) + njaiΩ j (ω j′ + iv j )0× – i(Ω j + u j′0 y – Z j ky )(ω j′ + ivj ) (ω j′ + iv j )(ω j′ + iv j – iZ j kx ) 0– i(u j′0 z – Z j k z ) (ω j′ + iv j )Ω j (u j′0 z – Z j k z )δ(16)δ = (ω ′j + i v j ) (ω ′j + i v j – i Z j k x ) – Ω j (Ω j + u j′0 y – Z j k y )(17)Ω j = q j B0 / m j
Zj =

(18)

dn j 0 dPj 01
γ Tj0–m j n j0 ω j′dxdx(19)v j = Σ n k0 〈σv 〉jk(20)

ω j′ = ω – k ⋅ u j 0(21)k≠j

γ Tj k ′j = γ Tj k + ω ′j m j u j 0 + i q j E0 + i m j Σ 〈σ v 〉jk n k 0 uk 0k ≠j↔

Fig. 1(a) Contour map of [log(∫ |E |2 dV )]–1 as a fuction ofcomplex frequency for TAE or low frequency modes
without pressure gradient, parallel current andcollisions (q0 = 1, qa = 2).

(22)

↔

In the collisionless limit, 〈σ v〉jk → 0, σ j,xyz = σ 0j .
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Akutsu T. and Fukuyama A., Full Wave Analysis of MHD Modes Using Multi-Fluid Dielectric Tensor in an Inhomogeneous Plasma

Eβα αFig. 1(b) with pressure gradient, without parallel current andcollisions

Fig. 2(a) TAE mode structure (Eβ) without pressure gradient,parallel current and collisions

EβFig. 1(c) with pressure gradient and collisions, withoutparallel current

α αFig. 2(b) With pressure gradient, without parallel current andcollisions

EβFig. 1(d) with collisions, pressure gradient and parallel current

α αFig. 2(c) With parallel current, without pressure gradient andcollisions

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Akutsu T. and Fukuyama A., Full Wave Analysis of MHD Modes Using Multi-Fluid Dielectric Tensor in an Inhomogeneous Plasmaand collisions, asymmetry of mode distribution becomesdominant and unstable mode disappears (Fig. 1(c)). Parallel
current does not make large effects on the modes distribution
in the complex frequency space (Fig. 1(d)).
The effects of the pressure gradient, parallel current, andcollisions on TAE radial mode structure are shown in Fig.
2(a) ~ (c). With parallel current, m = –2 electric field becomeslarger compared with other cases.

4.2 Analysis of internal kink modeFig. 3

Internal kink mode is analysed with q0 = 0.7 and qa = 3.
In Fig. 3, the internal kink mode corresponts to the hollow
point in f ~– 0.08i MHz. And the displacement of the mode isshown in Fig. 4.
In Fig. 5, the dependence of the internal kink growthrate γ on pressure is compared with the formula obtained by
An. Martynov et al. [4]:Contour map of [log(∫ |E |2 dV )]–1 as a fuction of complex
frequency for internal kink mode (q0 = 0.7, qa = 3)

γ τ A = 0.5ε1 (κ 1 – 0.5)

ξx× β bu – 0.5 – εε a1 (κ 1 – 1.5 |δ 1 + 0.04|)

Displacement in x-direction for internal kink mode (q0 =0.7, qa = 3)

5. Conclusions
We derived the dielectric tensor including the effects ofpressure gradient, the parallel current, and the collisional
friction.
These effects in the TAE and the lower frequency region
are investigated using the code TASK/WA. With pressure
gradient, two new symmetric branches appear in the regionof Re(f ) < 0.1 MHz (Fig. 1(b)). Because the effects ofcontinuum damping and radiative damping are not includedin the dielectric tensor, quantitative analyses of the damping
rate is out range of the present study.
While the parallel current does not make large effects on
the mode distribution in the complex frequency space, mode
structure is strongly affected and m = –2 electric field becomeslarger compared with the cases without the parallel current.
The growth rate of the internal kink mode obtained byTASK/WA is compared with the previous computation result
Eq. (26). The relative dependence of the growth rate on thepressure βbu agrees well with each other, but the positive offsetof the growth rate may be attributed to some destabilizingγ τΑβ buFig. 5(26)where τA is the Alfvén time, βbu = 2µ0(〈p〉1 – p1)/B2pl is the socalled beta Bussac, 〈p〉1 is the volume-averaged pressure overthe inside of the q = 1 surface, p1, Bpl, ε 1, κ1, δ1 are thepressure, poloidal magnetic field, inverse aspect ratio,elongation, and triangularity on q = 1 surface. εa is the inverse
aspect ratio on plasma edge. In the pressent calulations, ε1 =0.12, εa = 0.33, κ1 = 1, and δ1 = 0. Though the slope ofgrowth rates obtained by TASK/WA is coincident with Eq.
(26), and the minimum point of TASK/WA (βbu = 0.168) isalmost coincident with 0.161 obtained from Eq. (26), γ τAdiffers by about 0.017 for wide rage of βbu. This discrepancy
may be attributed to the other destabilizing mechanism suchas parallel current, but not yet understood well.

α αFig. 4(1.23 – 1.26ε 1)

Growth rate of internal kink mode as a function of βbu .
Growth rate obtained by TASK/WA (solid line) and Eq.
(26) (dotted line).

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Akutsu T. and Fukuyama A., Full Wave Analysis of MHD Modes Using Multi-Fluid Dielectric Tensor in an Inhomogeneous Plasma[2] J.A. Wesson, Nucl. Fusion 18, 87 (1978).
[3] A. Fukuyama and T. Akutsu, Proc. of IAEA Fusion
Energy Conference (Lyon, France) IAEA-CN-94/TH/P314.
[4] An. Martynov and O. Sauter, Proc. Joint VarennaLausanne Int. Workshop (Varenna 2002) p. 297.

mechanism such as parallel current, which is left for furtherstudy.

References[1] C.Z. Cheng, Liu Chen and M.S. Chance, Ann. of Phys.
161, 21 (1985).

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