US-Japan JIFT Workshop on Progress of Extended MHD Models NIFS, Toki,Japan 2007/03/27 MHD Linear Stability Analysis Using a Full Wave Code T. Akutsu and A. Fukuyama Department of Nuclear Engineering, Kyoto University Present affiliation: SGI Japan, Ltd
Outlines Introduction TASK/WA code Full Wave Analysis Derivation of resistive MHD dielectric tensor Benchmark tests Resistive wall mode including the effect of ferromagnetism Dispersion relation Comparisons with TASK/WA code Effect of plasma rotation Summary
Introduction Various kinds of extended MHD models Ideal MHD Resistive MHD Hall MHD Finite-electron-mass MHD Two-fluid MHD Kinetic models Uniform kinetic model: ω ω,ω d,ω B No FLR, Reduced FLR, Differential OP, Integral OP, Spectral Gyrokinetic model: ω ω B Hamiltonian model: three (adiabatic) constants of motion
Full Wave Analysis Electric field E with a complex frequency ω Maxwell s equation and the dielectric tensor 1 μ(r) [ E(r)] ω2 ɛ 0 dr ɛ (r, r ) E(r ) = i ω j ext vr) Type of analyses Antenna excitation: ω: real (boundary-value problem) Analysis of wave heating and current drive Loading impedance analysis (weakly stable eigenmode) Spontaneous excitation: ω: complex (eigen-value problem) Analysis of linear stability
Previous Analysis TASK/WM Fourier mode expansion in poloidal and toroidal directions Maxwell s equation and the dielectric tensor 1 μ 0 [ Emn (ρ) ] ω 2 ɛ 0 m n ɛ m m,n n (ρ) E m n (ρ) = i ω j ext(r) (Uniform kinetic + gyrokinetic) dielectric tensor Plasma dispersion function for parallel interaction Electronfinite mass Nofinite Larmor radius effects Finite difference method in radial direction Alfvén eigenmode analysis Excitation by energetic particle No coupling with drift waves
Present Analysis TASK/WA Fourier mode expansion in poloidal and toroidal directions Maxwell s equation and the dielectric tensor 1 [ Emn (ρ) ] ω 2 ɛ μ(ρ) 0 ɛ m m,n n (ρ) E m n (ρ) = i ω j ext(r) m n Resistive MHD dielectric tensor Finite element method in radial direction Higher numerical accuracy Comparison with previous Resistive MHD analyses Analysis of Resistive Wall Mode (RWM)
TASK/WA: Coordinates Flux coordinate: (s, Θ,ϕ) = (x 1, x 2, x 3 ) ϕ: Toroidal angle s = ψ p : Radial coordinate Θ: Poloidal angle (Boozer coordinate) k R = n B ϕ B + mr B Θ r B : constant k = n R B ϕ B + m r B Θ B : not constant k = 0 on a rational surface
TASK/WA: Finite Element Method Weak form of Maxwell s equation: (w: weighting function) ( w) 1 μ ( E) dv + ω2 ɛ 0 w ɛ E dv + i ω w j ext dv Interpolation function In plasma: second-order Lagrange polynomial In vacuum: linear function Boundary conditions On magnetic axis: E 2 se θ 0 for s 0 Finiteness of i ωb = E On plasma surface: Surface term due to a derivative of discontinuous parameters. On perfectly conducting surface: E 2 = 0, E 3 = 0
TASK/WA: Eigenmode analysis Excitation, j ext, proportional to the electron density Find complex ω which maximize the volume-integrated electric field amplitude Inverse of volume-integrate wave amplitude, phase and amplitude of the determinant of coefficient matrix as a function of the growth rate ω
Resistive MHD Dielectric Tensor Resistive MHD equation Linearization: X = X 0 + X 1 Laplace Transform: complex ω E = B t dp dt +Γp u = 0 ρ m du dt + p = j B E + u B = η j Flux coordinate: (s, Θ,ϕ)=(x 1, x 2, x 3 ) Co-variant and contra-variant expression
Matrix Expression (I) Definitions ū 0 = (0, u 2 0, u3 0 ), ū 1 = (u 1 1, u2 1, u3 1 ) B 0 = (0, B 2 0, B3 0 ), Ē 1 = (E 1,1, E 1,2, E 1,3 ) g = {g ij }, = (d/dx 1, d/dx 2, d/dx 3 ) u i 0 u j 1 Λα ij ū 0 Λ ū 1, ū 1 ū 0 + 2ū 0 Λ ū 1 U ū 1 j 1 B 0 B axis R j 1 1 ( ) k e j ω 0 k e j 0 I K ( j 0 ), k = ( id/dx 1, m, n): wave number ω = ω ū 0 k
Matrix expression (II) Adiabatic equation of state i ω p 1 = dp 0 dx 1u1 1 +Γp 0 u 1 Equation of motion ρ 0 ( i ω + ) U ū 1 + g p 1 = JB axis g R j 1 + g K ( j 0 ) Ē 1 Ohm s law g Ē 1 + JB axis g R ū1 + g K (ū 0 ) Ē 1 = η j 1 Finally we obtain the linear relation between ū 1 and Ē 1 [ηρ 0 ( i ω + U ) (JB axis ) 2 g R g R ] = ū 1 + η i ω g ( dp0 dx 1u1 1 +Γp 0 u 1 [ JB axis g R g + JBaxis g R g K (ū0 ) + η g K ( j 0 ) ) ] Ē 1
Dielectric Tensor Induced current: J 1 = σ E 1 Electrical conductivity tensor σ = σ 0 + d σ x dx 1 + d σ 1 R dx 1 0 + d d σ 1 R dx 1 1 dx 1 + d d σ 2 L dx 1 22 R dx 1 0 + d d d σ 2 L dx 1 22 R dx 1 1 dx 1 where σ 0 and σ x are 3 3 matrixes, σ 1 and σ 2 are 3 2 matrixes, R 0 and R 1 are 2 3 matrixes, L 22 is a 2 2 matrix. Dielectric tensor ɛ = I + i ωɛ 0 s σ s
Circular Waveguide Analytic solution k 2 z + ( j ml )2 a 2 = ω2 c 2 46.7MHz, 128.2MHz, 204.3MHz for k z = 1/3m 1, a = 2mCm = 1 Inverse of volume-integrated electric field amplitude 1.40E+01 log 10 (1/E 2 ) 1.00E+01 6.00E+00 2.00E+00 0.000E+00 5.000E+01 1.000E+02 1.500E+02 2.000E+02 2.500E+02 Re(ω)
Spectrum in a Cylindrical Plasma Square of frequency normalized by the Alfvén time τ A = R 0 /v A, Ω 2 = (ωτ A ) 2 From high frequency: fast wave, shear Alfvén wave, slow wave Good agreement with Fig. 5 in M. Chance, et. al, Nucl. Fusion 17 (1977) 65 Parameters Uniform density, B z, and q Poloidal mode number m = 2 nq = 1.9 Ω 2 10 6 10 4 10 2 1 10 2 10 4
External Kink Mode in a Cylindrical Plasma Flat current profile Analytic solution [ γ 2 q 2 aτ 2 A = 2(m nq a) 1 m nq ] a 1 (a/b) 2m a = 1m, b = 1.2aCL = 2πR 0 = 40π mcm = 2Cn = 1 Growth rate as a function of nq 0.30 0.25 Analytic solution TASK/WA 0.20 γ 2 τa 2 0.15 0.10 0.05 0.00 1.0 1.2 1.4 1.6 1.8 2.0 nq a
m = 1 Internal Kink Mode and Tearing Mode Growth rate as a function of plasma resistivity 10 1 TASK/WA Ideal internal kink τ H -2/3τ R -1/3 γτa 10 2 10 3 10 10 10 9 10 8 10 7 10 6 10 5 10 4 10 3 η = 5 10 10 Ω m η plasma (Ω m) E 2 / g 22 E 2 / g 22 4.0E+04 4.0E+03 η = 1 10 4 Ω m 3.0E+04 2.0E+04 1.0E+04 0.0E+00 0.0 0.2 0.4 0.6 0.8 1.0 s(ψ) 3.0E+03 2.0E+03 1.0E+03 0.0E+00 0.0 0.2 0.4 0.6 0.8 1.0 s(ψ)
Spectrum in a Tokamak Plasma Spectrum of Incompressible plasma in a Solovev equilibrium cf. Fig. 2 in M. Chance, et. al, J. Comput. Phys 28 (1978) 1 ɛ = a/r 0 = 1/20, κ = 1, q 0 = 0.08591, n = 10, m = 0, 1, 2 10 4 TASK/WA Chance et al. 10 3 10 2 Ω 2 10 1 10 1 10 2
plasma vacuum I wall vacuum II Dispersion Relation of RWM Cylindrical plasma: (radius: a) Axial length: 2πR 0 (periodic) Poloidal mode number: m Toroidal mode number: n, k = n/r 0,(k 2 r 2 1) Surrounded by a resistive and ferromagnetic wall: (radius: b) r a b b + d 0 Wall thickness: d Wall permeability: μ W,(ˆμ μ W /μ 0 ) Wall resistivity: η W
Dispersion Relation of RWM (I) Response of the vacuum regions and the wall [ ] λ 2 (b + d) m2 ˆμ 2 (1 e 2dλ ) λmˆμ(1 + e 2dλ ) d b b ( a 2m α = [ ] λ 2 (b + d) + m2 ˆμ 2 ( (1 e 2dλ ) + λmˆμ(1 + e 2dλ ) 2 + d ) b) b b where λ 2 = ˆμγτ W bd Response of the plasma α = 1 + + m2 b 2 + k2 2m (nq a + m)/(nq a m) +Δ a (γ)(1 + ˆγ 2 ) + ˆγ 2 m where the normalized growth rate is ˆγ = γτ A q a /(m nq a )
Dispersion Relation of RWM (II) Value of Δ a (γ) When the q profile is flat: Δ a (γ) = m 1 When the rational surface q(r mn ) = m/n is close to the plasma surface, r mn a: Δ a (γ) = Δ a (0)(1 + ˆγ 2 ) 1 1 Δ a (0) ( 1 1 (y2 + ˆγ 2 ) 1 dy ), Δ a (0) where ξ is a solution of the Euler equation f = rf2 k 2 0 d dr ( f dξ dr ), g = 2 k2 k0 2 (μ 0 p) + gξ = 0 ( k 2 0 r 2 ) 1 rf 2 + 2 k2 k 2 0 r2 rk 4 0 ( ) rξ ξ ˆFF a F = kb z + m r B θ, k 2 0 = k 2 + m2 r 2, ˆF = kb z m r B θ
Numerical Results for RWM Comparison between the dispersion relation and the numerical results by TASK/WA Plasma resistivity η plasma is non-zero in TASK/WA Ideally conducting wall at r = c in TASK/WA 2.0 1.5 1.0 0.5 0.0 Er /10 5 (AU) -0.5-1.0 0.0 0.4 Typical Electric Field Profile (Re/Im) for b/a = 1.23, c/a = 2.0, m = 2, n = 1 E r E θ E θ in the wall 0.8 1.2 1.6 2.0 r/a 3.0 2.0 1.0 Eθ/10 4 (AU) 0.0 0.0 0.4 0.8 1.2 1.6 2.0 r/a -0.4-0.8-1.2-1.6 1.23 Eθ (AU) 1.24 r/a 1.25
Dependence on the Wall Radius b/a γτw ratio q profile: Flat q profile: Wesson type q(r) = q a = 1.35 q 0 = 0.8, q a = 1.8 for various values of d/a q(r) = q a r 2 /a 2 800 700 600 500 400 300 200 100 0 1.01 1.00 dispersion relation (μˆ =1) (μˆ =10) TASK/WA (μˆ =1) (μˆ =10) (DR) / (WA code) (μˆ=1) (μˆ =10) 0.99 1.14 1.16 1.18 1.2 1.22 1.24 1.26 1.28 1.3 b/a γτw 200 150 100 50 1 (1 r 2 /a 2 ) q a/q 0 dispersion relation (μˆ =1) (μˆ =10) TASK/WA (μˆ =1) (μˆ =10) 0 1.14 1.16 1.18 1.2 1.22 1.24 1.26 1.28 1.3 There is a soft threshold of the wall radius b/a. The dispersion relation correctly predicts γ. b/a
Destabilization by µ Wall > 1 (b/a = 1.23) γτw ratio q profile: Flat q profile: Wesson type q(r) = q a = 1.35 q 0 = 0.8, q a = 1.8 for various values of d/a wall thickness d/a = 0.02 90 80 70 60 50 40 30 20 10 0 1.02 1.01 1.00 0.99 dispersion relation d/a=0.02 d/a=0.01 d/a=0.005 TASK/WA d/a=0.02 d/a=0.01 d/a=0.005 (DR)/(WA code) d/a=0.02 d/a=0.01 d/a=0.005 0 2 4 6 8 10 μˆ γτw 60 50 40 30 20 10 0 dispersion relation TASK/WA 0 2 4 6 8 10 γ increases with the increase of μ Wall. For flat q profiles, the dispersion relation correctly predicts γ. μˆ
Stabilization by Toroidal Rotation Plasma rotation stabilizes RWM when plasma dissipation exists It was confirmed that the kinetic dissipation is not important when the plasma is incompressible near the plasma edge. q profile: Wesson type (q 0 = 0.8Cq a = 1.8) Toroidal rotation dependence of the RWM growth rate for various values of η edge γτw 40 30 20 10 0-10 η edge (Ω m) = 1 10-9 1 10-8 1 10-7 1 10-6 4.11 10-6 1 10-5 0 0.025 0.05 0.075 0.1 0.125 0.15 Ωτ A
Comparison of Dissipation Mechanisms Resistivity only TASK/WA Resistivity and perp. viscosity R. Fitzpatrick and A. Aydemir Nucl. Fusion 36 11 (1996) γτw 70 60 50 40 30 20 10 0-10 -20 0 0.025 0.05 0.075 0.1 0.125 0.15 Ωτ A r w =1.090 r w =1.200 r w =1.275 Perpendicular viscosity reduces the growth rate for faster rotation and longer wall distance.
Summary We have developed a full wave code TASK/WA in order to study the applicability of the full wave analysis to the linear stability study of global modes in a tokamak. Resistive MHD dielectric tensor was used in the present analysis and the results are compared with previous MHD studies. Good agreement was shown for cylindrical plasmas and for incompressible modes in tokamak plasmas. We analyzed the ferromagnetic effect on the resistive wall mode in a cylindrical plasma using a resistive MHD dielectric tensor. The growth rate of RWMs calculated from the dispersion relation agrees well with the results of the TASK/WA code. The critical plasma rotation which stabilizes RWMs increases almost linearly with the increase of ˆμ.