A theory for localized low-frequency ideal MHD modes in axisymmetric toroidal systems is generalized to take into account both toroidal and poloidal

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1

2 MHD spectra pre-history (selected results I

3 MHD spectra pre-history (selected results II

4 Abstract A theory for localized low-frequency ideal MHD modes in axisymmetric toroidal systems is generalized to take into account both toroidal and poloidal equilibrium plasma ows. The general set of equations describing the coupling of shear Alfv en and slow (sound modes and dening the continuous spectrum of rotating plasmas in axisymmetric toroidal systems is derived. The equations are applied to study the continuous spectra in large aspect ratio tokamaks. The unstable continuous modes in the case of predominantly poloidal plasma rotation with the angular velocity exceeding the sound frequency are found. Their stabilization by the shear Alfv en coupling eect is studied.

5 Starting Equations A standard set of ideal MHD equations: ( ρ t + v v = p + ( B B, B t p t = (v B, B = 0, ρ t + (ρv = 0, + v p + Γp v = 0. OUR OBJECTIVE: to describe the eect of general (poloidal and toroidal rotation on the continuous MHD spectrum.

6 Axisymmetric Equilibrium Tokamak magnetic eld and plasma velocity: B = F (R, Z ϕ + ψ(r, Z ϕ, v = κ(ψ ρ B + R Ω(ψ ϕ, ( 1 κ ρ F ( 1 κ ρ κωr = I(ψ, κ B R Ω + Γ p ρ Γ 1 ρ = H(ψ, p = S(ψρΓ, ( ψ+r p +F ψ R ψ = R R ( ( F κ ψ κ ψ ρ +ρ κ ψ ρ ψ = 0, R R ( 1 ψ + ψ R R Z. The equilibrium conguration is dened by ve functions, which are constant on the magnetic surfaces: Ω(ψ and κ(ψ describe the stationary plasma velocity eld; I(ψ corresponds to the eective poloidal current; S(ψ the entropy and H(ψ the enthalpy in absence of plasma rotation.

7 General equations set ρ( iω + v v ψ + C ψ,s v s + C ψ, v + C ψ,ρ ρ = P ψ ψ + B Q ψ + (S γ sq s + α sk ψ Q, ρ( iω + v v s + C s,ψ v ψ + C s, v + C s,ρρ = B ψ P + B Q B s + γs ψ Q ψ + K sb Q, α s [ ( ] v ρ iωv + ρv + C,ψ v ψ + C,s v s + C,ρ ρ ρ = B p α sq sb F Q ψ (j B ψ, B = Q ψ ψ + Q s B ψ ψ + Q B, v = v ψ ψ + v s B ψ ψ + v B B. Notations: α s = B / ψ γ s = B B/ ψ K ψ = K ψ K s = K (B ψ/b K 1 B B ( B B S = [ ] B ψ B ψ ψ ψ

8 Generalized Continuum Mode Equation I P / ψ = 0 ξ ψ = (B ψ η B 1 Γp ρ Γp + B [ ( ( ] ρζ ρf B B η B B This implies that ξ/ ψ terms are of the same order as the terms proportional to η and ζ. The normal plasma perturbation, described by ξ, is small compared to plasma perturbations lying on the magnetic surface and described by η and ζ. Therefore, the terms proportional to ξ (but not to ξ/ ψ! can be neglected.

9 Generalized Continuum Mode Equation II ( ( ψ ψ ρˆω ˆω η + B B η + B B B Γp + B 1 F [ ( ( ] Γp F ρ B (v B + B R κ B ψ B B ρ B B { ( ( } [ ρζ ρf B η B + 1 (v BB R B B F ( ] { κb ψ ( ρζ ρ B i ˆω + κ ( } ρf B B ρ η B = 0 B Straight eld line coordinates: J = ( ψ ϕ θ, dϕ dθ = B ϕ = F q(ψ. B θ JR ˆω ω + iv Displacement components: ξ = ξ ψ perpendicular to the magnetic surfaces, η = ξ (B ψ perpendicular to the magnetic ψ eld, ζ = ξ B along the equilibrium magnetic eld.

10 Generalized Continuum Mode Equation III [ ρ B ( ] [ ρζ Γp ˆω ρ ˆω B ( ] ρζ + ρb B ρ Γp + B B B ( ΓpB ( η ρf ρb Γp + B ρ B (B F B η B [ ( ] iρ (v BB R κb ψ F ρ B ˆωη B [ κ B ( ] ρf +ρη B B = 0 ρ 3 B The only assumption: the mode is localized around the particular magnetic surface: f ψ f ψ ψ.

11 Approximation for spectrum analysis Slow rotation M P, M T 1 Large aspect ration r/r 1 Circular magnetic surfaces: ψ = ψ(r : R R 0 + r cos θ, z r sin θ Three mode approximation used: η = η m δ(ψ ψ 0 exp [i(mθ + nφ ωt], ζ = (ζ m 1 e iθ + ζ m+1 e iθ exp [i(mθ + nφ ωt]. In the absence of mode coupling: η m the Alfv en mode, ζ m 1 and ζ m+1 two slow, sound waves.

12 Equilibrium in Large Aspect Ratio Limit: R 0 /a 1 λ ρ and λ F describe the density and poloidal current stream function oscillations on the magnetic surfaces caused by the centrifugal and Coriolis eects: 1 λρ B ρ = ρ R B R, 1 F B F = λ F R B R. In term of angular velocity of rotation ( ] Here [ λ ρ = 1 1 D Ω T [ Γp λ F = 1 D 1 Ω P ω A B (qω P Ω T q Ω P Ω P v θ = κj/ρ, Ω T v φ = Ω + qω P, ( D 1 Ω P /ω ω A s q Ω P. ω s = c s /R (c s = Γp/ρ, qω P Ω T + q Ω P Ω T Ω P ω A ω A = v A /qr (v A = B /ρ ]., Low plasma pressure: Γp/B 1 Slow poloidal rotation: Ω P/ω A 1 λ ρ = M P M PM T + MT / 1 MP, λ F = Γp B M P M PM T + M T M P / 1 M P M T Ω T /ω s, M P qω P /ω s Far away of resonance (MP = 1 the poloidal angle dependent parts of p, ρ, F are assumed to be small compared to their poloidal angle independent parts: 1 MP 1 In general the poloidal current stream function F is not a function of poloidal ux ψ that is important for the case of electromagnetic perturbations with (m, n 0 (m poloidal, n toroidal wave numbers.

13 Dispersion Relation for Continuous Spectrum Here Dispertion Relation : ω k v A ω s M 1 ω s M 1 = 0 D 1 D 1 ] ( M ±1 = ω [(1 + λ ρ(1 M P M T + M T (3 + λρ + ω {k c s M T 1 + M T [ ] ]} [ M T M P (1 + λ ρ + M T (1 + λ ρ(1 M P and ±k c s ω s q [( 1 + M T + k c s M P M T (1 + λ ρ + ± ωs q ( [( 1 + M T 1 M P M T + M T ( M T λ ρ ] ( M T λ ρ M P (1 + λρ +M P M T (1 + λ ρ M T D ±1 = ( ω Ω P ( k ± 1 qr c s, ω ω mω P nω T, k (m + nq/qr ]

14 Perturbations under analysis where [ (1 MP Λ( ω = ω 4 ωs M T + M ω T 1 MP { + ω4 s q 4 (m, n = 0 electrostatic perturbations and (m, n 0, but m + nq(ψ = 0 perturbations localized at the rational magnetic surfaces obey the same dispersion law: ω Λ( ω ˆω 4 4M P ω ω s /q = 0, (1 M P + q [ ( 1 M P M T + M T 4M P M T (1 M P M T + M T ] + MT M P q ( + M T (1 MP ]} 1 + 4M P 1 M P, ˆω ω ω s q (1 M P

15 Solutions for Electrostatic Perturbations ordinary GAM modied by rotation. Three branches of the continuous spectrum: ω 1 = ω s [ + 1q + 4MT 4M PM T (1 1q +M P ( 1q + q 4 ] ( ] ω = ω s [1 q 4M P M T + MP 3 q acoustic mode induced by poloidal rotation (in the case of very slow poloidal rotation MP 1 it transforms into the sound wave. When the poloidal rotation is suciently fast this mode is unstable. Instability criterion: [ ( A (1 M P + q 1 M P M T + M T 1 + 3M P 1 M P ] + M T (1 MP 4M P M T (1 M P M T + M T < 0. ω 3 = 0 neutrally stable zonal ow.

16 Electromagnetic perturbations Dispersion relation: ω Λ( ω k v A ( ˆω 4 4M P ω ω s /q = 0. In case k v A ω s perturbations localized close to the rational surface. Four roots of dispersion relation are dened by the equation Λ( ω = 0. The remaining two roots are described by the expression ω = k v A (1 MP. A Zonal ow solution ω = 0 transforms into the Alfv en wave, when A > 0, or into the linearly unstable convective cell mode, when A < 0.

17 Poloidal Rotation: Special Cases Electrostatic Perturbations Dispertion Relation: ω 4 ω s ω [ 1 + M P q M P + ω4 s q 4 [ (1 M P + q (1 + 3M P 1 M P ] ] = 0. Electromagnetic Perturbations Dispertion Relation: ω 6 ω [ω 4 s { [ ω ω s ω s q q ( 1 1 M P ] M P q + k v A + ] (1 M P + q (1 + 3M P 1 M P Solutions: ω 1, ω s ± ( 1 { 1 + M P q + 1 M P 1 1 M P ω 1 GAM with upshifted eigenfrequency; M P q + 4M P (1 M P q 4 ω sound mode: unstable if A p = (1 M P + q (1 + 3M P /(1 M P < 0 (the instability arises only when Ω P > c s /qr; } } + k v A (1 + M P k v ωs 4 A q 4 (1 M P = 0. Solutions, case k v A ω s : ω 3 (zonal ow transforms into ω = k v A (1 M P (1 M P +q (1+3M P /(1 M P unstable if A p < 0., which is Solutions, case 1/q R k v A ω s : ( ω 1, = ω s q 1 ω s k v ± M P, ω 3 = k v A A ω 3 = 0 zonal ow.

18 Slow Poloidal Rotation: General Case k v A ω s α q k v A /ω s all the solutions are real ω and ω 3 are the modications of the electrostatic modes due to the coupling with the Alfv en wave ω 1 is the mode, which transforms into the zonal ow at the rational magnetic surfaces frequencies of all modes are growing with the increase of the distance from the rational surface.

19 Fast Poloidal Rotation: General Case k v A ω s with the increase of α up to the point α.39 the growth rate of sound mode decreases due to the coupling with the Alfv en mode and the growth rate of Alfv en mode still increases due to the increase of the parallel wave number in the interval.39 < α < 10.0 the squares of the frequencies of modes, which are the continuation of the sound and Alfv en modes are complex numbers. when α > 10.0 all three modes become purely oscillating due to the coupling of Alfv en and slow modes. α q k v A /ω s

20 Toroidal Rotation; Isentropic Magnetic Surfaces k 0 Dispertion Relation: ( [ ( ] ω k v A ω ω s q k c s 4 ω s q k c s = {[ ( ( ] ω + ωk c s M T ω s ( 1 + M T + M4 T 4 ω ω s q k c s + 4 ω ω s q k cs M T Six Solutions near a rational surface (k 1/qR and k v A ω s : ( 1 + M T 1 + M T } GAM: ( ω = ω s k = q + 4M T + M 4 T zonal ow: ω = 0 ω TGAM. The sound mode for perturbations with k = 0 does not exist. Limits k v A ω TGAM : Two sound waves ( ω = nω T ± ωs q M T (1 + M T / qk 1 + M T + M4 T /4 k cs + O c s ω s Coupling between the GAM and the Alfv en wave [ ω = (ω 1 TGAM + k v A ± (ω TGAM + k v A 4k v ωs A q ] 1/ GAM: ω = ω TGAM Alfv en wave: ω = k v A ω TGAM : shear Alfv en wave sound wave: ω = ωs /q k v A 1+q (+4M T +M4 T /

21 Conclusions A theory of low frequency localized, ideal MHD modes in large-aspect ratio tokamaks has been developed to include the eects of both toroidal and poloidal stationary plasma ows. It describes the coupling of the slow (sound branch and the shear Alfve n branch through the curvature, plasma pressure, and rotation eects. The general dispersion relation for continuous spectrum in the case of low-pressure large aspect ratio tokamaks with circular magnetic surfaces has been analytically derived. There are three branches of electrostatic, axisymmetric perturbations and of perturbations located at the rational surface of solution exist: zonal ow, GAM modied by rotation eects and the acoustic mode induced by poloidal rotation. The mode induced by poloidal rotation is unstable when the poloidal rotation is suciently fast. The corresponding instability criterion is derived. Away from the rational surface in the general case of electromagnetic perturbations (m, n 0 the zonal ow mode transforms into the shear Alfv en or into linearly unstable convective cell mode. Stabilization by the shear Alfv en coupling eect is demonstrated.