Nonlinear hydrid simulations of precessional Fishbone instability

Transcription

1 Nonlinear hydrid simulations of precessional Fishbone instability M. Faganello 1, M. Idouakass 1, H. L. Berk 2, X. Garbet 3, S. Benkadda 1 1 Aix-Marseille University, CNRS, PIIM UMR 7345, Marseille, France 2 Institute for Fusion Studies, University of Texas, USA 3 CEA, IRFM, 13115, St Paul lès Durance, France

2 Outline Introduction The Precessional Fishbone instability Tokamak geometry and particle motion A reduced Fishbone model Linear theory Analytic results Numerical results Numerical code Linear benchmark Nonlinear results Conclusions

3 Context: the Fishbone instability First observation on the PDX tokamak, with near perpendicular neutral beam injection active Bursts of electromagnetic instability, associated with losses of fast particles Mode frequency consistent with trapped fast ion precession frequency Dominant m = n = 1, kink-shaped, mode structure Observation of frequency chirping Figure : [McGuire et al., PRL, 1983]

4 Trapped particle trajectories (passing are confined too!) ω c ω bounce > ω precession possible decoupling of the three dynamics: Full kinetic, Gyrokinetic or Bounce-averaged descriptions For elegant derivation, using Lie algebra, see [Littlejohn, PS, 1982] (x, v) (X gc, v, (J c, ζ c)) ω ωc (X gc, v ; J c) (X gc, v ; J c) (α, β, (J, ζ ); J c) ω ω b (α, β; J, J c) ((J p, ζ p); J, J c))

5 The small inverse aspect ratio Tokamak ɛ = a/r 0 1 ( B = B T ˆϕ ) + ψ ϕ R B T = B 0 0 R B0 1 r R 0 cos θ & ψ = RA ϕ R 0A ϕ(r, θ, ϕ, t) with ψ eq = ψ eq(r) (concentric circular cross sections at equilibrium) Safety factor: q(r) = rb θ,eq R 0 B T where B θ,eq = R 1 0 r ψ eq = R 1 0 ψ eq i.e. magnetic winding number Note: B θ B T & q = O(1)

6 Equilibrium trajectories: bounce motion Let s take ( r ) 1 ρ c J c µ = 1 2 mv 2 gyro/b T cst E = 1 2 mv mv mv 2 + µb T (r, θ) }{{}}{{} potential well, r cst E kin along field line Circular cross section A classical pendulum :-)) ( ) v 2 = (r 2 + R0 2 q 2 (r)) θ 2 R0 2 q 2 (r) θ 2 & B T = B 0 1 r R 0 cos θ E mr = 1 0 2q2 2 θ 2 µb0r mr 3 0 q2 }{{} = g/l Trapping parameter: k 2 = e+1, e = E µb 0 R 0 2 µb 0 r particles) J = 1 2π Bounce period: T b = 2πω 1 b cos θ + µb0 mr 2 0 q2 }{{} =cst (k 2 [0, 1] for trapped mv ds = 8 π R 1/2 0 q ] mµb 0 [(k 2 1)K(k) + E(k) mr = 2π J / E = 4R 0q 0 µb 0 K(k)

7 Banana width & precessional motion P ϕ = mrv ϕ + e ψ is an exact invariant (ϕ is cyclic at the equilibrium) c v ϕ v changes along the trajectory ψ (i.e. r) must change too P ϕ = mr 0v θ=0 + e c ψeq(r) + e c ψ eqδr θ=0 = P ϕ turning point = e c ψeq(r) Banana orbit width: δr θ=0 = qr 0 ω c r v θ=0 Two possible reasons for toroidal precession: 1) Magnetic shear: forward/backward motions are not along the same line ϕ = dϕ = qdθ }{{} =0 + q δrdθ q (r)δr θ=0 ω D,1 q (r) mω cr J ω b 2) The magnitude of v is not the same during forward/backward motion δe k, θ=0 = µb 0δr θ=0 ω D,2 δv θ=0 R 0 = µb0q mω cr 0r It is possible to obtain ω D rigorously ( ω D = ω D,1 + ω D,2 ) using ω D = E J ω b J p E = J ; J p = P ϕdϕ = P ϕ = e J p c ψeq(r) ω D = c e 1 ψ eq J r

8 Deeply trapped particles Let s take v 0, thus: J = 0 and stays zero (ω ω b ), as well as v µ and J = 0 are parameters (ϕ, Pϕ ) are the natural canonical variables: 2D phase space :-) P ϕ e c ψ, i.e. P ϕ r & Pϕ r ( The Hamiltonian H = 1 P e A ) 2 2m c + µb T + eφ reduces to H(ϕ, P ϕ) = µb T (P ϕ) + eφ(ϕ, P ϕ, t) }{{}}{{} equilibrium mode The coupling is done only via the electric potential φ ϕ = H P ϕ = µb T P ϕ + eφ = µ r B T (r) P ϕ P ϕ r }{{} ω D = µb 0 q mωc R 0 r P ϕ = H φ(r, ϕ, t) = e ϕ ϕ +e r φ(r, ϕ, t) P ϕ r

9 A reduced Fishbone model: Fast particle response Take into account only deeply trapped particles (v = 0) with a single value for the magnetic moment, µ = µ Reduction of the phase space from 6D to 2D (ϕ, P ϕ ) All fast particles are contained well inside the q = 1 surface ( core region ) The dominant mode has a kink-like shape (mode numbers m = n = 1, electric potential φ/r cst well inside of the q = 1 surface) f t + [H, f ] = 0 ; H(ϕ, Pϕ) = µ B T (P ϕ) + eφ(ϕ, P ϕ) µ B T P α = ω D (P α) = µb T q(p ϕ) ω C mrr(p ϕ) ; φ = r(p φ0(t) ϕ) e iϕ r 0 Note: particles interact with a single mode. All the other ones vanish far quicker in the core region where particles are.

10 A reduced Fishbone model: Bulk plasma response Fluid description for the bulk of the plasma, neglecting the thermal pressure effects and density variations Reduced-MHD description. φ t ψ t + {φ, ψ} = 0 + {φ, φ} {ψ, ψ} = ρ = α normalization [( ˆϕ κ) P,h ] Toroidal effects are retained only for the contribution given by fast particles (κ is the toroidal curvature and P,h the fast particle pressure) Cylindrical geometry As before we set φ = r φ 0(t) r 0 e iϕ iθ well inside the q = 1 surface On the contrary strong variations are allowed across the q = 1 surface and finally φ 0 for r a One mode evolution: only Fast particle (kinetic) nonlinearities are retained, MHD nonlinearities are neglected

11 Linear theory: Analytic results I Let s take f = F eq + δf, with δf F eq, and t iω. The mode equation reads ω 2 ( φ r ) + v 2 A,T R 2 0 ( 1 1 ) 2 ( ) φ = iω 1 r d r r 2 ρ(δf ) q(r) r r 3 0 where δf = er φ 0 df eq r 0 dr dr dp ϕ ω D (r) ω Note as a spatial gradient corresponds to the usual velocity gradient. Here a decreasing density is equivalent to a bump on the tail. Finally a general dispersion relation is obtained : ( ) 1 y 2 q(y) 2 dfeq dy i = K dy (1) q(y) 0 yω ω D (y=1) where K takes into account the energetic content for the fast particles, the MHD and geometric parameters and y = r/r (r being so that q(r ) = 1).

12 Linear theory: analytic results II Let s take: F eq = n 0 2 δ(µ µ )δv (1 erf(β(y y 0)) analytic values the threshold condition and mode frequency at the threshold K 0 = 1 ( ; ω πβy 3 0 = ω D (y 0) 1 1 ) 0 β 2 y0 2 where y 0 is the position of the highest radial gradient in the distribution function Close to the threshold K = K 0 + δk ; ω = ω 0 + δω + iγ we obtain a growth rate and a correction for the real frequency: γ δk π ω 0 ; δω 1 π γ K 0 2βy 0 2 βy 0

13 Nonlinear numerical code Based on the same domain decomposition: 1) The core region I Nonlinear kinetic description: Semi-lagrangian code assuming φ = r φr bound I Linearized MHD response, including the fast particle pressure: i.e. an evolution equation for r φr bound plus the frozen-in equation for ψ bound 2) Thin annular region around q = 1 surface I No fast particle here I Cylindrical slab description I Semi-spectral MHD code (only one mode at present time) φ & ψ bound as B.C. r bound I Uses r I Provides ψ & φ, in particular φ bound (the Hamiltonian)

14 γ Linear benchmark With numerical simulations, the linear results are recovered Good agreement with frequency value Good agreement with growth rate and mode shape 2.5 x x φ K/K x Figure : Growth rate as a function of K/K 0. Stars are the numerical values, the green line is the analytic prediction. Figure : φ profile in the annular layer. In blue the real part, in red the imaginary part.

15 Nonlinear results: Mode saturation level The first local maximum of the amplitude is proportional to γ 2 i.e when the phase-space island width φ reaches the resonance width γ [Zonca et al., NJP, 2015] Amplitude oscillations are far larger compared to the usual Bump on Tail case, with [Berk et al., PLA, 1997] or without dissipation [O Neil, PF, 1965] x 10 5 Figure : Evolution of the kinetic energy of the mode versus time, in logscale

16 Frequency chirping and particle ejection Chirping is observed during the saturated phase (case studied here: K/K 0 = 1.2, giving γ/ω = 2.8%) Phase space structure motion matches frequency change Asymmetric system Higher amplitude for the down chirping mode, i.e. particle ejection Outgoing particles continue to interact with the mode but actually are not trapped into the mode well Figure : On the left : Evolution of the distribution function averaged over angle. On the right : spectrogram of the mode.

17 e e e x e e Structures in phase-space Dynamics close to the first maximum and minimum of the mode amplitude Only partial folding of f inside the phase-space island Strong stretching in ϕ & P ϕ directions Figure : Distribution function at different stages of the saturation.

18 Island contraction and slippage

19 Contraction and slippage: The Fishbone peculiarity The dispersion relation of the Fishbone is really different from the usual Bump on Tail one: BoT: marginally stable plasma wave + particle driver leading to instability i.e. RD(ω, k) 0 & a small ID gives the instability Thus when the energetic driver drops, the mode response is not dramatic. Fishbone exists only because fast particles are there, indeed ω R = ω D,h It is a genuine energetic particle mode [Zonca et al., NJP, 2015] The mode response to the energetic driver variation can be strong: 2 t 2 ( ) φ + ωa(r) 2 r ( ) φ = i r t 1 r d r r 2 ρ(f ) = RHS r 3 0 In the core ω 2 A,bound tt ω 2 R thus the mode is slave, compared to the particle driver: ( ) φ r bound RHS ωa 2 bound giving φ bound = φ 0 RHS Amplitude and phase simply follow: the mode is slave to the energetic driver.

20 Conclusions The Precessional Fishbone instability can be described by an hybrid Fluid-Hamiltonian model where the phase-space coordinates are quite unusual. This reduced model is able to catch the qualitative dynamics of the mode: the frequency downchirping of the mode and the gradual ejection of fast particles. It has the great advantage, over more complete models, of permitting an easier analysis of the structure dynamics in phase-space. Future perspectives: 1) Allowing MHD nonlinearities to develops around the q = 1 surface. 2) Looking at a more general approach for energetic particle modes.