Nonlinear MHD effects on TAE evolution and TAE bursts

Transcription

1 Nonlinear MHD effects on TAE evolution and TAE bursts Y. Todo (NIFS) collaborating with H. L. Berk and B. N. Breizman (IFS, Univ. Texas) GSEP 3rd Annual Meeting (remote participation / Aug. 9-10, 2010)

2 Outline MEGA code Nonlinear MHD effects on TAE evolution Simulation of TAE bursts using time dependent f 0 2

3 MHD Equations with EP effects EP effects 3

4 Energetic particle current density Energetic ion current density without ExB drift: parallel + curvature drift + grad-b drift magnetization current 4

5 Simulation Study of Nonlinear Magnetohydrodynamic Effects on Alfvén Eigenmode Evolution and Zonal Flow Generation

6 Outline Introduction - simulation of TAE bursts Numerical investigation of nonlinear MHD effects on TAE (n=4) evolution Comparison of linear and NL MHD simulations MHD nonlinearity suppresses the TAE saturation level Suppression mechanism of TAE saturation level Spatial profiles and evolution of nonlinearlygenerated n=0 mode Excitation of geodesic acoustic mode (GAM) Summary 6

7 Reduced Simulation of Alfvén Eigenmode Bursts [Todo, Berk, Breizman, PoP 10, 2888 (2003)] Nonlinear simulation in an open system: NBI, collisions, losses The experimental results were reproduced quantitatively. Store of energetic ions Destabilization of AEs Transport and loss of energetic ions Time evolution of energetic-ion density profile. 7 Stabilization of AEs

8 consistency and inconsistency consistent with the experiment: synchronization of multiple TAEs drop in stored beam energy at each burst burst time interval inconsistent in saturation amplitude simulation δb/b~10-2 inferred from the experiment δb/b~10-3 8

9 Numerical investigation of nonlinear (NL) MHD effects on a TAE (n=4) evolution 9

10 Initial Plasma Profile f (v,µ,σ,p ϕ ) g(v)h(ψ) 1 1 g(v) = v tanh v v b Δv + v c slowing down distribution isotropic in velocity space 10 v b = 1.2v A, v c = 0.5v A, Δv = 0.1v b

11 TAE spatial profile (n=4) The main harmonics are m=5 and 6. 11

12 Comparison between linear and NL MHD runs (j h is restricted to n=4) ρ t = (ρ eq v) + ν n Δ(ρ ρ eq ) ρ eq t v = p + (j eq j heq ) δb + (δj δ j h ) B eq (νρ eq v) (νρ eqω) B t = E p t = ( p eqv) (γ 1) p eq v + ν n Δ( p p eq ) +ηδj j eq E = v B eq + η(j j eq ) j = 1 µ 0 B ω = v ρ t = (ρv) + ν nδ(ρ ρ eq ) ρ t v = ρω v ρ ( v2 2 ) p + ( j j h ) B + 4 (νρ v) (νρω) 3 B t = E p t = ( pv) (γ 1) p v + ν n Δ( p p eq ) +(γ 1)[νρω νρ( v)2 + ηj ( j j eq )] E = v B + η( j j eq ) j = 1 µ 0 B ω = v The viscosity and resistivity are ν=ν n =10-6 v A R 0 and η=10-6 µ 0 v A R 0. The numbers of grid points are (128, 64, 128) for (R, φ, z). The number of marker particles is 5.2x φ π/2 for the n=4 mode. 12

13 Comparison of linear and NL runs β h0 =1.5% Sat. Level (linear) ~ 3x10-3 Sat. Level (NL) ~ 3x10-3 β h0 =2.0% Sat. Level (linear) ~ 1.6x10-2 Sat. Level (NL) ~ 8x10-3 The saturation level is reduced to half 13 in the nonlinear run.

14 Suppression mechanism of TAE saturation level (1/2) Energy of each toroidal mode number n (n=0, 4, 8, 12, 16) E n 1 2 ( ρ n=0 v 2 2 n + (B B eq ) ) dv n Energy dissipation of each toroidal mode number n (n=0, 4, 8, 12, 16) D n [νρ n=0 ω n νρ n=0 ( v n )2 + ηj n ( j j eq ) n ] dv Damping rate of each toroidal mode number n (n=0, 4, 8, 12, 16) γ d n = D n / 2E 4 Total damping rate of all the toroidal mode numbers (n=0, 4, 8, 12, 16) γ d ALL = D n / 2E 4 14 n

15 Suppression mechanism of TAE saturation level (2/2) β h0 =1.7% Sat. Level (linear) ~ 1.2x10-2 Sat. Level (NL) ~ 6x10-3 The total damping rate (γ dall ) is greater than the damping rate in the linearized MHD simulation (γ d lin ). 15

16 Schematic Diagram of Energy Transfer Energetic Particles Drive Linearized MHD n=4 TAE n=0 and higher-n modes Dissipation NL coupling Dissipation Thermal Energy Thermal Energy NL coupled modes 16

17 Effects of weak dissipation β h0 =1.7% The viscosity and resistivity are reduced to 1/4, ν=ν n = v A R 0 and η= µ 0 v A R 0. The nonlinear MHD effects reduce the saturation level also for weak dissipation. 17

18 Excitation of GAM Evolution of TAE and zonal flow Spatial profile of zonal flow After the saturation of the TAE instability, a geodesic acoustic mode is excited. 18

19 Summary of NL MHD effects on a TAE instability Linear and nonlinear simulation runs of a n=4 TAE evolution were compared. The saturation level is reduced by the nonlinear MHD effects. The total energy dissipation is significantly increased by the nonlinearly generated modes. The increase in the total energy dissipation reduces the TAE saturation level. The zonal flow is generated during the linearly growing phase of the TAE instability. The geodesic acoustic mode (GAM) is excited after the saturation of the instability. The GAM is not directly excited by the energetic particles but excited through MHD nonlinearity. 19

20 Simulation of Alfvén eigenmode bursts with nonlinear MHD effects

21 Simulation with source, loss, collisions and NL MHD Time dependent f 0 is implemented in MEGA particle loss (at r/a=0.8 for the present runs) for marker particles to excurse outside the loss boundary and return back to the inside phase space inside the loss boundary is well filled with the marker particles particle weight is set to be 0 during r/a>0.8 if the simulation box is extended to include the vacuum region, the loss boundary can be set at more realistic location 21

22 δf method with time-dependent f 0 (1/2) t f + { H, f } ν v 2 v v3 3 + v c ( ) f = S(v) When f 0 satisfies t f 0 + { H 0, f 0} ν v 2 v ( v3 3 + v ) f c 0 = S(v), the evoution of δf is given by t δf + { H 0 + H 1,δf } + { H 1, f 0 } ν v 2 v v3 3 + v c With a definition d dt δf t δf + H 0 + H 1,δf the evoution of δf is expressed by d dt δf + { H 1, f 0} 3νδf = 0. ( ) δf = 0. { } νv 1 + v c 3 v 3 v δf, 22

23 δf method with time-dependent f 0 (2/2) The evolution of phase space volum V that each particle occupies should be considered. Comparison of two eqs.: d dt and ( fv) = SV d dt f 3νf = S gives d dt V = 3νV We solve the evolution of both δf and V of marker particles. 23

24 Time-dependent f 0 A solution of t f 0 ν ( v 2 v v3 3 + v ) f c 0 = S(v), is f 0 (v, t) = 1 1 ν v v erf v'-v b erf v - v b c Δv Δv with ( ) v c 3 v' = ( v v ) exp 3ν(t + t c inj) S(v) = 2 1 π v 2 Δv exp v - v b Δv 2, 1/ 3 v b : injection or birth velocity of energetic particle t inj : injection starts at t = -t inj < 0, Note : Here we have neglected {H 0, f 0 } term for parallel beam injection (zero magnetic moment) and w/o finite orbit width effect. 24

25 Physics condition of simulation similar to the previous reduced simulation of TAE bursts with a few exceptions parameters a=0.75m, R 0 =2.4m, B 0 =1T, q(r)= (r/a) 2 NBI power: 10MW beam injection energy: 110keV (deuterium) v b =v A parallel injection (v // /v=-1 or 1) slowing time: 100ms The time-dependent f 0 is similar to but slightly different from that in the previous slide. no pitch angle scattering 25

26 Evolution of TAE amplitude and stored beam energy n=2 TAE n=3 TAE Energetic particle losses due to TAE bursts. Synchronization of multiple TAEs. (Amplitude is measured at the mode peak locations.) 26

27 Comparison of NL and linear MHD runs stored beam energy n=2 TAE Linear MHD NL MHD NL MHD effects: reduction of saturation level, rapid damping after saturation, and suppression of beam ion loss. 27

28 Spatial profile and evolution of zonal flow n=0 poloidal flow profile at t=8.03ms. evolution of zonal flow at r/a=0.28 (top) and r/a=0.53 (bottom). 28

29 Summary of TAE burst simulation TAE bursts are successfully simulated with NL MHD effects using time-dependent f 0. synchronization of multiple TAEs beam ion loss at each burst (maximum saturation level: δb/b~ ) NL MHD effects: reduction of saturation level rapid damping after saturation suppression of beam ion loss 29