H.B. Park

T. Sato

Y. Todo

National Institute for Fusion Science (NIFS-Repository)

§3. Fokker-Planck Simulation of Alpha-particle-driven Toroidicity-induced Alfven Eigenmode Todo, Y., Park, H.B., Sato, T. Alpha-particle-driven Toroidicity-induced Alfven eigenmodes (TAEs) were first observed in the Tokamak Fusion Test Reactor (1FIR) [1]. It should be noted that the estimated amplitudes of the TAEs were very small DB I B ::::: 10-5 , and they persisted much longer than the typical damping time. We carried out five dimensional Fokker-Planck simulation of the alpha-particle-driven TAE in 1FIR. A perturbative model [2] is adopted to investigate time evolution of the n=4 TAE, where the TAE is described by a linear eigenmode structure and by amplitudes of the sine part and the cosine part. We use coordinates, major radius R, vertical coordinate z, toroidal angle qJ, total velocity v, and the pitch-angle variable S = vII I v. We consider concentric circular magnetic surfaces as the equilibrium magnetic field for simplicity. Electromagnetic field is a superposition of this equilibrium field and the TAE field. We simulate the n=4 mode observed at TFTR shot #103101. We take an approximate q-profile which is shown in Fig. 1. The eigenmode equations (e.g. Eqs. (3) and (4) of Ref. [3]) for two poloidal harmonics m=6 and m=7 are solved to obtain n=4 TAE structure. The plasma density is taken to be uniform for simplicity and the boundary condition that we imposed for this eigenvalue problem is zero amplitude at r>0.45a to avoid an intersection with the Alfven continuum spectrum in this simplified model. The mode structure is shown in Fig. 1 with the q-profile. The parameters are taken to be consistent with the experiment; Ro=2.52 [m], a=O.90 [m], Bo = 5 [T], and the mode frequency oY'21C=214 [kHz]. We take an alpha distribution simplified from Eq. (51) of Ref. [4]. The central alpha pressure is f3a(O) = 6 X 10-4. The algorithm to advance amplitude and phase of the TAE mode is that developed in Ref. [2]. We take the plasma number density n = 3.5 X 1019 [m-3] with the effective mass of 2.04 for the calculated eigen frequency to be consistent with the observed TAE frequency. It is in good agreement with the experimental electron number density around r=O.3a where the TAE localizes spatially. The simulation was carried out with realistic collisional rates, the pitch-angle scattering rate v d= 1.0 [1/s] and the slowing-down rate V= 2.8 [1/s]. The mode damping rate is chosen Y d = 5 xl 0-3 (J). The amplitude evolution is shown in Fig. 2. We would like to discuss the linear behavior before moving on to the nonlinear stage. Linear alpha drive is estimated Y L = 9.9 X 10-3 (J) from the linear growth rate and the damping rate. This linear alpha drive is in good agreement with the NOVA-K analysis result YL =8xl0-3 {J) [5]. 312 It is interesting to investigate the detail of the alpha-TAE resonance, since the cut-off velocity is lower than the Alfven velocity. It is known that the particles with velocity lower than the Alfven velocity can resonate with TAE at v = v A 13. We confirmed the distribution function is consistent with the resonance condition at VII = -v A 13 and VII = V A 13. Let us turn to the nonlinear behavior. It can be seen in Fig. 2 that the amplitude decreases monotonically after saturation. The TAE never stays at steady amplitude. This is inconsistent with the experimental results. Convergence check in number of grid points is required, but it has not been done due to computational restriction. It needs more effort to resolve this discrepancy. 3 10-3 2.5 10-3 - - 4.8 m=6 2 10-3 4 -~ 3.2 :i 1.5 10-3 ~ 2.4 ...0 l- I 10-3 1.6 =a-5 10-4 0.8 o 10° 0 -5 10-4 -0.8 0 0.2 0.4 0.6 0.8 1 ria FIG. 1. The q-profile and the m=6 and m=7 poloidal harmonics of the n=4 TAE. o 1000 2000 3000 4000 5000 (Ot FIG. 2. Amplitude evolution of the n=4 TAE with the realistic pitch-angle scattering rate and Yd IYL =0.5. REFERENCES [1] R. Nazikian et ai., Phys. Rev. Lett. 78,2976 (1997). [2] 1. Candy et at., Phys. Plasmas 4, 2597 (1997). [3] G. Y. Fu and 1. W. Van Dam, Phys. Fluids B 1, 1949 (1989) [4] Y. Chen et ai., Phys. Plasmas 6, 226 (1999). [5] G. Y. Fu et ai., Phys. Plasmas 5, 4284 (1998). 

Fokker-Planck Simulation of Alpha-particle- driven Toroidicity-induced Alfv?n Eigenmode

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