1. Excitation of internal kink mode by barely trapped suprathermal electrons* Youwen Sun, Baonian Wan, Shaojie Wang, Deng Zhou, Liqun Hu and Biao Shen * Sun Y.W. et al., Phys. Plasmas 12, 092507(2005)

2. Outline 1. Background 2. Dispersion relation 3. Threshold condition 4. Application to experiment 5. Conclusion

3. 1. Background Fish-bone oscillations have been observed in the neutral-beam injection experiments on several tokamaks and two models have been proposed to explain the these bursts. In recent experiments on the DIII-D Tokamak and the HL-1M Tokamak, an internal kink instability driven by barely trapped suprathermal electrons produced by high-field-side off-axis Electron Cyclotron Resonance Heating (ECRH) has been observed. A recent paper has already investigated the sawtooth stabilization by barely trapped energetic electrons.

4. 2. Dispersion relation In presence of the energetic particles, the dispersion relation for the internal kink mode is Minimized ideal variational energy (γ I = -ω A δW c ) The kinetic contribution coming from the energetic trapped particle Inertial term (1) (2) ω d =K 2 E/(K b mrRω c ),α=μ/E, K 2 =(2/εα) 1/2 [2E(k 2 )-K(k 2 )]/π, K b =(2/εα) 1/2 K(k 2 )/ π, k 2 =sin 2 (θ b /2)=(1/αB 0 -1+ε)/2ε, ε=r/R,

5. Two different models for the distribution function of the suprathermal electrons (3a) (The slowing-down distribution) (3b) (The exponential distribution) n(r)= p h (r)/( 2 3/2 πB 0 mK b0 E m ) n(r)=2 1/2 p h (r)/(3π 3/2 B 0 mK b0 T h 5/2 ) * * The kinetic contributions (4a) (4b) K 20 =K 2 (α=α 0 ), Ω=ω/ω d,(ω d =ω dm for the slowing-down distribution and ω d =ω dT for the exponential distribution and ω dm (ω dT ) is the bounce averaged toroidal precessional frequency of the electrons with energy E m (T h ) at K 2 /K b = K 20 / K b0 and r=r s ),ε s =r s /R, and Z is the plasma dispersion function.

6. Because ω c 0 for the barely trapped electrons, the barely trapped condition is K 20 /K b0 <0 (the numerical calculation shows that it is equivalent to 0.8261 ≤ k 2 ≤ 1, or 2.2813 (130.7 o ) ≤ θ b ≤ π (180 o )). For K 20 /K b0 <0, the sign and the form of δW k in Eqs. (4a) and (4b) are the same as that of the deeply trapped energetic ions. Substituting Eqs. (4a) and (4b) into Eq.(1) respectively, the dispersion relations are given by (5a) (5b)

7. 3. Threshold condition From the imaginary parts of Equations (5a) and (5b) ， the value of β ’ h at threshold for these two models are given by The equations for the real frequencies at threshold are found through setting Ω i =0 and replacing β ’ h by β ’ h, crit in the real parts of Equations (5a) and (5b), (6a) (6b) (7a) (7b)

8. Comparing the threshold conditions for these two models, we find ω rb /ω ra ~3.9ω dT /ω dm ~3.9T h /E m, β ’ h,crit,b /β ’ h, crit,a ~3.4ω dT /ω dm ~3.4 T h /E m, and ω ib /ω ia ~1.6 T h /E m. Consequently, there is no essential difference between the threshold conditions given by these two distribution models, if E m ~3T h, which is the same as the result for energetic ions Writing β ’ h =β ’ h, crit +∆β ’ h and substituting the real frequencies at threshold, from the imaginary parts of Equations (5a) and (5b) the growth rates are given by (8a) (8b)

9. 4. Application to experiment For the barely trapped electrons with a single value pitch angle distribution, K 20 /K b0 can be estimated to be -0.4. For typical D Ⅲ -D parameters [7]: major radius R=1.76m, minor radius a=0.62m, normalized singular layer radius ρ s =r s /a ≈ 0.2, B 0 =1.77T, n e ≈ 3.0*10 19 m -3, T e ≈ 2.5keV, ω r /2π ≈ 10kHz, ω i ≈ 5*10 3 s -1, ω A ≈ 8.2*10 6 s -1 (assuming s ≈ 0.2, Z eff ≈ 2), the ECRH power P eff ≈ 1.1MW. On the deposition radius surface, 1.4% of the electrons are the suprathermal electrons with 7.9% of the total electron energy and 0.27% electrons have energy above 36 keV, and they possess 3.4 %of the electron energy. For the slowing-down energy distribution f(E) ∝ E -3/2, E m is estimated to be 110keV by solving the equation, and the slowing down time is estimated to be τ s ≈ 3ms. Assuming β h (r)= β 0 exp[-(r-r p ) 2 /δ r 2 ] and choosing δ r /r s ~ 0.1 and r p =r s, we obtain β ’ h ≈ 0.91β 0. Using the energy balance condition for energetic electrons, we obtain β 0 ≈ 1.4%and β ’ h ≈ 1.3%. In high-field-side ECRH experiments, the beta value of the barely trapped energetic electrons is one order less than that of the total energetic electrons. Consequently, the beta value of the barely trapped suprathermal electrons in the experiment can be estimated to be β ’ h,exp ≈1.3‰.

10. For the slowing down model, we find ω dm = -K 20 E m /(K b0 r s RB 0 ) ≈ 114*10 3 s -1. Then the threshold conditions are given by ω r ≈ ω dm /2 ≈ 57*10 3 s -1 (ω r /2π ≈ 9.1kHz)and β ’ h, crit ≈ 2.7‰, which are of the same order as the experimental data. The value of ∆β ’ h /β ’ h can not be found from the experimental data. However, the required ∆β ’ h /β ’ h value can be estimated to be about 6% from Eq. (8a) for the experimental growth rate. For the exponential energy distribution f(E) ∝ exp(E/T h ), T h is estimated to be 20keV using the same method. Then, we find ω dT =- K 20 T h /(K b0 r s RB 0 )≈21*10 3 s -1. The threshold conditions are given by ω r ≈1.94ω dT ≈40*10 3 s -1 (ω r /2π≈6.4kHz)and β ’ h, crit ≈1.7‰, which are also of the same order as the experimental data. The required ∆β ’ h /β ’ h value can be estimated to be about 19% from Eq. (8b) for the experimental growth rate.

11. 5. Conclusion In summary, the barely trapped suprathermal electrons can also destabilize the internal kink mode, when their density gradient is positive within the rational surface and the beta value of them exceeds a threshold. With the assumption of two different models of energy distribution function of the suprathermal electrons, the threshold beta value of the barely trapped suprathermal electrons, the real frequency and growth rate of the fishbone mode are found in this paper. The threshold condition is insensitive to the form of the energy distribution function of the suprathermal electrons similar to the result of energetic ions. The calculated threshold beta value of the barely trapped suprathermal electrons and the real frequency of the mode are in reasonable agreement with the experimental observations on DIII-D. Since the contribution of the deeply trapped electrons can be neglected only for high-field-side ECRH experiments, this phenomenon cannot be observed in the low- field-side ECRH experiments.

12. Thank you !