1 ASIPP Sawtooth Stabilization by Barely Trapped Energetic Electrons Produced by ECRH Zhou Deng, Wang Shaojie, Zhang Cheng Institute of Plasma Physics, Chinese Academy of Sciences, Hefei, China

2 ASIPP Introduction Dispersion Relation and Stabilization of Sawtooth Application to Explanation of Experiments Conclusion Contents

3 Introduction Sawtooth oscillation is associated with internal kink modes. It is of significant interest to control sawtooth in tokamak experiments. In experiments, a variety of methods, including NBI, LHCD, ECRH…, have been utilized to control MHD activities in tokamak. ECRH is an ideal choice to stabilize sawteeth for its advantageous features, e. g. accessibility, highly localized wave absorption and high absorption rate.

4 Introduction(Ctd) The main experimental results of ECRH on sawtooth stabilization 1. Sawtooth can be completely stabilized if the heating is on the high field side q=1 surface and the launched power is large enough. (Y. Liu, et al, PST, 4, 1153 (2002) and K. Hanada, et al, PRL 66, 1974 (1991)) 2. Little effect was found when heating was outside q=1 surface. (K. Hanada’ 91, ) 3.Effect on sawtooth weakens when the heating location shifts from q=1 surface to magnetic axis on HFS. (K. Hanada’91, Z. Pietrzyk et al, NF, 39, 587(1999)) 4. When heating is on magnetic axis or on q=1 surface of LFS, sawtooth shapes change ( Y. Liu’ 02) or period of sawtooth extended (K. Hanada’ 91)

5 Introduction(Ctd) There are two possible mechanisms of the sawtooth stabilization by ECRH. 1. ECRH modifies the local plasma parameters, e. g. the current profile (K. Hanada’ 91), the pressure profile (W. Park, et al, Phys. Fluids 30, 285 (1987)). This model fails to explain the difference between the HFS and LFS heating. 2. The energetic electrons produced by ECRH interact with internal kink modes and may stabilize or destabilize sawtooth. The stabilization of sawtooth by high energy ions has been under wide investigation, both theoretically and experimentally (F. Porcelli, PPCF, 33, 1601(1991)). So far, little attention has been paid to energetic electrons interacting with internal kink modes.

6 Dispersion Relation and Sawtooth Stabilization General Variational Principle Consider a large aspect ratio tokamak plasma consisting of a core ( c ) background component and a hot ( h ) energetic electrons. The stability analysis of internal kink modes is given using a general variational principle

7 Dispersion Relation and Sawtooth Stabilization The usual minimization procedure for analyzing internal kink modes, with assumption of a parabolic q profile, gives According to Chen L. et al ( PRL 52, 1122, 1984), we obtain the energy contribution from the energetic electrons where pitch angle variableenergy of energetic electrons magnetic moment

8 Dispersion Relation and Sawtooth Stabilization According to Chen L. et al ( PRL 52, 1122, 1984), we obtain the energy contribution from the energetic electrons where pitch angle variable energy of energetic electrons magnetic moment with the mode frequency, F 0h the equilibrium distribution of high energy electrons. the bounce average toroidal processionl frequency.

9 Dispersion Relation and Sawtooth Stabilization Through straightforward integration, we obtain for two models of equilibrium energetic electron distribution in energy 1. Slowing down distribution 2. Exponential distribution in energy The average is

10 Dispersion Relation and Sawtooth Stabilization When ideal kink modes can be stabilized. For two models of energy distribution, we obtain the same So far, all the derivations are valid for both barely trapped and deeply trapped electrons, and we did not assume a specific spatial distribution of energetic electrons. where E(k 0 2 ) and K(k 0 2 ) are complete elliptic integrals and

11 Dispersion Relation and Sawtooth Stabilization To proceed, we should specify a spatial distribution of energetic electrons. The absorption of the electron cyclotron wave is localized in narrow space in ECRH experiments, we assume a Gaussian distribution of energetic electrons with respect to minor radius where This assumption is valid only when the energy confinement time is much longer than the slowing-down time of energetic electrons.

12 Dispersion Relation and Sawtooth Stabilization With this assumption, we have where reflects the influence of bounce angle. The barely trapped with have a positive contribution.

13 Dispersion Relation and Sawtooth Stabilization The influence of radial distribution is included in It peaks at, when the heating location shifts from surface, it decreases quickly. The negative contribution is physically meaningless since the model Gaussian distribution causes less energetic electrons inside r=r p than outside it.

14 Dispersion Relation and Sawtooth Stabilization So we conclude that 1.The trapped energetic electrons with have a positive contribution to to stabilize sawtooth oscillations. Those with have a negative contribution to destabilize sawtooth oscillations. 2.The stabilization is most effective when the heating location is on the q=1 surface on HFS.

15 Application to Explanation of Experiments The total energy store in energetic electrons, normalized by magnetic energy, is The energy balance condition is Then we obtain

16 Application to Explanation of Experiments For the K. Hanada experiment, the parameters are cm, cm, T, m -3, the slowing down time is ms estimated using and assuming, the effective power absorption kW is needed on q=1 surface HFS heating to achieve which is of the order of, the EC power in experiment to completely stabilize sawtooth was ~200 kW, the discrepancy may be ascribed to the following to causes:

17 Application to Explanation of Experiments 1.The influence of resistivity. should be larger than a positive critical value, depending on plasma profiles to completely stabilize resistive kink modes. 2. The diffusion of energetic electrons reduced their radial pressure gradient and hence weaken their effect on sawtooth. In this case, ms, ms, the narrow width Gaussian distribution is not applicable. We can estimate using spatial diffusion equation

18 Application to Explanation of Experiments Assuming a linear distribution with respect to r within q=1 surface, we estimate that ~200kW effective power absorption is needed to achieve in rough agreement with the experimental results.

19 Application to Explanation of Experiments For HT-7 shot 5255 T n c =2.1*10 19 m -3 We estimate cm ms and assume and then we have kW is needed to completely suppress sawtooth oscillations when heating is on q=1HFS surface. R=122 cm

20 Application to Explanation of Experiments If then MW is required to completely stabilize sawtooth.

21 Conclusions 1.The barely trapped energetic electrons produced by ECRH in tokamaks can effectively stabilize sawtooth oscillations if the heating location is on the HFS and close to the q=1 rational surface. The required effective power is in rough agreement with experimental results. 2.When the heating location shifts away from the q=1 rational surface, the stabilization effect decreases quickly. 3.Deeply trapped energetic electrons produced by ECRH on LFS may destabilize sawtooth oscillations. 4.Another theoretical model for sawtooth stabilization by ECRH is related to the modification of background plasma profiles, e. g. pressure and current density etc., to form a kink mode stabilizing. configuration. However, the explanation of the difference between the HFS heating and the LFS heating on q=1 surface is unsatisfied.

22 Conclusions(Ctd) 5. Our model can qualitatively explain the asymmetry between the HFS heating and LFS heating effect observed in experiments. In the real situation, the two mechanism are expected to work simultaneously to explain both the asymmetry and the extension of sawtooth period on LFS.