B. Breizman

Dam, J." Van

H. Berk

N. "Nakajima

National Institute for Fusion Science (NIFS-Repository)

§7. Theoretical Analysis of Burst Modes in CHS Nakajima, N., Berk, H.*, Breizman, B.*, Van Dam, J.* * IFS, Univ. of Texas, USA Recently, fishborn-like burst modes (Burst modes) have been observed in L = 2/M = 8 Compact Helical System (CHS) experiments. In order to clarify the properties of such Burst modes theoretically, the formulation has been developed. Since the Burst modes are considered to be driven by energetic fast ions, the treatment of fishborn instabilities in tokamaks [1] are ex-tended. In the fishborn cases, (m, n) = (1,1) internal kink modes near the marginally sta-ble state are destabilized by energetic fast ions. Thus, the dispersion relation is obtained by the variational method of the following functional: where ~r is the radial displacement of (m, n) = (1, 1) internal kink modes at the marginally stable state, and 8W F, 8W/, and 8W K are the potential energy of MHD fluid, the inertial en-ergy, and the kinetic energy, respectively. Since such a ~r at the marginally stable state has, dis-continuity at the mode rational surface satisfy-ing t = n/m = 1 (the approximate form is a step function), the inertial energy is introduced only vicinity of the mode rational surface, in order to construct the well-behaved continuous eigenfunction. The kinetic term is assumed not to contribute to determining the structure of the eigenfunction ~r, but to determining the eigenvalue w through the dispersion relation. Thus, the functional given by Eq. (1) is rewrit-ten as D[~r] 8W;'[~r] + 8Wj[~r] + 8WF[~r] + 8W~[~r] (2) where superfixes sand e denote inertial layer near the mode rational surface and external re-gion, respectively. This procedure are reason-able, because the driving source of (m, n) = (1, 1) internal kink modes exist in the external region. In the case of Burst modes in CHS ex-periments, MHD modes associated with them are considered to be ideal interchange modes. Since interchange modes have the tendency to be localized around their mode rational sur-faces, the stability is determined only by the equilibrium quantities around their mode ratio-nal surfaces. This tendency becomes stronger near the marginally stable state or in the high-mode-number limit, and the stability is de-termined by the Mercier criterion (in three-dimensional systems) or the Suydam criterion (in one-dimensional systems) at their mode ra-tional surfaces. Thus, for interchange mode, all the terms of the functional given by Eq.(l) should be considered simultaneously in the sin-gular layer near their mode rational surfaces: On the basis of a stellarator-expansion, the linear growth rate of ideal interchange modes is obtained in Ref. [2] by using the asymptotic matching method of two solutions (one is a so-lution in the inertial layer, and the other is a solution outside of the inertial layer. Note that the singular layer consists of both regions). The linear growth rate is expressed in terms of the Suydam criterion. On the basis of the method mentioned above, the dispersion relation is J-w(w - W*i) 16r: exp {~ [3argr (1 + ~iU) -argr(l + iu) - tan-1 (exp-1r"/2) - ~7r]} (4) where u = J4(DF + DK(W)) - 1, and DF and D K (w) are the Suydam criterion and a kinetic contribution, respectively. References [1] Chen. L and White. R. B, Phy. Rev. Lett., 52 (1984) 1122. [2] Kulsrud. R. M, Phy. Fluids, 6 (1961) 904 191 

Theoretical Analysis of Burst Modes in CHS

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