H. Sugama

W. Horton

National Institute for Fusion Science (NIFS-Repository)

§20. On the Saturation of Multihelicity Resistive Interchange Modes Sugama, H. Horton, W. (Inst. Fusion Studies, Univ. Texas) Nonlinear amplitude equations of the multi-helicity modes near marginally stable states are given by [1 ,2] dAn/ dt = /An- :2: Vn-n1 ,n-n2 An-n1 +n2 Anl An2 n1 ,n2 where An(t) (n = 0,±1,±2,···) denote the amplitudes of the multi-helicity modes with the lowest poloidal mode number m = 1 and the toroidal mode number n for the time t. The potential fluctuation is approximately given in terms of the linear eigenfunctions ¢>1 for the dominant m = 1 modes as J = L~=-ooAn4>t(x + n~)sin[27r(y/Ly + nz/Lz)] where x, y, and z are the coordinates in the radial, poloidal, and toroidal directions. Here m = 1 and n = 1 corresponds to the largest poloidal and toroidal wavelengths not in the whole toroidal region but in the local slab sys-tem, which are denoted by Ly and Lz. Thera-dial interval between the adjacent m = 1 mode rational surfaces is given by ~ _ LsLy/ Lz ( L s: the magnetic shear length) while ·all the m = 1 modes have the same linear growth rate 1(> 0). The coefficients Vmn of the nonlinear term is calculated from the linear eigenfunc-tions. The coefficient V00 represents the inten-sity of the self-interaction or the nonlinear cou-pling of the modes with the same helicity and Vmn for (m,n) :f. (0,0) represents that of the interaction between the modes with different helicities. If we use T = 1t, Xn = (Voo/!)A~, ,\ = (Vat + Vit) /Voo and retain only the self-interaction and interaction between adjacent modes, we obtain 1 dXn 2 dT = Xn(1 - Xn- ,\Xn-1 - -XXn+d· Here the parameter ,\ measures the strength of the nonlinear interaction between the adjacent m = 1 modes and is an increasing function of the ratio of the radial width W of the m = 1 mode structure ¢>1 ( x) to the interval ~. We easily obtain the following nontrivial station-ary solutions of the above equations: (I) (II) X n = 1 / ( 1 + 2,\) (for all n) X _ { 0 (for even n) n - 1 (for odd n) Linear analysis shows that, for the case (a) ,\ < 1/2, the solutions (I) and (II) are sta-ble and unstable, respectively, while, for the case (b) ,\ > 1/2, the solutions (I) and (II) are unstable and stable, respectively. Radial profiles of the mode structures corresponding to these solutions are schematically shown in Figs.l (a) and (b). Thus, when the multihe-licity interaction becomes strong enough, one of the two adjacent mode amplitudes vanishes. This tendency is also confirmed by the non-linear simulation of the multihelicity resistive interchange modes [2]. (a) >. < 1/2 (b)>->1/2 f' I' 1'"\ ("\ t'"'\ I \1 \I \I \1 \ I A Y 1. l \ 1 1\ I\ I\ 1\ \ < :( ~~~ )(' Fig. 1. Radial profiles of the mode structures corresponding to the solutions (I) (left) and (II) (right) for the cases (a),\< 1/2 and (b) ,\ > 1/2. Solid (dashed) curves correspond to stable (unstable) solutions. References 1) Sugama, H., Nakajima, N., and Wakatani, M. : Phys. Fluids B 3. (1991) 3290. 2) Sugama, H. and Horton, W.: Comments Plasma Phys. Controlled Fusion 17 (1996) 277. 201 

On the Saturation of Multihelicity Resistive Interchange Modes

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