Quasi-Coherent Modes as Localized Viscous Ballooning Modes*

The quasi-coherent mode associated with the enhanced-$D_\alpha$ H-Regime produced by the Alcator C-Mod machine has been observed [1] as being well localized in the deep $E_r$ well at the edge of the plasma column and propagating in the ion diamagnetic direction in the plasma reference frame. This is explained as resulting from the excitation of a ``viscous ballooning mode'' with frequency close to $\omega_{di}$, where $\omega_{di}\equiv - k_\phi c/e B_\theta n dp_i/dr$ is the ion diamagnetic frequency. The deep $E_r$ well is shown to be responsible for the mode localization. For this we start our analysis from the model dispersion relation [2, 3], $\omega+i\gamma_\mu \omega-\omega_{di}=- \gamma^2_{RT}+ k^2_{stp} v^2_A 1+i D_m k^2_{\perp} \omega-\omega_{*e}$, in a plane geometry, where $\gamma_\mu$ is proportional to the ion transverse viscosity, $\gamma_{RT}$ is the Rayleigh-Taylor growth rate simulating the effects of the magnetic field line curvature, $v_A$ is the Alfv\'{e}n velocity, $D_m\equiv c^2\eta/4\pi$, $\eta$ is the plasma resistivity and $\omega_{*e}\equiv cT_e/e B r_0 d \ln n/dr$ is the electron drift wave frequency. *Sponsored in part by the U. S. DOE. \\[4pt] [1] I. Cziegler, Private communication (2010).\\[0pt] [2] G. Ara, et al., \textit{Annals Phys}. {\bf 112}, 443 (1978).\\[0pt] [3] B. Coppi, et al., \textit{Annals Phys.} {\bf 121}, 1 (1979).